Recent Developments In Theoretical Physics 9789814287333, 9789814287326
This volume covers recent developments in the major areas of theoretical physics. The scope of the book ranges from smal
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Recent Developments in Theoretical Physics
Statistical Science and Interdisciplinary Research Series Editor: Sankar K. Pal (Indian Statistical Institute) Description: In conjunction with the Platinum Jubilee celebrations of the Indian Statistical Institute, a series of books will be produced to cover various topics, such as Statistics and Mathematics, Computer Science, Machine Intelligence, Econometrics, other Physical Sciences, and Social and Natural Sciences. This series of edited volumes in the mentioned disciplines culminate mostly out of significant events — conferences, workshops and lectures — held at the ten branches and centers of ISI to commemorate the long history of the institute.
Vol. 2
Advances in Intelligent Information Processing: Tools and Applications edited by B. Chandra & C. A. Murthy (Indian Statistical Institute, India)
Vol. 3
Algorithms, Architectures and Information Systems Security edited by Bhargab B. Bhattacharya, Susmita Sur-Kolay, Subhas C. Nandy & Aditya Bagchi (Indian Statistical Institute, India)
Vol. 4
Advances in Multivariate Statistical Methods edited by A. SenGupta (Indian Statistical Institute, India)
Vol. 5
New and Enduring Themes in Development Economics edited by B. Dutta, T. Ray & E. Somanathan (Indian Statistical Institute, India)
Vol. 6
Modeling, Computation and Optimization edited by S. K. Neogy, A. K. Das and R. B. Bapat (Indian Statistical Institute, India)
Vol. 7
Perspectives in Mathematical Sciences I: Probability and Statistics edited by N. S. N. Sastry, T. S. S. R. K. Rao, M. Delampady and B. Rajeev (Indian Statistical Institute, India)
Vol. 8
Perspectives in Mathematical Sciences II: Pure Mathematics edited by N. S. N. Sastry, T. S. S. R. K. Rao, M. Delampady and B. Rajeev (Indian Statistical Institute, India)
Vol. 9
Recent Developments in Theoretical Physics edited by S. Ghosh and G. Kar (Indian Statistical Institute, India)
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Platinum Jubilee Series
Statistical Science and Interdisciplinary Research — Vol. 9
Recent Developments in Theoretical Physics Editors
Subir Ghosh Guruprasad Kar Indian Statistical Institute, India
Series Editor: Sankar K. Pal
World Scientific NEW JERSEY
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Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.
Statistical Science and Interdisciplinary Research — Vol. 9 RECENT DEVELOPMENTS IN THEORETICAL PHYSICS Copyright © 2010 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
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ISBN-13 978-981-4287-32-6 ISBN-10 981-4287-32-6
Printed in Singapore.
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Foreword
The Indian Statistical Institute (ISI) was established on 17th December, 1931 by a great visionary Professor Prasanta Chandra Mahalanobis to promote research in the theory and applications of statistics as a new scientific discipline in India. In 1959, Pandit Jawaharlal Nehru, the then Prime Minister of India introduced the ISI Act in the parliament and designated it as an Institution of National Importance because of its remarkable achievements in statistical work as well as its contribution to economic planning. Today, the Indian Statistical Institute occupies a prestigious position in the academic firmament. It has been a haven for bright and talented academics working in a number of disciplines. Its research faculty has done India proud in the arenas of Statistics, Mathematics, Economics, Computer Science, among others. Over seventy five years, it has grown into a massive banyan tree, like the institute emblem. The Institute now serves the nation as a unified and monolithic organization from different places, namely Kolkata, the Headquarters, Delhi, Bangalore, and Chennai, three centers, a network of five SQC-OR Units located at Mumbai, Pune, Baroda, Hyderabad and Coimbatore, and a branch (field station) at Giridih. The platinum jubilee celebrations of ISI have been launched by Honorable Prime Minister Prof. Manmohan Singh on December 24, 2006, and the Govt. of India has declared 29th June as the “Statistics Day” to commemorate the birthday of Prof. Mahalanobis nationally. Professor Mahalanobis, was a great believer in interdisciplinary research, because he thought that this will promote the development of not only Statistics, but also the other natural and social sciences. To promote interdisciplinary research, major strides were made in the areas of computer science, statistical quality control, economics, biological and social sciences, physical and earth sciences. The Institute’s motto of ‘unity in diversity’ has been the guiding principle of all its activities since its inception. It highlights the unifying role of statistics in relation to various scientific activities. v
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In tune with this hallowed tradition, a comprehensive academic programme, involving Nobel Laureates, Fellows of the Royal Society, Abel prize winner and other dignitaries, has been implemented throughout the Platinum Jubilee year, highlighting the emerging areas of ongoing frontline research in its various scientific divisions, centers, and outlying units. It includes international and national-level seminars, symposia, conferences and workshops, as well as series of special lectures. As an outcome of these events, the Institute is bringing out a series of comprehensive volumes in different subjects under the title Statistical Science and Interdisciplinary Research, published by the World Scientific Press, Singapore. The present volume titled Recent Developments in Theoretical Physics is the ninth one in the series. The volume consists of twenty chapters, written by eminent physicists from different parts of the world. These chapters provide an in-depth study of the recent developments in various branches of theoretical physics, namely, (i) Relativity, Gravitation and Astro-Particle Physics; (ii) High Energy Physics, Nuclear Physics & Quantum Mechanics; (iv) Condensed Matter Phenomena; (iv) Nonlinear Dynamics; and (v) Quantum Information. Both reviews and contributory articles in cutting edge areas are included. The volume has five sections in the aforesaid specialized topics, preceded by an introductory article on the present status of theoretical physics. I believe the state-of-the art studies presented in this book in different advanced topics of theoretical physics will be very useful to both young researchers as well as senior practitioners. Thanks to the contributors for their excellent research contributions, and to the volume editors Prof. Subir Ghosh and Dr. Guruprasad Kar for their sincere effort in bringing out the volume nicely in time. Initial design of the cover by Mr. Indranil Dutta is acknowledged. Sincere efforts by Prof. Dilip Saha and Prof. Barun Mukhopadhyay for editorial assistance are appreciated. Thanks are also due to World Scientific for their initiative in publishing the series and being a part of the Platinum Jubilee endeavor of the Institute.
June 2009 Kolkata
Sankar K. Pal Series Editor and Director
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Preface
This volume is an in depth study of the recent developments in various branches of theoretical physics. Both review style articles and cutting edge research works appear in this peer reviewed collection under the Platinum Jubilee Volume series of Indian Statistical Institute. Senior experts as well as promising young scientists have contributed in this issue. Most of the articles are expanded versions of lectures delivered by the respective authors in a conference, Recent Developments in Theoretical Physics, held at I.S.I., Kolkata, during January 4–8, 2008. The volume starts off with introductory remarks by Avinash Khare on the present status of theoretical physics, delivered at the conference. He set the tone by a critical analysis of the famous 1980 lecture by Stephen Hawking that suggested that in about twenty years most of the key issues in Theoretical Physics will be understood. Khare reviews the developments in the last two decades, (some of which in fact appear in this volume), and concludes happily that contrary to Hawking’s “prediction” Theoretical Physics is ever thriving with new discoveries and new outlooks of existing unsolved problems. The volume covers five broad areas of Theoretical Physics: (I) Relativity, Gravitation and Astro - Particle Physics; (II) High Energy Physics, Nuclear Physics & Quantum Mechanics; (III) Condensed Matter Phenomena, (IV) Nonlinear Dynamics (V) Quantum Information In (I) the articles revolve around theoretical aspects of Black Holes, both classical and quantum, with emphasis on very recent ideas. Parthasarathi Majumdar discusses the holographic principles gravitational physics, in the context of black hole thermodynamics. The connection with two dimensional conformal field theories leads to a corrected form of BekensteinHawking entropy of black holes. Rabin Banerjee’s article deals with the recent excitement in Hawking Radiation from black holes as derived from gravitational anomalies. It includes some of the recent original contribuvii
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tions of Banerjee and collaborators in this area. On the other hand, A. S. Majumdar studies the effects of extra dimensions on classical black holes and their evolution in an intriguing braneworld scenario. The work also probes the role of Dark Matter and suggests observational signatures for braneworld black holes. The last article in this section by C. S. Uunnikrishnan encompasses the basic principles of dynamics and relativity. It advocates a non-conventional view that the matter frame of the universe provides a preferred absolute frame and that its gravity determines the laws of motion. In the quantum regime, the vastly different collective behaviors of fermions and bosons are also linked to the novel idea of cosmic gravity. The author also suggests experiments to establish some of these new ideas. In (II) the first article by Giovanni Amelino-Camelia on Doubly-Special Relativity (DSR) is in fact in the borderline of (I) and (II). Giovanni Amelino-Camelia, being the originator of the idea of this extension of Einstein’s Special Theory of Relativity discusses key open issues of the theoretical and phenomenological consequences of the observer-independent small-length/large-momentum scale (Planck scale?), that is associated with DSR. In the exciting area of Astro-Particle physics, Neutrinos have captured the attention of the scientific community in the past decade. Amitava Raychaudhuri concentrates on this topic and introduces the readers, with his firsthand experience, to the ongoing India-based Neutrino Observatory (INO) project. The few years old proton spin problem is still not satisfactorily solved and A. N. Mitra analyzes a novel way of tackling this serious issue by using a three quark wavefunction, obtained recently in the high momentum regime of QCD. Afsar Abbas reviews the present status of Nuclear Physics and delves in to the recently amassed experimental data on Exotic Nuclei and the rich new physics that these nuclei are hinting at. Non-Hermitian Hamiltonians with real energy eigenvalues for a specific range of parameters have created a lot of interest in recent years. A Sinha and P Roy exploits the principles of supersymmetric quantum mechanics in their study of the Generalized Swanson Model along with its Pseudo supersymmetric Partners. Section (III) is devoted to some specialized areas in Condensed Matter Physics where Berry’s geometrical phase is playing an ever increasing role. Pratul Bandyopadhyay introduces the readers to Berry’s Phase in Quantum Physics where the non-trivial topology of the parameter space spanned by the cyclic parameter, in which the quantum system evolves, plays an essential role. Pierre Gosselin, Alain Berard & Herve Mohrbach describe their very recent work on the direct effects of Berry’ phase on
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particle dynamics in vastly different settings such as in the Bloch electrons in solids, spin Hall effect in semiconductors, relativistic Dirac particles in strong external fields and the gravitational birefringence of photons propagating in a static gravitational field. Anyons, planar excitations having arbitrary spin and statistics, constitute a distinct phase of matter and these exotic states have been observed experimentally in Ga-As semiconductors. B. Basu explains the intrinsic spin Hall effect in Ga-As alloys by considering spinning model of anyons in external electromagnetic field. In this scheme a non-commutative configuration space is induced and once again Berry’s phase plays a major role. The last article in this section by Arnab Das & Bikas K. Chakrabarti is multi-disciplinary in nature. It surveys the recent success in annealing or optimizing the cost functions of complex systems utilizing quantum fluctuations through mapping of such computationally hard problems to classical spin glass problems, quantum spin glasses and consequent annealing or analog quantum computation. Section (IV) begins with the article by D. Grumiller and R. Jackiw on the emergence of Liouville gravity from pure Einstein gravity in a scheme that incorporates spherical reduction and dualization. Features of Einstein gravity, such as interactions with matter and the Bekenstein-Hawking entropy appear in a natural way. Avinash Khare, in his contribution, compares and contrasts various features of the discrete and the corresponding continuum field theory models in 1 + 1 dimensions. The discussion is based on the generalized versions of discrete Φ4 field theory and discrete nonlinear Schrodinger equation and translationally invariant, static as well as time-dependent, exact solutions of these models including kink, pulse, periodic and short-period solutions are constructed. In the fluid dynamics area Supriyo Paul, Krishna Kumar, Pinaki Pal & Mahendra K Verma describe the construction of a model for flow reversal in two-dimensional Rayleigh-Benard convection where a change in flow patterns occurs with increasing values of Rayleigh numbers. In Euclidean networks there are nodes that have well defined position coordinates and the linking probabilities are dependent on these coordinates. Parongama Sen reviews the studies concerning the effective dimensionality of a Euclidean network from a modern perspective. Section (V), the last section, constitutes of studies in Quantum computation. In the first article, Preeti Parashar obtains the largest ensemble of qubits which satisfy the general transformation of equal superposition. This is achieved in different frameworks, namely, linearity, no-superluminal signaling and non-increase of entanglement under LOCC. The author also
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considers the associated quantum random walk problem. In local Cloning of Entanglement Sujit K. Choudhary and Ramij Rahaman shows the (im) possibility of exact cloning of orthogonal but equally entangled quantum states where local operations and classical communication are concerned. They also compute the amount of entanglement necessary in blank copy for various examples. Throughout the conference and afterwards during the process of compiling this edition, we have been very fortunate in having the active support of our Director, S. K. Pal, and all possible cooperation from our departmental colleagues, research scholars and technical staff. In particular we are very grateful to Ramij Rahaman, Senior Research Scholar, Physics and Applied Mathematics Unit, ISI, for his participation at all stages of our work and to Indranil Dutta, Machine Intelligence Unit, ISI for preparing the camera ready copy. It is our hope that both the seasoned practitioners and young researchers of physics will get a panoramic view of the recent exciting happenings in diverse branches of physics and will enjoy and benefit from this volume. Subir Ghosh Guruprasad Kar EDITORS
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Contents
Foreword
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Preface
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1. Is the End of Theoretical Physics Really in Sight?
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A. Khare Relativity, Gravitation and Astro-Particle Physics 2. Holography, CFT and Black Hole Entropy
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P. Majumdar 3. Hawking Radiation, Effective Actions and Anomalies
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R. Banerjee 4. Probing Dark Matter in Primordial Black Holes
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A. S. Majumdar 5. Physics in the ‘Once Given’ Universe C. S. Unnikrishnan
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High Energy Physics, Nuclear Physics and Quantum Mechanics 6. Doubly-Special Relativity
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G. Amelino-Camelia 7. Nuances of Neutrinos
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A. Raychaudhuri 8. Dynamics of Proton Spin
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A. N. Mitra 9. Whither Nuclear Physics?
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A. Abbas 10. Generalized Swanson Model and its Pseudo Supersymmetric Partners
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A. Sinha and P. Roy Condensed Matter Phenomena
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11. The Relevance of Berry Phase in Quantum Physics
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P. Bandyopadhyay 12. Quantum Hamiltonian Diagonalization
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P. Gosselin, A. B´erard and H. Mohrbach 13. The Hall Conductivity of Spinning Anyons
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B. Basu 14. Quantum Annealing and Computation A. Das and B. K. Chakrabarti
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Nonlinear Dynamics
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15. Liouville Gravity from Einstein Gravity
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D. Grumiller and R. Jackiw 16. Exact Static Solutions of a Generalized Discrete φ4
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A. Khare 17. A Model for Flow Reversal in Two-Dimensional Convection
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K. Kumar, S. Paul, P. Pal and M. K. Verma 18. Euclidean Networks and Dimensionality
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P. Sen Quantum Information
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19. Equal Superposition Transformations and Quantum Random Walks
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P. Parashar 20. Cloning Entanglement Locally S. K. Choudhary and R. Rahaman
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Chapter 1 Is the End of Theoretical Physics Really in Sight?
Avinash Khare Institute of Physics Bhubaneswar, Orissa 751005, India I critically analyze the claim made by Hawking in his famous 1980 lecture that the end of Theoretical Physics is in sight and that in about twenty years one would have understood the key issues in Theoretical Physics. I review the progress made in the last twenty seven years since his claim and conclude that his prophesy has turned out to be incorrect. In fact, looking at the several unsolved problems as well as exciting new discoveries, it is highly unlikely that there will be an end of Theoretical Physics, at least in the foreseeable future. At the beginning, I also address Research vs Teaching syndrome that has seriously affected Indian Science since independence.
Contents 1.1 1.2 1.3 1.4
Research vs Teaching instead of Research and Teaching in Is End of Theoretical Physics in Sight? . . . . . . . . . . . Remarkable Developments in Physics from 1900 to 1979 . Developments From 1980 to 2007 . . . . . . . . . . . . . . 1.4.1 Revolution in Information Technology . . . . . . . 1.5 Was Hawking Right In His Assertion? . . . . . . . . . . . 1.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1.1. Research vs Teaching instead of Research and Teaching in India I am honored to have been asked to give the Keynote address at this prestigious conference, to mark seventy five years of Indian Statistical Institute Based on the Keynote Address Given on Dec. 4, 2007 at THEOPHYS-07 Conference, on the occasion of 75 years of Indian Statistical Institute, Kolkata 1
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(ISI). It is time to pay our respect to its founder Director, Prof. P.C. Mahalanobis. His monumental contribution in building this institute and starting front ranking research in statistics is well known. I would like to point out one remarkable contribution of Prof. Mahalanobis which some how has not received enough appreciation. I have no doubt that it is this crucial decision which put ISI on the world map and helped in producing many outstanding experts in Statistics. As you all know, in India we have the great divide between teaching and research. While most institutes exclusively do only research and (almost) no teaching, in most universities most of the scientists have to spend substantial amount of time in teaching. Further, in most universities, the infrastructure and other facilities are poor and bureaucracy has caused havoc in the administration. This great divide has affected Indian Science immensely and there is a serious crisis facing the country. In this scenario, what Mahalanobis did, clearly stands out. In 1960 he started the B.Stat. and M.Stat. programs (Undergraduate and postgraduate degrees in Statistics) in ISI. In the last 47 years, this programme has produced several outstanding researchers and teachers, who are occupying key positions in many institutions all over the world. In fact, before the B.Math. programme started in Bangalore and Chennai around 2000, the B.Stat. programme at ISI Kolkata was also the best programme in India for the B.Math. students. It is worth emphasizing here that this teaching has, in no way, affected the research quality at ISI and ISI is very well known throughout the world as a premiere centre of research in statistics. As far as I know, by and large, the faculty members teach about one course in a year which I think is a very reasonable teaching load. I would like to compare here Mahalanobis with Homi Bhabha. There is absolutely no doubt that apart from his famous work (i.e. Bhabha scattering, Bhabha wave equation etc), Bhabha also laid the foundation of the Atomic Energy programme in India and also started Tata Institute of Fundamental Research (TIFR). However, unlike Mahalanobis, Bhabha never started either B.Sc. or M.Sc programme in TIFR. Bhabha perhaps felt that research and teaching cannot go hand in hand and top researchers should be left free to do research. And once TIFR set this example, almost all other institutes in India followed Bhabha’s example and exclusively concentrated on research and did almost no teaching. With the infrastructure and other facilities too dwindling, soon better scientists preferred research institutes over universities and by and large the quality of the university science departments suffered (I must make it clear that there are still some
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good teachers and researchers in the universities and it is only because of them that good students are still coming to science). This surely was a big blunder on the part of Bhabha, which is one of the key reason for the decline of science in India. From my own experience of teaching at the Predoctoral level in Institute of Physics Bhubaneswar for the last 33 years (of course I must confess that this is no substitute for teaching at B.Sc. or M.Sc. level), I can definitely say that teaching one course per year is in fact extremely beneficial (rather than hindrance) to research. I have no doubt that most of (whatever) physics that I know, has been learnt because of the teaching that I did. It is sad to see that even after sixty years after independence, the country is not ready to rectify this mistake. It is worth pointing out another contribution of Mahalanobis. In 1931 he started an international journal in Statistics named “Sankhya” and remarkably, even after seventy seven years, it continuous to be a reputed international journal. Compare that with physics. We still do not have a high quality journal from India. I think one should critically examine why we have failed while Mahalanobis has been successful.
1.2. Is End of Theoretical Physics in Sight? Let me now come to the main theme of my talk. You may recall that 27 years ago, on April 29, 1980, when he took over as the Lucasian Professor at Cambridge, Hawking1 gave a talk entitled “Is the End In Sight for Theoretical Physics?” As is well known, in this talk he advocated that in about twenty years (from 1980), one would have understood the basic laws of nature and what will be left can be taken care off by computers. The purpose of this address is to critically examine this assertion of Hawking. In this context, it is worth remembering that this is not the first time that some eminent scientist has made such a strong claim. At the end of the nineteenth century too, physicists (including people like Michelson) had claimed that one had understood the basic laws of physics and what was left was merely the matter of details. In order to properly judge Hawking’s claim, it might be worthwhile to first quickly look at the progress made in physics in the first eighty years of the twentieth century and judge how correct was the claim of Michelson and others. After that, I shall examine in some detail the progress made in physics in the last twenty seven years and then point out several open
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problems which makes it abundantly clear that the end is nowhere in sight for theoretical physics. 1.3. Remarkable Developments in Physics from 1900 to 1979 As is well known by now, the prophesy of Michelson and others was rather premature. As it turned out, there was tremendous progress in physics in the first eighty years of the last century, perhaps more progress than in the earlier 300 years. That is why many people have termed it as the golden period of physics. At conceptual level I believe that three of the most remarkable developments of the twentieth century have been Relativity, Quantum Mechanics and Chaos, since all three cut away at the basic tenets of Newtonian physics. While relativity cuts away at the idea of absolute space and time, quantum mechanics killed the idea of controllable measurement process. Finally, chaos put to end the idea of deterministic predictability. Besides, quantum mechanics and relativity have led to so many new discoveries, many of which have tremendous technological applications which have changed the very fabric of our civilization. Some of these applications are nuclear fusion and fission, superconductivity, semiconductors, lasers and space technology. In human history, normally it has taken hundreds of years just to discover one layer of matter, but in these eighty years we have uncovered two layers of matter. By mid-thirties it was first established that the basic constituents of nature are proton, neutron and electron. And by mid-seventies we had discovered the next layer of matter and established that the basic constituents of nature are in fact quarks and leptons (electron is one of the lepton) and that all hadrons (including protons and neutron) are made out of quarks. One had also established that the various interactions between quarks and leptons are well described by the so called “Standard Model” with the gauge group SU 3C SU 2L U 1Y . 1.4. Developments From 1980 to 2007 Let us now examine the claim of Hawking and see if indeed one has obtained answers to the fundamental problems in theoretical physics in the last twenty seven years. There is no doubt that there are several areas where remarkable progress has been made in the last twenty seven years, but without doubt amongst them one area stands out. I have in mind the area of Astrophysics and
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Cosmology which now has become precision science. The idea of big bang cosmology along with inflation has received strong experimental support. We now know that approximately 70 percent of all the matter in the universe is in the form of dark energy, 26 percent is in the form of cold dark matter while only 4 percent is ordinary baryonic matter. While we have almost no idea about the nature of dark energy, what is perhaps clear is that dark energy is distributed homogeneously in the universe, and that it does not cluster like ordinary matter. Further, it acts as repulsive force, i.e. as anti-gravity and seems to increase the rate of expansion of the universe. It is estimated that dark energy has been present for at least nine billion years. In high Energy Physics, W and Z 0 were experimentally discovered precisely at the value predicted by the standard model. Further,one also discovered the top quark. However, without doubt the most remarkable discovery has been the detection of neutrino oscillations and hence the confirmation that the neutrinos (at least some of them) are not massless but have tiny nonzero mass. On the theoretical front in high energy physics, one remarkable development has been “String Theory” which to date is the only consistent (perturbative) quantum theory of gravity and is also a consistent quantum theory unifying all four basic interactions. In the areas of Condensed Matter and Material Science, one remarkable development is in the area of Nano science and technology which has the potential for tremendous applications. Some of the other developments have been high temperature superconductivity, integer and fractional quantum Hall effect, and soft condensed matter physics which has strong overlap with complex systems and biophysics. There have been progress in the other areas too. For example, there has been remarkable progress in the area of quantum optics and optical communication, quantum information and nonlinear science. Besides these, there has been remarkable development in bringing together various areas of science and showing once again the interdisciplinary nature of science in general and physics in particular. For example, in the last twenty seven years the realization has come that to find answers to deep questions in cosmology and astrophysics, which essentially deals with physics at long distance, require answers to fundamental questions in high energy physics (which addresses issues at very short distance) and vice a versa. In fact a whole new area has emerged in the last twenty seven years called “Astroparticle Physics” which tries to address questions bordering both these areas.
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Another remarkable collaboration has been between the areas of theoretical physics and mathematics. For a long time, the relationship between physics and mathematics was rather one sided with mathematics providing technique and justification for physics ideas and models. However, in the last twenty seven years, physics has made significant contributions to mathematics. The best illustration of that is the work of Witten for which he was awarded the Fields Medal (which is equivalent to four Noble prizes in physics, since it is only awarded once in four years!) by mathematics community in 1990. This was awarded for his three contributions in physics (i) proof of positive mass theorem based on supersymmetry (ii) supersymmetry and Morse theory (iii) Jones polynomials from Chern-Simons theory, all of which have important implications for some of the areas of mathematics. The other significant contribution of physics to mathematics was the work of Seiberg and Witten regarding Donaldson invariants. It might be recalled here that Donaldson, who was awarded Fields medal in 1986 for this work, had obtained these invariants while working on the differential topology of four-manifolds. Seiberg-Witten, while working on the N 2 supersymmetric Yang-Mills theory, realized that it has not only Donaldson invariants but using duality they predicted the existence of duals of Donaldson invariants, now known as Seiberg-Witten invariants. What was also remarkable about their work was that whereas one requires rather nontrivial effort to compute the Donaldson invariants in mathematics, Seiberg-Witten were able to compute the same with almost 1/1000’th of the effort! Similarly, the areas of Condensed Matter and High Energy Physics continue to enrich each others areas even more than before. In the recent years, the ideas of conformal field theory has played significant role in such diverse areas as string theory, critical phenomenon and even two-dimensional turbulence. Further, the explanation of fractional quantum Hall effect needs ideas of conformal field theory as well as abelian and nonabelian gauge theories. In fact the area of low dimensional field theory and condensed matter physics has come up in rather big way during these twenty seven years. 1.4.1. Revolution in Information Technology Finally, one cannot talk about important developments in the last twenty seven years in physics without mentioning about the revolution in Information Technology. I believe that it is as big as the great industrial revolution and like the earlier revolution, this one too is going to have profound implications for all human activities including physics. I feel that most people
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have still not appreciated its deep significance and the profound changes that are already taking place in countries like India which are at the forefront of this revolution. I think our generation is lucky that it is able to witness this great revolution and get a feel of how society must have reacted when it went through industrial revolution. Let me now mention few important contributions which this revolution has already made in physics and has profoundly changed the way we carry out research. Undoubtedly, the most important discovery is that of world wide web (WWW). It is worth mentioning here that this discovery was made by a high energy physicist, Tim Berners-Lee, in 1989-90 while he was addressing the problem of how to transmit data from CERN experiments to 400 experimentalists sitting over five continents. Soon came the discovery of arXiv by Paul Ginsberg as well as that of electronic mail. These three discoveries have profoundly changed the way research is being done. The younger generation cannot even imagine how research was carried out before 1990, and how one could have a long distance collaboration. Because of the information technology revolution, tremendous progress is being made in many areas of science including physics. It has helped in providing better insight of many concepts and has given better visualization of dynamical processes. Several areas like Gnome project, bio-informatics, could not have started without this revolution. For almost three hundred years, we are categorizing physics in terms of theoretical and experimental physics. But now there is a third branch, i.e. “ Computational Physics”. I do not mean here problems dealing with number crunching, but I am talking about issues which can neither be addressed theoretically nor experimentally but can only be addressed by using computers. The first example of that was the famous work of Fermi, Pasta and Ulam (with significant help from Mary Tsingou) in 1954 where they wanted to answer if weak nonlinear interaction between large (though not infinite) number of particles can lead to ergodic behavior of the system or not. It is worth remembering that the results of this paper inspired the subsequent discovery of KdV solitons and the whole industry of solitons! Further, using computers one has been able to guess several exact results and subsequently prove them analytically. I will mention here two examples where I have first hand experience. The first is concerning the new identities for the Jacobi elliptic functions2 which Sukhatme and I discovered in 2001. While we could prove these identities analytically in few simpler cases, we had no clue how to prove these in general. So we did the next best thing. We tried to check the validity of the identities in the general case numerically
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using Maple and to our satisfaction we found that the identities are indeed correct to an accuracy of six-seven decimal places. We mentioned this in our paper and conjectured the validity of the identities in the general case. Only later, in collaboration with Arul Lakshminarayan, we were able to prove these identities rigorously in the general case. The second example I have in mind concerns the problem of three noninteracting anyons. I might recall here that anyons are particles in two dimensions, which unlike three or higher space dimensions, obey any quantum statistics. In 1990 we were trying to compute the third virial coefficient of a noninteracting anyon gas (which to this day remains an unsolved problem). Unfortunately, it was not possible to separate mixed symmetry states from symmetric and anti-symmetric states and so as a first approximation, we numerically computed the third virial coefficient in the Boltzmann basis and while doing so numerically obtained a neat relation between the interacting part of the two and three anyon partition functions.3 Subsequently, Virendra Singh proved our result analytically. The point I would like to emphasize here is that in both cases, only because one could numerically obtain or verify certain relations that one even thought about how to prove these results analytically. The information revolution has also raised some interesting questions like “what constitutes a proof?” Take for example, the celebrated Riemann hypothesis about the nontrivial zeros of the zeta function being only on the half-line. By now, one has numerically checked that the first 15 billion zeros of the Riemann zeta function are indeed on the half-line. The question is, does this constitute a proof of the Riemann hypothesis? While I believe that the answer is NO, it is not that easy to argue against those who would like to regard it to be as good as a proof. There is one more aspect of the information technology revolution which has hardly been explored so far, and that is its use as a valuable aid in teaching. I have no doubt that the future generations will mock at us saying how backward we were in our methods of teaching. I have absolutely no doubt that in coming few decades, information technology is going to revolutionize the way we teach right from primary school to research level. Its potential for (i) visualization of very many difficult concepts and dynamical processes (ii) as a substitute for experiments (specially in underdeveloped countries) (iii) exposure to excellent lectures has hardly been realized so far.
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1.5. Was Hawking Right In His Assertion? Finally, let us ask ourselves if indeed the end of theoretical physics is in sight? The answer is unequivocally NO. So many basic questions have remained unanswered and one does not see any light at the end of the long, dark tunnel. For example, we still do not have any idea about what is beyond the standard model of high energy physics. In the last twenty seven years, several theoretical ideas have been discussed including string theory, but it is not at all clear if these ideas or right or wrong. Physics being an experimental science, I strongly feel that whether these ideas are right or wrong can only be decided by experiments. The Large Hadron Collider (LHC) at CERN is being built to look for the elusive Higgs particle of the Standard Model. There is hope that LHC may also discover some of the supersymmteric particles thereby indicating direction beyond the standard model. On the other hand, in case LHC does not find any new physics (apart from the Higgs particle), then there is a real danger that one may not be able to obtain large scale funding required for electron-positron linear collider and the progress of the field may be in danger. In any case, a major breakthrough in accelerator technology is badly needed if we are to explore physics at distances less than 1020 meter. As far as string theory is concerned, it is fair to say that while the ideas are attractive, it is too early to say if it is indeed TOE (theory of everything) as was widely proclaimed by several supporters of the string theory. As of now, string theory has really no predictions which can be tested in present day laboratories, and it is not even clear if in foreseeable future too this will be possible. Coming back to big bang model of the early universe, the present picture crucially depends on the assumption that the baryon number is not an exact symmetry of nature. One important consequence of baryon number violation is proton decay. However, till today there is absolutely no evidence for proton decay. I strongly feel that unless one observes proton decay, the big bang picture of the early universe would remain suspect. Similarly, while approximately seventy percent of all matter in the universe is in the form of dark energy, one has no idea about the nature of this dark energy. Unveiling this mystery may reveal new physics and could also shed light on future direction in high energy physics. Besides, while twenty six percent of all matter in the universe is in the form of cold dark matter, we still do not have any definite candidate for the dark matter. Finally, in spite of many many attempts, it has still not been possible to have a consistent quantum
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theory of gravity, and this I believe is one of the major unsolved problem of modern day physics. In nonlinear physics, one major unsolved problem is that of turbulence and in general that of spatio-temporal chaos. Similarly, the area of condensed matter physics is undergoing a conceptual change with a recent emphasis on soft condensed matter physics and complex systems which has strong interface with several issues in biology. This has opened up several new challenges and it is increasingly becoming clear that the standard courses in condensed matter physics that are being offered in graduate and undergraduate stage are inadequate to attack these challenging issues. 1.6. Conclusions In view of what has been said above, there is no doubt that the prophesy of Hawking was rather premature and the end of theoretical physics is no where near in sight. Nature is far more subtle than what we human beings can imagine it to be. In this connection, I would like to recall a conversation that I had with t’Hooft in 1994 at Puri regarding TOE. He strongly believed that there is TOE and he felt that once we have found it, we would know. However, he was not sure, how long it would take. However, I had and still have a slightly different view point. I feel that even if we find TOE, it will raise further deeper questions. Nature is very subtle and human explorations are like opening of different onion layers. As we uncover one layer, there is next one to be explored. I feel that our attempts at understanding TOE is like trying to uncover yet another sari of Draupadi. References 1. S. Hawking, Is the End in Sight for Theoretical Physics?, Cambridge Univ. Press (Cambridge, 1980). 2. A. Khare and U.P. Sukhatme, J. Math. Phys. 43 (2002) 3798; A. Khare, A. Lakshminarayan and U.P. Sukhatme, J. Math. Phys. 44 (2003) 1822; Pramana (J. of Phys.) 62 (2004) 1201. 3. R.K. Bhaduri, R.S. Bhalerao, J. Law and M.V.N. Murthy, Phys. Rev. Lett. 66 (1991) 523. 4. Virendra Singh, Private Communication; His proof is given in A. Khare, Fractional Statistics and Quantum Theory, Second Edition (World Scientific, Singapore, 2005).
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PART 1
Relativity, Gravitation and Astro-Particle Physics
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Chapter 2 Holography, CFT and Black Hole Entropy
Parthasarathi Majumdar Theory Group, Saha Institute of Nuclear Physics, Kolkata 700064, India Aspects of holography or dimensional reduction in gravitational physics are discussed with reference to black hole thermodynamics. Degrees of freedom living on Isolated Horizons (as a model for macroscopic, generic, eternal black hole horizons) are argued to be topological in nature and counted, using their relation to two dimensional conformal field theories. This leads to the microcanonical entropy of these black holes having the Bekenstein-Hawking form together with finite, unambiguous quantum spacetime corrections. Another aspect of holography ensues for radiant black holes treated as a standard canonical ensemble with Isolated Horizons as the mean (equilibrium) configuration. This is shown to yield a universal criterion for thermal stability of generic radiant black holes, as a lower bound on the mass of the equilibrium isolated horizon in terms of its microcanonical entropy. Saturation of the bound occurs at a phase boundary separating thermally stable and unstable phases with symptoms of a first order phase transition.
Contents 2.1 2.2 2.3 2.4 2.5
Introduction . . . . . . . . . . . . . . . . . . Holographic Hypothesis . . . . . . . . . . . . Weakly Isolated Horizons . . . . . . . . . . . Loop Quantum Gravity : Spin Network Basis Quantum Isolated Horizon . . . . . . . . . . . 2.5.1 Counting of CS states . . . . . . . . . 2.6 Low-tech way : It from Bit . . . . . . . . . . 2.7 Radiant Black Holes . . . . . . . . . . . . . . 2.7.1 Saddle Point Approximation . . . . . . 2.7.2 Schwarzschild Black Hole . . . . . . . . 2.7.3 AdS Schwarzschild Black Hole . . . . . 2.8 Questions Yet to be Resolved . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . 13
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2.1. Introduction The laws of black hole mechanics1 relate changes in the area Ahor of a spatial section of the event horizon (EH) of stationary black hole spacetimes to variation in parameters like the change in the ADM mass M and the surface gravity κhor on the EH, δAhor κhor δM
0 const κhor δAhor
,
(2.1)
These laws appear to have a curious analogy with the zeroth, first and second laws of standard thermodynamics, with the area of the EH. Yet, black hole spacetimes like Schwarzschild and Kerr emerge as exact solutions of the vaccum Einstein equation in complete absence of matter or energy. There is no conceivable source for any kind of microstates usually thought to be as the origin of thermodynamical behavior. The originators of these laws, understandably, did not venture beyond their formulation and derivation from general relativity. Bekenstein2 was the first to argue that these laws must signify thermodynamic behavior of spacetimes, beyond mere analogy. If an object falls through an EH, there is a net reduction in the entropy of the part of spacetime external to that EH. To be consistent with the second law of standard thermodynamics, these spacetimes themselves must carry entropy, whose increase compensates for the reduction mentioned above, such that the sum of the black hole and external entropies never decreases. This, essentially, is Bekenstein’s statement of the so-called Generalized second law of thermodynamics. In order for this black hole entropy to be consistent with eq. (2.1), Bekenstein proposed that Sbh
Ahor kB 4lP2
1 .
(2.2)
where, lP Gc3 12 1033 cm is the Planck length, usually taken to be scale at which quantum gravitational effects dominate physics. The factor 4 in (2.2) emerges from Hawking’s formulation3 of black holes in presence of ambient quantum matter as radiating like a thermal black body with a temperature given by κhor , thus abundantly substantiating Bekenstein’s hypothesis. However, the natural emergence of the Planck length in (2.2)
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goes to suggest that thermodynamics associated with classical spacetimes really stems from microstates arising from the quantum ‘atoms’ of such spacetimes, as would be found in a theory of quantum gravitation. What remains a puzzling observation however, is the dependence of black hole entropy upon an area rather than on three dimensional volume as in standard thermodynamics. How and why that happens are the key issues we wish to address in this article. The next section deals with a discussion of the Holographic hypothesis as formulated by ’t Hooft and our proposal that it may be a consequence of the huge invariance group associated with spacetime in general relativity. This is followed by a description of Weakly Isolated Horizons (WIH) as an inner boundary of spacetime which serves as the protptype of a large class of horizons including but not only, stationary black hole event horizons. How holography emerges in this description is also pointed out, especially in connection with the topological description of the boundary degrees of freedom. A very brief description of Loop Quantum Gravity is next given, with emphasis on the spectrum of certain geometrical observables. How this leads to a quantum theory of WIH and eventually to the microcanonical entropy of generic, macroscopic, eternal black holes (in 4 dimensions) is in the next section. Another version of holography in spacetime physics is discussed next, within the context of a standard equilibrium statistical mechanics approach to a canonical ensemble of radiant (hence non-isolated) black holes. A rather universal criterion of thermal stability of radiant black holes is discussed, in which the mass of the equilibrium configuration (chosen to be a WIH) is bounded from below by the microcanonical entropy of this configuration. The point of saturation of this bound corresponds to a ‘phase boundary’ between a thermally stable and an unstable phase; the transition has tell-tale signs of a first order phase transition. Brief comparison with the pioneering work of Hawking and Page is given. We end with a list of issues yet unresolved. 2.2. Holographic Hypothesis The Holographic hypothesis was postulated by ’t Hooft4 as one way of understanding how in gravitational physics, information about the full black hole spacetime is encoded on the EH - a three dimensional null hypersurface (which is also an ‘outer trapped surface’and a boundary of the black hole spacetime). The main idea of this hypothesis is best stated in ’t Hooft’s words,4
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... Given any closed surface, we can represent all that happens inside it by degrees of freedom on this surface itself. This ... suggests that quantum gravity should be described by a topological quantum field theory in which all degrees of freedom are projected onto the boundary. The questions that immediately arise are
Is Holography in (quantum) general relativity a consequence of diffeomorphism and local Lorentz invariance of spacetime in presence of boundaries (like the EH)? If it does arise naturally in spacetime physics, one need not postulate it as an additional hypothesis. Is there a link to a topological field theory on a boundary of spacetime? Is there any relation to a two dimensional conformal field theory? How does one compute the entropy of black holes on this basis? Is there any implication for thermal stability of radiant black holes?
The above list is somewhat biased toward the order of topics in this article, and hence can be construed as a list of contents. We shall show, not always rigorously, that the answer to each of the queries above is most likely in the affirmative, thus obtaining most of what ’t Hooft hypothesized a decade and a half ago. We begin with the oft-made observation that local gauge invariance is not a statement of symmetry but rather of redundancy of some of the degrees of freedom used to formulate the theory. E.g., in vacuum Maxwell electrodynamics, the photon field A admits the decomposition A
A grad d4 x G x x div A x AT
G x x
where Aω A grad ω
grad
d4 x G x x div A x
(2.3)
aL
δ 4 x x . Under a gauge transformation A AT aL
AT ω aL
ω
AT
aL
ω,
(2.4)
thus clearly revealing the unphysical nature of the longitudinal degree of freedom aL . The 4-vector field AT is 4-divergence free (transverse in spacetime). Integral curves of this field are closed spacelike curves. It is trivial
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to show that the Maxwell field strength tensor is uniquely determined by AT and is completely independent of aL . Yet, the standard formulation of the Maxwell theory continues to use this redundancy, perhaps because most feel it is convenient to do so. What is also obvious is the nonexistence of a gauge invariant tensorial conserved N¨ other current for vacuum electrodynamics, corresponding to gauge transformations. If gauge invariance were a symmetry, surely such a current would be in existence. A similar redundancy exists in spacetime physics as portrayed in general relativity. Spacetime is diffeomorphic to itself under coordinate diffeomorphisms, thus underlining the redundancy of coordinate frames. The clinching evidence for this is the nonexistence of a covariantly conserved energy momentum tensor for spacetime. All attempts to construct such a tensor at best yield non-covariant expressions that can hardly be given a true physical meaning. There is thus no such thing as a local ‘gravitational’ energy density in the spacetime of general relativity. This state of affairs becomes clearer in a canonical formulation of vacuum general relativity. In this formalism, diffeomorphism invariance of the theory leads to a ‘bulk’ Hamiltonian Hbulk
N H N P ,
M
lapse
(2.5)
shif t
where M is a three dimensional spacelike hypersurface (partial Cauchy surface) on which initial data are specified. This means that the Lapse and Shift functions are Lagrange multipliers, enforcing the First Class constraints (infinitesimal spacetime diffeomorphismn generators), H 3 g, 3 Π 3
3
P g, Π
0
0.
Here, phase space is spanned by the 3-metric 3 g and 3-momenta 3 Π. On the constraint surface then one has Hbulk 0. There is thus no notion of a ‘bulk’ energy associated with spacetime in general relativity, signifying that diffeomorphism invariance is not a symmetry but a redundancy. In a formulation of general relativity using orthonormal tetrads as local Lorentz frame variables, there is in addition, another first class constraint corresponding to local Lorentz transformations. Is there any notion at all of ‘gravitational energy’ in general relativity? For spacetimes that are asymptotically flat, i.e., those that in some sense approximate Minkowski spacetime infinitely far away from the EH,
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both in space and time, it is possible to define globally conserved quantities like energy, momentum or angular momentum at the boundary of spacetime. These definitions of global generators depend crucially on how one describes the asymptopic structure of spacetime. For spacetimes with additional ‘inner’ boundaries like the WIH, one can define an energy associated with the WIH, which is distinct from the energy defined at the asymptopic boundary. It is obvious in any case that the total Hamiltonian for a general relativistic system is given on the constraint surface in phase space by H
Hbu;l
Hbdy
Hbdy ,
(2.6)
where ‘boundary’ may consist of several disconnected hypersurfaces embedded in spacetime. Our main interest in what follows will be on WIHs as inner boundaries of spacetimes. In any event, the very fact that any notion of gravitational energy or momentum or indeed angular momentum of spacetime refers to the boundary of spacetime rather than the bulk is ample evidence of holography at play. 2.3. Weakly Isolated Horizons Event horizons of stationary black holes are excessively global. This is implied in the following features of such horizons:
Event horizons are determined only after the entire spacetime is known. Stationarity implies that the black hole metric has global timelike isometry with the corresponding Killing vector field generating time translations at spatial infinity. Event horizons are usually treated separately from cosmological horizons like de Sitter horizons. A unifying treatment is desirable. The ADM mass featuring in the First law of black hole mechanics in eq. (2.1) is defined at spatial infinity and is in no way associated with the event horizon.
These features make is necessary to seek generalizations which are characterized locally. The particular generalization which we adopt here is known as the weakly Isolated Horizon (IH), developed in.14 The properties of an IH can be summarized as follows:14
It is a null inner boundary of spacetime with topology R S2 .
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H’
H’
H
M4
H M3
M’
M2 H
i0
M1
M
Fig. 2.1.
Weakly Isolated Horizon
The area of the IH AhorS 2 const. This is what is inherent in the isolation. It also means that the IH is never crossed, even though there may exist matter and radiation arbitrarily close to it. It is a marginal outer trapping surface. There is no global timelike isometry associated with the IH; this implies that nonstationary generalization of stationary black hole horizons. On IH one can define a surface gravity κl , which however is not defined outside of the IH, since there is no global timelike Killing vector field allowing such a definition. The ‘Zeroth law of IH mechanics’ κl IH const can be demonstrated, although the norm of κl is not fixed since there once again there is no timelike Killing vector. such that On IH, one can define mass a MIH MADM Erad δMIH κl δAhor . . . ; this may be termed as First law of Isolated Horizon Mechanics Such horizons correspond thermodynamically to a microcanonical ensemble with fixed Ahor .
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Now, because any IH is an inner boundary, the variational principle cannot be applicable without an appropriate boundary term S
SEHL
SIH
(2.7)
where, the second term is chosen such that its variation cancels the surface term arising from the variation of the Einstein-Hilbert-Lorentz action. On the other hand, since an IH is null, the 3-metric on the IH is degenerate : 3 g 0. This implies that on the IH one cannot have an action of the
3 g L. Rather, the quantum field theory describing the IH usual type IH dof must be a 3 dimensional Topological Field Theory (TFT). The issue that must be addressed now is : which TFT ? This question is of considerable importance because the microcanonical black hole entropy Sbh log dim HT F T , where HT F T is the Hilbert space corresponding to the theory describing the dynamics of the IH. For this one needs to briefly touch upon the matter of appropriate canonical variables for general relativity. So far we have written eqn.s (2.5), (2.6) and (2.6) symbolically in terms of functions of the 3-metric 3 g and its conjugate 3-momenta. Through canonical transformations and fixing the gauge invariance associated with local Lorentz boosts (time gauge), a more convenient formulation is seen to emerge15 for a real SU 2 connection ASU 2 on the spacelike Cauchy surface M chosen to supply Cauchy data, with its conjugate momenta being the densitized triads E, the pullbacks of tetrads (local frame fields) to M . With IH boundary conditions, the solution space of vacuum Einstein equation admits a closed two-form (symplectic structure) Ω δ1 , δ2
1 tr δ1 A δ2 Σ 16πlP2 M AS tr δ1 A δ2 A 8πγlP2 S Ωblk Ωbdy ,
1 2
1 2 (2.8)
where, S is spatial foliation of IH by M and Σ E E. The solution space corresponding to the Ωbdy is the one corresponding to the SU 2 Chern Simons equation with the pullback of Σ playing the role of source on the IH. Further pulled back to the foliation S of the IH, this equation is the Chern Simons Gauss law k FCS A Σ S 0 (2.9) 2π where, k is the nearest integer value of As 8πlP2 . Thus, the entire role of bulk spatial degrees of freedom characterized by Σ (determined by solving
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Einstein equation) is as source for the Chern Simons degrees of freedom (given by FCS ) characterizing boundary (IH) geometry. It is clear that this is a version of a Holographic picture, albeit in a somewhat subtle way. Certain aspects of the Holographic Hypothesis have already been realized such as primacy of boundary degrees of freedom, a TFT on the inner boundary (IH) and so on. Note that this realization of the holography paradigm has been crucially dependent on the diffeomorphism and local Lorentz invariance of spacetime and the IH dynamics. We shall see another aspect of this picture later.
2.4. Loop Quantum Gravity : Spin Network Basis in brief This is a background-independent, non-perturbative approach to quantum general relativity.16 Quantum three dimensional space is pictured as a network with edges carrying spin j n2 , n Z and vertices consisting of invariant SU 2 tensors (‘intertwiners’). The operators on this space are nonlocal, being holonomies of the SU 2 connection along edges of the network and smeared densitized triads. The network is not a rigid network but a floating one more like a 3 dimensional fishnet. The length of the edges is not fixed to any scale; nor are the edges required to remain straight. Arbitrary knottings of edges are allowed. The vertices can have any valence consistent with conservation of net spin on the edges meeting at a vertex. Local Lorentz invariance and spatial diffeomorphism invariance require that the network be a closed one with no ‘hanging’ edges. Each state in this basis is required to be annihilated by the local Lorentz and spatial diffeomorphism generators, and the set of all spinnet states span a kinematical Hilbert space. The physical Hilbert space, consisting of the kernel of the Hamiltonian constraint, is yet to be worked out in detail. The spinnet basis is the eigenbasis for geometrical operators like length, area and volume in three dimensions. Consider for instance a two dimensional spacelike surface of classical area Acl embedded in a spin network; links of the network will intersect this surface. Assume that the surface is divided into tiny patches such that each patch is pierced by only a single link, whose spin is encoded in the puncture on that patch. The eigenvalues of the area operator are then given, in terms of the spins ji , i 1, 2, . . . , N
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at these punctures (or equivalently, the spin numbers ni ) by a n 1 , . . . , nN
N 1 2 γlP np np 2 4 p1
(2.10)
lim a n1 , ....nN
Acl O lP2
(2.11)
N
The area operator thus has a bounded, discrete spectrum, even though there is a certain degree of arbitrariness in the scale of discreteness.16 Also, the rate of convergence of the discrete eigenvalues to the continuum value is approached rather rapidly.16 2.5. Quantum Isolated Horizon Now, the IH embedded in a spatial geometry represented by spin networks; the spatial section of an IH, assumed spherical here for simplicity, is a two dimensional surface punctured by spin network links which transmit their spins to the punctures. A cartoon depicting this is shown in the following diagram. With the kind of penetration of spin network links discussed above, the Consistency Condition eq.(2.9) can be expressed as a condition in terms of the relevant quantum operators operating on the kinematical Hilbert space, ˆ E ˆ Ψ 0 , k FˆCS E (2.12) S
where Ψ Hbulk Hbdy . The object now is to compute dim Hbdy which is basically the dimensionality of the CS Hilbert space on an IH with N punctures on the S 2 having spins ji acting as pointlike sources. 2.5.1. Counting of CS states Having argued that the boundary (IH) degrees of freedom constitute those of an SU 2 Chern Simons theory with pointlike spin-valued sources, we now relate the Hilbert space of the Chern Simons theory to conformal blocks of the SU 2k WZW model that lives on the foliation S of the IH, following Witten and others.17 According to this relationship, dim HCS sources Ω j1 , . . . , jN where Ω is the number of conformal blocks of the SU 2k WZW model ‘living’ on S 2 with point sources carrying spins j1 , . . . , jN . Using the Verlinde formula, this number can be computed exactly.6 Alternatively, one can solve the Chern Simons theory and calculate Ω j1 , . . . , jN directly.5 Our interest is in macroscopically
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Quantum Isolated Horizon
large black holes (IHs), i.e., those IHs whose areas AIH lP2 . Thus the deficit angles of punctures made are being required to together represent a smooth S 2 . Intuitively, this means that one must maximize the number of punctures for a given fixed classical area, and this requires the spin on each puncture to be as small a possible, i.e., spin 1/2. With this choice, Ω 12, 12, . . . , 12 can be easily computed and its logarithm extracted, giving the microcanonical entropy,7 Ahor 3 Ahor Ahor 1 log Smicro kB O . (2.13) 4lP2 2 4lP2 4lP2 where, the correct normalization for the leading oder Bekenstein-Hawking result emerges as a fit for a real parameter (Barbero-Immirzi parameter15 invariably multiplying the Planck scale.5 Observe, however, that this is the only ambiguity in the entire infinite series each of whose terms are finite. Admittedly, the counting presented above is rather crude! One should actually consider a set of N punctures with a spins ji , i 1, . . . , N , count Ω j1 , . . . , jN and sum over all possible spins and punctures. Setting all
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spins equal to 1/2 is not guaranteed to give the same number, although in view of our argument regarding deficit angles, it may be a reasonable approximation for asymptotically large horizon areas. Various computations of Ω by this procedure have been reported. The one that appears to resemble a truly statistical mechanical computation is that of 18 which is quite consistent with our final result (2.13) in so far as leading logarithmic corrections to the area law are concerneda . 2.6. Low-tech way : It from Bit The idea of It from bit 4,19 essentially is to think of a floating two dimensional lattice covering a two dimensional sphere S 2 which we take to be the horizon. Each plaquette of the lattice is taken to be of Planck area size, so that the area of the horizon in Planckian units is given by an integer p Ahor lP2 which is also the number of plaquettes covering the horizon. Place in each plaquette a spin 1/2 object so that two states can be associated it, the ms 12 states. If all states of the horizon with random orientation of spin 1/2 variables are considered, the number of states Ω p 2p . However, as in the previous subsection, we are interested in states of this system with net spin jtot 0 so that p p Ω p (2.14) p 2 1 p 2
mtot 0
mtot
1
which yields the same degeneracy and Smicro upon using the Stirling approximation for the factorials in eq. (2.14). Without having made any use of the Holographic hypothesis, all aspects of it have thus been seen to be realized in the ab initio computation of the microcanonical entropy of IHs which we believe serve as good prototypes of black hole event horizons. The only lacuna in the computation is that this is actually a computation of the dimensionality of the kinematical Hilbert space rather than the physical one. There is thus the crucial assumption that the states we count also belong to the physical Hilbert space. This remains a grey area because of the difficulties associated with discerning the semiclassical behavior of the states spanning the kernel of the bulk Hamiltonian operator. On the positive side, the calculation we present yields an asymptotic series of terms of decreasing powers of the area of the a The coefficient of the logarithmic correction depends upon the particular Chern Simons theory, as explained in detail in9
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It from Bit
IH, of which only the first, i.e., the term proportional to AIH had been anticipated by Bekenstein and Hawking. Each of the quantum spacetime correction terms is robust, unambiguous and finite, requiring no ad hoc regularization or indeed renormalization of coupling constants. However, the system under consideration corresponds to a nonradiant (or nonaccreting) isolated black hole and therefore is unphysical. It is not as yet clear how the foregoing formalism can be extended or generalized to realistic situations. In the following we adopt a pragmatic approach based on equilibrium statistical mechanics of canonical Gibbs ensembles including Gaussian thermal fluctuation corrections, to reach some understanding of radiant black holes, based on our approach to quantum spacetime geometry. 2.7. Radiant Black Holes Black holes undergoing Hawking radiation (or indeed thermal accretion) show a runaway behavior in that as they radiate, their Hawking temperature increases. This is once again in contradistinction with standard thermodynamic systems which radiate and cool down as they approach thermal equilibrium. This instability due to thermal radiation appears to occur for all asymptotically flat spacetimes13 but not necessarily for asymptotically anti-de Sitter spacetimes. In this part of the article, we analyze the sit-
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uation from our standpoint of equilibrium statistical mechanics within a ‘mean field’ approach incorporating thermal fluctuations. We also choose the mean field or equilibrium configuration to be an IH whose quantum behavior we believe is on firm footing as explained in the previous sections. Beyond this assumption, our approach is not exactly semiclassical, in contrast to most of the literature on black hole thermodynamics, because we do not use any aspect of classical spacetime geometry, like the form of the metric or even its asymptotic structure. This allows us to derive a general criterion for thermal stability of black holes. In canonical general relativity, we mentioned that the bulk Hamiltonian is a sum of first class constraints. The total Hamiltonian for any spacetime with boundary is thus H
Hbulk
Hbdy
(2.15)
where, Hbdy is the Hamiltonian corresponding to all boundaries including the one at spatial (i.e., the ADM Hamiltonian) such that Hbulk
0
(2.16)
In any theory of quantum general relativity, one expects ˆ H
ˆ bulk H
ˆ bdy H
(2.17)
such that ˆ bulk ψN blk H
0
where ψN blk are states characterizing bulk space. Choose as basis eigenstates of the full Hamiltonian
Ψ cN,α ψN blk χα bdy
(2.18)
(2.19)
N,α
With these properties, the canonical partition function Z
ˆ blk H ˆ bdy T r exp β H
2 ˆ blk ψN χα exp β H ˆ bdy χα cN,α ψN exp β H N,α
ˆ bdy χα c ˜α χα exp β H 2
α
const. Zbdy ,
ˆ bdy T rbdy exp β H (2.20)
where we have used (2.18) for the bulk states ψN . This has the rather remarkable ramification that black hole thermodynamics is completely
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determined by the boundary partition function This is the holographic picture reappearing in another guise, once again originating from the Hamiltonian constraint characterizing temporal diffeomorphism invariance. Consistent with this picture is the preeminence of the area of the horizon rather than the volume in determining the entropy of black hole spacetimes, although this is not directly follow from our arguments. 2.7.1. Saddle Point Approximation Assume now that
Boundary in question is a black hole event horizon ˆ bdy : En E an where, an 4π 3γl2 n Eigenvalues of H P
These imply that Zbdy
g E an exp βE an
(2.21)
n
dn g E a n exp βE a n f or n 1
dE exp Smicro E βE
log
dE , dn
(2.22) (2.23)
where g E an is the degeneracy associated with the area eigenstate state labeled by an . We now make the saddle point approximation, with saddle point chosen to be E MIH , i.e., equilibrium configuration is chosen to be an isolated horizon with a mass M Ahor . In saddle point approx using standard formulae of equilibrium statistical mechanics including Gaussian thermal fluctuations, Scanon where, ∆
Smicro Ahor
1 log ∆ 2
M Ahor Smicro Ahor M Ahor Smicro Ahor 2 M Ahor S Ahor
(2.24)
(2.25)
For thermal stability, Scanon real ∆ 0; it turns out that this is also a necessary condition which guarantees the positivity of the heat capacity. This ∆
M Ahor Smicro Ahor
0
M Ahor Smicro Ahor
(2.26) (2.27)
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Upon integrating this inequality with respect to area and choosing appropriate dimensional constants for simplicity, we get M Ahor
Smicro Ahor ,
(2.28)
a criterion8–11 described entirely in terms of quantities well-understood within the IH-LQG framework. Since this criterion has been obtained without any reference to classical spacetime geometry, it is perhaps worthwhile to check that indeed it holds for known semiclassical situations. 2.7.2. Schwarzschild Black Hole In this case, the area and mass are related by the well-known relation M Ahor
1 2
Ahor
(2.29)
It is obvious that, since the Smicro Ahor in this case, for large Ahor the bound is not satisfied, and we expect a thermally unstable situation, as indeed it is. This is consistent with the heat capacity C 0. Unfortunately, this sort of instability is endemic to all asymptotically flat black hole spacetimes, as has been discussed in detail in.13 2.7.3. AdS Schwarzschild Black Hole Asymptotically anti-de Sitter spacetimes have a timelike infinity requiring specification of incoming data. The incoming radiation has to precisely cancel the outgoing one in order to completely specify Cauchy data, for a range of black hole parameters (mass M and cosmological constant Λ), thereby guaranteeing a stable thermal equilibrium. This range is given by l Λ12 Ahor 4π 12 . How do we see that in the criterion derived above? For this we need only use the mass-area relation for AdS Schwarzschild black holes 12 1 Ahor Ahor 1 . (2.30) M Ahor 2 4π 4πl2 So long as the area is within the range specified above, it is obvious that 32 M Ahor Smicro Ahor which means that the inequality (2.28) is satisfied. As one approaches the endpoints of the range, i.e., when l approaches Ahor 4π 12 , the system looks more and more like an asymptotically flat Schwarzschild spacetime, and thermal instability begins to set in. It is obvious however that enroute to instability, the inequality (2.28) must saturate for certain values of the parameters. This can be shown12 to
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correspond to the heat capacity blowing up from the positive side. When the inequality reverses, the heat capacity is definitely negative, exhibiting a sort of behavior reminiscent of first order phase transitions in statistical mechanics. As emphasized earlier, no classical metrics have been used anywhere in this analysis, and in this sense the treatment here can be thought of as a generalization of the pioneering semiclassical treatment of Hawking and Page.13 Observe also that the criterion for thermal stability involves the domination of the black hole mass over the microcanonical entropy which has to do with the disorder associated with the degrees of freedom of the quantum isolated horizon. Thus the transition from a thermally stable to an unstable phase, although not a typical phase transition in statistical mechanics, is still not without similar symptoms. 2.8. Questions Yet to be Resolved In this section we list a list of pending issues which await satisfactory resolution:
The origin of the assumed mass-area relation remains somewhat obscure, although an approach may be to derive it from an analogue of the relation in LQG between the area operator and the bulk Hamiltonian.20 An important issue is the possibility of Hawking radiation from an IH. It is to be seen what precisely among the characterizations of isolation can best be discarded to enable this. Can one go beyond effective description of black hole as IH (inner boundary of sptm) and consider realistic dynamical collapse? Is the thermal nature of the Hawking radiation spectrum an artifact of the semiclassical approximation? Does the lowest area quantum lP2 have implications for the information loss problem?
References 1. J. Bardeen, B. Carter and S. W. Hawking, Comm. Math. Phys. 31, 161 (1972). 2. J. Bekenstein, Phys. Rev. D7, 2333 (1973). 3. S. W. Hawking, Comm. Math. Phys. 43, 199 (1975).
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4. G. ’t Hooft, “Dimensional Reduction and Quantum Gravity” in Salamfstschrift edited by A. Alo and S. Randjbar-Daemi, ICTP, Trieste, 1993; see also R. Bousso, Rev. Mod. Phys. 74 825-874, 2002 for a more recent review. 5. A. Ashtekar, J. Baez, A. Corichi and K. Krasnov, Adv. Theor. Math. Phys. 4, 1 (2000) and references therein. 6. R. Kaul and P. Majumdar, Phys. Lett. B439, 267 (1998). 7. R. Kaul and P. Majumdar, Phys. Rev. Lett. 84, 5255 (2000). 8. S. Das, P. Majumdar and R. K. Bhaduri, Class. Quant. Grav. 19, 2355 (2001). 9. A. Chatterjee and P. Majumdar, Arxiv:hep-th/0303030. 10. A. Chatterjee and P. Majumdar, Phys. Rev. Lett. 92, 141301 (2004). 11. A. Chatterjee and P. Majumdar, Phys. Rev. D 72, 044005 (2005). 12. P. Majumdar, Class. Quant. Grav. 24, 1747 (2007). 13. S. W. Hawking and D. N. Page, Comm. Math. Phys. 87 577 (1983). 14. A. Ashtekar, C. Beetle and S. Fairhurst, Class. Quant. Grav. 17, 253 (1996) and references therein. 15. G. Immirzi, ArXiv:gr-qc 97010052 and references therein. 16. A. Ashtekar and J. Lewandowski, Class. Quant. Grav. 21 R53 (2003). 17. E. Witten, Commu. Math. Phys. 18. A. Ghosh and P. Mitra, Phys. Rev. D 74, 064026 (2006). 19. J. A. Wheeler, “It from Bit” in Sakharov Memorial Lectures Vol. II, Nova Publishing, Moscow, 1992. 20. T. Thiemann, Phys. Lett. B380, 257 (1999).
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Chapter 3 Hawking Radiation, Effective Actions and Anomalies
Rabin Banerjee S.N. Bose National Centre for Basic Sciences, JD Block, Sector III, Salt Lake, Kolkata-700098, India We give a derivation of Hawking radiation from charged black holes using only covariant gauge and gravitational anomalies. These calculations are reinforced by an effective action based approach which also reveals a connection with the Unruh vacuum. While many of the results were previously discussed in1 and,2 there are some new ones including novel interpretations.
Contents 3.1 3.2 3.3 3.4 3.5 3.6
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . General Discussion on Covariant and Consistent Anomalies . . Covariant Gauge Anomaly and Charge Flux . . . . . . . . . . . Consistent Gauge Anomaly and Charge Flux . . . . . . . . . . Covariant Gravitational Anomaly and Energy-Momentum Flux Effective Actions and Unruh Vacuum . . . . . . . . . . . . . . 3.6.1 Charge and Energy Flux . . . . . . . . . . . . . . . . . . 3.6.2 Connection with Unruh vacuum . . . . . . . . . . . . . . 3.7 Higher Spin Anomaly and Hawking Flux . . . . . . . . . . . . 3.8 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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31 34 36 38 39 41 44 45 47 48 50
3.1. Introduction Hawking radiation is an important quantum effect in black hole physics. Specifically, it arises in the background spacetime with event horizons. The radiation has a spectrum with Planck distribution giving the black holes one of its thermodynamic properties that make it consistent with the rest of physics. Hawking’s original result3 has since been rederived in different 31
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ways35,36 thereby reinforcing the conclusion to a certain extent. However, the fact that no one derivation is truly clinching has led to open problems leading to alternative approaches with fresh insights. An anomaly in quantum field theory is a breakdown of some classical symmetry due to the process of quantization (for reviews, see27,28,30 ). Specifically, for instance, a gauge anomaly is an anomaly in gauge symmetry, taking the form of nonconservation of the gauge current. Such anomalies characterize a theoretical inconsistency, leading to problems with the probabilistic interpretation of quantum mechanics. The cancelation of gauge anomalies gives strong constraints on model building. Likewise, a gravitational anomaly24,25 is an anomaly in general covariance, taking the form of nonconservation of the energy-momentum tensor. There are other types of anomalies but here we shall be concerned with only gauge and gravitational anomalies. The simplest case for these anomalies which is also relevant for the present analysis, occurs for 1 1 dimensional chiral fields. Specifically, the matter sector can be either Weyl fermions or complex scalars. Long back Christiansen and Fulling4 reproduced Hawking’s result by exploiting the trace anomaly in the energy momentum tensor of quantum fields in a Schwarzschild black hole background. The use of anomalies, though in a different form, has been powerfully resurrected recently by Robinson and Wilczek.5 They observed that effective field theories become two dimensional and chiral near the event horizon of a Schwarzschild black hole. This leads to a two dimensional gravitational anomaly. The existence of energy flux of Hawking’s radiation is necessary to cancel this anomaly. The method of 5 was soon extended to charged black holes6 by using the gauge anomaly in addition to the gravitational anomaly. Further advances and applications of this approach may be found in a host of papers,7–22 including a recent review.23 The approach of 5,6 is based on the fact that a two dimensional chiral (gauge and/or gravity) theory is anomalous. Such theories admit two types of anomalous currents and energy momentum tensors; the consistent and the covariant.27,28,30 The covariant divergence of these currents and energy-momentum tensors yields either the consistent or the covariant form of the gauge and gravitational anomaly, respectively.24–30 The consistent current and anomaly satisfy the Wess Zumino consistency condition but do not transform covariantly under a gauge transformation. Expressions for the covariant current and anomaly, on the contrary, transform covariantly under gauge transformation but do not satisfy the Wess Zumino condition.
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Similar conclusions also hold for the gravitational case, except that currents are now replaced by energy-momentum tensors and gauge transformations by general coordinate transformations. In5,6 the charge and the energy momentum flux of the Hawking radiation are obtained by a cancelation of the consistent anomaly. However the boundary condition necessary to fix the parameters are obtained from a vanishing of the covariant current at the event horizon. In a couple of papers,1,2 together with Kulkarni, we have reformulated the analysis of 5,6 such that only covariant expressions are used throughout and consistent expressions are completely bypassed. The point is that since covariant boundary conditions are mandatory, it is conceptually clean to discuss everything from the covariant point of view. There are two distinct advantages of such an approach. First, the calculations become very brief and compact. Secondly, fluxes of higher spin currents are easily computed in this method. The point is that covariant anomalies for the higher spin currents are known.32 Hence the covariant anomaly approach becomes quite natural in this context.31 An effective action based method has also been developed to obtain the Hawking flux.2 The starting point is the structures for the two dimensional effective actions which are known both for the chiral (anomalous) and usual (anomaly free) cases. The relevant expressions for the covariant gauge current and the covariant energy momentum tensor are obtained by taking appropriate functional derivatives of the chiral effective action. The free parameters are fixed by imposing the vanishing of these covariant forms at the event horizon. The Hawking flux from charged black holes is correctly reproduced by taking the asymptotic infinity limit of the covariant gauge current and the covariant energy momentum tensor. Finally, by referring to the anomaly free effective action, we establish the connection of our approach with calculations based on the Unruh vacuum.7,38 It should be pointed out that the diffeomorphism anomaly, subjected to the covariant boundary condition, is a convenient way to obtain the off diagonal term of the stress tensor that determines the Hawking flux. No other information is necessary. In the usual trace anomaly approach, both the expression for this anomaly as well as the normal Ward identity are required to obtain a structure of the stress tensor which contains undetermined parameters. These parameters are then fixed by imposing conditions appropriate for the Unruh vacuum states, eventually leading to the flux. In this sense, therefore, the present approach is considerably simpler.
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3.2. General Discussion on Covariant and Consistent Anomalies Here we briefly summarise some results on anomalies highlighting the peculiarities of two dimensional spacetime. First, the consistent gauge anomaly as taken in5,6 is considered, ∇µ J µ
e2 ρσ
¯ 4π
ρ Aσ
e2
4π
g
ρσ
ρ Aσ
(3.1)
where corresponds to left(right)-handed fields, respectively. Here gµν is the two dimensional r t part of the complete Reissner-Nordstrom metric given by5,6 ds2
f rdt2
1 dr2 r2 dΩ2d2 . f r
(3.2)
so that g detgµν 1 and dΩ2d2 is the line element on the d 2 sphere. The gauge potential is defined as A
Q dt. r
(3.3)
Now a word regarding our conventions. As is evident from (3.1) the antisymmetric tensor ¯ρσ differs from its numerical counterpart ρσ ( 01 01 1) by the factor g. Since here g 1, the two get identified. Henceforth we shall always use ρσ , omitting the g factor. The function f r vanishes at r r which defines the event horizon of the black hole. The current J µ in (3.1) is called the consistent current and satisfies the Wess-Zumino consistency condition. Effectively this means that the following integrability condition holds;26,27 δJ µ x δAν y
δJ ν y . δAµ x
(3.4)
The covariant divergence of the consistent current yields the consistent anomaly. The structure appearing in (3.1) is the minimal form, since only odd parity terms occur. However it is possible that normal parity terms appear in (3.1). Indeed, as we now argue, such a term is a natural consequence of two dimensional properties. To fix our notions, consider the interaction Lagrangian for a chiral field ψ in the presence of an external gauge potential Aµ in 1 1 dimensions, 1 γ5 ¯ γµ Aµ ψ. (3.5) LI ψ 2
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Using the property of two dimensional γ- matrices, γ5 γ µ
µν
γν ,
(3.6)
it is found that Aµ couples as a chiral combination g µν µν Aν . Note that the usual flat space identity (3.6) holds due to the specific structure of the two dimensional metric. Hence the expression for the anomaly in (3.1) generalises to, e2 αβ αβ g Aβ . (3.7) α
4π This is a non-minimal form for the consistent anomaly dictated by the symmetry of the Lagrangian, and has appeared earlier in the literature.25 It is clear that if J µ is a consistent current then J¯µ , which is given by, ∇µ J¯µ
µ J¯ µ
e2 µ A (3.8) 4π is also a consistent current since the extra piece satisfies the integrability condition (3.4). It is possible to modify the new consistent current (3.8), by adding a local counterterm, such that it becomes covariant, J¯µ
J˜µ
Jµ
e2 J¯µ ! Aα αµ g αµ . 4π
(3.9)
The current J˜µ yields the gauge covariant anomaly, e2 αβ
Fαβ . (3.10) 4π Note that the covariant current (3.9) does not satisfy the Wess-Zumino consistency condition since the counterterm violates the integrability condition (3.4). Moreover the gauge covariant anomaly (3.10) has a unique form dictated by the gauge transformation properties. This is contrary to the consistent anomaly which may have a minimal (3.1) or non-minimal (3.7) structure. Now we will concentrate our attention on the gravity sector. If we omit the ingoing modes the energy momentum tensor near the horizon will not conserve, while there is no difficulty in the region outside the horizon. The analysis5,6 for obtaining the flow of energy momentum tensor was done by using the minimal form of the consistent d 2 anomaly,24,25,28,29 for right handed fields, ∇µ J˜µ
∇µ Tνµ
1 βδ
96π
α δ α Γνβ ,
(3.11)
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Here we consider the general form for d 2 consistent gravitational anomaly. It is worthwhile to point out that the consistent gravitational anomaly and the consistent gauge anomaly are analogous satisfying similar consistency conditions. This is easily observed here by comparing (3.11) with (3.1) where the affine connection plays the role of the gauge potential. We therefore omit the details and write the generalized anomaly by an inspection of (3.7) on how to include the normal parity term. The result is, ∇µ T¯νµ
1 96π
δ α
βδ g βδ Γα νβ
Aν .
(3.12)
The covariant energy momentum tensor, on the other hand, has the divergence anomaly, ∇µ T˜ µ
ν
1
νµ 96π
µ
R A˜ν .
(3.13)
This is called the covariant anomaly as distinct from the consistent anomaly (3.11). Furthermore, it was shown by us1 that the analysis of 5,6 remains unaffected if, instead of taking the minimal anomaly (3.1,3.11), the more general structure (3.7,3.12) is considered.
3.3. Covariant Gauge Anomaly and Charge Flux The current is conserved outside the horizon so that ∇µ J˜µo µ J˜µo ˜r r Jo 0. Near the horizon there are only outgoing (right-handed) fields and the current becomes (covariantly) anomalous (3.10), ˜r rJ H
e2 Frt 2π
e2 2π
r At .
(3.14)
The solution in the different regions is given by, J˜ro J˜rH
co ,
(3.15)
e At r At r , 2π
(3.16)
2
cH
where co and cH are integration constants. The current is now written as a sum of two contributions from the two regions, J˜µ J˜µo Θ r r J˜µH H, where H 1 Θ r r . Note that the region near the horizon (H) is defined by the coordinates r
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to r while outside (O) the horizon is characterized by r to . Then by using the conservation equations, the Ward identity becomes, 2 e µ r ˜ ˜ At H µJ r J r 2π e2 r r ˜ ˜ At . δ r r Jo JH (3.17) 2π To make the current anomaly free the first term must be canceled by quantum effects of the classically insignificant ingoing modes. This is the WessZumino term induced by these modes near the horizon. Effectively it ime2 At H which is anomaly plies a redefinition of the current as J˜r J˜r 2π free provided the coefficient of the delta function vanishes, leading to the condition, e2 (3.18) At r . 2π The coefficient cH is fixed by requiring the vanishing of the covariant current at the horizon r r . This yields cH 0 from (3.16). Hence the value of the charge flux is given by, co
co
cH
e2 e2 Q At r . 2π 2πr
(3.19)
This is precisely the current flow of the Hawking blackbody radiation with a chemical potential.6 One might feel slightly uneasy of partitioning spacetime into two regions by means of the step function θ. This is however only an artefact and the results may be obtained without this prescription. To see this the expression for the covariant current (3.16) subjected to the boundary condition cH 0 is written as e2 J˜rH At r At r . (3.20) 2π Now the flux co in (3.19) just corresponds to the asymptotic value of the anomaly free current J˜ro (3.15) at infinity. Moreover, from (3.14) we observe that, in the asymptotic infinity limit r , the anomaly vanishes in the gauge (3.3). This implies that the Hawking flux can as well be calculated by taking the asymptotic limit of the anomalous current (3.20). We find, using (3.3) in (3.20), J˜rH r
which exactly reproduces (3.19).
e2 At r . 2π
(3.21)
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3.4. Consistent Gauge Anomaly and Charge Flux The analysis of Hawking radiation using the consistent gauge anomaly was previously done in.6 However we give a different presentation which makes contact with the covariant analysis done in the preceding section. The minimal consistent anomaly (3.1) differs from the covariant anomaly (3.14) by factor of 2 so that, e2 r At . 4π The solution to this equation is given by,
r rJ H
JrH
c¯H
(3.22)
e2 At r At r 4π
(3.23)
which is the analogue of (3.16). Here c¯H is an integration constant that corresponds to the value of the consistent current at the horizon r r . We now determine c¯H . Observe that the covariant and minimal consistent currents are related by, J˜r
Jr
e2 tr
At 4π
Jr
e2 At 4π
(3.24)
which follows from (3.9). At the horizon, therefore, e2 J˜rH r JrH r At r 0 (3.25) 4π where the last equality is a consequence of the boundary condition; namely, the vanishing of the covariant current at the horizon. Hence, c¯H
JrH r
e2 At r . 4π
(3.26)
Inserting this in (3.23) we find,
J rH
e2 1 At r At r . 2π 2
(3.27)
As happened for the covariant case, the consistent anomaly (3.22) also vanishes in the asymptotic r limit. Hence the asymptotic limit of (3.27) will yield the anomaly free current at infinity that can be identified with the Hawking flux. This flux is therefore given by, JrH r which reproduces (3.21).
e2 At r 2π
(3.28)
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The Hawking flux derived either from the consistent or the covariant current yield identical results. However the boundary condition must be covariant. The fact that the flux is determined by either current is connected with a general result of anomalies which states that, in the absence of an anomaly, the covariant and consistent currents are identical. Since the anomaly vanishes at asymptotic infinity, the Hawking flux is determined by either current. The equivalence of the two currents at r is also seen from (3.24), using the gauge (3.3). 3.5. Covariant Gravitational Anomaly and Energy-Momentum Flux In the presence of a charged field the classical energy-momentum tensor is no longer conserved but gives rise to the Lorentz force law, ∇µ T˜µ ν Fµν J˜µ . The corresponding anomalous Ward identity for covariantly regularized quantities is then given by, ∇µ T˜ µ
ν
Fµν J˜µ A˜ν ,
(3.29)
where A˜ν is the covariant gravitational anomaly (3.13). Since the current J˜µ itself is anomalous one might envisage the possibility of an additional term in (3.29) proportional to the gauge anomaly. Indeed this happens in the Ward identity for consistently regularized objects.6 Such a term is ruled out here because there is no such covariant piece with the correct dimensions, having one free index. For the metric (3.2) the covariant anomaly is purely time-like A˜r 0 while, A˜t
˜ rN t
r
˜r ; N t
f f
f 2
2 . (3.30) 96π Next, the Ward identity is solved for the ν t component. In the exterior region there is no anomaly and the Ward identity reads,
˜r r Tt o
Frt J˜ro .
(3.31)
Using (3.15)this is solved as T˜tro
ao co At r,
(3.32)
where ao is an integration constant. Near the horizon the anomalous Ward identity, obtained from (3.29), reads ˜r r Tt H
Frt J˜rH
˜r r Nt ,
(3.33)
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Inserting J˜rH from (3.16) yields the solution r e2 2 ˜r . At N T˜trH aH dr r co At t 4π r
(3.34)
Writing the energy-momentum tensor as a sum of two combinations T˜νµ T˜νµo Θ r r T˜νµH H we find 2 e 2 µ r r ˜ ˜ ˜ A Nt H ∇µ Tt r Tt co r At r r 4π t e2 2 ˜ r δ r r . T˜tro T˜trH At N (3.35) t 4π The first term is a classical effect coming from the Lorentz force. The second term has to be canceled by the quantum effect of the incoming modes. As before, it implies the existence of a Wess-Zumino termmodifying 2 µ µ e 2 ˜ ˜ ˜ the energy-momentum tensor as Tt Tt 4π At Ntr H which is anomaly free provided the coefficient of the last term vanishes. This yields the condition, e2 2 ˜ r r . A r N (3.36) t 4π t where the integration constant aH is fixed by requiring that the covariant energy momentum tensor vanishes at the horizon. From (3.34) this gives aH 0. Hence the total flux of the energy momentum tensor is given by ao
Since f r
aH
ao
e2 2 ˜ r r . A r N t 4π t
(3.37)
˜ r r 0 we find from (3.30) that N t 2π β
f r
the surface gravity of the black hole κ expressed in terms of the inverse temperature β as ao
e2 Q2 2 4πr
π . 12β 2
2
f 2 r
192π
. Using
, the final result is
(3.38)
This is just the energy flux from blackbody radiation with a chemical potential.6 As shown for the charge flux derivation, it is possible to avoid the use of step functions to obtain the energy momentum flux. The point is that the covariant gravitational anomaly (3.30) vanishes in the asymptotic r limit. Hence the energy momentum flux, defined
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by the anomaly free T˜tro , can as well be obtained from the r limit of T˜ r . Now , as already explained, the boundary condition yields aH 0.
t H
Hence, T˜trH which, at r
,
r
dr
r
co At
r
e2 2 ˜tr A N 4π t
(3.39)
yields,
T˜tr r
co At
r
e2 2 ˜ r r A N t 4π t
e2 2 ˜ r r A r N t 4π t
(3.40) (3.41)
which is just the Hawking flux (3.37). Following the technique discussed in section 4, it is also feasible to obtain the Hawking flux from the consistent gravitational anomaly. 3.6. Effective Actions and Unruh Vacuum For a two dimensional theory the expressions for the effective actions, whether anomalous (chiral) or normal, are known.7,40 Both these are required in our analysis. For deriving the Hawking flux, only the form of the anomalous (chiral) effective action, which describes the theory near the horizon, is required. The currents and energy momentum tensors are computed by taking appropriate functional derivatives of this effective action. Next, the parameters appearing in these solutions are fixed by imposing the vanishing of covariant currents (energy momentum tensors) at the horizon. Once these are fixed, the Hawking fluxes are obtained from the asymptotic r limits of the currents and energy momentum tensors. To show the connection with the Unruh vacuum the form of the usual effective action, which describes the theory away from the horizon, is necessary. The currents and energy momentum tensors, obtained from this effective action, are solved by using the knowledge of the corresponding chiral expressions. The results reproduce the expectation values of the currents and energy momentum tensors for the Unruh vacuum. First, we consider the effective theory away from the horizon. This is defined by the standard effective action Γ of a conformal field with a central charge c 1 in this blackhole background.7 Γ consists of two parts; the gravitational (Polyakov) part and the gauge part. Adding the two
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contributions we obtain, Γ
gR
1 96π
d2 xd2 y
e2 2π
d2 xd2 y µν
x
1 x, y gR y ∆g
x
µ Aν
1 x, y ρσ ∆g
ρ Aσ
y .
(3.42)
Here R is the two-dimensional Ricci scalar given by R f , and ∆g ∇µ ∇µ is the laplacian in this background. The energy-momentum tensor Tµν o in the region outside the horizon is defined as, Tµν o
δΓ . g δg µν
2
(3.43)
The explicit form for Tµν o is thus given by Tµν o
1 2gµν R 2∇µ ∇ν G 48π e2 1 ρ ∇µ B∇ν B gµν ∇ B∇ρ B π 2
(3.44)
Similarly, the form for the gauge current can be obtained, δΓ δAµ
e2 µν
π
Jµo
G x
d2 y ∆g1 x, y
B x
d2 y ∆g1 x, y µν
ν B.
(3.45)
Here
gR
y ,
µ Aν
y .
(3.46) (3.47)
From now on we would omit the g 1 factor from all the expressions. Hence we work with the antisymmetric numerical tensor µν defined by
tr 1. B x and G x satisfy, ∇µ ∇µ B
r At
; ∇µ ∇µ G R f ,
(3.48)
respectively. The solutions for B and G are now given by B
Bo r at b ;
G Go r 4pt q ;
r Bo
1 At c, f
r Go
1 f z , f
(3.49)
(3.50)
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where a, b, c, p, q and z are constants. Also note that Bo and Go are functions of r only. The current (3.45) and the energy momentum tensor (3.44) satisfy the normal Ward identities, ∇µ Jµo
0 ; ∇µ Tνµo
Fµν Jµo
(3.51)
Note that in the presence of an external gauge field the energy momentum tensor is not conserved; rather the Lorentz force term is obtained. In the region near the horizon we have gravitational as well as gauge anomaly so that the effective theory is described by an anomalous (chiral) effective action which is given by,40 1 z ω z A (3.52) 3 where Aµ and ωµ are the gauge field and the spin connection, respectively, and, ΓH
z v
1 4π
d2 xd2 y µν
µ vν
x∇1 x, y
ρ
ρσ g ρσ vσ y
(3.53)
From a variation of this effective action the energy momentum tensor and the gauge current are computed. To get their covariant forms in which we are interested, however, appropriate local polynomials have to be added. This is possible since energy momentum tensors and currents are only defined modulo local polynomials. The final results for the covariant energy momentum tensor and the covariant current are given by,40 1 2 µν µ δgµν T δAµ J δΓH d x (3.54) l 2 where the local polynomial is given by, l
1 4π
d2 x µν Aµ δAν
1 wµ δwν 3
1 Rea δea 24 µ ν
(3.55)
The covariant energy momentum tensor T µν and the covariant gauge current J µ are obtained from the above relations as,a
1 4π
e2 Dµ BDν B 4π 1 µ 1 D Dν G δνµ R 24 24 Tνµ
1 µ D GDν G 48
(3.56)
a matter of notation, we are no longer using J˜µ or T˜µ ν for the covariant forms, as done in section 3 or 4. It was necessary there for distinguishing it from the consistent forms.
a As
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e2 µ D B. 2π Note the presence of the chiral covariant derivative, Jµ
Dµ
∇µ µν ∇ν
µν D
ν
,
(3.57)
(3.58)
instead of the usual one that occurred previously in (3.44), (3.45). The definitions of B and G are provided in (3.46), (3.47). By taking the covariant divergence of (3.56) and (3.57) we get the anomalous Ward identities, ∇µ J µ
e2 ρσ
2π
ρ Aσ
e2 2π
r At
(3.59)
1
νµ µ R. (3.60) 96π The anomalous terms are the covariant gauge anomaly and the covariant gravitational anomaly, respectively. These Ward identities were also obtained from different considerations in the previous sections. ∇µ Tνµ
Fµν J µ
3.6.1. Charge and Energy Flux Here we calculate the charge and energy flux by using, respectively, the expressions for the covariant current (3.57) and the covariant energy momentum tensor (3.56). We will see that the results are the same as that obtained by the anomaly cancelation (consistent or covariant) method,1,2,6 outlined previously, where the starting point was the anomalous Ward identity (3.59) or (3.60). First, we derive the charge flux. Using (3.49) and (3.58) we have from (3.57), e2 At r c a (3.61) 2π We now impose the boundary condition that the covariant current J r vanishes at the horizon, implying J r r 0. This leads to a relation, Jr
c a At r
(3.62)
Hence the expression for J r takes the form, e2 At r At r (3.63) 2π which, expectedly, has the same form as (3.20). The charge flux, obtained by taking its asymptotic r limit reproduces (3.19) or (3.21). We Jr
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next consider the energy momentum flux by adopting the same technique. After using the solutions for B x and G x, the r t component of the covariant energy momentum tensor (3.56) becomes, 2 e2 1 1 r 2 Tt At r At r p f z 4π 12π 4 1 1 1 pf f f f f z . (3.64) 24π 4 4 Now we implement the boundary condition; namely the vanishing of the covariant energy momentum tensor at the horizon, Ttr r 0. This condition yields, p
1 ; f z f
4
f r
r .
(3.65)
Using either of the above solutions in (3.64) we get, e2 2 At r At r 4π 1 2 f f 2 2f f . 192π
Ttr
(3.66)
This expression is in agreement with that given in.7 As already discussed the energy flux is abstracted by taking the asymptotic infinity limit of (3.66). This yields, Ttr r
e2 2 1 At r f 2 . 4π 192π
(3.67)
which correctly reproduces the Hawking flux (3.37). 3.6.2. Connection with Unruh vacuum Here we compute the anomaly free current and the energy momentum tensor, which describe the theory away from the horizon, and show that these agree with the expectation values of these observables for the Unruh vacuum. We consider the expression for the current Jµo in the region outside the horizon. From (3.45) and (3.49) we obtain, Jro
e2 a, Jto π
e2 At r c . πf
(3.68)
At asymptotic infinity (where the anomaly vanishes) the result for Jro and JrH (3.21) must agree. Taken together with (3.62) this implies a c
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and hence the currents outside the horizon are given by,
At r 2
J ro
e2 At r ; Jto 2π
e2 πf
At r
1 At r . 2
(3.69)
This is also the expectation value of the current for the Unruh vacuum in the d 2 RN black hole.7,38 Now we consider components of the anomaly free energy momentum tensor defined in (3.44). The r t component of Tνµo is given by Ttro
e2 2 e2 At r At rAt r 4π 2π 1 zp, 12π
while the t t component becomes, e2 Ttto A2t r At r At r 2πf 1 2f f f f z 8p2 48πf
1 2 At r 2 f z 2 . 2
(3.70)
(3.71)
The asymptotic form of (3.70) must agree with that of (3.67). A simple 1 2 f . Substituting this in (3.65) yields two inspection shows that zp 16 1 1 ; z 1 f . Using either of solutions p 8 f ; z 2 f and p 18 f
2 these solutions in (3.70) and (3.71) we obtain, Ttro
e2 2 e2 At r At rAt r 4π 2π 1 f 2 , 192π
while the t t component becomes, e2 1 A2t r At r At r A2t r Ttto 2πf 2 1 1 2 4f f f 2 f
. 96πf 2
(3.72)
(3.73)
Likewise Trro can be computed either directly or from noting the trace R that follows from(3.44) and then using (3.73). These are also Tµµo 24π the expressions for the expectation values of the various components of the energy momentum tensor found for the Unruh vacuum.7,38
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3.7. Higher Spin Anomaly and Hawking Flux In this section we compute the Hawking flux obtained from higher spin anomaly. It corresponds to the higher spin moments of Hawking flux. These results have earlier appeared in31 using the step function that enter in the anomaly calculation approach. We present an alternative derivation avoiding the step functions. This is more in line with our analysis given earlier (see section 3, 4). µ the divergence anomaly is given by,32 For a spin 3 current Jνρ ∇µ J µ νρ
µ
µ
Fνµ T ρ Fρµ T ν
1 ∇ν RJρ 16
1 1 ∇ρ RJν gνρ ∇µ RJ µ 16 16 h σ µ σ µ νσ ∇ ∇µ F ρ ρσ ∇ ∇µ F ν 24π
gνρ ασ ∇
σ
∇µ F µα . (3.74)
This is the covariant anomaly. Here T µ ν and J µ are the covariant energy momentum tensor (current) already encountered. The Planck constant is explicitly inserted to show the anomalous (quantum) correction to the classical result. This piece is the spin 3 generalization of the divergence anomalies in the electric and gravitational backgrounds. The spin 3 current appearing in (3.74) is regarded as the appropriate covariant current near the event horizon of the black hole. Furthermore, since the expectation value of the current depends only on r in the black hole background taken here, the relevant ν ρ t component of (3.74) becomes, 1 1 r r gtt ∇µ RJµH r J H tt 2Ftr T H t ∇t RJH t 8 16 1 r µ σ µα 2∇ ∇µ F t gtt ασ ∇ ∇µ F . (3.75) 24π where we have included the suffix H to imply that the various objects are defined on the event horizon. Also, the h factor is now deleted. The solution to the above equation, subject to the boundary condition JrH tt r 0, is given by,31 r 1 1 JrH tt dr r 2Co2 At Co A2t Co1 f f f 2 16 r A3t f 2 r2 At f f r At f f f 2 At (3.76) 6π 96π 48π 32π
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1
2
where Co and Co are (3.19 ) and (3.37), respectively, corresponding to the charge flux and energy momentum flux. As before we notice that the anomaly piece in (3.75) vanishes in the asymptotic r limit. Hence the Hawking flux is just given by the asymptotic infinity expression for (3.76), 3 k2 1 3 k2 Q 1 Q r At r A r JH tt r 24π 6π t 24π r
6π r
(3.77)
1
2
obtained after substituting the explicit values of Co and Co , and taking the gauge (3.3). This result coincides with the appropriate n 2 moment of the Hawking flux.31 3.8. Discussions This paper is an expanded account of 1,2 with some new results and interpretations. The flow of charge and energy momentum from charged black hole horizons were obtained by a cancelation of the covariant anomalies. Since the boundary condition involved the vanishing of the covariant current at the horizon, all calculations involved only covariant expressions. Neither the consistent anomaly nor the counterterm relating the different currents, which were essential inputs in,6 were required. Consequently our analysis was economical and, we feel, also conceptually clean. We would here like to mention that the interplay of covariant versus consistent anomalies, as occurring in,5–7 has been specifically discussed in the appendix of.12 It should be pointed out that the flux is identified with Jro or Ttro which are the expressions for the currents exterior to the horizon. Here these currents are anomaly free implying that there is no difference between the covariant and consistent expressions. Actually the germ of the anomaly lies in this difference.26,27 Hence it becomes essential, and not just desirable, to obtain the flux in terms of the covariant currents. This is consistent with the universality of the Hawking radiation and gives further credibility to the anomaly cancelation approach. We further reiterate that, as already stated in the literature,5,6 this anomaly canceling approach is valid for any dimensions, although explicit expressions are required only for the two dimensional anomalies. It was shown, performing a partial wave decomposition, that physics near the horizon is described by an infinite collection of massless 1 1 dimensional fields, each partial wave propagating in spacetime with a metric given by
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the ‘r t sector of the complete spacetime metric (3.2). This simplification, which effects a dimensional reduction from d-dimensions to d 2 is exploited. A reason in favor of working with covariant anomalies is the fact that their functional forms are unique, being governed solely by the gauge (diffeomorphism) transformation properties. This is not so for consistent anomalies. This feature becomes important when discussing the Hawking flux for higher spin currents. The calculations, based on conformal field theory techniques, naturally yield the covariant forms of the anomalies.31 We have used these expressions to compute the higher moments of the Hawking flux. Although explicit calculations were presented for the spin 3 case, further higher spins pose no problems. Beside the anomaly canceling technique,5,6 we provide an alternative way of computing the Hawking flux. This does not require the splitting of space into two regions (near to and away from the horizon) by the introduction of step functions. This is mandatory in the anomaly canceling approaches.1,5,6 The covariant anomaly equation is solved by using the vanishing of the covariant current at the horizon as a boundary condition. The flux is given by the asymptotic infinity limit of the current/stress tensor. We have also given a derivation of the Hawking flux from charged black holes, based on the effective action approach, which only employs the boundary conditions at the event horizon. It might be mentioned that generally such approaches require, apart from conditions at the horizon, some other boundary condition, as for example, the vanishing of ingoing modes at infinity.4,37,38 The latter obviously goes against the universality of the Hawking effect which should be determined from conditions at the horizon only. In this we have succeeded. Also, the specific structure of the effective action from which the Hawking radiation is computed is valid only at the event horizon. This is significant since other effective action based techniques do not categorically specify the structure of the effective action at the horizon. Rather they use the nonvanishing trace of the energy momentum tensor to derive the form for the effective action and are usually restricted to two dimensions only.37 Further, we have exploited the information from the chiral effective action, which describes the theory near the horizon, to completely fix the form of the normal effective action that describes the theory away from the horizon. The expressions for the currents and energy momentum tensors obtained from the latter reproduce the results obtained by using the Unruh vacuum approach.7,38
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Our analysis can be easily extended to other (e.g. rotating) black holes with Kerr-Newman metric. Although we have given the analysis for det g 1, this scheme is equally valid for more general cases where det g " 1. It is a straightforward generalization with inclusion of g factors (see, for instance, Refs. 33 and 34). Acknowledgment I thank Shailesh Kulkarni for discussions. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27.
R. Banerjee, S. Kulkarni, Phys. Rev. D 77, 024018 (2008) [hep-th/0707.2449]. R. Banerjee, S. Kulkarni, Phys. Lett. B 659, 827 (2008) [hep-th/0709.3916]. S. Hawking, Commun. Math. Phys. 43, 199 (1975) S. Christensen and S. Fulling, Phys. Rev. D 15,2088 (1977). S. P. Robinson and F. Wilczek, Phys. Rev. Lett. 95, 011303 (2005) [grqc/0502074]. S. Iso, H. Umetsu and F.Wilczek, Phys. Rev. Lett.96, 151302 (2006) [hepth/0602146]. S. Iso, H. Umetsu and F. Wilczek, Phys. Rev. D 74, 044017 (2006) [hepth/0606018]. K. Murata and J. Soda, Phys. Rev. D 74, 044018 (2006) [hep-th/0606069]. E. Vagenas and S. Das, JHEP 0610, 025 (2006) [hep-th/0606077]. M. R. Setare, Eur. Phys. J. C 49, 865 (2007) [hep-th/0608080]. Z. Xu and B. Chen, Phys. Rev. D 75 024041 (2007) [hep-th/0612261]. S. Iso, T. Morita and H. Umetsu, JHEP 0704, 068 (2007) [hep-th/0612286]. Q. Q. Jiang and S. Q. Wu, Phys. Lett. B 647, 200 (2007) [hep-th/0701002]. Q. Q. Jiang, S. Q. Wu and X. Cai, [hep-th/0701048]. Q. Q. Jiang, S. Q. Wu and X. Cai, Phys. Rev. D 75, 064029 (2007) [hepth/0701235]. X. Kui, W. Liu and H. Zhang, Phys. Lett. B 647, 482 (2007) [hepth/0702199]. H. Shin and W. Kim, arXiv:0705.0265 [hep-th] J. J. Peng and S. Q. Wu, arXiv:0705.1225 [hep-th]. Q. Q. Jiang, arXiv:0705.2068 [hep-th]. B. Chen and W. He, arXiv:0705.2984 [gr-qc]. U. Miyamoto and K. Murata, arXiv:0705.3150 [hep-th]. S. Iso, T. Morita and H. Umetsu, arXiv: 0705.3494 [hep-th]. S. Das, S.P Robinson and E. C Vagenas, arXiv: 0705.2233 L. Alvarez-Gaume and E. Witten, Nucl. Phys B 244, 421 (1984). W. A Bardeen, B. Zumino, Nucl. Phys. B 244, 421 (1984). H. Banerjee and R. Banerjee, Phys. Lett. B 174, 313 (1986). H. Banerjee, R. Banerjee and P. Mitra, Z. Phys. C 32, 445-454 (1986).
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28. R. Bertlmann, “Anomalies In Quantum Field Theory,” (Oxford Sciences, Oxford, 2000). 29. R. Bertlmann and E. Kohlprath, Ann. Phys. (N.Y.) 288, 137 (2001). 30. K. Fujikawa and H. Suzuki, “Path Integrals and Quantum Anomalies,” (Oxford Sciences, Oxford, 2004). 31. S. Iso, T. Morita and H. Umetsu, arXiv:0710.0456 [hep-th]. 32. S. Iso, T. Morita and H. Umetsu, arXiv:0710.0453 [hep-th]. 33. S. Gangopadhyay, S. Kulkarni, arXiv:0710.0974 [hep-th]. 34. S. Q. Wu, Z. Y. Zhao, arXiv:0709.4074 [hep-th]. 35. G. Gibbons, S. Hawking, Phys. Rev. D 15, 2752 (1977). 36. M. Parikh, F. Wilczek, Phys. Rev. Lett. 85, 5042 (2000). 37. For a review, see, A. Wipf, in Black Holes: Theory and Observation, edited by F. W. Hehl,C. Kiefer and R. Metzler (Springer, Berlin, 1998). 38. W. Unruh, Phys. Rev. D 14, 870 (1976). 068 (2007) [hep-th/0612286]. 39. J.J. Peng, S. Q. Wu, arXiv:0709.0167 [hep-th]. 40. H. Leutwyler, Phys. Lett. B 153, 1 (1985).
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Chapter 4 Probing Dark Matter in Primordial Black Holes
A. S. Majumdar S. N. Bose National Centre for Basic Sciences Block JD, Sector III, Salt Lake, Calcutta 700098, India We consider several black hole solutions in standard gravity as well as alternate theories where the geometry of the black hole spacetimes are modified due to extra-dimensional effects. We study the cosmological evolution of primordial black holes in standard cosmology as well as in the braneworld scenario. The process of accretion of radiation in the radiation dominated era could under certain conditions counterbalance the effect of Hawking evaporation, and this leads to longer lifetime for the black holes. We investigate the survival till present times of primordial black holes in the braneworld scenario which could act as candidates for non-baryonic cold dark matter. Observational constraints from several cosmological eras could be used to impose bounds on the initial mass spectrum for the black holes. A mechanism for the formation of binaries with primordial braneworld black holes is also discussed. We show how possible observational signatures of modified gravity could be obtained from strong gravitational lensing of black holes.
Contents 4.1 Introduction . . . . . . . . . . . . . . . . . . . . 4.2 Primordial Black Holes in Standard Cosmology . 4.3 Black Holes in Alternate Gravity Theories . . . . 4.4 Braneworld Cosmology with Black Holes . . . . . 4.5 Observational Constraints and Binary Formation 4.6 Observables of Strong Gravitational Lensing . . . 4.7 Conclusions . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .
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4.1. Introduction Primordial black holes are potentially fascinating cosmo-archeological tools,1 viz., they could capture the physics of the early universe during their formation eras, and relay it to later times through the end products of their evaporation, and possibly through other means of detecting them. Primordial black holes have numerous implications on processes occurring during diverse cosmological epochs. One of the major unsolved riddle in present cosmology is to ascertain the constituents of the dark matter present in our universe. The most significant consequence of the existence of primordial black holes could be their survival till present times as components of dark matter. The present article investigates this possibility first from theoretical motivations, and then discusses mechanisms of obtaining observational signatures of such a scenario. Black holes could be formed in the early universe through a variety of mechanisms.2 A widely studied mechanism is that of the cosmological density perturbations generated through inflation.3 Inflation leads to a scale invariant spectrum of primordial density perturbations which tightly constrains the production rate and the initial mass spectrum of primordial black holes formed through the collapse of density perturbations.4 Recent observational results on the Cosmic Microwave Background Radiation5 favoring a primordial power spectrum with a spectral index very close to one (ns 1) have motivated the detailed investigation of the production of primordial black holes in inflationary models with distinguished scales or broken scale invariance.6 It is possible for PBHs to have ramifications through their evaporation products on diverse cosmological processes such as on baryogenesis and nucleosynthesis,7 on the cosmic microwave background radiation,8 and on the growth of perturbations as well.29 PBHs could act as seeds for structure formation10 and could also form a significant component of dark matter through efficient early accretion in braneworld scenarios.11,12 Mechanisms for growth of supermassive black holes by PBHs accreting dark energy have been proposed,13 though recent results by Harada et al.14 indicate the lack of self-similar growth of black holes by accreting quintessence. Due to the fact that black holes can accrete their surrounding energy as well as radiate energy through the process of Hawking evaporation, the interest in the study of the evolution of primordial black holes15 ranges over a very big time span starting from their formation and running right through their lifetimes which, in certain cases could even exceed the present age
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of the universe. A key question concerning primordial black holes is their longevity which depends crucially on the effectiveness of various accretion processes. Upon formation in the early universe, the black holes are surrounded by radiation which is the dominant form of available energy at that time. The debate as to how significant the process of accretion of radiation is for primordial black holes in the standard cosmological scenario, ranges till date,15,16 though numerical simulations have indicated considerable increase of black holes lifetimes with notable consequences.17,18 Primordial black holes could also accrete the energy of background scalar fields that are present in a variety of scalar-tensor models in cosmology.20 If the present day acceleration of the universe results from a dominant scalar field energy, as in models of quintessence,21 then primordial black holes could grow up enormously in size by accreting such dark energy. Indeed, the observational evidence in the favor of the abundance of supermassive black holes in galactic centres22 presents a challenge for theoretical astrophysicists to formulate formation mechanisms for such huge black holes, and primordial black holes could be potential candidates.13 Most of the studies concerning the implications of primordial black holes on cosmology have focussed on the products emanating from Hawking radiation. If primordial black holes are formed with a spectrum of initial masses, they are likely to have a range of lifetimes corresponding to different epochs of the universe. During their later stages of evolution, the Hawking temperature exceeds the threshold of heavy particle production. It was realized that the baryon number violating and CP-violating decay of heavy particles radiated by them could lead to baryogenesis.23 If a fraction of the population of primordial black holes have lifetimes of the order of or somewhat exceeding the electroweak phase transition time, then the baryons produced would evade the sphaleron wash-out.24 This mechanism could be responsible for generating the observed baryon asymmetry in the universe.17,19 But, in general the decay of primordial black holes would produce an excess of light particles and photons. Nucleosynthesis results can therefore be used to constrain the abundance of primordial black holes in the relevant mass range.25 Further bounds on the primordial black hole abundance comes through the Hawking emission of photons that could distort the CMBR spectrum at high frequencies.26 A comprehensive discussion of the constraints on the primordial black holes abundances imposed by observational requirements at various cosmological epochs can be found in Ref. 27.
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An exciting possibility featuring primordial black holes is their capacity to be a significant portion of cold dark matter. It has been claimed recently that a scenario with primordial black holes as cold dark matter could be viable for particular models of broken scale invariance for the primordial power spectrum.28 Primordial black holes also produce isocurvature perturbations with interesting consequences.29 The probe of black holes as dark compact objects in galaxy haloes is the target of much recent observational activity.30 A large number of these black holes are expected to be in the form of binaries. Different observational evidences as well as theoretical arguments ranging from lensing effects to hypervelocity star ejections have been put forward to bolster the claim in support of the existence of black hole binaries31 in a wide mass spectrum. Recently, it has been proposed that sub-lunar mass binaries comprised of braneworld black holes could produce detectable gravitational waves in their coalescing stage.32 Gravitational lensing experiments do indeed leave open the possibility of the existence of sub-lunar mass black holes in several mass ranges.33 Primordial black holes have been studied extensively in the standard cosmological model, though there remain several gaps in our understanding of their formation, evolution and implications. Recent resurgence of interest in the physics of primordial black holes comes also from the advent of braneworld models of the universe. Indeed, the entire scenario of cosmology with primordial black holes gets modified in the context of braneworld gravity. There has been widespread activity in braneworld gravity recently.34 The braneworld scenario of our universe as enunciated in the Randall-Sundrum (RS-II) model35 opens up the fascinating possibility of the existence of a large extra spatial dimension. Consistency of the RS-II model demands that the standard model fields are confined to the brane, except for gravity which could also propagate into the bulk. This is ensured by the positive tension brane, and the Ads5 bulk with a negative cosmological constant and curvature radius l. The resultant modification of the Newtonian gravitational potential is being probed by current experiments which have set the upper limit of the scale of the extra dimension in the sub-millimeter region, i.e., l 0.2mm.36 Standard Schwarzschild black hole solutions have to be modified in the context of this model.37,38 Furthermore, the evolution of primordial black holes and their multifarious consequences require completely new investigation in braneworld cosmology. In this article we will first present a review of the general features of cosmology with a population of primordial black holes. This will be done
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in Section 3.2 completely within the context of standard cosmology, with the aim of highlighting the essential features of primordial black hole evolution taking into account accretion and Hawking evaporation. We will then discuss various black hole solutions in alternate gravity models such as in braneworld models, and string-inspired models in Section 3.3. In Section 3.4 we will discuss the evolution of primordial black holes in braneworld cosmology. We will highlight the fact that accretion of radiation is a significant effect for braneworld black holes,11 which leads to mass growth and prolonged lifetimes for the primordial population. The modified cosmological evolution of the braneworld era impacts most of the observational constraints7,8 on the density of primordial black holes that are obtained in standard cosmology. In Section 3.5 we will present a formalism which could be used to work out the modified constraints in braneworld cosmology.39 In this section we further outline a scenario for the formation of black hole binaries.40 A fascinating prospect for future observations is the detection of black holes through strong gravitational lensing. In Section 3.6 we will briefly review the formalism for strong gravitational lensing applied to spherically symmetric static as well as non-static metrics. We will then show how this formalism could be used to obtain lensing observables for black holes in various modified gravity models.41–43 In Section 3.7 we conclude with a summary of the main results reported in this article. 4.2. Primordial Black Holes in Standard Cosmology We begin by considering a population of primordial black holes in the early radiation dominated era of the universe. These black holes could be formed by various mechanisms2 operative in the post-inflationary stage of the universe. In the simplest analysis we do not take into account effects of the initial mass distribution4,6 of black holes, but work with the uniform black holes mass M . The time evolution of the FRW scale factor a t of the universe is governed by the equation a 2 a2
8πG M tnBH t ρR t 3
(4.1)
where nBH t is the number density of the black holes at the epoch t, and ρR t is the energy density of radiation. The number density of black holes scales as a3 , so that nBH t
NBH a 3 t
(4.2)
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where the total number of black holes formed at time t0 is NBH nBH t0 a3 t0 . The mass of a black hole will change with time due to two factors. First, the process of Hawking radiation causes a loss of mass. Secondly, accretion of the surrounding matter by the black hole leads to an increase in its mass. The rate of change of mass of a single black hole can be written as 2 4 M 4πRBH (4.3) σTBH f ρR where RBH is the Schwarzschild radius, σ is the Stefan-Boltzmann constant, TBH is the Hawking temperature, and f is an O 1 accretion efficiency factor. Relativistic particles and photons surrounding the black hole dominate the energy density of the universe. The thermalysed plasma has density ρR and a temperature T given by ρR
π 2 gT 4 30
(4.4)
where g is the number of relativistic particle species. The first term on the right-hand side of Eq.(4.3) corresponds to the Hawking radiation effect with Hawking temperature given by TBH
M42 8πM
(4.5)
with M4 denoting the Planck mass. The second term in Eq.(4.3) corresponds to the accretion rate of the surrounding radiation ρR which during the radiation dominated era falls off as 1t2 . Thus, the rate equation for an isolated black hole in the radiation dominated era can be written as M
As M44 M2
Bs M 2 M42 t2
(4.6)
where As and Bs are dimensionless numbers.17,18 The cross-section of the capture of relativistic matter by the black hole 2 could be somewhat different from the geometrical cross section 4πRBH due to the effects of gravitational deflection of he surrounding matter towards the black hole and the finite mean free path of particles being captured. The size of the hubble radius which grows as RH t limits the amount of surrounding energy that could be causally accreted by the black hole, thereby affecting the rate of accretion for black holes of the similar size. The actual capture cross-section also depends on the relative motion of the black hole with respect to the cosmic frame. A black hole moving with
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velocity v sees the background radiation temperature T to have an angular dependence T T θ (4.7) γ 1 vcosθ with the Lorentz γ factor impeding Hawking radiation for black holes moving with relativistic velocities. This implies that in the cosmic frame the rate at which the mass of the black hole changes is17 T4 1 1 2 M 4πRBH σ BH 4γ T 4 (4.8) γ 3 γ A high value of γ will enhance the mass gain due to accretion and also inhibit mass loss due to Hawking radiation, leading to longer life for the black hole.17 The process of accretion entails several different complexities.15–18 For the sake of keeping the present analysis as uncomplicated as possible, we will work with the unknown accretion efficiency factor f introduced in Eq.(4.3). The interplay of accretion and Hawking radiation for a black hole balance at a particular mass Mc . Setting M 0 in Eq.(4.3), one can obtain Mc using Eqs.(4.4) and (4.5) to be σM42 (4.9) 8π f ρR 14 If a black hole is formed with an initial mass lesser than Mc , it will be shortlived as a result of the Hawking evaporation process, the rate for which will keep increasing with time. Accretion will never be able to keep pace with this runaway evaporation. However, a black hole formed with an initial mass M0 Mc will keep growing due to accretion of radiation till enough radiation density is available. The critical mass Mc corresponds to the Hawking temperature TBH c given by Mc
π2 f g T (4.10) 30σ Thus a black hole with Hawking temperature lesser than TBH c , will be in an accreting phase. As stated earlier, the duration of this accreting phase will depend on the energy density of radiation which in turn is affected by the number density of black holes present. The energy density of the relativistic matter ρR will evolve both due to the expansion of the universe, as well as because of the absorption or emission by the black holes. The time-variation of ρR is thus given by d ρR ta4 t M NBH a t (4.11) dt TBH c
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When the right-hand side of Eq.(4.3) vanishes, it yields the usual result ρR a4 . This happens when either no black holes are formed, i.e., NBH 0, or when the individual black hole masses do not change with the cosmic time. In our simplistic analysis let us assume that at time t0 a certain fraction β of the total energy density of the universe is trapped in NBH number of black holes, all of which have approximately the same mass M0 . The remaining fraction 1 β of the energy goes into relativistic particles. We assume a small value of β corresponding to a radiation dominated era at t0 . Denoting ρBH as the sum of the energy densities of all the black holes, one has ρ t0 ρR t0 ρBH t0
(4.12)
π2 g 4 T 30
(4.13)
ρBH t0 βρ t0 M0 nBH t0
(4.14)
ρR t0 1 β ρ t0
Let us now write down together the equations governing the cosmological evolution of the universe beyond the time t0 given by the hubble expansion Eq.(4.1), the evolution of radiation density Eq.(4.11), and the rate of change of black hole mass Eq.(4.3), using Eqs.(4.12–4.14), i.e., a 2 8π M βρ t0 a3 t0 ρ (4.15) R a2 3M42 M0 a3 ρR
4ρR
M
a a
As M44 M2
M βρ t0 a3 t0 M0 a3
16πf M 2 ρR M44
(4.16)
(4.17)
The above coupled Eqs.(4.15),(4.16) and (4.17) can be numerically integrated for relevant initial conditions at t0 . This analysis has been performed by several authors.17,18,23 The generic results can be summarized as follows. If one chooses a sufficiently small value of β, and the initial black hole masses greater than the critical mass Mc , then accretion of radiation is the dominant effect initially. This causes the energy density of radiation to decrease rapidly. Hence the temperature of the universe falls down at a faster rate than that arising out of the usual expansion of the scale factor. (For larger values of β, the universe may enter an early and
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non-standard matter (black hole) dominated epoch. During such a phase accretion by the black holes becomes increasingly insignificant because of very rapidly falling radiation density. We will describe such an early black hole dominated expansion in some details within the context of braneworld cosmology in Section 3). Though the accretion of radiation is unable to increase substantially the mass of the black holes (unless one has an extremely small β), it can cause considerable increase of black hole lifetime. Depending on the initial mass M0 , at some stage the black holes enter a quiescent phase where their mass stays nearly constant since both the accretion and the evaporation rates are negligible. During this stage the rate of hubble expansion is slightly increased compared to the standard radiation dominated expansion. The black hole evaporation rate then slowly begins to rise and this causes reheating of the universe. Towards the end of their lifetimes the black holes emit heavy particles which could produce the baryon asymmetry of the universe.17–19,23 However, if the black holes are formed with suitably larger initial mass, they could survive up to later eras, possibly even contributing a fraction to the cold dark matter density of the universe.28 The initial mass M0 of course determines the actual cosmological epoch when the black holes evaporate out. It is expected that black holes are produced with an initial mass spectrum in any realistic scenario of black hole formation.3,4,6 A mass distribution could introduce interesting peculiarities into the dynamics at particular epochs over and above the simplified scenario considered by us. Heavier black holes that are formed later are likely to hinder accretion of the smaller one produced earlier. (This specific feature will be discussed in the context of braneworld black holes in Section 4). A distribution function could be defined in general quantifying the mass and energy fraction in black holes formed at different times. The effect of a mass distribution can be incorporated into the cosmological evolution (Eqs.(4.15),(4.16) and (4.17)) by introducing an additional differential equation for the distribution function and specifying further initial conditions related to the distribution function (see Ref. 18 for details). Of course, the mass distribution should be such as to ensure that the closure of the total energy density has not taken place at any epoch. This consideration leads to bounds on the initial mass spectrum. Moreover, since the black holes radiate most of their energies at the very end of their lifetimes, cosmological processes operating at that particular epoch are affected. Therefore known observations can be used to impose further bounds on the initial distribution. A number of such analysis have been carried out,27 but without
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taking into account the process of accretion. The accretion of radiation by primordial black holes affects the dynamics of cosmological evolution even more significantly in the braneworld context, as we will see in Section 3.4. 4.3. Black Holes in Alternate Gravity Theories Obtaining the gravitational field due to a localized matter distribution on the brane has been an involved and challenging task right since the inception of braneworld models. This question is the forbearer of the problem of finding the final state of gravitational collapse on the brane, which is of central importance concerning the existence of black hole solutions in the braneworld scenario. The process of gravitational collapse in the braneworld scenario is much complicated compared to general relativity because in the former case whereas matter is confined to the brane, the gravitational field can also access the extra dimension. The corrections to the Newtonian potential of a point mass M at large distances due to the extra dimension were calculated to be44,45 2M 2l2 V r 2 1 2 (4.18) M4 r 3r The failure of current experiments to detect such corrections at submillimeter scales have set the upper limit on the curvature radius l of the 5-th dimension as l 0.2mm.36 The effect of Kaluza-Klein modes on the metric exterior to a static and spherically symmetric matter distribution on the brane was considered by Garriga and Tanaka.45 They obtained a solution in the weak field limit given by 2M 4M l2 2M 2M l2 2 dt 1 dr2 r2 dΩ2 dS42 1 2 M4 r 3M42 r3 M42 r 3M42 r3 (4.19) Note that this solution is quite different from the Schwarzschild metric and that the gravitational potential obtained from this metric (4.19) has 1r3 corrections compared to the Newtonian potential. Further perturbative studies46,47 have also established that the first weak field correction to the Newtonian potential on the brane is proportional to 1r3 . The projected Weyl term Eµν on the brane carries the imprint of Kaluza-Klein modes that could be relevant in the process of gravitational collapse. If the Weyl term vanishes, then the standard Schwarzschild solution in 4 dimensions can be assumed as the simplest black hole solution on
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the brane by ‘stacking’ it into the extra dimension. Such a vacuum solution of the 4-dimensional Einstein equation is of the ‘black string’ type,48 and can be generalized to the case of a cosmological constant in 4 dimensions as well.49 Subsequently, it was shown that the black string is unstable to large-scale perturbations.50 Another solution to the vacuum 4-dimensional Einstein field equations is obtained by setting the 4-dimensional cosmological constant to zero thus obtaining a relation between the brane tension and the Ads radius. Since the projected Weyl tensor on the brane is divergence free for the vacuum case, one gets for static solutions a closed system of equations given by51 Rµν
Eµν
Rµµ ∇ Eµν µ
0 0
(4.20)
Dadhich et al.54 have prescribed the mapping of the 4-dimensional general relativity solution with traceless energy momentum tensor of the EinsteinMaxwell type to a vacuum braneworld solution in 5 dimensions with the correspondence κ2 Tµν
Eµν
(4.21)
An exact black hole solution to the effective field equations on the brane of the Reissner-Nordstrom type was given with the above correspondence (4.21) as54 1 2M Q 2M Q dS42 1 2 2 dt2 1 2 2 dr2 r2 dΩ2 (4.22) M4 r r M4 r r where Q 0 is not the electric charge of the conventional ReissnerNordstrom metric, but the negative ‘tidal charge’ arising from the projection on to the brane of the gravitational field in the bulk. Since the black hole mass M is the source of the bulk Weyl field, the tidal charge Q could be viewed as the reflection back on the brane of the gravitational field of M by the negative AdS5 bulk cosmological constant. In the limit r l, it can be shown that54,55 Q
Ml M42
(4.23)
The bulk tidal charge thus strengthens the gravitational field of the black hole. It has been further argued56 that since the back reaction of the bulk onto the brane strengthens gravity on the brane, the formation of a
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black hole as result of gravitational collapse is favored as against a naked singularity. The metric with negative tidal charge (4.22) has a spacelike singularity and one horizon given by M QM44 (4.24) rh 2 1 1 M4 M2 which is larger than the Schwarzschild horizon. So the bulk effects are seen to increase the entropy and decrease the temperature of the black hole. A more general class of spherically symmetric and static solutions to the field equations with a 5-dimensional cosmological constant can be derived by considering a general line element of the type ds2
A
rdt2 B rdr2
r
2
dΩ2
(4.25)
and relaxing the condition A r B 1 r used while obtaining the Schwarzschild or the Reissner-Nordstrom metrics. Casadio et al.57 obtained two types of solutions by fixing either A r or B r, and then demanding the correct 1r asymptotic behavior for the other in terms of the post Newtonian (PPN) parametrization. In the first case, the choice A r 1 2M M42 r leads to the metric ds24
1
2M 2 dt M42 r 1
3M 1 2M 2r 4 M 4β 1 2M 1 2 2 M r 2M r 4
r
2
dΩ2
(4.26)
4
in terms of the PPN parameter β which impacts the deflection and time delay of light.58 Note that the above metric was also derived as a possible geometry outside a star on the brane.59 The solution (4.26) is of the temporal Schwarzschild form having a horizon rh 2M M42 . The corresponding Hawking temperature is given by57 1 6 β 1 (4.27) TBH 8πM Thus, in comparison with Schwarzschild black holes, the black hole (4.26) will be either hotter or colder depending upon the sign of β 1. Alternately, the choice for B r of the form B 1 r 1 2γM M42r, in terms of the PPN parameter γ, yields the line element 2 1 2M dr2 2 2 2 dt2 r dΩ (4.28) dS4 2 γ 1 1 2 2M γ M4 r 1 M 2r 4
This form of the metric represents a nonsingular wormhole, and has been discussed earlier in the literature in the context of 4-dimensional general
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relativity.60 Wormhole solutions in the braneworld context have been discussed by Bronnikov et al.61 Furthermore, a class of static, spherically symmetric and non-singular braneworld solutions with horizon have been obtained62 by relaxing the vanishing scalar curvature condition (4.20) used to obtain the solutions (4.22),(4.26) and (4.28). Stationary solutions representing charged rotating black holes have also been found recently.63 The arbitrariness of the projected bulk Weyl term Eµν and its geometric origin is at the root of the variety of braneworld black hole and wormhole solutions since both the functions A r and B r in Eq.(4.25) have to be determined by it.64 A specific configuration for the Weyl term with a negative equation of state has been considered and the resultant geometry with a singular horizon has been worked out to provide one more example of a possible braneworld black hole solution.65 The black hole solution (4.22) exhibits a 1r2 correction to the Newtonian potential on the brane in contrast to the weak field correction of 1r3 as in the solution (4.19). The solution with tidal charge (4.22) is reflective of the short distance or strong gravity limit where the 1r2 correction to the gravitational potential may even dominate over the 1r. This corresponds to the fact that at short distances, braneworld gravity is truly 5-dimensional. For short distances r # l it is natural to consider the 5dimensional Schwarzschild solution as a braneworld black hole candidate given by66 ds25
1 2 r2 rBH 2 1 BH dt 1 dr2 r2 dΩ23 2 2 r r
(4.29)
where the horizon size r0 is so small (r0 # l) so that the black hole effectively “sees” all the spatial dimensions on the same footing. A generalization to higher dimensions37,67 of the hoop conjecture leads to the above form of the metric as a static solution to collapsing matter on the brane. Near the event horizon, the black hole would have no way of distinguishing between the bulk dimension and the braneworld ones. Numerical simulations for scales sufficiently small compared to the AdS scale l seem to also support the existence of static solutions satisfying the AdS5 boundary conditions.68 The induced 4-dimensional metric on the brane near the event horizon of the 5-dimensional black hole (4.29) is obtained by integrating out the extra dimension to be 1 2 2 rBH rBH 2 2 dt 1 2 dr2 r2 dΩ2 (4.30) dS4 1 2 r r
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This 4-dimensional metric is different from the standard 4-dimensional Schwarzschild solution as it reflects the 5-dimensional character of the strong gravitational field near the black hole horizon in the form of the 1r2 gravitational potential. It is however expected that far from the event horizon the metric (4.30) would approach the standard 4-dimensional Schwarzschild form, as was shown explicitly in 2 1 dimensional braneworld framework.69 The properties of small black holes with the geometry given by Eq.(4.30) have been studied extensively by Argyres et al.37 In general these black holes have lesser temperature and a longer lifetime compared to the standard 4-dimensional Schwarzschild black holes. Also since the 5-dimensional Planck mass could be much lower compared to the 4dimensional Planck mass (M5 # M4 ), these black holes could be produced in particle accelerators70 and cosmic ray showers.71 They could thus provide one avenue of testing higher dimensional or braneworld physics. Of course, the consequences of a population of primordial black holes of the type (4.30) are potentially rich during many different cosmological eras, and these will be described in details in section 4. The study of black holes in the background of a cosmological constant, i.e. in the de Sitter space time, have also attracted some interest for their conjectured relevance in AdS/CFT correspondence,72 and due to the phenomenon of black-hole anti-evaporation.73 The dilaton black hole solution in a de Sitter universe has been recently derived.74 The dilaton of the GMGHS black hole is a scalar field occurring in the low energy limit of the string theory. It has an important role on the causal structure and the thermodynamic properties of the black hole. Though it was shown earlier that a dilaton black hole solution in the de Sitter background is not possible with a simple dilaton potential,75 Gao and Zhang74 circumvented this problem with a dilaton potential being a combination of Liouville type terms. Such a black hole solution is interesting since it combines stringy features with the backdrop of the present accelerated expansion of the universe. The action for a dilaton-de Sitter black hole can be written as 4 S d4 X g R 2 µ φ µ φ λ 3 λ 2φφ0 2φφ0 . e e (4.31) 3
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In terms of the Hubble parameter H, with H 2 λ3 , the metric is given by 2M 2 r r 2D H dt2 dS 2 1 r 1 2M 2 1 r r 2D H dr2 r r 2D dΩ2 (4.32) r For H 0 the metric goes to GMGHS black hole.76 For both D 0 and H 0 it reduces to the well known Schwarzschild metric. Also when φ φ0 0 the action of this space-time reduces to the action of a ReisserNordstrom-de Sitter black hole. It may be noted here that the solution for a dilaton-de Sitter black hole corresponds to a particular choice of the dilaton potential. Other solutions corresponding to other potentials with minima φ0 at which V φ0 0 are possible, but here we only consider the solution which is written74 analytically in a closed form. 4.4. Braneworld Cosmology with Black Holes The cosmology of the RS-II model entails a modified high energy phase in the early radiation dominated era of the universe during which the right hand side of the Einstein equation contains terms that are quadratic in the brane energy momentum tensor.34 Other modifications include the so-called “dark-energy” term which is given by the projection of the bulk Weyl tensor. Transition to the standard radiation dominated era takes place when t tc l2. Such a modified high energy evolution has rich consequences for the physics of the early universe.79 In particular, the inflationary scenario is altered, allowing the possibility of steep inflaton potentials to accomplish the desired features.80 Constraints on the duration of the brane dominated high energy phase are enforced by the necessity of conforming to the standard cosmological observational features such as nucleosynthesis and density perturbations. Let us first review briefly some of the important features of the RS-II braneworld cosmology. The effective 4-dimensional Einstein tensor on the brane is given by34 Gµν
8π τµν M42
κ
4
Πµν
Eµν
(4.33)
where τµν is the brane energy-momentum tensor; Πµν is quadratic in the brane EM tensor; and Eµν is the projection of the 5-d Weyl tensor. The
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4-dimensional Planck’s mass M4 is related to the gravitational coupling con8π κ2 stant κ and the AdS length l by M 2 l . For the Friedmann-Robertson4 Walker metric on the brane, the Friedmann equation is given by 2 8π ρ Λ4 k ρ ρKK (4.34) H2 3M42 2λ 3 a2 with H being the Hubble constant, ρ the energy density, and k 1, 0, 1 representing open, flat and closed branes, respectively. ρKK is the effective energy density coming from the bulk Weyl tensor, λ 3M56 4πM42 is the brane tension, and Λ4 the effective 4-d cosmological constant on the brane. The AdS curvature radius l is given by the bulk cosmological constant Λ5 and 5-d Planck mass M5 as Λ5 3M53 4πl2 . The induced 4-d cosmological constant Λ4 is given by 6 M5 1 (4.35) Λ4 3 M44 l2 Setting Λ4 0, one obtains a relation between the brane tension and AdS radius given by 14 12 4π l l4 (4.36) λ14 3 l4 Nucleosynthesis constrains the “dark energy” term ρKK to be negligible compared to the radiation density ρ.81 In the following analysis we will neglect ρKK . Assuming a radiation dominated equation of state, the k 0 solutions for the Friedmann equation are given by ρR
for the energy density, and
a a0
3M42 32πt t tc
t t tc t0 t0 tc
(4.37)
14 (4.38)
for the scale factor a during the radiation dominated era, and where tc l2 effectively demarcates the brane dominated “high energy” era from the standard radiation dominated era. For times earlier that tc , i.e., t tc (or ρ λ), one has the non-standard high energy regime during which the radiation density and the scale factor evolve as ρR
3M42 32πtc t
(4.39)
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14 (4.40)
respectively. As a consequence, the time-temperature relation also gets modified during the brane dominated high energy era, i.e., T t14 . But, for times much later than tc , i.e., t tc (or ρ λ), one gets back the standard radiation dominated cosmological evolution given by ρR and
a a0
3M42 32πt2 t
t0 tc 12
(4.41)
12 (4.42)
Our purpose here is to explore the evolution of primordial black holes in this non-standard braneworld scenario. The formation and evolution of black holes is an interesting and complex issue of investigation in braneworld cosmology. Horizon sized density perturbations in the high energy phase could lead to the formation of black holes on the brane by the process of gravitational collapse. The geometry of such black holes is inherently 5-dimensional.66 There have been several recent efforts on understanding different types of black hole solutions on the brane.37 In particular, black holes that are formed out of the collapse of horizon sized density perturbations, have been argued to have a 5-dimensional Schwarzschild metric, which when projected on the brane takes the form of a modified 4-dimensional Schwarzschild solution.38 The horizon radii of such black holes is proportional to the square root of their masses, a feature that modifies the Hawking temperature, and consequently slows down the evaporation process.38 Note however, that the behavior of large black holes that are super-horizon sized having radius much greater than the Ads curvature radius could be completely different inasmuch as they may evaporate out rapidly as a consequence of Ads-CFT correspondence.52 We will restrict our attention here to small sub-horizon sized black holes only. Since the 5-dimensional fundamental scale could be several orders of magnitude below the Planck scale, a lot of current excitement stems from the possibility of braneworld black holes being produced in high energy particle collisions either in the laboratory, or in cosmic ray showers.70 Such processes could also be responsible for producing primordial black holes out of the thermal plasma in the early universe. In this
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article, we will not explore the details of any particular mechanism of primordial black hole production. In fact, a comprehensive analysis of the formation of primordial black holes in the braneworld scenario remains to be performed. Our analysis here deals with the evolution of black holes once they are formed in the high energy braneworld regime. To this end, let us first consider the evolution of a single primordial black hole which is formed with a sub-horizon mass in the high energy radiation dominated era. Since we are considering the 4-d projection of the 5-d Schwarzschild metric, such a black hole will have an altered mass-radius relationship given by38 12 12 12 8 l M rBH l4 (4.43) 3π l4 M4 The black hole will accrete the surrounding radiation with a rate proportional to the surface area of the black hole times the energy density of radiation. The Hawking evaporation rate, as for the case of standard black holes, is proportional to the surface area times the fourth power of temperature.53 The Hawking temperature is given by TBH
1 2πrBH
(4.44)
Taking into account these effects of accretion and evaporation together, the rate of change of mass M of a braneworld black hole is given by 2 4 M 4πrBH (4.45) gbrane σTBH f ρR where gbrane is effective number of particles that can be emitted by the black hole (we assume that the black holes can emit massless particles only and take gbrane 7.2538 ), f is the accretion efficiency (0 f 1),12 and σ is the Stefan-Boltzmann constant. The black hole also evaporates into 2 5 gbulk TBH . However, this term the bulk, with a rate proportional to 4πrBH 38 is subdominant even for very small black holes, and has negligible effect on their lifetimes. Substituting the expressions for the black hole radius (Eq.(4.43)), the temperature-radius relation (4.44), and the energy density of radiation (4.39), the black hole rate equation (4.45) in the radiation dominated high energy braneworld era can be written as11 M
AM42 M tc
BM t
(4.46)
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where A and B are dimensionless numbers given by 3 A 163 π 2f B π The exact solution for the black hole rate equation is given by11 2B 12 2AM42 t0 t 2AM42 t 2 M t M0 2B 1 tc t0 2B 1 tc
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(4.47) (4.48)
(4.49)
with M0 being the formation mass of the black hole at time t t0 . If the black hole is formed out of the collapse of horizon or sub-horizon sized density perturbations, the formation time and mass are related by38 12 12 t0 1 M0 l (4.50) t4 4 M4 l4 It was argued in Ref. 11 that a black hole so formed continues to grow in size by the accretion of radiation during the high energy radiation dominated era, with its mass given by11,12 B M t t (4.51) M0 t0 One needs to next consider, as in the case of primordial black holes in standard cosmology, a certain number density of primordial black holes exchanging energy with the surrounding radiation by accretion and evaporation. As done in Section 2, we will again use the simplifying assumption that all black holes are formed with an average initial mass M0 . We thus assume that at a time t0 the fraction of the total energy in black holes is β, and the number density of black holes is nBH t0 . Hence, one has to now solve the coupled cosmological equations for the radiation density ρR t, the matter density in the black holes M tnBH t, and the scale factor a t (the counterpart of the Eqs.(4.15), (4.16), and (4.17) in Section 2) for the braneworld scenario. Notwithstanding the absence of such an analysis in the literature till date, one could still gain certain interesting insights into the cosmological evolution, by making a few simplifying assumptions,11,12 as we now elaborate. The number density of black holes nBH t scales as a t3 , and thus for a radiation dominated evolution on the brane, one gets nBH tnBH t0 t0 t34
(4.52)
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since a t t14 . The net energy in black holes grows since accretion dominates over evaporation. The condition for the universe to remain radiation dominated (i.e., ρBH t ρR t) at any instant t can be derived to be11 β
t0 tB 14 1 t0 tB 14
(4.53)
If the value of β exceeds the above bound, there ensues an era of matter (black hole) domination in the high energy braneworld phase. Such a phase of matter domination should definitely be over by the time of nucleosynthesis for the cosmology to be viable. We shall first explore the situation when the cosmology stays radiation dominated up to the time when brane effects are important, i.e., t tc . From Eq.(4.53), this requires B 14 β t0 (4.54) 1 β tc Further, the black holes should remain small enough, i.e., M M4 3π 4 tt4 for the 5D evaporation law to be valid.38 These criteria can be used to put an upper bound to the average initial mass11 22B M0 3π tc (4.55) B M4 t4 4 2 2 The growth of the black holes in the radiation dominated era due to accretion slows down with time, since the surrounding radiation density gets diluted. The rate of evaporation is also insignificant for a wide range of M0 at this stage since the black hole masses could have grown by several orders of magnitude from their initial values. It is expected that there ensues an era during which the black hole mass stays nearly constant over a period of time, as is the case for standard cosmology (Section 2). The accretion rate is smaller for the braneworld case since the surface area is M instead of M 2 for 4D black holes. Moreover, the evaporation rate is ( M 1 ) instead of M 2 . Hence, the black hole mass will stay for while near a maximum value Mmax reached at time tt before evaporation starts dominating. The expression for the lifetime tend of a black hole in this scenario is given by11 2B B tend 4 B M0 tc t2t 2 2 (4.56) t4 A M4 t4 tc t4
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It is important to note that the modified evaporation law also contributes to the increased lifetime for braneworld black holes.38 However, the effect of accretion is more significant, as can be seen by comparing the lifetime of a 5-d black hole in the presence of accretion to the the lifetime due to purely an altered geometry.11 Depending upon the values taken by the parameters tc and l, one could obtain several interesting examples of primordial black holes surviving till various cosmologically interesting eras.11,12 One particular choice worth mentioning is that of a black hole formed with an initial mass M0 108 M4 103 g, that will survive up to the present era if one chooses ll4 1030 . Let us now explore the case of matter (black hole) domination in the high energy phase. The onset of such a matter dominated era is derived to be11 4B41 theq 1 β γB (4.57) t0 β B The mass of a black hole at theq is given by M theq M0 γB . For t theq the Hubble expansion is essentially driven by the black holes (p 0) which dominate over radiation. Since the number density of black holes scales as matter (nBH t a3 ), for t tc one has H ρBH , and thus the scale factor grows as
a t t13
(4.58)
During this era, the radiation density ρR is governed by the equation d 4 ρR ta t M tnBH ta t (4.59) dt where the contribution from accretion black holes is comparable to the normal redshiting term (ρR a4 ) because at this stage the black holes dominate the total energy density. Some further analysis leads to the following expression for the radiation density11 1B theq βB t0 14 1 γ ρ t0 ρR t γB ρ t0 (4.60) t B 1 3 B t The black hole mass grows as 13 B C B 14 theq t 14 C B M t M0 γB exp 3B γB 3B γB (4.61) B t B t0
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where C BβB
13 . As expected, the accretion regime lasts for a brief duration in the matter dominated phase, beyond which the black hole evaporation starts to play a significant role. The universe gets reheated as ρR t increases with time. The stage of black hole domination lasts up to a time tr (ρR tr nBH tr M tr ). Subsequently, radiation domination takes over once again. One can derive 2 3 A M4 tc
(4.62) 2B tr 2δγB M0 4γB with
δ
B 1 3 1 β
3β31 (4.63)
A stringent restriction is imposed by demanding that the standard low energy cosmology (for t tc ) should emerge as radiation dominated. By requiring that the era of black hole domination be over before tc , i.e., tr tc , one gets a lower bound on β from Eq.(35), i.e., B 14 3B31 4t0 B 1 3 (4.64) β 3tc The evaporation time of black holes in this scenario has also been calculated.11 The black hole lifetime is given by 2 tend M0 tc 2B
γ (4.65) t4 M4 t4 B The effect of accretion is less significant in prolonging black hole lifetimes in this case, as borne out by the following example. For instance, taking the value of the AdS radius to be ll4 1020 , and β 103 , black holes with M0 1012 g evaporate during the present era. This is only to be expected, since early matter domination results in larger hubble expansion rate which further restricts the availability of radiation for the black holes to accrete. 4.5. Observational Constraints and Binary Formation We have seen that it is possible to have primordial black holes formed in the high energy era of the braneworld scenario to survive up to several cosmologically interesting eras. Accretion of radiation during the radiation dominated era is primarily responsible for the increased longevity of
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braneworld black holes, though their 5-d geometry also contributes to a slower rate of Hawking evaporation. It is also worth recalling here that primordial black holes in certain models could also accrete the energy of a cosmological scalar field.13 Such an effect could lead to further growth of these black holes beyond the early radiation dominated era, thus pushing up their lifetimes further. As mentioned repeatedly, the implications for a population of primordial black holes in cosmology are diversely manifold. At any particular era, the surviving black holes would contribute a portion to the total energy density as dark matter in the universe. If the black holes are produced with an initial mass spectrum, then one would have evaporating black holes at different eras. Hawking radiation from these evaporating black holes would on one hand produce all kinds of particles including heavier ones which could lead to baryogenesis, and on the other, could contribute significantly to the background photons, thus diluting the baryon to photon ratio. Observational constraints impacting different cosmological eras could be used to impose restrictions on the initial mass spectrum of braneworld black holes in a manner similar to the primordial black holes in standard cosmology. Clancy et al.39 have shown how standard constraints are modified in the case of braneworld cosmology. There have been further studies on the impact of braneworld primordial black holes on the high energy diffuse photon background and cosmic ray antiprotons.77 An interesting question is whether primordial braneworld black holes could contribute to a significant fraction of cold dark matter. To simplify the treatment, one can assume that radiation domination persists up to tc , and that the accretion of radiation is possible only up to tc . A black hole mass fraction αM in terms of the radiation density can be defined (related to the mass fraction βBH ) as ρBH t ρT βBH 1 βBH
αM0 t αM0 t0
(4.66)
Similarly, a ‘final’ mass fraction α tevap is defined at the end of the black hole life-cycle, since the black holes radiate most of their energy towards the very end of their lifetimes. Observational constraints on α tevap are considered at different cosmological epochs, given by α t L4D t
(4.67)
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for standard 4-dimensional black holes, or α t L5D t
(4.68)
for braneworld black holes. Then these constraints are evolved backwards to constrain initial mass spectrum. Accretion in the high-energy phase leads to α t MBH ta t. Therefore the constraints on the initial black hole mass fraction are given by a0 L04D αi 4D L4D t (4.69) a t for standard cosmological evolution, and a0 5D L5D t L05D αi a t
(4.70)
in the braneworld scenario. Any astrophysical or cosmological process to be constrained at certain epoch is dominantly affected by the PBHs with lifetimes of that epoch. The constraints on primordial black holes in standard cosmology thus get modified to39 51616B 8B L05D L5D tevap l (4.71) 0 L4D L4D tevap lmin
1 3
where lmin tevap . The departure from standard constraints is sensitive to the accretion efficiency, as is expected. A more detailed study on how the initial mass spectrum of the black holes is distorted due to braneworld accretion has been undertaken by Sendouda et al.77 The diffuse photon background emitted by the spectrum of black holes has been shown to be modified in accordance with the mass spectrum of the black holes. These results have been compared to the observed diffuse photon background to obtain bounds on the initial black hole mass fraction, the scale of the extra dimension, and the accretion efficiency by these authors. The observed number density of massive particles could also be used to obtain bounds on the initial mass fraction of the black holes, as in the case of Schwarzschild primordial black holes in standard cosmology. Further constraints on the scale of the extra dimension have been derived77 by considering the recent observation of sub-Gev galactic antiprotons as originating from braneworld black holes present in our galaxy. A relevant issue is to investigate whether primordial braneworld black holes could contribute to a significant fraction of cold dark matter. If so, the direct searches of cold dark matter compact objects through gravitational lensing might be able to reveal their presence in galactic haloes.
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We now discuss a scheme of binary formation for primordial braneworld black holes. We consider sub-horizon sized black holes in the RSII model formed during the high energy phase. We study the interaction of two or more neighbouring black holes mediated by the surrounding radiation in the radiation dominated high energy era. These black holes exchange energy via the processes of evaporation and accretion with the radiation bath, and through it, with each other. The physical distance between two such neighbours increases initially with the Hubble expansion. Forming out of horizon collapse, the initial mass ratio of two such black holes is on average proportional to the square of the ratio of their formation times (From Eq.(4.50)). By sudying the effect of the interaction term it is possible to show that an initial mass difference between two such black holes can never decrease during the radiation dominated era.40 Such mass differences facilitate the formation of binaries during the standard low energy phase via three-body gravitational interactions. A formation mechanism for primordial black hole binaries in standard cosmology has been investigated by Nakamura et al.78 They showed that if equal mass primordial black holes are inhomogeneously distributed in space, three-body gravitational interactions could lead to the formation of binaries. The scheme of binary formation discussed here could become more effective when combined with the scheme based on spatial inhomogeneities.78 Consider the evolution of two neighbouring black holes, BH1 and BH2, having masses M1 and M2 respectively, with initial masses M10 M20 , or t10 t20 . BH2 forms at a physical distance d0 from BH1. Their physical separation d grows at a rate t14 . We shall assume that d0 l, and that the gravitational potential φ M M42d1 2l2 3d2 # 1
(4.72)
so that exchange of energy between them via gravitational waves can be neglected during the high energy regime. Both of these black holes are immersed in the radiation bath whose density ρR falls off as given by Eq.(4.39). In isolation, the change of mass of each black hole is effected by the net sum of its Hawking evaporation depending on its temperature, and the accretion of radiation depending on the radiation density in the universe as given by Eq.(4.46). However, due to the presence of BH2, the net rate of change of the mass M 1 of BH1 gets a further contribution proportional to the product of M 2 and the solid angle subtended by BH2 on BH1, and viceversa. In other words, BH1 feels the local difference in radiation density as a result of an accreting (or evaporating) BH2, and similarly for BH2. Thus
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the evolution equation for each of them can be written as
BMi AM42 ri2 M j (4.73) t M i tc 4d2 for i, j 1, 2, where the Schwarzschild radius at the time of formation is given in terms of the formation mass by Eq.(4.43), and A and B are given by Eqs.(4.47). The first and second terms on the r.h.s of Eq.(4.73) represent accretion from the average radiation density, and evaporation respectively (section 3), and the last term arises from the local inhomogeneity in radiation density due to the j-th black hole. Considering both BH1 and BH2 to be in their accreting phases at the formation time t02 of BH2, it is possible to see form Eqs.(4.73) that for BH1 the first term is comparable to the interaction (third) term when dr2 t t10 t t20 12 . Thus, for a while after its formation, BH2 is able to suppress the growth of BH1. If on the other hand, BH2 forms at a time when BH1 begins to evaporate, BH2 registers enhanced growth due to the locally denser radiation coming from the evaporating BH1. The effect of two or more interacting black holes on their evolution has been studied in Ref. 40. It was shown that interaction with the surrounding radiation always leads to disequilibration of masses, viz., an initial small mass difference between two neighbouring black holes always increases in the high energy radiation dominated era. Some initial mass differences are always to be expected since the black holes formed out of density perturbations at different times have masses given by Eq.(4.50). Further, any realistic spectrum of initial masses will automatically lead to mass differences for two neighboring black holes formed even during the same era. It was shown40 that their mass difference at the end of the high energy era always exceeds their initial mass difference. This is a key result that facilitates binary formation, as follows. When the universe exits the high energy phase (t tc ), let the masses of BH1 and BH2 be M1 and M2 respectively. Consider a spherical region of radius d in which the average matter density (contributed by BH1 and BH2) is given by M i
3 M1 M2 (4.74) 4πd3 As radiation density in this region falls off like a4 , the spherical region of radius d will become matter dominated at a time tf given by (using Eqs.(4.39), (4.40) and (4.74)) 3 6 2B 12 8 d0 t0 tc (4.75) tf 2 3π r20 tc ρ¯BH
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When t tf the matter dominated region with radius d gets cuts-off from the surrounding radiation dominated expansion of the universe which continues up to teq $ tc . For t tf , the black holes BH1 and BH2 will form a bound system. The size of this bound system df is related to the initial black hole separation d0 by40 3B 3 1 tc d0 df d0 (4.76) 16 t0 l Nakamura et al78 formulated a scheme of binary formation based on the 3-body gravitational interaction of primordial black holes which are spatially inhomogeneeously distributed in standard cosmology. The scheme that we discuss here is based on mass differences for primordial braneworld black holes. It is worth noting that in any realistic situation there will both be mass and spatial inhomogeneities, and binary formation proceeds as a result of 3-body interaction aided by the combination of both kinds of inhomogeneities. However, to simplify the analysis here we consider the effect of mass differences only and assume a uniform spatial distribution of black holes. Recall that interaction via the surrounding radiation results in mass differences between neighboring black holes. Consider three neighboring black holes to have mass ratios at tc given by M1 δ12 M2 and M2 δ23 M3 with δ12 , δ23 1. Such a scenario will arise generically from initial mass differences due to horizon sized collapse at different eras, and possible interactions with other black holes. The region encompassing BH1 and BH2 will decouple from the background Hubble expansion at a time that can be computed using Eq.(4.75), when BH1 becomes gravitationally bound to BH2 with an elliptical orbit whose major axis af is af df
(4.77)
BH3 provides the required tidal force for a stable orbit. The minor axis bf of the binary is proportional to the tidal force times the squared free fall time, which is given by 3 2δ12 δ23 df bf af (4.78) 1 δ12 dP where dP is the distance of BH3 from the mid-point of the BH1-BH2 axis. Although BH1 binds to BH2 at time tf , mass differences hamper the possibility of BH3 being gravitationally bound to BH1. In fact, even if all the black hole separations are equal before binary formation, the total region encompassing the binary and BH3 will be matter dominated at a time much
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later than tf . BH3 continues to move away from the BH1-BH2 binary with the Hubble flow while dP registers growth by a factor tmd 13 tf 12 .40 The gravitational potential of the binary at BH3 is much weaker compared to the BH1-BH2 interaction. The role of the third black hole BH3 is to simply provide the tidal force necessary to prevent headlong collision of BH1 and BH2. The eccentricity of the binary orbit can be obtained from Eqs. (4.77) and (4.78). For uniformly distributed black holes formed around the era t0 with masses M0 , the eccentricity is given by
e¯ 1 η 2
2δ12 δ23 1 δ12
2 12 (4.79)
where η is a geometrical factor of order 1.78 The Newtonian approximation that we have used for describing the gravitational interaction between black holes forming a binary is valid if the gravitational potential for the BH1BH2 binary is such that M1 M42 # df 1 e
(4.80)
From Eqs.(4.76) and (4.79) it is possible to show that the Newtonian analysis is suitable for black holes which are formed at times greater than a certain formation time t0 given by t4 t0 2B 1
#
43 δ12 δ23 1 δ12 2 η 2 αm 0
(4.81)
where αm0 ρBH t0 ρR t0 . Present interest in the study of binaries stems from the emission of gravitational radiation during their coalescing stage. It is expected that future gravitational wave detectors would be sensitive to gravitational waves coming from binaries in the sub-lunar range.32 The above mechanism for binary formation is suitable for the formation of such binaries within a reasonable range of parameters, as was shown in Ref. 40. In addition, the scheme based on inhomogeneous spatial distribution78 of black holes could also act in conjunction to aid binary formation. The range of parameters could be further widened by considering accretion of radiation in the standard radiation dominated phase (section 2) and/or accretion of scalar field dark energy.13,20 This calls for further work in this area including accurate determinations of the coalescence time for various models and parameters.
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4.6. Observables of Strong Gravitational Lensing The bending of light due to the gravitational potential of a massive object is one of the first predictions of the general theory of relativity. Its application in the phenomenon of gravitational lensing82 has potentially diverse possibilities. Gravitational lensing in the weak field limit83 is till date one of the most widely used tool in observational astrophysics and cosmology. On the other hand, strong field gravitational lensing, though limited in observational utility because of presently inadequate instruments, remains our ultimate scope for exploring the physics of strong gravitational fields. The general technique for analysing strong gravitational lensing for spherically symmetric metrics has been developed by Bozza84,85 who has also formulated useful connections between observational quantities like fluxes and resolutions and the metric parameters. Strong gravitational lensing is also endowed with richer phenomenological features like relativistic images86,87 and retrolensing.88–90 It would be thus worthwhile to investigate if the results of strong gravitational lensing could be used for probing the modifications to general relativity made in braneworld geometries. Such studies are also motivated from the possibility of producing TeV scale black holes in future accelerators.70 Analysis of the trajectories of light and massive particles in various braneworld and higher dimensional metrics have been undertaken recently. Kar and Sinha91 obtained the bending angle of light for several brane and bulk geometries. Frolov et al92 found certain non-trivial features about the propagation of light in the Myers-Perry66 metric for a 5-dimensional black hole solution. This solution represents primordial black holes that could be produced with size r l in the early high energy era of the RS-II braneworld scenario.38 It was shown that such black holes could grow in size due to accretion of radiation, and consequently survive till much later stages in the evolution of the universe.11 With a suitable choice of parameters, some of these black holes could also exist in the form of coalescing binaries in galactic haloes at present times.40 The weak field limit of gravitational lensing was studied for the Myers-Perry metric and certain notable differences from the standard Schwarzschild lensing were found.41 Thereafter, Eiroa93 analysed the strong field lensing and retrolensing effects for the Myers-Perry black hole. Strong field gravitational lensing in a couple of other braneworld metrics has been discussed by Whisker94 and some lensing observables have been computed using parameters for the galactic centre black hole.
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For a spherically symmetric metric ds2
A
rdt2 B rdx2
C
r dΩ2
(4.82)
where the asymptotic forms of the functions A r and B r have the standard 1r form, and C r r2 asymptotically, the general formalism of strong field gravitational lensing has been worked out by Bozza.85 It is required that the equation C r C r
A r A r
(4.83)
admits at least one positive solution the largest of which is defined to be the photon sphere rph . A photon emanating from a distant source and having an impact parameter u will approach near the black hole at a minimum distance r0 before emerging in a different direction (see Figure 4.1). The closest approach distance is given in terms of the impact parameter by C0 (4.84) u A0 where the functions C and A are evaluated at r0 . The deflection angle of the photon in terms of the distance of closest approach is α r0 I r0 π
I r0
2 Bdr
r0
(4.85)
C
C A0 C0 A
(4.86) 1
The weak field limit is obtained by expanding the integrand in Eq.(4.86) to the first order in the gravitational potential. This limit however is not a good approximation when there is a significant difference between the impact parameter u and the distance of closest approach r0 , which occurs when A r0 significantly differs from 1, or C r0 from r02 . By decreasing the impact parameter, and consequently the distance of closest approach, the deflection angle increases beyond 2π at some stage resulting in one or more photonic loops around the black hole before emergence. Further decrease of the impact parameter to a minimum value um corresponding to the distance of closest approach r0 rph results in the divergence of the deflection angle (integral in Eq.(4.86)), which means that the photon is captured by the black hole. Strong field gravitational lensing is useful for studying the deflection of light in a region starting from just beyond
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the photon sphere up to the distance where the weak field approximation approaches validity. In the general analysis of strong field gravitational lensing it becomes necessary to extract out the divergent part of the deflection angle. In order to do so two new variables y and z are defined as85 A r y y0 z 1 y0
y
(4.87) (4.88)
where y0 A0 . In terms of these variables, the integral (4.86) in the deflection angle is given by 1
I r0
R z, r0 f z, r0 dz 0
R z, r0 f z, r0
(4.89)
2 By 1 y0 C0 CA 1 y0 1 y0 z y0 CC0
(4.90) (4.91)
where all functions without the subscript 0 are evaluated at r A1 1 y0 z y0 . The function R z, r0 is regular for all values of z and r0 , while f z, r0 diverges for z 0. The deflection angle can be written as a function of θ uDd , where θ is the angular separation of the image from the lens, and Dd is the distance S
I
D ds
α
Ds
u
M r0 Dd
δ
θ
O
Fig. 4.1. Gravitationl lensing for point like mass object M . A light ray from the source S passes the lens with an impact parameter u, and is deflected by an angle α. The observer sees an image I of the source at the angular position θ.
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between the lens and the observer (See Figure 4.1). In order to do so, an integral 1
bR
g z, rm dz
(4.92)
0
is defined, where g z, rm R z, rm f z, rm R 0, rm f0 z, rm The expression for the deflection angle is given as a function of θ as85 θDd α θ a log 1 b um R 0, rm a 2 βm 2βm b π bR a log ym
(4.93)
u D d
(4.94) (4.95) (4.96)
where R 0, rm and bR are given in Eqs.(4.90) and (4.92), respectively, and βm is defined as βm
Cm 1 ym
2
ym Cm A rm Cm 2 C 2 2ym m
(4.97)
In strong lensing there may exist n relativistic images given by the number of times a light ray loops around the black hole. The positions of these are obtained as solutions of the lens equation given by tanδ
tanθ
Dds tanθ tan α θ Ds
(4.98)
for specific positions of the source and the lens respective to the observer, and using the the value of the deflection angle from Eq.(4.94). The relativistic images formed by light rays winding around the black hole are highly demagnified compared to the weak field images. When the source, the lens and the observer are highly aligned, it is possible to obtain the most prominent of the relativistic images.86 Hence, the analysis of strong lensing is usually restricted to the case when both δ and θ are small,85 though the general case for arbitrary positions can also be analysed.90 With the above restriction on the values of δ and θ, a light ray will reach the observer after winding around the lens n number of times only if the deflection angle α is
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very close to a multiple of 2π. Substituting α one gets δ
θ
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2nπ ∆αn in Eq.(4.98),
Dds ∆αn Ds
(4.99)
The position of the n-th relativistic image θn can hence be obtained as a solution of the lens equation (4.99) as85 en um en δ um D1
Ds d um θn 1 en (4.100) Dd aDds Dd where um is the minimum impact parameter, and en is given by en
eb2nπa
(4.101)
The magnification µn of the n-th relativistic image is given by85 µn
1 δ θ δ θ
θ n
u2m en 1 en Ds aδDds Dd2
(4.102)
The above formula for magnification is valid under the approximation of a point source. However, for an extended source the magnification at the image position can also be derived by integrating over the luminosity profile of the source.89 It is useful to obtain the expressions for the various lensing observables in terms of the metric parameters. For n an observable θ can be defined85 representing the asymptotic position approached by a set of images. The minimum impact parameter can then be obtained as um
D d θ
(4.103)
In the simplest situation where only the outermost image θ1 is resolved as a single image, while all the remaining ones are packed together at θ , two lensing observables can be defined as85 S
θ1 θ
(4.104)
representing the separation between the first image and the others, and µ1 R (4.105) µn
n 2
corresponding to the ratio between the flux of the first image and the flux coming from all the other images.
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In terms of the deflection angle parameters a and b, these observables can be written as85 S
θ eba2πa
R e2πa
(4.106) (4.107)
The above equations (4.106) and (4.107) can be inverted to express a and b in terms of the image separation S and the flux ratio R. Therefore the knowledge of these two observables can be used to reconstruct the deflection angle given by Eq.(4.94). The aim of strong field gravitational lensing is to detect the relativistic images corresponding to specific lensing candidates and measure their separations and flux ratios. Once this is accomplished, the observed data could be compared with the theoretical coefficients obtained using various metrics. A precise set of observational data for strong gravitational lensing, if obtained, could therefore be able to discriminate between different models of gravity. In the braneworld scenario the computation of the above parameters has been performed for several metrics41,94 taking the black hole at the centre of our galaxy as a potential candidate. We will now consider examples of the various lensing quantities defined above for some of the possible braneworld black hole geometries discussed earlier. It is instructive to compare the braneworld lensing quantities with the standard Schwarzschild ones which for strong gravitational lensing are given as follows.85 Choosing the Schwarzschild radius rs 2M M42 as the unit of distance, the photon sphere is given by 3 (4.108) rph 2 for Schwarzschild lensing. The corresponding minimum impact parameter is 3 3 (4.109) um 2 The coefficients a and b defined in Eqs.(4.95) and (4.96) are given by a1 b
π 2log 6
2
3 log 6
(4.110) (4.111)
The deflection angle (4.94) is obtained in terms of the parameters a and b to be85 2θDd 1 log 216 7 4 3 π α θ log (4.112) 3 3
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In the weak field limit, the expression for the bending angle is given by 4M αweak 2 (4.113) M4 r0 For the analysis of gravitational lensing by braneworld metrics, let us first consider the Garriga-Tanaka weak field solution45 given by Eq.(4.19) in isotropic coordinates. The metric in terms of the standard coordinates can be written as 2 Ml2 2 1 MM2 r 3M 2 r3 2M 4M l 4 4 dr2 r2 dΩ2 ds24 1 2 dt2 2M 2Ml2 M4 r 3M42 r3 1 M 2 r 3M 2 r 3 4
4
(4.114) The formal expression for the radius of the photon sphere rph can be obtained as a function of M and l using Eq.(4.83) to be
313 rs2 2 3rs3 20l2 rs 2 10 10l4 rs2 3l2 rs4 13 3rs3 20l2 rs 2 10 10l4rs2 3l2 rs4 13 (4.115) 2313 where rs 2M M42 . However, it can be seen from Eq.(4.115) that no real solution for rph exists for admissible values of l and rs . This is to be expected since the metric (4.19) represents a weak field solution. The weak field limit of the bending angle was obtained to be91 rph
rs 2
αweak
4M M42 r˜0
4M l2 M42 r˜03
(4.116)
where r˜0 in this case is the isotropic coordinate equivalent of the distance of closest approach r0 in standard coordinates. Gravitational lensing in the weak field limit by the Myers-Perry66 braneworld black hole has been worked out.41 When the impact parameter u exceeds a few times the horizon radius given by Eq.(4.43), but is still lesser than l (u l), the application of weak field lensing could be relevant. The weak field limit of the deflection angle was calculated to be41 2M ll4 αweak (4.117) M4 r02 In order to satisfy the requirement that u l, and also obtain non-negligible magnification at the image location, the mass of the black hole should be such that41 M l (4.118) M4 l4
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Using the maximum allowed value for l by present experiments,36 one then obtains that M 108 M . So weak field gravitational lensing for such a black hole could be applicable only for masses in the sub-lunar range. As discussed above the braneworld scenario is conducive to the existence of primordial black holes in the sublunar mass range.11,12 If such black holes exist in our galactic halo, then the magnification of their weak field images turns out to be diminished compared to the standard Schwarzschild black holes of similar mass.41 The analysis of strong field gravitational lensing can however be performed in other braneworld geometries. For example, let us consider lensing in the metric with tidal charge (4.22) given by54 1 2M Q 2M Q dS42 1 2 2 dt2 1 2 2 dr2 r2 dΩ2 M4 r r M4 r r (4.119) which resembles the Reissner-Nordstrom metric, but with Q 0 in the braneworld context. Again using units of distance 2M M42, it is straightforward to obtain the expressions for the photon sphere and the minimum impact parameter given by85 3 9 32Q (4.120) rph 4 2 3 9 32Q (4.121) um 4 2 3 8Q 9 32Q The coefficients a and b in the deflection angle are given by85 rph rph 2Q a (4.122) 2 9Qr 2 3 rph rph ph 8Q 8Q 3 4 log 6 2 3 b π 2log 6 2 3 9 2 rph Q2 3 rph rph 9Qrph 8Q2 alog 2 (4.123) 2 r rph 2Q3 rph ph Q In terms of the above coefficients one obtains the complete expression for the deflection angle using Eq.(4.94). The weak field limit of the bending angle was derived to be91 1 3πQ 4M αweak (4.124) M42 16M r0 r0
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Note that the bending angle is always positive because of negative tidal charge Q unlike the electric charge of the Reissner-Noredstrom metric. We next consider the braneworld solution (4.26) in terms of the PPN parameter β given by57 ds24
1
3M 1 2M 2r 4 M 4β 1 2M 1 2 2 M r 2M r
2M 2 dt M42 r 1
4
r
2
dΩ2
(4.125)
4
For the above metric the expressions for the radius of the photon sphere rph and the minimum impact parameter are of course similar to those for Schwarzschild lensing given by Eqs.(4.108) and (4.109), as is easy to see using Eqs.(4.129), (4.83) and (4.84). The expression for the deflection angle can be derived in terms of the coefficients a and b which for the metric (4.125) are given by (setting the unit of distance as 2M M42)
3 6 4β 1
a
b π
(4.126)
2 3 3 log 6 2 3 log6 (4.127) 6 4β 1 6 4β 1
Kar and Sinha91 obtained the weak field limit of the bending angle for the metric (4.125) to be αweak
2M 1 β r0
(4.128)
Note that the standard Schwarzschild expressions are recovered in Eqs.(4.126), (4.127) and (4.128), as should be, for β 1. The dilaton-de Sitter metric in the Schwarzschild coordinate system given by Eq.(16.1) can be written in the form of a general spherically symmetric and static metric useful for the analysis of gravitation lensing as (4.129) ds2 A rdt2 B rdx2 C r dΩ2 where A r
ξ 2 H r 1 1 ξ 2 2 r 1 H r r ξ C r r2 1 r
1 1 r
B r 1
r
2
1
(4.130)
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e with rs 2M as the measure of distance and ξ Q2M 2 , and H is the Hubble parameter now measured in Schwarzschild radius units. For the dilaton-de Sitter metric we get radius of photon sphere to be rps 3 ξ4 η , where η 9 10ξ ξ 2 . The impact parameter calculated at r0 rps is the minimum impact parameter which for the dilaton-de Sitter metric is given by43
um
3 ξ η 3 η 3ξ 16 3 ξ η 64 H 2 3 ξ η 3 4ξH 2 3 ξ η 2 (4.131)
It is easy to check that at H 0, and ξ 0, um 3 2 3 , which is the value calculated for Schwarzschild metric, and for H 0, um goes to GMGHS black hole value.95 The expression for the strong field limit of the deflection angle may be written as85 αθ
a ¯ln
θDol urps
1 ¯ b,
(4.132)
where, a ¯
R 0, rps 2 βrps
(4.133)
R
3ηξ 3 η 3ξ
0, rps 2
¯b π bR a ¯ln
(4.134)
2βrps 1
1 2 rps rps
1
ξ 2 rps H
(4.135) bR ln
2βrps γrps βrps
R 0, rps 2ln 4 βrps 2 βrps 2 βrps γrps δrps
(4.136)
Note that the expression for ¯b given in Eq.(16.23) contains a H-dependent term which introduces a small correction to the lensing angle. The lensing observables S and R corresponding to the ratio between the flux of the first image and the flux coming from all the other images as defined in Eqs.(4.106) and (4.107) respectively, can be computed using
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the expressions for um , a, and b for particular geometries. One can obtain the magnitudes of S and R for particular lensing candidates using the known values for their masses and distances. For the black hole located at the centre of our galaxy at a distance of Dd 8.5kpc and with mass M 2.8 106M , the position of relativistic images in Schwarzschild strong lensing was first computed by Virbhadra and Ellis,86 and the observable parameters S and R were computed by Bozza.85 Using the above black hole as a candidate lens the values for these observables for several possible braneworld gemetries,42,94 as well as for the dilaton-de Sitter geometry43 have been calculated given by the metric (4.119) with tidal charge54 and another solution.65 The measurement of some of the observables involves a microsecond resolution which should in principle be attainable by the very long baseline interferometry projects planned for future. The actual identification of faint relativistic images would be extremely difficult in practice due to the inherent disturbances.86 However, an accurate measurement of these would be able to distinguish between various geometries. In order to unambiguously determine the exact nature of the black hole through the lensing angle coefficients a and b, one has to measure the observables S and R. Since this involves the resolution of two faint images separated by 0.02µ arc sec, such an observation would need a leap of technological development over the present astronomical facilities.85,94 4.7. Conclusions The subject of primordial black holes is a fascinating one that has received the attention of a number of researchers over the last few decades. Black holes could be formed in the universe through several mechanisms2 and in a wide mass range. Their impact on cosmological evolution could be diverse, and is yet to be investigated in full details for different models of black hole formation and cosmological evolution. Among the notable cosmolological consequences of surviving primordial black holes, to name a few one could mention the possibility of generation of the baryon asymmetry in the universe,17–19,23 and their viability as candidates of cold dark matter.28,29 Primordial black holes could also have interesting astrophysical consequences, such as the seeding of supermassive black holes,13 and a possible explanation for the observed emission of the 511 Kev line from the galactic bulge.96 In the present article we have studied various aspects of the idea of survival of primordial black holes as dark matter from theoretical as well as
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from observational perspectives. We have discussed the cosmological evolution of the early universe with a population of primordial black holes. The effects of Hawking radiation and accretion of energy from the surrounding radiation counteract on the black holes, and thereby impact the whole cosmological evolution. We have carried out this exercise in the context of standard cosmology and also in the braneworld scenario separately. There are key differences between the above two scenarios. Several possible black hole geometries have been described motivated from string theoretic and braneworld models. We have also provided an overview of the formalism for strong gravitational lensing through which it might be possible in future to observationally distinguish between various black hole models. In standard cosmology the effectiveness of accretion in the growth of black hole mass is restricted because of the fact that the cosmological Hubble radius grows at the same rate as the individual black holes. The role played by accretion here is essentially to counterbalance Hawking evaporation. Therefore, although the net mass increase for a black hole is not quite significant, its lifetime could undergo notable increase.17,18 In the braneworld scenario, the first difference stems from the modified geometry of 5-dimensional black holes. This leads to a lower evaporation rate thus increasing the lifetime.38 The braneworld cosmological evolution is also modified from the standard one in the early high energy phase.34,79 In the radiation dominated high energy phase the rate of mass change for a black hole can be integrated to give mass growth at a rate lower than the rate of increase of the Hubble radius. Accretion is clearly the dominant effect in the early radiation dominated high energy phase.11,12 This could lead to significant rise in the lifetime for primordial black holes with multifarous consequences. One such consequence that we have discussed in this article is that of binary formation by primordial black holes. If black holes form a significant part of cold dark matter, then a large number of them are expected to be in the form of binaries. Gravitational interaction between neighboring primordial black holes could start the formation of binaries at early cosmological eras.78 Braneworld evolution amplifies initial mass differences between neighboring black holes,40 and this effect favors binary formation. Binaries emit gravitational waves during their coalescing stage, and therefore are of observational relevance. So there exists an exciting possibility of obtaining the signature of the physics of primordial black holes in the early universe via gravitational waves from binaries.32 A population of primordial black holes impacts all cosmological eras due to their evaporation
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products and modifies the energy balance at each stage. Thus the observational constraints on the presence of different species of particles at any era is impacted by the initial and consequently the time evolved population of primordial black holes surviving up to that era. We have outlined a method39 for constraining the initial distribution in modified brane cosmology. Such an approach needs to be reformulated for other alternate models of gravity, such as the Jordan-Brans-Dicke model in which primorial black holes have interesting evolution, as shown recently.97 From the observational perspective, if primordial black holes are present in galactic haloes, they should be amenable to detection by gravitational lensing experiments. For braneworld black holes the standard lensing results seem to be modified even in the weak-field limit.41 On the other hand, strong gravitational lensing through its characteristic features such as relativistic images offers the best hope of probing the strong character of gravity in modified models in future. We have discussed the formalism for obtaining the lensing observables as fucntions of the metric coefficients in spherically symmetric geometries. The actual observables for certain candidate geometries motivated from string and braneworld models have already been calculated42,43,94 using the galactic centre black hole as a candidate lens. More detailed analysis of the lensing phenomena together with improved ideas of detection is required. In conclusion, the study of primordial black holes is an active branch of research with a potential to shed light on several fundamental problems of comology. References 1. B. J. Carr, Lect. Notes Phys. 631, 301 (2003). 2. B. J. Carr, in Observational and Theoretical Aspects of Relativistic Astrophysics and Cosmology, eds. Sanz J. L., and Goicoechea L. J. (World Scientific, Singapore 1985). 3. B. J. Carr and J. E. Lidsey, Phys. Rev. D48, 543 (1993); A. M. Green and A. R. Liddle, Phys. Rev. D56, 6166 (1997). 4. K. Jedamzik and J. C. Niemeyer, Phys. Rev. D59 124014 (1999); A. M. Green and A. R. Liddle, Phys. Rev. D60, 063509 (1999); G. D. Kribs, A. K. Leibovich and I.Z. Rothstein, Phys. Rev. D60, 103510 (1999). 5. For a recent review, see, D. Scott and G. Smoot, Phys. Lett. B592, 1 (2004). 6. A. A. Starobinsky, JETP Lett 55, 489 (1992); T. Bringmann, C. Keifer and D. Polarski, Phys. Rev. D65, 024008 (2002); D. Blais, T. Bringmann, C. Keifer and D. Polarski, Phys. Rev. D67, 024024 (2003). 7. A. R. Liddle and A. M. Green, Phys. Rept. 307, 125 (1998). 8. J. H. MacGibbon and B. J. Carr, Astrphys. J. 371, 447 (1991).
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Chapter 5 Physics in the ‘Once Given’ Universe: New Perspectives on Dynamics, Relativity, Quantum Spectra and the Spin-statistics Connection C. S. Unnikrishnan Gravitation Group, TIFR, Homi Bhabha RoadMumbai 400005, India The fact that the fundamental theories of physics were completed well before any significant knowledge about the real universe and its enormous gravity was available necessitates a re-examination of these theories, especially that of dynamics and relativity. The results of this analysis, along with several experimental facts, reveal that the matter frame of the universe provides a preferred absolute frame and that its gravity determines the laws of motion. Newton’s law of motion arises as a relativistic gravito-magnetic effect, the equivalence principle as a natural consequence of the gravitational reaction to motion in the massive universe, and the familiar dilation of the rates of clocks as a cosmic gravitational time dilation with absolute cosmic frame velocity as a key factor. In the quantum regime, partial contribution to the fine structure splitting of spectral lines, and the phases that determine the vastly different collective behavior of fermions and bosons are also linked to cosmic gravity; phenomena like Bose-Einstein condensation and Fermi pressure seem to have a cosmic connection. For the first time we seem to be able to grasp the physical basis of the spin-statistics connection as linked to a fundamental universal interaction. These ideas of cosmic relativity unify kinematics and dynamics, answering outstanding fundamental questions. A new experiment to compare the true one-way speed of light in different directions relative to an inertially moving observer, as well as a reanalysis of several earlier experiments on the propagation of light confirm the importance of the cosmic absolute frame. These results, with firm empirical support, imply important revisions in the theoretical physics of motion and relativity.
Contents 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.2 Cosmic Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 5.2.1 The gravitational potentials of the universe . . . . . . . . . . . . . . . 102 99
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5.2.2 Cosmic potentials and the gravitational metric . . . . . . . 5.2.3 Rates of clocks and clock comparison experiments . . . . . . 5.3 The Laws of Motion, Equivalence Principle and the Pseudo-Forces 5.4 Quantum Mechanics and the Cosmic Gravitational Potentials . . . 5.4.1 Spectral fine structure and matter-wave interferometry . . 5.4.2 The Spin-Statistics Connection . . . . . . . . . . . . . . . . 5.5 The One-Way Speed of Light . . . . . . . . . . . . . . . . . . . . . 5.6 General Relativity and Cosmic Relativity . . . . . . . . . . . . . . 5.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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5.1. Introduction Almost all of the fundamental theories of physics were completed well before observational cosmology made any significant progress. When theories of relativity were completed, for example, our universe consisted of a small number of nearby galaxies and novae, and sparsely distributed stars. Naturally, the gravity of such a universe in our local neighborhood was thought to be insignificant. In spite of the early warning by E. Mach in the context of Newtonian pseudo-forces, and Einstein’s claimed attempts at incorporating Machian ideas into general relativity, it is generally considered that the universe and its properties are unimportant in the determination of local physics and its laws. Nothing could be farther from truth. A simple calculation shows that the gravitational potential of all the matter in the universe, with its present density close to the critical density of about 1 1029 g/cm3, is a billion time larger than the gravitation potential of the earth, even though its gradient over space is essentially zero. Further, moving through the universe results in the generation of large gravito-magnetic potentials due to the relative motion of the charge of gravity, or the current of matter, and this in turn affects motion itself, and also rates of clocks and quantum phases. Rigorous estimates can be made, and they show that the cosmic gravitational effects on clocks moving through the cosmic matter frame is just as large as observed in experiments. Also, the prediction that the time dilation factors actually depend on the absolute velocity with respect to the cosmic frame, rather than on relative velocities as it is believed today, has strong empirical support from clock comparison experiments. It is important to stress the fact that the universe, essentially in its present form, has always been there and will remain as it is during the entire evolution of science, and every test of a physical theory has to be performed
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and validated with this ‘once-given’ universea , its matter, and its gravity as a pre-existing background. Clearly, if cosmic gravity has anything to do with local physics, we have completely missed that in our fundamental theories. One of the serious implications is for the theory of propagation of light. If the universe is a preferred frame for motion, then naturally the velocity of light is not an invariant constant relative to moving observers, just as was argued during the days of the aether as a hypothetical medium for the propagation of light. The fact that there is not even a single experimental evidence that support directly the hypothesis that the one-way speed of light is an invariant constant also means that it is still only a belief that could be overthrown when a suitable experiment is devised. It is also a general belief that such an experiment could not be done since all one-way speed measurements seems to require two clocks that are spatially separated, and synchronized by some means. This turns out to be wrong, and a simple experiment can be devised and performed. Finally, incorporating the unavoidable gravitational presence of the universe into the Einstein’s equations of general relativity completes what was missing in the equations, and renders the theory Machian in the correct sense, consistent with all experimental observations. In the rest of the paper I will rigorously prove that we have indeed missed incorporating the important and large effects of cosmic gravity in the fundamental theories of motion and relativity. The consequence of acknowledging the gravitational presence of the universe naturally yields a theory called Cosmic Relativity1 with strong empirical support, and several remarkable results. Surprises will include a direct derivation of Newton’s law of motion as a relativistic gravito-magnetic effect, establishing the weak equivalence principle as a consequence of cosmic gravity, as well as the demonstration that the Newtonian pseudo-forces arise as gravito-magnetic effects in a noninertial frame moving in the universe, as hypothesized by Mach. Then we will proceed to examine the empirical evidence to make a judgment and choice of the correct physical theory of relativity of motion. Empirical evidence regarding clock comparison and the speed of propagation of light relative to moving observers will be examined in particular. We will see that clocks transported relative to reference clocks stationary in a laboratory can run faster than the reference clock, rather than slower as predicted by special relativity, in some situations when the reference laboratory is moving relative to the cosmic frame. This effect is supported by aA
deeply meaningful and logically precise expression by E. Mach
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experimental evidence.1,2 In short, empirical rigor, combined with logical necessity, will allow a clear demarcation and choice of the correct physical theory of motion and relativity. Motion in the presence of a uniform distribution of matter, as opposed to motion in empty space, is affected by the current of matter, with important physical effects on the phases of quantum systems. Two important consequences are splitting of spectral lines3 and gravitational phases in quantum scattering, both arising from the spin-gravity coupling.4 We will identify the Thomas precession contribution in fine structure splitting with this cosmic gravitational contribution, and then proceed to show that the cosmic gravitational phase arising in the scattering of identical particles depends in the correct way on the spin of particles to give the discrete phases required for the spin-statistics connection for fermions and bosons. If this is true, then we are faced with the astonishing possibility that the local physics of Bose-Einstein condensation, superfluidity, and superconductivity as well as the remarkable properties of solids, condensed matter and the Fermi pressure in nuclei and stars have a direct link to cosmic gravity. Propagation of light is inseparably linked to the development of the relativity theories. I will discuss the idea behind a new measurement of the genuine one-way speed of light relative to an inertial reference frame without the need to use spatially separated clocks, and will establish that the untested assumption of its absolute invariance is unfounded.5,6
5.2. Cosmic Relativity 5.2.1. The gravitational potentials of the universe A calculation of the Newtonian gravitational potential of all the matter in the universe is difficult except when the universe is finite in space or time or both. This difficulty is avoided when one deals with the gravitational metric of the Universe, as to be done in a rigorous treatment. However, we will start with an estimate of the Newtonian potential and its relativistic modification before re-examining the argument in the language of the metric. This is possible since the standard model of cosmology, supported by some observational evidence, indicates that the age of the universe TU is finite, at about 14 billion years. Since the average density is measured to be around 1 1029g/cm3, and the spatial range of integration is approximately cTU , we get for the gravitational potential,
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ΦU
4πr2 drGρr
0
2πGρc2 TU2
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c2
(5.1)
The surprise, known for some time now, is that the Newtonian potential is numerically equal to the square of the speed of light. This is not a coincidence, and it already indicates a connection between cosmology and local physics. If I claim, purely phenomenologically, that it is the gravitational potential of the Universe that appears in all relativistic formulae instead of the square of the speed of light, this claim cannot be refuted empirically. This immediately answers why this factor appears universally in relativistic formulae. However, we will continue to use the factor c2 to preserve familiarity. It is worth pointing out at this stage that this potential is about a billion times larger than the gravitational potential of the earth at its surface. Now I consider motion with respect to the average matter frame of the universe. (Note that the first order Doppler shift or the dipole anisotropy of the Cosmic Microwave Background Radiation (CMBR) provides a convenient, practical and precise method for the measurement of the velocity of such motion). In any relativistic theory of gravity one expects that the current of the charge of the interaction would generate magnetic potentials. In the first approximation, this vector potential is just the product of the velocity and the usual Newtonian gravitational potential. We get for the gravitational vector potential (usually called the gravito-magnetic potential),
AiU
vi
ΦU c2
vi
(5.2)
It is important to note that such a potential can arise only for motion through a matter-filled universe, and not for motion through empty space. Also, a gravitational vector potential necessarily arises for motion through the universe. The consequences are remarkable. Before we examine these, let me also note that along with this vector potential in a moving frame, the Newtonian potential itself gets modified while moving. Treating the total potential approximately as the components of a four-vector potential this modification can be easily derived as
ΦU
Φ U 1 v 2 c 2
(5.3)
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5.2.2. Cosmic potentials and the gravitational metric Present treatments of the physics of motion appeals to special relativity in empty space when gravitational effects of the earth, sun etc. can be ignored and uses the Minkowski space-time as the fundamental structure. This is grossly inconsistent with known cosmology since even far away from local gravitational sources, the metric is the Robertson-Walker metric with its absolute and universal time coordinate and slowly varying spatial distances. Thus, ignorance about the real presence of the once-given universe has given rise to the widespread use of a wrong metric and the situation never got corrected. The inconsistency can be stated simply. Empty space remains empty, homogenous and isotropic for all observers in uniform motion. It is Lorentz invariant and uniform motion cannot be detected. However, homogenous and isotropic non-empty universe becomes anisotropic with a large matter current for an observer in motion. It is not Lorentz invariant since motion can be absolutely detected by a local measurement of dipole anisotropy of the CMBR. Even though the slowly varying scale factor of the FRW metric can be set to a constant over time scales relevant for laboratory experiment, or even over the time scales of the progress of civilizations, it is only for a class of absolutely stationary observers the metric resembles the Minkowski metric – observers who are at rest in the CMBR frame. For others moving with uniform velocity, the metric is anisotropic since space itself is anisotropic. The issue is whether this situation has implications to our fundamental theories. It turns out that it is precisely these anisotropies in the metric that reflect the gravito-magnetic potentials, as one could have guessed. The FRW metric for the spatially flat universe, consistent with recent observations, is ds2
2
2
dt a t
dx2 dy 2 dz 2
(5.4)
The scale factor a tcan be written as a t a0 1 a a0 where a a0 1018 m/s/m, the Hubble parameter at present. For observers at rest relative to the CMBR frame, and only for them, this resembles the Minkowski metric if we ignore the slow change in the scale factor. But that does not mean that the metric is universally of the Minkowski type for all inertial observers! It is clear from physical considerations of anisotropy of space for all inertially moving observers in the matter-filled universe that a coordinate transformation that keeps the metric invariant and isotropic is not
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physically reasonable. This is already clear from the fact that real universe is not Lorentz invariant in the sense mentioned earlier. So, Lorentz transformation is not the physically appropriate transformation for observers moving in the universe. The Galilean transformation, however, gives the right anisotropic metric. With x
x vt, y
y, z
z; t
t
We have ds2
c
2
1 v 2 c2 dt2 dx2 dy 2 dz 2 c v cdx dt
(5.5)
This Galilean transformation is like the Langevin transformation of the angular coordinate in a 2-dimensional rotating frame. The modified metric coefficients are
g00
1 v 2 c2 , g01
v c
(5.6)
We see that the metric components that represent the gravitational potentials match with what was derived from a post-Newtonian consideration of the matter-filled universe.1 In particular, the gravito-magnetic vector potential is v/c, where v is now the ‘absolute’ velocity with respect to the CMBR frame and not the relative velocity between different inertial observers. The impossibility to detect the local potential translates to the impossibility to detect motion without external material reference, and to the usual Lorentz invariance. But this does not imply the equivalence of the situation with one in which there is no potential. The modified metric immediately gives the gravitational time dilation in the rates of clocks transported in the cosmic frame as dτ
g00 dt
1 v 2 c2 dt
(5.7)
Also, the lengths of moving length scales are shortened. The physical length measured in the moving frame is given by dl2
g11 dx
2
2 g01 2 dx g00
dx 2
v 2 c2 dx 2 2 2 1 v c
dx 2 1 v 2 c 2
(5.8)
Since the physical length in the stationary frame is same as the coordinate length (g11 1, g01 0), we get
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dx dl
1 v 2 c2
(5.9)
We see that the standard Galilean transformation along with the modified gravitational metric reproduces all experimentally relevant relativistic features, with absolute velocity as the parameter. This is completely consistent with all known experimental evidence. Note that the result on length contraction is reproduced because of the presence of matter, giving rise to the vector potential. The usual myth that the Galilean transformations do not result in relativistic features consistent with experiments of the Michelson-Morley type is then incorrect – the metric (gravitational potentials) is in fact non-invariant under motion, as it should be when there is matter around, and this is the source of the relativistic modifications of time and length. All time dilations usually attributed to kinematical effects of relative motion in special relativity, like the muon life-time dilation, are gravitational time dilations arising from motion in the matter-filled universe. This result clarifies that all time dilations are gravitational – an important unifying insight about an important physical phenomenon. 5.2.3. Rates of clocks and clock comparison experiments The cosmic gravitational time dilation can be understood in terms of eq. 5.3 or eq. 5.7. In the language of potentials, the moving clock is in a larger potential ΦU ΦU 1 v 2 c2 12 and hence would run slower compared to a clock that is stationary in the cosmic frame. For small velocities, the difference in the gravitational potentials is approximately v 2 2c2 that gives the usual experimentally observed time dilations in clock comparison experiments. In the language of the metric, the physical time dilation involves the coordinate time and the metric coefficient, as given in eq. 5.7. Since the time dilation factors in Cosmic Relativity depend on the absolute cosmic frame velocity rather than on the relative velocity between different observers, whether or not the reference laboratory is moving in the cosmic frame becomes important in determining the rates of clocks. The most dramatic of predictions of Cosmic Relativity in this context is that a clock that is transported relative to a reference clock that is stationary in the laboratory can run faster, instead of being slower, in some situations when the laboratory itself is moving relative to the matter frame of the Universe.1,2 This is in complete variance from the special relativistic prediction in which the transported clock runs slower because the theory is based on relative velocities. The crucial difference is easy to see from figure 5.1.
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Fig. 5.1. A clock comparison experiment in a laboratory moving in the cosmic frame with velocity V
If the laboratory is in motion through the Universe at velocity V, then what is considered as a stationary clock (T0) is in reality moving at velocity V through the Universe, and the clock that is transported leftward (T2) at velocity u is actually moving at velocity u V , and the clock that is transported rightward (T1) is moving at velocity u+V. We get for the rate difference between the laboratory clock and the rightward clock T1
1 V
2
u c
2 1 2
1 V 2 c2
12
u2 2c2
uV c2
(5.10)
and for the difference between the laboratory clock and the leftward clock T2
1 V
2
u c
2 1 2
1 V 2 c2
12
u2 2c2
uV c2
(5.11)
We see that there is an asymmetry between the time dilations of T1 and T2 given by 2uV c2 in spite of the fact that they have equal relative speeds with respect to the reference clock. What is usually referred to as the ‘Sagnac term’ in GPS and clock comparison terminology is in fact the cosmic gravitational effect we have derived here. We finally have the correct physical identification of the source of this asymmetry routinely incorporated into the GPS clocks phenomenologically. The situation when the speed V u2 is very interesting. Then uV c2 u2 2c2 , and the clock T2 should be running faster than the laboratory clock T0 in spite of the fact that it is being transported relative to T0 at speed u! Thus, if cosmic gravity is responsible
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for the time dilation of clocks, then the rate of T1 has to be faster than that of the laboratory clock, and the integrated accumulated time will be more (older), rather than less (younger), in contradiction of the usual special relativistic wisdom. This would be the case if clocks are transported westward on the earth’s surface (equator) with speeds smaller than about 900 m/s. What is perhaps surprising is that this is exactly what is observed in clock comparison experiments, even though it is never widely discussed in this form. The earliest such experiment was done by Hafele and Keating around 1970, in which Cesium atomic clocks were transported around the globe in jet planes westward and eastward after synchronizing with a laboratory clock.7 The typical ground speeds were about 220 m/s. Since all motion through the universe except the one due to the rotation of the earth relative to the matter frame of the universe are common for the laboratory clock and the flight clocks, we need to compare only the absolute velocity effects due to the earth’s rotation. At the latitude of 39 (Washington) the laboratory clock moves at 360 m/s, whereas the westward flight clocks moved at 190 m/s estimated with average latitude of 30 . With about 49 hours of duration for the experiment this results in 127 ns of cosmic gravitational time dilation for the laboratory clock, and 35 ns for the flight clock. Clearly, the westward clock runs faster by about 92 ns. The experimental data revealed a motional time advance, instead of dilation, of 94 19 ns (residuals after subtracting out the gravitational time dilations due to the height of the jet planes). The clocks that were transported westward came back ‘older’ instead of being ‘younger’ by about 94 ns, exactly as estimated in cosmic relativity.1,2 The westward and eastward clocks collectively also showed the asymmetry given by eqns. 5.10 and 5.11. The evidence for the correctness of the absolute frame time dilation is unambiguous. It is only a strong adhesion to beliefs and lack of empirical rigor that can ignore such an important and definite rejection of a theory based on relative speeds.
5.3. The Laws of Motion, Equivalence Principle and the Pseudo-Forces Since the metric in a frame that is moving with respect to the massive cosmic frame has the off-diagonal anisotropic term proportional to the velocity, there are several physical effects that have analogues in electrodynamics when there is a vector potential present. We can write
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v i c
(5.12)
If this vector potential varies either in time or in space, then the gravitational interaction is not a constant and these manifest as gravitational forces. Since we are dealing with a vector potential these forces have the character of magnetic forces in the case when the curl of the potential is nonzero, and the force is like an electric field when the potential is time dependent. First we consider the case when the vector potential is time dependent. This happens when the velocity is not constant, and there is an acceleration.
mg
dAi dt
i
mEg
m g ai
(5.13)
This resembles the Lenz’s law in electrodynamics, indicating an inductive electric field. The mass that appears here is the gravitational mass, since we are talking about a gravitational coupling (see below for further explicit discussion). We can identify this as the Newton’s law of motion. It is necessary to apply a force, equal to the gravitational mass times the acceleration to move a body in this matter-filled universe. The distinction between inertial mass and gravitational mass has disappeared, and we have the fundamental universal law of motion as a gravito-magnetic relativistic law linked to cosmic gravity! In the language of the potentials, the vector potential can be written as Ai
v i Φ U c 2
(5.14)
The Newtonian potential contains the gravitational coupling constant. Then,
mg
dAj dt
j
mEg
a j m g Φ U c 2
aj m i
(5.15)
It is the combination mg ΦU c2 that we call the inertial mass. Therefore, the ratio of the inertial and gravitational mass is a property of the universe, being the cosmic gravitational potential, and hence universal. We have the equivalence principle for the gravitational and inertial masses emerging out of cosmic relativity.8 There is indeed only one type of mass – the gravitational mass. This ‘dressed’ by the cosmic gravity is what we call
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the inertial mass. Since ΦU c2 1, we get mg mi without any further adjustment, though this is not necessary for the validity of the equivalence principle. In fact, the result is deeper than at the level we discussed so far. From the derivation of equation 13, we can see that acceleration in the massive universe generates a real inductive gravitational field that is directly proportional to the acceleration. Therefore, it is not just that acceleration is equivalent to a gravitational field – it is equal to a real local gravitational field. Acceleration is not pseudo-gravity; it is gravity itself in which the cosmic gravity appears as a gravito-magnetic reaction. The full equivalence principle emerges naturally from cosmic relativity, and it is no more an assumption, but an inseparable consequence of motion in a matter-filled universe. This has another important consequence. Naively, one might think that if the gravitational coupling constant itself is material dependent, the equivalence principle will be violated. Indeed, this is one of the main motivations that drive precision tests of the equivalence principle. The surprise from cosmic relativity is that even when the gravitational coupling is material dependent, the universality of free-fall is respected! To see this, consider the free-fall of two objects of different materials and of the same mass, which couple gravitationally with coupling constants Ga and Gb . The interaction of the masses with the cosmic potential can be written as the product of this coupling and a function fu that depends only on cosmological parameters. From eqn. 5.15, we have, mga Ga fU c2
mia ; mgb Gb fU c2
mib
(5.16)
Therefore, the ratios of the inertial to the gravitational masses of the two objects now depend on the coupling constants. Since the acceleration towards a gravitating object in an experiment is also proportional to the same coupling constants, we get for the acceleration of the two objects,
aa Ga mga mia
c 2 f U ;
ab Gb mgb mib
c2 fU
(5.17)
The acceleration is independent of the coupling constants and universality of free-fall is respected – a remarkable result.8 Hence we conclude that the experiments searching for violations of the equivalence principle would see a violation of the universality of free fall only when short range material dependent couplings are involved, but neither the standard long range
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gravitational interaction nor its modification with a material dependent coupling constant will show any such violation. Next we examine the consequence of a nonzero curl for the vector potential. In a frame that rotates relative to the matter in the universe, or when motion is non-inertial with constant speed, the matter current is solenoidal, and the curl of the vector potential is ∇ v A ∇
g B 2Ω
(5.18)
This is a gravito-magnetic field, familiar in Lense-Thirring precession and other subtle gravitational phenomena. Here, since the entire matter in the universe is the source, the field is very large. A test particle of mass m g moving with velocity u will experience a transverse acceleration u B 2u Ω which is the Coriolis force. Surprisingly, it is a relativistic gravitomagnetic force of the same type that is being studied with great effort in satellite experiments like the GP-B.9 Both arise due to coupling of matter current to matter current. The GP-B signal is tiny compared to Coriolislike effects because the matter in the earth is minuscule compared to that in the entire universe. In a frame rotating at angular velocity Ω, the velocity at the radial r. This is also the gravitational vector potential. This coordinate r is Ω is time dependent even for a constant radial distance and angular velocity since its direction is changing, and we get another kind of gravito-magnetic r changes by 2π T per force in rotating frames. Since the direction of Ω ˆ projected perpendicular to Ω r, the rotation, about the unit vector Ω acceleration is Ω Ω r – the centrifugal acceleration. Therefore, the centrifugal force is also of gravito-magnetic origin, due to the interaction with the matter-filled universe as prophesied by Mach.10 Some of these results are contained in the early work by D. Sciama.11 I find it ironical that one struggles with complex experiments to detect the tiny gravito-magnetic effects in relativistic gravity near the earth, in complete ignorance about these large cosmic gravito-magnetic effects that are everyday experience. 5.4. Quantum Mechanics and the Cosmic Gravitational Potentials 5.4.1. Spectral fine structure and matter-wave interferometry It is of course no surprise that the cosmic gravitational vector potential in a frame that is moving in this matter-filled universe will modify the phases
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of quantum systems through the modification of the momentum by the relation g p p mA
(5.19)
A related quantum effect is that the coupling of a spin s to the gravito g results in quantized values of the coupling energy. If magnetic field B we consider the motion of an electron in orbits with non-zero angular momentum, the cosmic gravitational vector potential in the orbiting frame is time dependent with a non-zero curl. The coupling of the electron spin to this gravito-magnetic field results in energy shifts that match with what is required to be added to the usual L-S coupling term between the electron and the charged nucleus to get the correct fine-structure splitting. In the standard view this is a kinematical term, called the Thomas precession term, but in reality it is a gravitational spin-orbit coupling.3 This implies new possibilities for probing the properties of the universe with precision laboratory spectroscopy. Coming back to the gravitational modification of quantum phases, we notice a connection to the modification of the rates of clocks. Once the frequency of an oscillator is fixed, as done in atomic clocks, time measurement is the same as phase measurement. Therefore, rates of clocks and phases are affected in the same way. The relevant frequency here is just the energy that couples gravitationally divided by the Planck’s constant. For light, the relation in equation (5.19) needs to be suitably modified, and the general g c2 . In a frame that has a cosmic relation applicable is then p p E A frame velocity parallel to the direction of propagation of light, we get g c2 p p hν A
hν c2 c v
(5.20)
This is consistent with the observer dependent one-way speed of light that we will discuss in section 5.5. The phase shift in a Sagnac interferometer in a rotating frame trivially follows from this, as analyzed from the frame itself (the second order corrections are nonzero, and follows when the metric is included in evaluating the physical phase shift). For a circular interferometer centered on the axis of a rotating frame, with total path length L, the phase shift is % 2 g dx EA c path
%
Ev dxc
! 2
S 2E Ω % curl v dS E c2 (5.21) c2
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is the angular velocity and S is the oriented area of the interferometer. Ω The first term on the left is the general formula, and it explicitly reveals that this is a gravitational Aharonov-Bohm phase.1,5 The general formula gives us the important result that inertial sensors like the Sagnac interferometer that use matter wave of particles with mass m, instead of light with frequency ω, will have a phase shift that is mc2 ω times larger for the same total path length! This result is very important for precision metrology, and has been confirmed in experiments. This is why atom interferometers with their total path several million times smaller than fiber-optic Sagnac interferometers can still surpass their sensitivity.
5.4.2. The Spin-Statistics Connection The relation between the spin of elementary particles and their quantum statistical behavior, called the spin-statistics connection is simply stated, but there is no physical understanding of this very fundamental connection.12 Usual ‘proofs’ depend on several assumptions that invoke features of relativistic quantum field theories, while the physical phenomena is manifest at the level of two-particle non-relativistic quantum mechanics. Also, the proofs are essentially pointing out the inconsistency of constructing theories in which integer spin particles (bosons) are linked to anti-commuting operators and half-integer spin particles (fermions) to commuting operators. The standard theory in which the spin-statistics connection works has the representations where bosons are linked to commuting field operators and fermions to anti-commuting field operators. In any case, there is no understanding of the spin-statistics connection in terms of its link a fundamental interaction, even though it is clear that spin itself is as important as mass in a fundamental theory. What is usually ignored is that all particle interactions take place in the presence of this matter-filled universe. Since it is the interaction energy that is important for the modification of quantum phases, and not its gradient or curvature, the large cosmic gravitational potentials significantly modify the quantum phases in the scattering of identical particles in identical quantum states, depicted in figure 5.2. There are two quantum mechanical amplitudes that contribute to this process. The statement of the spin-statistics connection in this context is that the exponential of the phase difference between these amplitudes is 1 for bosons, and 1for fermions. There is no fundamental derivation of this fact.
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Fig. 5.2. Quantum scattering of identical particles, resolved into the two relevant amplitudes.
Note that the only difference between the two scattering amplitudes is the amount by which the momentum of the scattered particles changes. The difference in the angles of deflection in the two cases is π. This has no special significance in empty space, but is of enormous importance in the presence of the matter in the universe, since the vector potential, proportional to the momentum, also changes by the same angle. This change results in the non-zero curl of the vector potential and the spin couples to this gravitomagnetic field – an essential consequence in the presence of matter. The interaction energy in the context of gravitational interaction is
g 2 s B
(5.22)
and we get the gravitational phase for the two particles as
2
g dt2 2s s B
Ωdt 2sθ
(5.23)
This immediately gives the phase difference between the two amplitudes as 2πs, which dictates that the amplitudes should be added directly for integer spin particles and should be subtracted for half-integer spin particles.4 This is the spin-statistics connection. If this result stands the scrutiny of rigor and thorough analysis, then we have the startling result that the complex and spectacular microscopic phenomena associated with a collection bosons or fermions, like the BoseEinstein condensation, Pauli exclusion and the Fermi pressure, and even the variety of atomic structure, are all linked to cosmic gravity.
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5.5. The One-Way Speed of Light The absolute constancy of the speed of light relative to any inertially moving observer is the postulated basis of the fundamental theory of motion and space-time. There is a general impression among most in the physics community that this is an experimentally established fact. However, it is only the two-way speed of light – the total time taken by light to traverse the distance between two reference points back and forth on the same path – that is shown to be independent of the velocity of the observer. It is an empirically established to high precision only that the two-way speed of light is isotropic relative to any inertial observer. The genuine one-way speed of light has never been compared in different directions relative to an inertially moving observer because, apparently, such a measurement requires two spatially separated clocks that needs to be synchronized and this requires the theory of propagation of a signal between the clocks and the entire effort becomes logically circular. Therefore, in the absence of a method to measure the one-way speed, one can get away with a theory of motion based on the unverified postulate that the one-way speed is an invariant. The two-way speed of all familiar waves (sound, water waves etc.) is independent of the velocity of an observer to first order in v/c. It is only in the second order there is a difference of time of propagation in different directions. However, even this second order anisotropy for light vanishes if there is real length contraction, as Lorentz and Fitzgerald postulated. Of course the other solution to the Michelson-Morley result is that there is no physical length contraction, but the speed of light is isotropic in any inertial frame, which forms the basis of special relativity. It is only when the one-way speed becomes measurable one can see a large difference in these two approaches – then there will an anisotropy that is first order in v/c in an absolute frame theory, whereas no anisotropy is expected in the special theory of relativity. Issues of length contraction and so on become irrelevant for first order effects and the experiment becomes a crucial demarcating decider. The cosmic frame defines a preferred frame and an inertially moving observer can no longer claim equivalence to an observer at rest. Absolute velocities can be measured from the dipole anisotropy of the CMBR with high precision. Also, the average temperature of the CMBR defines a local time that is universally synchronized. Universe is not Lorentz invariant, and there is no issue of lack of simultaneity. Is this consistent with the postulate
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of the invariance of the one-way speed of light? Only an experiment can resolve the issue, and it turns out that there is a simple way of performing the experiment without spatially separated clocks; in fact with no clocks at all! It is possible to measure the one-way speed of light and other waves, and particles with just one clock, without the need to synchronize two distant clocks. The idea is extremely simple.5,6 To see the point, consider the experiment discussed in the context of figure 5.3.
Fig. 5.3. The left panel depicts conventional thinking about one-way speed measurements. The right panel shows a common variation that measures the genuine one-way speed. The reference frame A can be moved inertially at different speeds.
In both cases, the wave travels a distance L, one-way from t1 to t2, and the fact that the one-dimensional path is bent around does not change the fact that it is the one-way speed that is measured. The results obtained are the same, as can be verified with particles or familiar waves. Now it is simple to see that I can bend the track around all the way to make the endpoint B coincide with the start-point A, and I will be able to measure the true one-way speed with only one clock at A, synchronized to itself. In fact, this is what is usually done in real running races to measure the one-way speed. The runner is never turned back on the same path, and the distance between the starting point and the runner monotonically increases (this distance can be measured accurate to first order, independent of theory, by a two-way measurement). To complete the task we also need to ensure that the reference point A itself can move inertially, and this is easy since it is on the linear section of the track. In fact, the clock can be avoided altogether if we send two wave-fronts in opposite directions (figure 5.4). If the speeds of the wave-fronts are equal in the two direction relative to A, then the distance to these wave-fronts from A should be the same at all times, and hence the two should reach back at the same time. The interference of these enable a precision measurement of the difference in the arrival times, and
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Fig. 5.4. Scheme for the measurement of the one-way speed relative to an inertial observer using only one clock (left). The right panel shows the comparison of speeds without using clocks.
an invariant and isotropic observer-independent speed implies an invariant interference fringe. If the experiment is done with sound or water waves one will see a fringe shift that is linear in the velocity of the reference point. This is precisely why the relative speeds of these waves are considered dependent on the velocity of the observer. The experiment can be done with light with sufficient precision despite the enormous velocity of light compared to the small velocity of the reference point A that is practical in the experiment. If the speed of light is anisotropic, c v and c vrelative to the frame A in which the source and the detector are contained, then the difference in arrival times is L c v
L c v
2Lv c2 1 v 2 c2
2Lv c2
(5.24)
showing the first order effect. This amounts to less than 0.01 femtoseconds for L of the order of a meter and velocities less than a meter/s, but is measurable with interferometry as Michelson demonstrated in his two-way speed comparisons. Figure 5.5 shows the schematic of the experiment and also the results.5,6 The linear dependence of the arrival time difference on the velocity of reference platform is clearly revealed in the measured results. The oneway speed of light is not an invariant constant relative to inertial reference frames. To first order in v/c, light behaves like any other waves as far as its relative velocity is concerned. The data fits the equation 5.22 well. The mythical pillar of relativity is uprooted by a hard experimental fact. I have argued elsewhere that some recent experiments that use two spatially separated clocks to look for anisotropies in the propagation of light returned null results precisely because the anisotropy in the time of propagation
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Fig. 5.5. Experimental set up used for the comparison of the one-way speed of light in opposite directions with respect to a inertial platform (left). The results of the measurements are shown in the right panel.
precisely canceled the gravitational effects on the clocks used during the noninertial motion of the reference frames, further proving the reality of the anisotropy of the speed of light relative to reference frames that are in motion through the cosmic frame.5,6 Special relativity was the wrong path taken to resolve the issue of the failure to detect uniform motion relative to the hypothetical aether, at a time when the real path was not lit enough by the stars of the universe. Replacing the hypothetical aether with the real matter-filled universe finally takes us to the correct theory of relativity. 5.6. General Relativity and Cosmic Relativity The usual wisdom is that in the limit of being far away from massive bodies like the earth and the sun, asymptotically, the metric that gives the correct physics is the Minkowski metric. This is clearly incorrect as we have seen. Even in a freely falling frame, the metric is the transformed FRW and not Minkowski metric. But this is not consistent with the Einstein’s equations since in the matter free region the field equations read Gµν 0. But space is not vacuum even in the absence of local massive sources, and the nearly homogeneous and isotropic universe is an ever-present background. This implies that the Einstein’s equation of general relativity is an approximate equation with physically incorrect limit and this needs to be corrected. Since the curvature terms that involves the universe are typically of order 2 where RH is the Hubble radius, and the typical curvature terms 1RH around massive local sources are of order GM R3 , the present usage of the approximate equation is still a very accurate approximation, except when dealing with the rates of clocks and phases for motion far away from
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massive objects. For example, the present general relativity does not solve the twin-clock problem, and it requires the realization that the motion of the clock is through the matter-filled universe.13 The correct equation for general relativity is 1 U Tµν Rgµν Tµν (5.25) 2 where I have written the energy momentum tensor of the ever present background matter of the universe on the left side to remind that this is a permanent and fundamental term in the equation – it cannot be removed. The curvature terms represent the total gravitational curvature. When U Tµν # Tµν as in most cases of solar system physics, we can ignore the cosmic matter and its curvature. But when these two terms are comparable U or when Tµν 0, the correct equation necessarily should retain Tµν , and only then the equation is consistent with the unavoidable presence of the matter-filled universe. Rµν
5.7. Conclusions All theories of physics are constructed and used, and their tests are done in the ever-present ‘once-give’ universe that is nearly homogeneous and isotropic with larger matter content. The cosmic gravitational potentials dominate any local potential and have important effects on clocks, quantum phases, and on laws of motion themselves. Since our fundamental theories were completed well before this was realized, before the progress in cosmology, a reconsideration of the theories of dynamics becomes necessary. Both theoretical considerations and experimental data point to the necessity of important revisions in our theories of motion and relativity. Experimental data confirm that the gravitational influence of the universe is the sole contributor of motional time dilations usually attributed to kinematics of motion in empty space, in special relativity. The prediction from cosmic relativity that a transported clock can run faster, instead of slower, relative to a stationary reference clock in the laboratory, if the laboratory itself is in motion relative to the matter frame of the universe, has been confirmed in experiments. An absolute preferred frame is a reality, and new measurements show that the true one-way speed of light relative to inertially moving of reference frames is not an invariant constant, contradicting the fundamental untested hypothesis of special relativity. The cosmic gravitational influence on quantum systems give rise to gravito-magnetic coupling
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to spin that helps explain the fine-structure splitting in atomic spectra and the spin-statistics connection. Newton’s law of motion, pseudo-forces in rotating frames and the equivalence principle itself emerge out naturally from Cosmic Relativity. A paradigm shift prompted by Cosmic Relativity is unavoidable in which there is a natural and satisfactory unification of kinematics and dynamics. References 1. C. S. Unnikrishnan, ‘Cosmic Relativity: The fundamental theory of relativity, its implications and experimental tests’, gr-qc/0406023 (2004); C. S. Unnikrishnan, ‘Cosmic Relativity: The Only Consistent Ontological Foundation for the Theory of Space-Time and Relativity’, in Foundations of Science, Editor, B. V. Sreekantan (PHISPC, New Delhi, 2006). 2. C. S. Unnikrishnan, “The effect of cosmic gravity on clocks moving through universe” to appear in the Proceedings of the first ESA Colloquium on scientific and fundamental aspects of the Galileo program, Toulouse, (2007) 3. C. S. Unnikrishnan, Mod. Phys. Lett. A 16, 429, (2001). 4. C. S. Unnikrishnan, “Spin-Statistics connection and the gravity of the Universe: The Cosmic connection”, gr-qc//0406043 (2004). 5. C. S. Unnikrishnan, “New measurements of the one-way speed of light and its relation to clock comparison experiments”, to appear in the Proceedings of the 11th Marcel Grossmann meeting, Berlin (2006). 6. C. S. Unnikrishnan, “Precision measurement of the one-way speed of light and implications to the theory of relativity”, in the Proceedings of the conference on ‘Physical Interpretations of Relativity Theory’, London, Ed. M. Duffy, (2006). 7. J. C. Hafele and R. E. Keating, Science 177, 166-170 (1972). 8. C. S. Unnikrishnan, “The equivalence principle and its tests in the context of gravity, quantum mechanics and cosmology” to appear in the Proceedings of the 11th Marcel Grossmann meeting, Berlin (2006). 9. Gravity Probe – B is a high precision space-based measurement of gravitomagnetic precession effects on a gyroscope, arising from earth’s rotation and from motion around the earth. See website http://einstein.stanford.edu/ for details. 10. E. Mach, Science of Mechanics, pp 277-306 (The Open Court Publishing Co. London, 1942). 11. D. Sciama, MNRAS 113, 34 (1953). 12. I. Duck and E. C. G. Sudarshan, Pauli and the spin-statistics theorem, World Scientific, Singapore, 1998. 13. C. S. Unnikrishnan, Current Science 89, 2009-2015 (2005).
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Chapter 6 Doubly-Special Relativity: Facts, Myths and Some Key Open Issues Giovanni Amelino-Camelia Dipartimento di Fisica, Universit` a di Roma “La Sapienza” and Sez. Roma1 INFN P.le A. Moro 2, 00185 Roma , Italy I report on the status of the development of “doubly-special” relativistic (“DSR”) theories with both an observer-independent high-velocity scale and an observer-independent small-length/large-momentum scale, with emphasis on some key open issues and some aspects that are particularly relevant for phenomenology. I also give a true-false characterization of the structure of these theories. In particular, I discuss a DSR scenario without modification of the energy-momentum dispersion relation and without the κ-Poincar´e Hopf algebra, a scenario with deformed Poincar´e symmetries which is not a DSR scenario, some scenarios with both an invariant length scale and an invariant velocity scale which are not DSR scenarios, and a DSR scenario in which it is easy to verify that some observable relativistic (but non-special-relativistic) features are insensitive to possible nonlinear redefinitions of symmetry generators.
Contents 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Relativity, Doubly Special . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Defining the concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 A falsifiable proposal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 More on the Concept (a true-false exercise) . . . . . . . . . . . . . . . . . . . 6.3.1 DSR not equivalent to Special Relativity . . . . . . . . . . . . . . . . . 6.3.2 Not necessarily involving the κ-Poincar´ e Hopf algebra . . . . . . . . . 6.3.3 Not any deformation, but a certain class of deformations of Special Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.4 A physics picture leading to DSR and the possibility of DSR approximate symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.5 Not any fundamental length scale . . . . . . . . . . . . . . . . . . . . . 6.3.6 And the same type of scale may or may not be DSR compatible . . . . 123
124 126 126 127 130 131 131 133 135 136 137 140
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6.3.7 Classical spacetime not a possibility when photon speed is wavelength dependent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 More on the Use of Hopf Algebras in DSR Research . . . . . . . . . . . . . . 6.4.1 A Hopf-algebra scenario with κ-Poincar´ e structure . . . . . . . . . . . 6.4.2 A Hopf-algebra scenario without κ-Poincar´ e and without modified dispersion relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 DSR Scenarios and DSR Phenomenology . . . . . . . . . . . . . . . . . . . . 6.5.1 A toy-model DSR-scenario test theory, confined to leading order . . . . 6.5.2 On the test theory viewed from an (unnecessary) all-order perspective 6.5.3 Photon stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.4 Weak threshold anomalies for particle reactions . . . . . . . . . . . . . 6.5.5 Wavelength dependence of the speed of light . . . . . . . . . . . . . . . 6.5.6 Crab-nebula synchrotron radiation data not significant . . . . . . . . . 6.6 Some Other Results and Valuable Observations . . . . . . . . . . . . . . . . . 6.6.1 A 2+1D DSR theory? . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.2 A path for DSR in Loop Quantum Gravity? . . . . . . . . . . . . . . . 6.6.3 Maximum momentum, maximum energy, minimum wavelength . . . . 6.6.4 Deformed Klein-Gordon/Dirac equations . . . . . . . . . . . . . . . . . 6.6.5 The “soccerball noproblem” . . . . . . . . . . . . . . . . . . . . . . . . 6.6.6 Curvature in energy-momentum space . . . . . . . . . . . . . . . . . . 6.6.7 A gravity rainbow? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Some Recent Proposals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.1 A phase-space-algebra approach . . . . . . . . . . . . . . . . . . . . . . 6.7.2 Finsler geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.3 A 5D perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.4 Everything rainbow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
141 142 142 145 148 148 149 151 153 155 156 157 157 159 159 160 160 161 162 163 163 164 164 165 166 166
6.1. Introduction The idea of “Doubly-Special-Relativity” (or “DSR”) is now about 7 years old,1 and more than 500 papers either fully focused on it or considered it alongside other possibilities (see, e.g., Refs. 1–71 and references therein). This allowed to achieve rapidly significant progress in exploring the DSR concept, but, as inevitable for such a large research effort over such a short time, it also affected negatively the process of establishing a common language and common conventions, and not enough was invested in comparing different perspectives. The pace of development was such that it could not be organized “in series”, with each new result obtained by one group being metabolized and used by other groups to produce the next significant result, but rather “in parallel”, with research groups or clusters of research groups obtaining sequences of results from within one approach, and then attempting to compare to similar sequences of results obtained by other
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research groups when already a barrier of “local dialects”, intuitions and prejudices has settled in. It might be useful at this stage of the development of DSR to pause and try to find some unifying features among the encouraging results found adopting different approaches, and confront from a perspective that might combine the different approaches some of the most stubborn “unsolved issues” which often appear, although possibly differently disguised, in all approaches. In the first part of these notes I follow my original discussion of Ref. 1 proposing the physics idea of a DSR theory. And to make the concept clearer I here also characterize it in the form of a “true-false exercise”, whose entries are chosen on the basis of the experience of these past few years in which some aspects of the DSR concept have been occasionally misunderstood. In particular, through the illustrative example of theories with “canonical” spacetime noncommutativity I discuss the possibility of a DSR scenario without modifications of the energy-momentum dispersion relation and without the κ-Poincar´e Hopf algebra. Through the illustrative example of some results originally obtained by Fock, I discuss the possibility of a relativistic theory with both an invariant length scale and an invariant velocity scale, which is not a DSR scenario. And I use some recent results on the relation between the conserved charges and the algebraic properties of the generators of a Hopf algebra as an example of cases in which some characteristic relativistic (but non-special-relativistic) features of a theory are insensitive to possible nonlinear redefinitions of the symmetry generators. Whenever possible I rely only on the physics concept of a DSR theory, without advocating one or another mathematical formalism, and I stress that this is at this stage still advisable since, although many encouraging results have been obtained, no mathematical formalism has been fully proven to be consistent with the DSR principles. (Actually for all of the candidate DSR formalisms that are under consideration the results obtained so far are not even sufficient to rigorously exclude the presence of a preferred framea , and it would therefore be rather dangerous to identify the DSR concept with one or another of these formalisms.) aI
will sometimes, for brevity, characterize violations of Galilei’s Relativity Principle as cases in which a “preferred frame” emerges. Of course, even when the Relativity Principle is violated there is a priori no reason to prefer one frame over another, but violations of the Relativity Principle render different inertial observers distinguishable so that a criterion to prefer one over another is devisable.
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When I do find useful to discuss some aspects of the DSR concept and some the “open issues” in terms of a candidate mathematical formalism, I shall primarily resort to the mathematics of Hopf-algebra spacetime symmetries, which at present appear to provide the most promising candidate for a formalism able to accommodate the DSR principles. But I shall also (although more briefly) comment on other candidate DSR formalisms, and in doing so I shall stress the need to characterize these different approaches in terms of (at least in-principle) observable features. It is not implausible that some of these formally different approaches actually describe the same DSR physical theory. I also offer some remarks (mainly in Section 6.5) on DSR phenomenology, primarily with the objective of showing that (in spite of the present, rather preliminary, stage of developments of DSR formalisms) we can already establish rather robustly certain general features of this phenomenology. 6.2. Relativity, Doubly Special 6.2.1. Motivation My proposal1 of the doubly-special-relativity scenario provided an alternative perspective on the studies of Planck-scale departures from Lorentz symmetry which had been presented in numerous articles (see, e.g., Refs. 72–81) between 1997 and 2000. These studies were advocating Planck-scale modifications of the energy-momentum dispersion relation, usually of the formb E 2 p2 m2 ηLnp p2 E n O Lpn 1 E n 3 , on the basis of preliminary findings in the analysis of several formalisms in use for Planck-scale physics. The complexity of the formalisms is such that very little else was known about their physical consequences, but the evidence of a modification of the dispersion relation was becoming robust. In all of the relevant papers it was assumed that such modifications of the dispersion relation would amount to a breakdown of Lorentz symmetry, with associated emergence of a preferred class of inertial observers (usually identified with the natural observer of the cosmic microwave background radiation). I was intrigued b The
relevant literature focuses on ultra-high energy (but still “sub-Plankian”) particles, p (and in any case one never expects experimental sensitivities that would so that E 2 n render meaningful correction terms of the type, say, ηLn p m E at ultra-high energies), 2 n 2 p2 m2 ηLn p2 so it is customary to write interchangeably E 2 p p E and E 2 n n 2 2 2 2 n 2 n m ηLp E . I am choosing to write E p m ηLp p E so that I can safely treat m as the rest energy of the particle.
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by a striking analogy between these developments and the developments which led to the emergence of Special Relativity, now more than a century ago. In Galilei Relativity there is no observer-independent scale, and in fact the energy-momentum relation is written as E p2 2m. As experimental evidence in favor of Maxwell equations started to grow, the fact that those equations involve a fundamental velocity scale appeared to require the introduction of a preferred class of inertial observers. But in the end we figured out that the situation was not demanding the introduction of a preferred frame, but rather a modification of the laws of transformation between inertial observers. Einstein’s Special Relativity introduced the first observer-independent relativistic scale (the velocity scale c), its dispersion relation takes the formc E 2 c2 p2 c4 m2 (in which c plays a crucial role for what concerns dimensional analysis), and the presence of c in Maxwell’s equations is now understood as a manifestation of the necessity to deform the Galilei transformations. I argued in Ref. 1 that it is not implausible that we might be presently confronted with an analogous scenario. Research in quantum gravity is increasingly providing reasons of interest in Planck-scale modifications of the dispersion relation, of the type mentioned above, and, while it was customary to assume that this would amount to the introduction of a preferred class of inertial frames (a “quantum-gravity ether”), the proper description of these new structures might require yet again a modification of the laws of transformation between inertial observers. The new transformation laws would have to be characterized by two scales (c and Lp ) rather than the single one (c) of ordinary Special Relativity. 6.2.2. Defining the concept The “historical motivation” described above leads to a scenario for Planckscale physics which is not intrinsically equipped with a mathematical formalism for its implementation, but still is rather well defined. With DoublySpecial Relativity one looks for a transition in the Relativity postulates, which should be largely analogous to the Galilei Einstein transition. Just like it turned out to be necessary, in order to describe high-velocity particles, to set aside Galilei Relativity (with its lack of any characteristic invariant scale) and replace it with Special Relativity (characterized by the invariant velocity scale c), it is at least plausible that, in order to describe c For
the vast majority of formulas, in these notes I am adopting conventions with c 1. I write c explicitly only when it seems appropriate to stress its role in a certain equation.
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ultra-high-energy particles, we might have to set aside Special Relativity and replace it with a new relativity theory, a DSR, with two characteristic invariant scales, a new small-length/large-momentum scale in addition to the familiar velocity scale. A theory will be compatible with the DSR principles if there is complete equivalence of inertial observers (Relativity Principle) and the laws of transformation between inertial observers are characterized by two scales, a high-velocity scale and a high-energy/short-length scale. Since in DSR one is proposing to modify the high-energy sector, it is safe to assume that the present operative characterization of the velocity scale c would be preserved: c is and should remain the speed of massless low-energy particles. Only experimental data could guide us toward the operative description of the second invariant scale Ldsr . These characteristics can be summarized1 of course in the form of some “DSR principles”. First the statement that Galilei’s relativity principle is valid:
(RP): The laws of physics take the same form in all inertial frames (i.e. these laws are the same for all inertial observers)
Then there must be a principle giving the operative definition of the length scale Ldsr (or a corresponding momentum/energy scale 1Ldsr ). Since at present we have no data one can only describe the general form of this law1
(La): The laws of physics, and in particular the laws of transformation between inertial observers, involve a fundamental/observerindependent small (possibly Planckian) length scale Ldsr , which can be measured by each inertial observer following the measurement procedure MLdsr
And finally one must have the speed-of-light-scale principle:1
d Notice
(Lb): The laws of physics, and in particular the laws of transformation between inertial observers, involve a fundamental/observerindependent velocity scale c, which can be measured by each inertial observer as the speed of light with wavelength λ much largerd than Ldsr (more rigorously, c is obtained as the infrared
that (Lb) is not really conceptually different from the standard speed-of-light postulate of Special Relativity. Actually the standard formulation of speed-of-light postulate of Special Relativity is phrased redundantly. Einstein could have adopted (Lb) and derive the wavelength independence from the absence of a fundamental length scale in Special Relativity. By straightforward dimensional analysis one indeed concludes that
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limit of the speed of light)
For the final element of the new relativistic theory, the description of the MLdsr measurement procedure for Ldsr we do not have enough experimental information to even make an educated guess. There are many physical arguments and theoretical models that predict one or another physical role for the Planck length, but none of these scenarios has any experimental support. It is still plausible that the Planck length has no role in space-time structure and kinematics (which would render DSR research of mere academic interest). Even assuming DSR actually does play a role in the description of Nature it seems likely that the correct formulation of MLdsr would end up being different from any proposal we can contemplate presently, while we are still lacking the needed guidance of experimental information, but through the study of some specific examples we can already acquire some familiarity with the new elements required by a conceptual framework in which the Relativity Principle coexists with observer-independent high-velocity and small-length scales. In light of this situation in Ref. 1. I considered a specific illustrative example of the postulate (La), which was also inspired by the mentioned 1997-2000 studies of Planck-scale modification of the dispersion relation:
(La ): The laws of physics, and in particular the laws of transformation between inertial observers, involve a fundamental/observerindependent small (possibly Planckian) length scale Ldsr , which can be measured by each inertial observer by determining the dispersion relation for photons. This relation has the form E 2 c2 p2 f E, p; Ldsr 0, where the function f is the same for all inertial observers and in particular all inertial observers agree on the leading Ldsr dependence of f : f E, p; Ldsr Ldsr cp2 E
In these past 7 years of course other examples of measurement procedure MLdsr have been considered by myself and others. an observer-independent law of wavelength dependence of the speed of light is not possible in theories that do not have an observer-independent length scale. The fact that I adopt (Lb) should not be interpreted as suggesting that the speed of light must necessarily have some wavelength dependence in a relativistic theory in which there is also an observer-independent length scale. The presence of an observer-independent length scale renders such a wavelength dependence possible, thereby requiring a more prudent formulation of the operative definition of c, but (as I already stressed in Ref. 1) there might well exist consistent relativistic scenarios in which an observer-independent length scale is introduced in such a way that the speed of light remains wavelength independent.
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6.2.3. A falsifiable proposal In closing this section on the “definition of doubly-special relativity” I intend to stress that, in spite of the fact that this definition does not specify a mathematical formalism, the DSR idea is falsifiable. Many alternative mathematical formalisms could be tried, and perhaps should be tried, but there is at least one test that can exclude DSR completely, without appeal. It is perhaps striking that an idea for Planck/scale physics which is not even at present formalized through a mathematical formalism is falsifiable, whereas so many ideas for Planck/scale physics that are fully formalized mathematically are still not falsifiable (the relevant formalisms still did not produce a “live-or-die” prediction). Of course any experiment providing evidence of a preferred frame would rule out DSR. And even just looking within the literature devoted to the fate of Lorentz symmetry at the Planck scale one finds a research programme which can be used to show explicitly the falsifiability of DSR. In fact, it has been recently realized (see, e.g., Refs. 82–84) that when Lorentz symmetry is broken at the Planck scale there can be significant implications for certain decay processes. At the qualitative level the most significant novelty would be the possibility for massless particles to decay. Let us consider for example a photon decay into an electron-positron pair: γ e e . And let us analyze this process using the dispersion relation E 2 2 2 2 p , (6.1) m E p η Ep where Ep 1Lp . Assuming then an unmodified law of energy-momentum conservation, one easily finds a relation between the energy Eγ of the incoming photon, the opening angle θ between the outgoing electron-positron pair, and the energy E of the outgoing positron, which, for the region of phase space with me # Eγ # Ep , takes the form cos θ A B A, with A E Eγ E and B m2e ηEγ E Eγ E Ep (me denotes of course the electron mass). For η 0 the process is still always forbidden, but for positive η and Eγ $ m2e Ep η 13 one finds that cos θ 1 in certain corresponding region of phase space. The energy scale m2e Ep 13 1013 eV is not too high for astrophysics. The fact that certain observations in astrophysics allow us to establish that photons of energies up to 1014 eV are not unstable (at least not noticeably unstable) could be used82,84 to set valuable limits on η. If following this strategy one did find photon decay then the idea of spacetime symmetries broken by Planck-scale effects would be strongly en-
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couraged. The opposite is true of DSR, which essentially codifies a certain type of deformations of Special Relativity. Any theory compatible with the DSR principle must have stable massless particles. A threshold-energy requirement for massless-particle decay (such as the Eγ $ m2e Ep η 13 mentioned above) cannot of course be introduce as an observer-independent law, and is therefore incompatible with the DSR principles. By establishing the existence of a threshold for photon decay one could therefore indeed falsify the DSR idea. And more generally any theory compatible with the DSR principle must not predict energy thresholds for the decay of particles. In fact, one could not state observer-independently a law setting a threshold energy for a certain particle decay, because different observers attribute different energy to a particle (so then the particle should be decaying according to some observers while being stable according to other observers). 6.3. More on the Concept (a true-false exercise) The definition of DSR given in Ref. 1, which (in order to render this review self-contained) I here repeated in the preceding section, is crisp enough not to require any further characterization. However, the fact that it does not come with a specification of mathematical formalism, also as a result of the large incoherent nature of the DSR literature that has developed in just a few years, has occasionally generated some confusion. This motivates the redundant task to which I devote the present section: drawing from the experience of these 7 years, I explicitly comment on some possible misconceptions concerning DSR. 6.3.1. DSR not equivalent to Special Relativity The DSR concept, as introduced in Ref. 1, is very clearly alternative to Special Relativity. How could it then be possible that some authors (see, e.g., Ref. 54) have instead naively argued the equivalence? I think this can be primarily traced back to the fact that some of the attempts to formalize the DSR concept (particularly within the Hopf-algebra formalism, which I shall discuss later in these notes) involve modifications of the commutators of Poincar´e generators, and they are such that one a Ta connecting the Poincar´ e-like genereasily finds a nonlinear map Tdsr a ators Tdsr adopted in one such attempt to construct a DSR theory and the standard generators of the Poincar´e algebra (i.e. the modification of com-
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a mutators found for the generators Tdsr can be eliminated by a nonlinear redefinition of the generators). Before I explain why in the specific frameworks where this modified commutators are found such a nonlinear redefinition of generators is futile, let me make a couple of a priori remarks:
If an attempted formalization of DSR turned out to be equivalent to Special Relativity (and therefore in particular did not truly manage to introduce an observer-independent short-distance scale in the laws of transformation of observables between different inertial observers) then of course one should simply conclude that the attempt failed, and avoid using such a failed attempt in characterizations of DSR. (ii) Within the relevant framework under consideration from a DSR perspective nonlinear redefinitions of generators are simply futile (see below), but there are other frameworks in which such redefinitions of generators are not futile and, when handled naively as “physics isomorphisms”, lead to misleading results. In general one must proceed with extreme caution before assuming the equivalence of two symmetry algebras linked by a nonlinear map. In particular, in the case of Lie algebras, which often provide an appropriate formalization of a physics symmetry, linear redefinitions of generators are allowed, but nonlinear redefinitions of the generators may destroy the Lie-algebra structure or anyway take us from a given Lie algebra to a completely inequivalent Lie algebra. A perhaps noteworthy example is the onee of the Galilei Lie algebra for 2D spacetimes which can be mapped (of course, nonlinearly) into the Poincar´e Lie algebra for 2D spacetimes.
Let me now turn to the point about the futility of nonlinear redefinitions of generators in the relevant attempts to find a DSR-compatible formalism. The unwise claims of equivalence with Special Relativity concern attempts to use the formalism of Hopf algebras in looking for theories compatible with the DSR principles. A key aspect of Hopf algebras is that besides the commutators they are also characterized by the so-called cocommutators (via the so-called coproduct rules), which affect the law of action of generators on products of functions. In fact according to Hopf algebras (unlike the case of Lie algebras) the action on products of functions is not obtainable by eI
learned about this map in conversations with Gianluca Mandanici.
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standard application of Leibniz rule to the action on a single function. This fits naturally with the properties of some structures used in Planck-scale research, such as noncommutative spacetimes (since the action of operators on products of functions of the noncommutative spacetime coordinates is naturally not subject to Leibniz rule). Nonlinear redefinitions of generators for Hopf algebras are admitted, just because they are futile: the form of the commutators of course changes but there is a corresponding change in the cocommutators, and the combined effect amounts to no change at all for the overall description of the symmetries. In Refs. 85 and 86 this result, which can be shown already at the abstract algebra level, was worked out explicitly, for two nonlinear redefinitions of two examples of Hopf algebra deformations of the Poincar´e algebra, finding that one obtains the same conserved charges independently of whether one analyzes the Hopf-algebra symmetries in terms of one set of generators or a nonlinear redefinition of that given set of generators. Essentially this comes from the fact that the Noether analysis of a theory with a given Lagrangian density will combine aspects described in terms of the action of generators on single functions and aspects which instead concern the action on product of functions. 6.3.2. Not necessarily involving the κ-Poincar´ e Hopf algebra The formalism of Hopf algebras which I just used to illustrate the weakness of some “equivalence claims” found in the literature, is also at present the most promising opportunity to find a formalism compatible with the DSR principles. However, it would be dangerous to identify the DSR idea with the mathematics of Hopf algebras, since at present we still know very little about the implications of Hopf-algebra spacetime symmetries for observable properties of a theory, and of course the DSR concept refers to the observables of a theory and their properties under change of inertial observer. Most attempts to construct a DSR theory using Hopf algebras focus on the so-called κ-Poincar´e Hopf algebra,87–89 but if indeed Hopf algebras prove to be usable in DSR theories then other Hopf algebras might well also be considered. I will mention in various points of these notes the case of the twisted Hopf algebras of symmetries of observer-independent canonical noncommutative spacetime, which appears to be equally promising. And of course it is at present fully legitimate (actually necessary) to look for realizations of the DSR concept that do not involve Hopf algebras. Even if Hopf algebras do eventually turn out to be usable in the construction of DSR theories it would be inappropriate to identify the math-
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ematics of (some) Hopf algebras with the DSR physics concept. This observation becomes obvious if one considers, for example, the nature of the works devoted to the κ-Poincar´e Hopf algebra before the proposal of the DSR concept. In the pre-DSR κ-Poincar´e literature one finds some warnings90 against attempts to integrate the boost generators to obtain a candidate for finite boosts (which appeared to lead only to quasi-group91 structure). But DSR-physicists needed of course finite transformations and guided by the DSR physical principles looked at the many ways in which κ-Poincar´e mathematics can be described, eventually finding one1,4 that allowed a consistent implementation of finite boosts on one-particle systems. Similarly, the law of energy-momentum conservation which is advocated by some κ-Poincar´e experts8 is incompatible with the DSR principles,33 but physicists guided by the DSR idea are looking for alternative descriptions of the law energy-momentum conservation among the many potentially suitable structures that are available within κ-Poincar´e mathematics. This point about the requirements for some mathematical formulas to be promoted to the status of a physics proposal has of course many sidesf , and its proper discussion goes well beyond the scopes of these notes. Let me just add one more (somewhat random) remark: if we are too quick in labelling mathematical formulas with the status of physics proposal then we might accordingly attribute to Bardeen, Cooper and Schrieffer (and their study of superconductivity) also the proposal of the Glashow-WeibergSalam Standard Model of particle physics. f To course students I often propose another illustration of the difficulties encountered in attempting to promote some mathematical formulas to the status of a physics proposal. For simplicity I choose the context of nonrelativistic quantum mechanics and I propose to the students the hypothetical situation of an author proposing canonical space iθ jk , without any further noncommutativity by simply writing the formula xj , xk qualification (of course analogous considerations may be formulated for any, even spacetime, noncommutativity, and even in relativistic settings, such as quantum field theory). I then ask: Would this formula amount to a physics proposal? Would that proposal be compatible with the DSR principles? Of course, the formula does not amount to a physics proposal. A lot of physical characterizations must be given to the formula before it carries any meaning for physics. Just to start one should notice that some entities playing the role of “space coordinates” appear in nonrelativistic quantum mechanics as arguments of the wave functions and other entities that could be described as “space coordinates” have the role of observables. One should specify which of the two entities is iθ jk . And what is the matrix θ jk ? In particular, does this to be described by xj , xk matrix transform under rotations of the axes of the observer’s reference frame? (in which case one would then expect the noncommutativity to break space-rotation symmetry) or is it intended as an observer-independent matrix? (in which case one might have to deform the laws of action of space-rotation generators on the space coordinates).
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6.3.3. Not any deformation, but a certain class of deformations of Special Relativity While, as I just stressed, it is clearly too early to associate with the DSR concept a specific mathematical formalism, it would also be disastrous to gradually transform (e.g. by gradual modifications of the definition of a DSR theory) the DSR concept into a large umbrella covering all scenarios for a length scale to enter spacetime symmetries. It is for this reason that I chose to do the redundant exercise of repeating here (in Section 6.2) exactly the definition of DSR theory originally given in Ref. 1. It actually did happen over these past 7 years that occasionally the DSR proposal was confused as the proposal that we should have some deformation of Special Relativity, with the idea that any deformation would be DSR-acceptable. The definition given in Ref. 1 (here repeated in Section 2) provides a physics picture which amounts instead to a rather specific class of deformations of Special Relativity, with which certain deformations are clearly incompatible. Consider for example the exercise reported by Fock in an appendix of Ref. 92. There Fock explores the role that each of the in-principle structures of Special Relativity plays in constraining the mathematics of the Special-Relativity framework. Unsurprisingly by removing one of the inprinciple structures Fock obtains De Sitter spacetime instead of Minkowski spacetimeg . De Sitter spacetime is a deformation of Minkowski spacetime and the De Sitter algebra is a deformation of the Poincar´e algebra, but of course these standard formalisms do not provide examples of DSR theories. Indeed De Sitter spacetime is a deformation of Minkowski spacetime by a long-distance scale (Minkowski obtained from De Sitter as the deformation length scale is sent to infinity), whereas one of the requirements for a DSR theory is that the deformation scale be a short-distance (high-energy) scale (Special Relativity obtained from DSR as the deformation length scale is sent to zero). Another example worth mentioning is the one of proposals in which one introduces a maximum-acceleration scale. Since already in Special Relativity acceleration is an invariant, such proposals do not a priori require a DSR formulation. Indeed often these proposals involve formalisms able to handle at once both inertial observers and essentially Rindler observers (of course acceleration changes when going from an inertial frame to a Rindler frame). At least in a generalized sense these are also “deformations” of g Fock does not appear to realize he has actually landed on De Sitter, but does handle all formulas correctly.
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Special Relativity, but typically they do not require a modification of the laws of transformation between inertial observers. As these two examples illustrate clearly not any “deformation” of Special Relativity provides a realization of the DSR concept. I guess the confusion some authors have on this point originates from a recent fashion to use “Deformed Special Relativity” as an equivalent name for Doubly-Special Relativity. Of course, there is no content in a name and one might consider it a free choice of the writer, but there are cases in which a certain choice of name may induce confusion, and this is certainly one of those cases, since “Deformed Special Relativity” invites a naive interpretation of the type “any deformation of Special Relativity will do”. And an additional source of confusion originates from the fact that in the literature, already before the proposal of the Doubly-Special Relativity idea, there was a research programme named “deformed special relativity” (see, e.g., Ref. 93 and references therein), which pursues physics motivation and physics objectives that are completely different from the ones of Doubly-Special Relativity. 6.3.4. A physics picture leading to DSR and the possibility of DSR approximate symmetries One other point which I stressed in Ref. 1 and has been largely ignored concerns the possibility that “DSR symmetries” might actually be only approximate symmetries, even within the regime of physics observations where they do (hopefully) turn out to be relevant. In order to clarify this point let me first discuss a certain “vision” for the structure of spacetime at different scales. While at present the status of development of research in quantum gravity allows of course some sort of “maximum freedom” to imagine the structure of spacetime at different distance scales, there are arguments (especially within the “emergent gravity” literature, but also beyond it) that would provide support for the following picture: (I) at superPlanckian distance scales the only proper description of spacetime degrees of freedom should be strongly “quantum”, so much so that no meaningful concept of spacetime coordinates could be introduced, (II) then at subPlankian but nearly Planckian scales some sort of intelligible geometry of spacetime might emerge, possibly allowing the introduction of some spacetime coordinates, but the coordinates might well be “quantum coordinates” (e.g. noncommutative coordinates) and the symmetries of this type of spacetime geometry might well be DSR symmetries, (III) then finally in the infrared our famil-
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iar smooth classical description of spacetime becomes a sufficiently accurate description. Besides providing a logical scheme for the emergence of DSR symmetries, this picture also explains in which sense I could argue that DSR symmetries might well be only approximate symmetries, even in the regime where they are relevant. Most authors (including myself) working on DSR insist on getting formulas that make sense all the way up to infinite particle energies. There is nothing wrong with that, but it might be too restrictive a criterion. If DSR is relevant only at energy scales that are subPlankian (but nearly Planckian), and if we actually expect to give up an intelligible picture of spacetime and its symmetries above the Plank scale, then perhaps we should be open to the possibility of using mathematics that provides an acceptable (closed) logical picture of DSR only at leading order (or some finite order) in the expansion of formulas in powers of the Planck length. What I mean by this will be made somewhat clearer later in this notes, when I set up the discussion of DSR phenomenology. There a candidate DSR test theory will be based on formulas proposed exclusively as leadingorder formulas. 6.3.5. Not any fundamental length scale Inadequate description/investigation of the laws of transformation between inertial observers is, as stressed in part in the previous subsections, a very serious limitation for an analysis being proposed as a DSR study. However, there is worse: there are (fortunately only very few) self-proclaimed “DSR studies” which are only structured at the level of proposing some “fundamental length/energy scale”, without any effort to establish whether this scale is of the type required by the DSR concept (in particular not providing any analysis of the role that this “fundamental scale” does or does not have in the laws of transformation between observers). Unfortunately, even outside the DSR literature, it is not uncommon to find in the physics literature a rather sloppy use of terms such as “fundamental scale”. In particular, “fundamental scales” are often discussed as if they were all naturally described within a single category. For DSR research it is instead rather important that these concepts be handled carefully. In this respect the DSR concept is conveniently characterized through the presence of a short-distance/high-energy which is “relativistically fundamental” in the same sense already familiar for the role of the scale c in Special Relativity.
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There are of course scales that are no less fundamental, but have properties that are very different from the ones of c in Special Relativity. Obvious examples are scales like the rest energy of the electron, which is of course a “fundamental” scale of Nature, but is relativistically trivial (a rest-frame property). Just a bit less obvious are cases like the one of the quantum-mechanics scale . Space-rotation symmetry is a classical continuous symmetry. One might, at first sight, be skeptical that some laws (quantum-mechanics laws) that discretize angular momentum could enjoy the continuous space-rotation symmetry, but more careful reasoning94 will quickly lead to the conclusion that there is no a priori contradiction between discretization and a continuous symmetry. In fact, the type of discretization of angular momentum which emerges in ordinary non-relativistic quantum mechanics is fully consistent with classical spacerotation symmetry. All the measurements that quantum mechanics still allows (a subset of the measurements allowed in classical mechanics) are still subject to the rules imposed by rotation symmetry. Certain measurements that are allowed in classical mechanics are no longer allowed in quantum mechanics, but of course those measurements cannot be used to characterize rotation symmetry (they are not measurements in which rotation symmetry fails, they are just measurements which cannot be done). A more detailed discussion of this point can be found in Ref. 94. Essentially one finds that is not a scale pertaining to the structure of the rotation transformations. The rotation transformations can be described without any reference to the scale . The scale sets, for example, the minimum non-zero value of angular momentum (L2min 32 4), but this is done in a way that does not require modification of the action of rotation transformations. Galilei boosts are instead genuinely inconsistent with the introduction of c as observer-independent speed of massless particles (and maximum velocity attainable by massive particles). Lorentz/Poincar´e transformations are genuinely different from Galilei transformations, and the scale c appears in the description of the action of Lorentz/Poincar´e generators (it is indeed a scale of “deformation” of the Galilei transformations). Both and c are fundamental scales that establish properties of the results of the measurements of certain observables. In particular, sets the minimum non-zero value of angular momentum and c sets the maximum value of speed. But has no role in the structure of the transformation rules between observers, whereas the structure of the transformation rules between observers is affected by c. I am describing c as a “relativistically
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fundamental” scale, whereas is a fundamental scale that does not affect the transformation rules between observers. A characterizing feature of the DSR proposal is that there should be one more scale playing a role analogous to c, and it should be a shortdistance/high-energy scale. One can try to introduce the Planck scale in analogy with rather than with c. In particular, Snyder lookedh for a theory95 in which spacetime coordinates would not commute, but Lorentz transformations would remain unmodified by the new commutation relations attributed to the coordinates. Such a scenario would of course not be a DSR scenario. In closing this subsection I should comment on one more type of fundamental constants. The electron-mass scale me , the quantum-mechanics scale and the (infrared-limit-of-the-) speed-of-light scale c are different types of fundamental constants, whose operative meaning is of course given through the measurements of certain corresponding observables. Another type of fundamental constants are the “coupling constants”. For example, in our present description of physics the gravitational coupling G is a fundamental constant. It does not impose constraints on the measurements of a specific observable, but it governs the laws of dynamics for certain combinations of observables. Also G is observer independent, although a careful analysis (which goes beyond the scopes of this note) is needed to fully characterize this type of fundamental scales. One can define G operatively through the measurement of static force between planets. In modern language this amounts to stating that we could define G operatively as the low-energy limit of the gravitational coupling constant. All observers would find the same value for this (dimensionful!) constant. This last remark on the nature of the fundamental constant G is particularly important for DSR theories. In our present description of physics the Planck scale is just the square root of the inverse of G rescaled through and c. The idea of changing the status of G (i.e. Ep ) from the one h In
his renowned paper in Ref. 95, Snyder explicitly states its objectives as the ones of introducing spacetime noncommutativity in a way that is compatible with Special Relativity. From the tone of the paper it appears that Snyder was certain to have succeeded, but actually it might be necessary to reexamine his proposal with the more powerful tools of symmetry analysis that have been recently developed. It appears to be not unlikely that some of Snyder’s conclusions might be based on a incorrect description of energy-momentum.33 From a physics perspective it remains possible that, while Snyder was proposing an ordinarily special-relativistic framework (and thought he succeeded), a more careful analysis of the mathematics of the Snyder noncommutative geometry might lead to structures suitable for a DSR theory.33
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of fundamental coupling scale to the one of relativistic fundamental scale might have deep implications.1,3 6.3.6. And the same type of scale may or may not be DSR compatible In the previous subsection I stressed the difference between scales that are potentially significant from a relativistic perspective (such as c) and scales that are not (such as and the electron rest energy). It is important to also stress that even when dealing with a scale that is potentially significant from a relativistic perspective, without a description/analysis of the laws of transformation between inertial observers any claim of relevance from a DSR perspective is futile. Consider for example the possibility of a “minimum wavelength”. This is clearly a type of proposal that is potentially significant from a relativistic perspective, and indeed one of the objectives of DSR research is to find a meaningful framework for the implementation of a “minimum wavelength principle” in an observer-independent manner. However, the proposal of a minimum wavelength does not in itself provide us with a DSR proposal. In particular, the minimum-wavelength bound could be observer dependent. A calculation within a given theory providing evidence of a minimumwavelength bound would of course not suffice to qualify the relevant theory as a DSR theory. One should at least also analyze what type of laws of transformation between observers apply in the relevant theory and verify that the minimum-wavelength bound (both the presence of a bound and the value of the bound) is observer-independent. Similar considerations apply for proposals of modified energymomentum dispersion relations and modified wavelength-momentum relations. It is well understood that some frameworks in which one finds modified dispersion relations are actually not compatible with the DSR principles: in the relevant frameworks the modification of the dispersion relation is observer dependent (at least because of observer dependence of the scale of modification of the dispersion relation), and Poincar´e symmetry is actually broken. Analogously in quantum-gravity scenarios (see, e.g. Refs. 96 and 97) in which one adopts a modified relation between momentum and wavelength (typically such that momentum is still allowed to go up to infinity but there is a finite minimum value of wavelength), before concluding in favor or against compatibility with the DSR principles an analysis of the laws of transformation between observers is of course necessary. One
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could imagine such a modification of the relation between momentum and wavelength to be a manifestation of a breakdown of Poincar´e symmetry, but there is also no in-principle obstruction for trying to implement it in a DSR-compatible way. 6.3.7. Classical spacetime not a possibility when photon speed is wavelength dependent The most studied possibility for obtaining a DSR-compatible theory involves noncommutative spacetime. Clearly at present we cannot exclude that there might be some DSR-compatible scenario based on classical spacetime or some other (non-noncommutative) type of nonclassical spacetime. However, in this subsection I want to stress (following the line of reasoning adopted in Ref. 98) that if one is proposing a DSR scenario in which there is a wavelength(/energy) dependence of the speed of photons (which, as mentioned, may or may not be the case) and the particles in the relevant framework are governed by quantum mechanics (which probably is the only safe assumption) then attempts to formulate the theory in a classical spacetime appear to be inadmissable. Since a detailed discussion is given in Ref. 98, I can here just sketch out the argument adopting a specific form for the energy dependence of the velocities of massless particles v
1
E , EQG
(6.2)
where EQG is a reference scale. The issue arises because in giving operative meaning to spacetime points (or, more rigorously, to distances between spacetime points) one inevitably resorts to the use of particle probes and spacetime can be meaningfully labeled “classical” only if the theory admits the possibility for such localization procedures to have absolutely sharp (“no uncertainty”) outcome, at least as the endpoint of a well-defined limiting procedure. Within ordinary quantum mechanics one could even contemplate the limit in which point particles have infinite inertial mass (so that the Heisenberg principle is ineffective) but that limit is not meaningful in theoretical frameworks, such as the DSR framework, in which the motivation originates from the quantum-gravity problem and therefore infinite inertial mass comes along with infinite gravitational charge (mass). In the opposite limit, the one of massless particles, one finds that the energy(/wavelength) dependence of the speed of light introduces an extra term
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in the balance of quantum uncertainties. Assume in fact that the measurement procedure requires some known time Tobs and therefore (in order to obtain measurement results compatible with the classical-spacetime idealization) we would like the particle probe to behave as a classical probe over that time. For that goal it is in particular necessary to keep under control two sources of quantum uncertainty, the one concerning the energy of the particle and the one concerning the time of emission of the particle. The uncertainty δx in the position of the massless probe when a time Tobs has lapsed since the observer (experimentalist) set off the measurement procedure will in general satisfy the following inequality δx δt δv Tobs
δt
δE Tobs , EQG
(6.3)
where δt is the uncertainty in the time of emission of the probe, and I used (6.2) to describe the uncertainty in the speed of the particle, δv, in terms of the uncertainty δE in the energy of the probe. Since the uncertainty in the time of emission of a particle and the uncertainty in its energy are relatedi by δt δE 1, Eq. (6.3) can be turned into an absolute bound on the uncertainty in the position of the massless probe when a time Tobs has lapsed since the observer set off the measurement procedure: 1 δE Tobs Tobs . (6.4) δx δE EQG EQG The right-hand side of (6.4) does exploit the fact that in principle the observer can prepare the probe in a state with desired δE (so it is legitimate to minimize the uncertainty with respect to the free choice of δE), but the classical behavior of the probe is not achieved in any case (in all cases δx is strictly greater than 0). 6.4. More on the Use of Hopf Algebras in DSR Research 6.4.1. A Hopf-algebra scenario with κ-Poincar´ e structure In trying to give a brief summary of some key aspects of the possibility of DSR scenarios based on the mathematics of Hopf algebras, let me start i It
is well understood that the δt δE 1 relation is valid only in a weaker sense than, say, Heisenberg’s Uncertainty Principle δx δp 1. This has its roots in the fact there is no self-adjoint operator canonically conjugate to the total energy, if the energy spectrum 1 does relate δt intended as uncertainty in is bounded from below. However, δt δE the time of emission of a particle and δE intended as uncertainty in the energy of that same particle, and therefore it applies in the context which I am here considering.
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with a rudimentary review of the most studied such scenario which is based on the κ-Poincar´e Hopf algebra and the associated κ-Minkowski noncommutative spacetime. The characteristic spacetime-coordinate noncommutativity of κMinkowski is given by xj , x0
iλxj
xk , xj
(6.5)
0,
(6.6)
where x0 is the time coordinate, xj are space coordinates (j, k &1, 2, 3'), and λ is a length scale, usually expected to be of the order of the Planck length. Functions of these noncommuting coordinates are usually conventionally taken to be of the form f x
d4 k f˜ k eikx eik0 x0 ,
(6.7)
where the “Fourier parameters” &k0 , ki ' are ordinary commutative variables. I shall here be satisfied with a brief review of a frequently used characterization of symmetries of κ-Minkowski, in which generators for translations, space-rotations and boosts are introduced on the basis of the following definitions (6.8) Pµ eikx eik0 x0 kµ eikx eik0 x0 ,
Rj eikx eik0 x0 jkl xk kl eikx eik0 x0 , Nj eikx eik0 x0
kj e
eik0 x0 x0
(6.9)
i k x
1 e2λk0 xj 2λ
λ2 k 2
λxl kl kj
(6.10) eikx eik0 x0 .
The fact that we are here dealing with a Hopf algebra (indeed the κ-Poincar´e Hopf algebra) is essentially seen by acting with these generators on products of functions (“coproduct”), for example Pµ eikx eik0 x0 eiqx eiq0 x0 λk0 1δµ0 qµ eikx eik0 x0 eiqx eiq0 x0 kµ e i k x ik0 x0 i " kµ qµ e e e qx eiq0 x0 , (6.11)
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Some authors naively focus their analyses exclusively on the properties of commutators of the generators of the κ-Poincar´e algebra, which turn out to be deformations99 of the commutators of the standard Poincar´e Lie algebra. And several (mutually incompatible) naive arguments have been proposed on the basis of such naive characterizations: in some cases the nonlinearities present in these commutators are taken as a full characterization of the κPoincar´e symmetries, while in other cases a large significance is incorrectly given to the fact that by nonlinear redefinition of the generators one can remove all anomalies of the commutators. However, a proper description of Hopf-algebra spacetime symmetries must clearly take into account both the commutators and the coproducts (and redefinitions of the generators affect simultaneously both commutators and coproducts). Consider for example the case of translation transformations. A translation transformation for a function f x should be described as df x i µ Pµ f x,
(6.12)
in terms of the translation generators and some transformation parameters
µ . And it turns out that the transformation parameters must reflect the properties of the coproduct. In fact, the transformation parameters must ensure that (if xµ is in κ-Minkowski) xµ µ is still a point in κ-Minkowski: xj j , x0 0
iλ xj
j
,
xi i , xj j
0.
(6.13)
So the transformation parameters must not be simple numbers but should instead be endowed with nontrivial algebraic properties. And it is easy to see85,100 that the form of these algebraic properties should reflect the properties of the coproduct in order to preserve Leibniz rule: d f g f dg df g .
(6.14)
Taking into account these observations it turned out to be possible85,100 to obtain conserved charges associated to the Hopf symmetries for a theory with classical fields in the noncommutative κ-Minkowski spacetime (while all previous attempts, which had naively ignored the role of the coproduct in the full characterization of symmetry transformations, had failed). The interested reader can find a rather detailed description of what we presently know about the κ-Poincar´e symmetries of κ-Minkowski in Refs. 85 and 100 and references therein. I shall be here satisfied with the few preliminary ingredients of this description that I just provided, as they are sufficient for me to discuss the relevance of this subject for DSR research. The idea that this mathematics might provide the basis for a DSR
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theory originates essentially in the observation that the spacetime noncommutativity of κ-Minkowski, as described in (6.6), is left invariant by the action of κ-Poincar´e generators, and the scale λ appears to be a reasonable candidate for a DSR-type second relativistic scale. It has also long been conjectured that with this κ-Minkowski/κ-Poincar´e recipe one might end up having a DSR theory with modified dispersion relation, but originally this suggestion was only based on the observation that the “mass Casimir” of the κ-Poincar´e Hopf algebra is a deformation of the mass Casimir of the standard Poincar´e Lie algebra. A definite statement about the status of the dispersion relation in this framework will require a meaningfully physical identification and characterization of concepts such as energy, spatial momentum, frequency and wavelength, and this task is perhaps now finally within reach, since we can now rely on actual derivations of conserved charges using the techniques developed in Refs. 85 and 100. However, a few residual issues must still be addressed85,100–102 before we can safely identify energy, spatial momentum, frequency and wavelength. And in general, as mentioned in earlier sections of these notes, some work remains to be done to fully establish that the κ-Minkowski/κ-Poincar´e formalism can be really used to construct a DSR theory. It will require still some work of digging through κ-Minkowski/κ-Poincar´e mathematics, guided by the DSR principles, looking for tools that are DSR compatible. But this is not trivial and not easily done without caution. For example, as mentioned, the κ-Poincar´e mathematics can inspire (and as inspired) a description of the kinematics of particle-reaction processes which is manifestly in conflict with the DSR principles (it would select a preferred frame), but looking around in the κ-Minkowski/κ-Poincar´e “zoo of mathematics”, if we indeed look around using the DSR principles as guidance, we might find other (possibly DSR-compatible) structures on which the kinematics of particle-reaction processes could be based. 6.4.2. A Hopf-algebra scenario without κ-Poincar´ e and without modified dispersion relations The fact that the κ-Poincar´e/κ-Minkowski framework briefly discussed in the previous subsection provides a promising path toward a DSR theory is appreciated by most authors involved in DSR research. So much so that some do not appear to even see the possibility of alternatives. Actually, one does not need to look too far to find an alternative which is equally promising: even within the confines of approaches based on spacetime non-
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commutativity one finds a framework which is indeed equally promising from a DSR perspective and appears to motivate investigation of a different set of candidate “DSR effects”. This is the case of the “canonical noncommutative spacetime” with characteristic spacetime-coordinate noncommutativity given by (µ, ν &0, 1, 2, 3') µ
x
, xν iθµν .
(6.15)
These noncommutativity relations (6.15) can be meaningfully considered endowing θµν with the properties of a Lorentz tensor, in which case one of course ends up with a scenario where Lorentz symmetry is broken, but they can also be meaningfully considered endowing θµν with the properties of an observer-independent matrix, which would of course be the case of interest from a DSR perspective. With observer-independent θµν the symmetries of this spacetime are described by a Hopf algebra which is significantly different from κ-Poincar´e, whose generators can be described as follows:
eikxc (6.16) c c c c Mµν eikx Mµν Ω eikx Ω ixµ ν eikx . (6.17) c c where xµ are auxiliary commutative coordinates, ν are ordinary deriva c ˜ k ) tives with respect to the auxiliary coordinates x , and (for any Φ Pµ eikx
Ω
Pµ Ω eikx
c
˜ k eikxc d4 k Φ
Ωi
c µ
µ
˜ k eikx . d4 k Φ
(6.18)
The fact that these are generators of a Hopf algebra manifests itself as usual immediately upon noticing that the action of (Lorentz-sector) generators does not comply with Leibniz rule: (6.19) Mµν eikx eiqx Mµν eikx eiqx eikx Mµν eiqx 1 αβ ikx θ ηαµ Pν e Pβ eiqx Pα eikx ηβ µ Pν eiqx , 2 As in the case of the κ-Poincar´e/κ-Minkowski framework, the idea that this framework based on canonical noncommutativity might provide the basis for a DSR theory originates essentially in the observation that the spacetime noncommutativity of canonical form, (6.15), is left invariant by the action of the Hopf-algebra generators (6.16)-(6.17), so that any physical consequence of that noncommutativity (such as “spacetime fuzzyness”) should be observer independent. To render explicit the presence of
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a short invariant length scale one can conveniently rewrite the observerindependent dimensionful matrix θµν in terms of an observer-independent length scale λ and an observer-independent dimensionless matrix τ µν (with an extra restriction, e.g. unit determinant, to avoid apparent overcounting of parameters upon introducing λ): θµν λ2 τ µν . And the length scale λ should indeed be small in order to ensure that one introduces an acceptably small amount of coordinate noncommutativity (we clearly have robust experimental evidence against large noncommutativity). For a detailed description of what we presently know about this alternative Hopf-algebra-based DSR scenario readers can look at Refs. 86 and 100 and references therein. One key point is that all evidence gathered so far (which however only concerns classical particles and fields in this “quantum” geometry) suggests that the dispersion relation is not modified in this framework, and therefore also from this perspective we might already have a framework providing complementary DSR intuition with respect to the more popular κ-Poincar´e/κ-Minkowski framework. Also for this canonical-noncommutativity framework it turned out to be possible86 to derive conserved charges within an analysis a la Noether, and in doing this it turned out to be necessary to endow the transformation parameters with nontrivial algebraic properties reflecting the coproduct structure. And the choice of ordering implicitly introduced in (6.18) in this case can be shown very explicitly to be inessential: a variety of alternative ordering conventions are easily considered and found86 to lead to the same conserved charges. Since by changing ordering convention for the spacetime coordinates one essentially ends up introducing nonlinear redefinitions of the generators of the Hopf algebra, the fact that the charges are independent of the choice of ordering convention also translates into yet another invitation for caution for those authors who naively assume that nonlinear redefinitions of the generators of a Hopf algebra might change the physical picture (in some extremely naive arguments it is stated that by nonlinear redefinition of the generators one might go from an incorrect to a correct picture, or that such a nonlinear redefinition of generators could eliminate the characteristic length scale of some relevant Hopf algebra, but here very explicitly one finds that such nonlinear redefinitions do not affect the conserved charges and their dependence on the characteristic scale of the Hopf algebra).
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6.5. DSR Scenarios and DSR Phenomenology Considering the very early stage of development of the Hopf-algebra scenarios for DSR, and the fact that the approach based on Hopf algebra is the best developed (the least undeveloped) attempt of finding DSR compatible theories, it is clear that we are not ready to do any “real” DSR phenomenology. In order to claim one was doing real DSR phenomenology a minimum requisite would be the availability of a theoretical framework whose compatibility with the DSR principles was fully established and characterized in terms of genuinely observable features. Since we are not ready for that, one might perhaps consider postponing all reasoning about phenomenology to better times (ahead?). But, at least from some perspectives the development of (however incomplete and however ad hoc) “toy DSR test theories” can be valuable. The exercise of developing such “toy test theories”, in the sense that will emerge from the next subsections, turns out to be valuable in providing a crisper physical characterization of the concept of a DSR theory (and in fact I found it useful to introduce one such “toy theory” already in my original papers in Ref. 1) and allows to clarify some general arguments (as in the case of the incompatibility of a decay threshold with the DSR principles, see later). 6.5.1. A toy-model DSR-scenario test theory, confined to leading order To illustrate what I qualify as a “toy DSR test theory” I can indeed use the example of such “toy theory” that I already used in my original papers in Ref. 1. This is a “limited theory” in that it only concerns the laws of transformation of the energy-momentum observablesj. It assumes that the energy-momentum dispersion relation is observer independent and takes the following form in leading order in the Planck length: E2
2 m2 λ p p2 E .
(6.20)
This dispersion relation is clearly an invariant of space rotations, but it is not an invariant of ordinary boost transformations. Its invariance (to leading order) is ensured adopting standard space-rotation generators Rj j Of
i jkl pk
pl
.
(6.21)
course, an analogous “limited theory” (with the same formulas) could be articulated for the frequency/wavelength observables and for laws governing the composition and onshellness of frequencies/wavelengths.
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and a deformed action for boost generators λ 2 λE 2 Bj ipj i E p E 2 pj
149
iλpj
pk
pk
.
(6.22)
For the limited “phenomenology of kinematics” that some authors have been doing with this limited “toy DSR test theory” the only remaining ingredient to be specified is the one linking incoming energymomenta to outgoing ones, as intended in a law of conservation of energymomentum. Let us start considering processes with two incoming particles, a and b, and two outgoing particles, c and d. The special-relativistic kinematic requirements for such processes are Ea Eb Ec Ed 0 and pa pb pc pd 0, but these clearly1 would not be observer-independent laws in light of (6.22). Working in leading order actually one finds several1 acceptablek alternative possibilities for the deformation of the law of conservation of energymomentum. In the following I will adopt Ea Eb λpa pb
Ec Ed λpc pd ,
pa pb λ Ea pb Eb pa pc pd λ Ec pd Ed pc .
(6.23) (6.24)
Analogous formulas can be obtained for any process with n incoming particles and m outgoing particles. In particular, in the case of a two-body particle decay a b c the laws Eb Ec λpb pc ,
(6.25)
pb pc λ Eb pc Ec pb .
(6.26)
Ea pa
provide an acceptable (observer-independent, covariant according to (6.22)) possibility. 6.5.2. On the test theory viewed from an (unnecessary) allorder perspective As stressed earlier in these notes, the physical picture that motivates the proposal of Doubly-special Relativity allows one to contemplate DSR symmetries as exact symmetries (applicable in a certain corresponding regime), k The
conservation laws that must be satisfied by physical processes should1 be covariant under the transformations that relate the kinematic properties of particles as measured by different observers (all observers should agree on whether or not a certain process is allowed).
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but it also invites one to consider the possibility that (even within the confines of the regime where DSR turns out to be applicable) DSR symmetries be just approximate symmetries. For example, formulas might not be exactly compatible with the DSR setup because of possibly including non-DSR terms that are negligible (but nonzero) in the DSR regime but become large at some even higher energy scales, where quantum-gravity effects might become so virulent not to allow even a DSR description. I should therefore correspondingly stress that it is not necessary (and not necessarily appropriate) to cast the “leading-order toy test theory” discussed in the previous subsection within some corresponding all-order DSR theory. Let me nonetheless briefly review some evidence we have that there is an “all-order toy test theory” from which the leading-order one discussed in the previous subsection can be derived. A first ingredient for such a theory could be the following all-order dispersion relation 2 cosh λE cosh λm p 2 eλE , λ2 in which case boost transformations could be generated by λ 2 1 e2λE p Bj ipj i iλpj pk E 2 2λ pj pk
(6.27)
(6.28)
(whereas, even in the all-order formulation, space-rotation transformations do not require deformation). Even for these “all-order boost generators” (and therefore, of course, also for their leading-order formulation) one manages to obtain explicit formulas4 for the finite boost transformations that relate the observations of two observers. These are obtained by integrating the familiar differential equations dE dξ
iBj , E
,
dpk dξ
iBj , pk
,
(6.29)
which relate the variations of energy-moomentum with rapidity (ξ) to the commutators between the boost generator (along the direction j of the chosen boost) and energy-momentum. The result is conveniently characterized by giving the formula expressing the amount of rapidity ξ needed to take a particle from its rest frame (where the energy is m) to a frame in which its energy is E: cosh ξ
eλE cosh λm , sinh λm
sinh ξ
peλE . λ sinh λm
(6.30)
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Since I am here only setting up a discussion of DSR phenomenology I shall not invest more energy in characterizing this all-order setup. Actually, aware of the fact that at least at present we could only (and only hoping for some luck) attempt a leading order DSR phenomenology (sensitivities presently available do not appear to allow us to go beyond that), some readers might wonder why I even briefly considered this all-order setup. Indeed it is not for planning an “all-order DSR phenomenology” but rather to stress that if one insists (and it is not necessary but possible to insist) on embedding the “leading order toy test theory” within an all-order setup, then at least one valuable constraint emerges: while within the leadingorder toy test theory nothing prevents the (dimensionful) parameter λ to be either positive or negative, within the all-order reformulation λ must be positivel . This comes about31 because attempting to find real energymomentum solutions for Eqs. (6.29) for all real values of ξ (which one ought to do in an all-order setup) and using the form (6.28) of the boost generators one actually finds the solutions codified in (6.30) only for positive λ, while the same equations for negative λ do not admit solution (not solutions for all values of ξ if the energy-momentum must be real31 ). 6.5.3. Photon stability The first possibility that I want to consider in this phenomenology section is the one of energy thresholds for particle decay, such as the possibility of the decay of a photon into an electron-positron pair, γ e e . I already discussed this process earlier in these notes, as a way to show that the DSR concept makes definite predictions, in spite of not being at present attached to any specific mathematical formalism. Any experiment providing evidence of a preferred frame would rule out DSR, and this implies that any theory compatible with the DSR principle must not predict energy thresholds for the decay of particles. In fact, one could not state observerindependently a law setting a threshold energy for a certain particle decay, because different observers attribute different energy to a particle (so then the particle should be decaying according to some observers while being stable according to other observers). This argument holds directly at the level of the logical structure of the DSR concept, but it is nonetheless useful to verify how our toy DSR test theory implements it. The key structure is the rigidity that the DSR concept introduces (for theories structured like our toy test theory) between l Meaning
that λ Lp is a positive real number.
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the structure of the energy-momentum dispersion relation and the structure of the energy-momentum-conservation law. The boost transformations must leave invariant the dispersion relation, and under those same boost transformations the energy-momentum-conservation law must be covariant. This led me in particular to introduce the energy-momentum-conservation law Eb Ec λpb pc ,
(6.31)
pb pc λ Eb pc Ec pb ,
(6.32)
Ea pa
for two-body particle decays a relation E2
b c, in association with the dispersion
2 m2 λ p p2 E .
(6.33)
If one was to combine (6.33) with an unmodified law of energymomentum conservation, as admissible in scenarios with broken Poincar´e symmetry (but not allowed in scenarios in which Poincar´e symmetry is deformed in the DSR sense), then it is well established that a threshold for the decay of a photon into electron-positron pairs can emerge. In the symmetrybreaking case the relation between the energy Eγ of the incoming photon, the opening angle θ between the outgoing electron-positron pair, and the energy E of the outgoing positron, takes the form cos θ A B A, where (for the region of phase space with me # Eγ # Ep ) A E Eγ E and B m2e λEγ E Eγ E (me denotes of course the electron mass), and for λ 0 (meaning λLp negative real number) one finds that cos θ 1 (and therefore the decay process is allowed) in certain corresponding region of phase space. Instead if the same analysis is done in the DSR-compatible framework of our toy DSR test theory, and therefore one adopts the modified dispersion relation (6.33) and the modified energy-momentum-conservation law (6.31)(6.32), one arrives at a result for cos θ which is still of the form A B A but now with A 2E Eγ E λEγ E Eγ E and B 2m2e . Evidently this formula is never compatible with cos θ 1, consistently with the fact that γ e e is always forbidden in our toy DSR test theory (so, in particular, there is no threshold for the decay). This discussion also shows at least one way in which toy DSR test theories such as the one I am considering, in spite of all their limitations for what concerns applicability (the one I am considering only gives a rough
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description of some aspects of kinematics) and motivation (not being derived from, and not even inspired by, a satisfactory comprehensive DSR theory), can be valuable in DSR research. For example, by analyzing decay processes within the framework of a toy test theory one can see explicitly the DSR principles at work in preventing the emergence of particle-decay thresholds, and this would be particularly valuable if one had not noticed the general DSR argument that forbids such decay thresholds. 6.5.4. Weak threshold anomalies for particle reactions In the recent wide literature on a possible Planck-scale breakdown of Lorentz symmetry there has been strong interest in the possibility of large “anomalies” in the evaluation of certain energy thresholds for particle reactions that are relevant in astrophysics. A simple way to see this is found in the analysis of collisions between a soft photon of energy and a high-energy photon of energy E that create an electron-positron pair: γγ e e . For given soft-photon energy , the process is allowed only if E is greater than a certain threshold energy Eth which depends on and m2e . In a broken-Lorentz-symmetry scenario this threshold energy could be evaluated combining the dispersion relation (6.33) with ordinary energy-momentum conservation, and this leads to the result (assuming # me # Eth # 1λ) Eth λ
3 Eth 8
m2e .
(6.34)
The special-relativistic result Eth m2e corresponds of course to the λ 0 limit of (6.34). The Planck-scale (λ) correction can be safely neglected as long as Eth λEth . But eventually, for sufficiently small values of and correspondingly large values of Eth , the Planck-scale correction cannot be ignored. This occurs for λm4e 13 (when, correspondingly Eth m2e λ13 ). This can be relevant for the analysis of observations of multi-T eV photons from certain Blazars.79,81 And performing a similar analysis for the photo-pion-production process, in which one has a highenergy proton and a soft photon as incoming particles and a proton and a pion as outgoing particles, one ends up finding74,78,81 the possibility of large effects for the analysis of the cosmic-ray spectrum in the neighborhood of the “GZK scale”. One might wonder whether something similar to these particle-reactionthreshold results that are so popular in the Lorentz-breaking literature could be found also in a DSR setup. In the case of photon stability we
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managed to draw a robust conclusion in spite of the present limited understanding and development of DSR theories, and our toy DSR test theory provided an explicit illustration of how the DSR principles intervene in the analysis. In the case of the analysis of particle-reaction thresholds the indications we can presently give from a DSR perspective are not as strong, but still rather valuable. The possibility of large particle-reaction threshold anomalies cannot be excluded simply on the basis of the DSR principles, since it does not conflict with any aspect of the structure of those principles, nonetheless, by trial and error, the DSR literature has provided evidence that large threshold anomalies are not naturally accommodated in a DSR framework. A key point to understand is that typically a DSR framework will produce smaller anomalies than a typical symmetry-breaking framework. This is to be expected simply because scenarios with deformation of symmetries produce of course softer departures from the original symmetries than scenarios in which those symmetries are broken (deforming is softer than breaking). It is difficult to convey faithfully here in a few pages how the DSR literature provides support for this intuition, but I can quickly discuss how our toy DSR test theory deals with particle-reaction thresholds, and this will to some extent allow readers to start forming their own intuition. The laws (6.23)-(6.24) for the case of γγ e e take the form1,30,31 E λP p E E λ p p , P p λEp λ P p p λE p λE p
(6.35)
where I denoted with P the momentum of the photon of energy E and I denoted with p the momentum of the photon of energy . Using these (6.35) and the dispersion relation of our toy test theory (6.33) one obtains (keeping only terms that are meaningful for # me # Eth # 1λ) Eth
m2e ,
(6.36)
i.e. one ends up with the same result as in the special-relativistic case. This indicates that there is no large threshold anomaly in our toy DSR test theory. Actually this test theory does predict a small threshold anomaly, but truly much smaller than discussed in some symmetry-breaking scenarios. If, rather than working in leading order within the approximations allowed by the hierarchy # me # Eth # 1λ, one derives a DSR threshold formula of more general validity within our toy DSR test theory,
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one does find a result which is different from the special-relativistic one, but the differences are quantitatively much smaller than in some known symmetry-breaking frameworks (and in particular in the DSR case, unlike some symmetry-breaking scenarios, the differences are indeed negligible for
# me # Eth # 1λ). By trial and error one ends up figuring out that the same outcome is obtained in other toy DSR test theories, if they are all based on dispersion relations roughly of the type of (6.33). Note however that by ad hoc choice of the dispersion relation one can modify our toy test theory in such a way to produce a large threshold anomaly. This is for example accomplished19 by adopting a dispersion relation which leads to the following energy-rapidity relation19 cosh ξ
2 2 4 6 2 E 2π E tanhm λ E E mλ , m
(6.37)
but it might be inappropriate to attach much significance to such an ad hoc setup. In closing this discussion of thresholds for particle-reaction thresholds let me just observe that in a theory that preserves the equivalence of inertial frames the threshold conditions must be written as a comparison of invariants. For example, it is no accident that in special relativity the threshold condition for γγ e e takes the form E m2e . In fact, m2e is of course a special-relativistic invariant and E is also an invariant (for the head-on collision of a photon with four momentum Pµ , such that P0 E, and a photon with four momentum pµ , such that p0 , one finds, also using the special-relativistic dispersion relation, that Pµ pµ 2E ). The fact that in constructing a DSR framework, by definition, one must also insist on the equivalence of inertial frames, implies that in any genuine DSR framework the threshold conditions must also be written as a comparison of invariants. 6.5.5. Wavelength dependence of the speed of light The structure I have introduced so far in our toy test theory does not suffice to derive an energy dependencem of the speed of photons, but this of course m The
title of this subsection mentions a wavelength dependence while the discussion considers an energy dependence. This may serve as a reminder of the two possibilities, which should however be treated as distinct possibilities.21 One may or may not have a DSR deformation of the energy-momentum relation and one may or may not have a DSR deformation of the wavelength-frequency relation. The two structures are linked by standard energy-frequency and momentum-wavelength relations in our ordinary (prequantum-gravity) theories, but those relations may also be DSR deformed, and there are
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will emerge if one assumes that the standard formula v dE dp applies. Assuming v dE dp our test theory leads to the following velocity formula (for m E # Ep 1λ): v
1
m2 2E 2
λE
,
(6.38)
and there is of course a rather direct way to investigate the possibility (6.38): whereas in ordinary special relativity two photons (m 0) with different energies emitted simultaneously would reach simultaneously a faraway detector, those two photons should reach the detector at different times according to (6.38). This type of effect emerging from an energy dependence of the speed of photons can be significant72,76 in the analysis of short-duration gammaray bursts that reach us from cosmological distances. For a gamma-ray burst it is not uncommon to find a time travelled before reaching our Earth detectors of order T 1017 s. Microbursts within a burst can have very short duration, as short as 103 s (or even 104 s), and this means that the photons that compose such a microburst are all emitted at the same time, up to an uncertainty of 103 s. Some of the photons in these bursts have energies that extend at least up to the GeV range. For two photons with energy difference of order ∆E 1GeV a λ∆E speed difference over a time of travel of 1017 s would lead to a difference in times of arrival of order ∆t λT ∆E Ep 102 s, which is significant (the time-of-arrival differences would be larger than the time-of-emission differences within a single microburst). 6.5.6. Crab-nebula synchrotron radiation data not significant For studies of scenarios in which Lorentz symmetry is broken by Planckscale effects another valuable opportunity is provided by observations of the Crab nebula which are naturally interpreted as the result of synchrotronradiation emission. This is part of the studies aimed at testing the possibility of energy dependence of the speed of particles of the type v
1
m2 2E 2
λE
,
(6.39)
arguments21 suggesting that there might be a deformation of the wavelength-frequency relation without an associated deformation of the energy-momentum relation (a scenario which of course would require deformation of the energy-frequency and momentumwavelength relations).
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within scenarios in which Lorentz symmetry is broken. Assuming that all other aspects of the analysis of synchrotron radiation remain unmodified at the Planck scale, one is led103 to the conclusion that, if λ is negative (λ Lp , λLp 0), the energy dependence of the Planck-scale (λ) term in (6.39) can severely affect the value of the cutoff energy for synchrotron radiation.104 In fact, for negative λ, an electron on, say, a circular trajectory (which therefore could emit synchrotron radiation) cannot have a speed that exceeds the maximum value 3 23 λme , (6.40) vemax 1 2 whereas in special relativity of course vemax 1 (although values of ve that are close to 1 require a very large electron energy). This may be used to argue that for negative λ the cutoff energy for synchrotron radiation should be lower than it appears to be suggested by certain observations of the Crab nebula.103 Only very little of relevant for this phenomenology can be said at present from a DSR perspective. One key concern is that we have at present a very limited understanding of what should typically characterize interactions in a DSR framework, and a proper analysis of synchrotron radiation requires a description of interactions.105 Moreover, the main indication from the Crab-nebula synchrotron-radiation analysis concerns the case of negative λ whereas, in the sense discussed in Subsection V.B, out toy DSR test theory appears to be admissible only for positive λ. 6.6. Some Other Results and Valuable Observations I have so far discussed only what from my perspective are the core facts (and myths) about DSR. While this is not intended as a well-rounded review (and omissions, even of deserving works, will be inevitable), especially for the benefit of DSR newcomers (DSR aficionados know these and more) I think I should mention a few examples of other studies that are representative of the range of scenarios that are under consideration from a DSR perspective. 6.6.1. A 2+1D DSR theory? At least in some formulations of quantum gravity in 2+1 spacetime dimensions a q-deformed deSitter symmetry algebra SO 3, 1q emerges106 for nonvanishing cosmological constant, and the relation between the cosmological constant Λ and the q deformation parameter takes the form ln q ΛLp
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for small Λ. It was observed in Ref. 28 that this relation ln q ΛLp implies (in the sense of the ”Hopf-algebra contractions” already considered in Ref. 87) that the Λ 0 limit is described by a κ-Poincar´e Hopf algebra. However the analysis only shows that the κ-Poincar´e Hopf algebra should have a role, without providing a fully physical picture. But, since, as mentioned earlier in these notes, there is some preliminary evidence that the mathematics of the κ-Poincar´e Hopf algebra might be used to produce a DSR theory, the observation reported in Ref. 28 generated some justifiable interest: besides being an opportunity to perhaps find a genuine DSR toy theory, this would also provide a striking picture for how the DSR framework might emerge from a quantum-gravity theory. It was then observed in Ref. 29 that some aspects of the formulation of Matschull et al107–109 of classical gravity for point particles in 2+1 dimensions are compatible with the DSR idea. The key DSR-friendly ingredients are the presence of a maximum value of mass and a description of energy-momentum space with “deSitter type” geometry (see later). However, several additional results must be obtained in order to verify whether or not a DSR formulation is possible. One key point is that Matschull et al. formulate107,108 the theory by making explicit reference to the frame of the center of mass of the multiparticle system. It may therefore be illegitimate to assume that the features that emerge from the analysis are observer independent. Moreover, rather than a deformation of the translation/rotation/boost classical symmetries of 2+1D space, many aspects of the theory, because of an underlying conical geometry, appear to be characterized by only two symmetries: a rotation and a time translation. And this description in terms of conical geometry is also closely related to the fact that the “observers at infinity” in the framework of Matschull et al. do not really decouple from the system under observation, and therefore might not be good examples for testing the Relativity Principle. Moreover, especially when considering particle collisions, Refs. 107–109 appear to describe frequently as total momentum of a multiparticle system simply the sum of the individual momenta of the particles composing the system, and, from a DSR perspective, such a linear-additivity law would be incompatible1 with a deformed dispersion relation. I should also stress that from a DSR perspective the 2+1 context might not provide the correct intuition for the 3+1 context. A key difference for what concerns the role of fundamental scales originates from the fact that in 3+1 dimensions both the Planck length and the Planck energy are
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related to the gravitational constant through the Planck constant (Lp Gc3 , Ep c5 G), whereas in 2+1 dimensions the Planck energy is obtained only in terms of the speed-of-light scale and the gravitational 2 1 c4 G2 1 . constant: Ep Still, even taking into account the “reasons for advancing cautiously” that I just gave, this should be considered one of the most exciting developments for DSR research, and indeed it continues to motivate several related studies (see, e.g., Refs. 39 and 40). 6.6.2. A path for DSR in Loop Quantum Gravity? Another exciting possibility that has received some attention in the DSR literature is the one of obtaining a DSR framework at some effective-theory level of description of Loop Quantum Gravity. An early suggestion of this possibility was formulated already in Ref. 28, taking as starting point the fact that the Loop Quantum Gravity literature presents some support110,111 for the presence of a q-deformation of the deSitter symmetry algebra when there is nonvanishing cosmological constant. As discussed in Ref. 112 (and preliminarily in Ref. 28), in the 3+1D context one expects a renormalization of energy-momentum which is still not under control, and for the analysis of Ref. 28 this essentially translates into an inability to fully predict the relation between the q-deformation parameter and the cosmological constant, which in turn does not allow us to firmly establish the potentialities of this line of analysis to produce a genuine DSR framework. While no fully robust derivation is available at present, the fact that over these past few years other arguments and lines of analysis have also suggested (see, e.g., Refs. 41 and 42) that a DSR framework might emerge at some effective-theory level of description of Loop Quantum Gravity is certainly exciting. 6.6.3. Maximum momentum, maximum energy, minimum wavelength Going back to the more humble (but presently better controlled) context of toy DSR test theories, it perhaps deserves stressing (in spite of the limited scope and limited significance of such test theories) that these test theories have been shown to accommodated rather nicely some features that are rather appealing from the perspective of a rather common quantumgravity intuition. The toy test theory I discussed in the phenomenology
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section would clearly predict a maximum Planckian (1λ) allowed value of (spatial) momentum. And clearly a correspondingly structured test theory for frequencies/wavelengths realizes a corresponding minimum-wavelength (λ) bound.1,21 Also much studied is a toy test theory that was first proposed by Magueijo and Smolin6 in which one arrives at a maximum Planckian (1λ) allowed value of energy. This is a test theory with structure that is completely analogous to the one of the DSR test theory I am here focusing on as illustrative example, but based on the dispersion relation
2
E2 p 1 λE 2
m2 , 1 λm2
(6.41)
rather than (6.27). 6.6.4. Deformed Klein-Gordon/Dirac equations Some progress has also been reached in formulating deformed Klein-Gordon and Dirac equations in ways that would be consistent with the structure of the toy DSR test theory I discussed in the phenomenology section. This is done primarily assuming that κ-Minkowski noncommutative spacetime provides an acceptable spacetime sector for our toy DSR test theory,11 but interestingly it can also be done fully within “energy-momentum space”. In the case of the energy-momentum-space Dirac equation this latter possibility materializes11,12 in the following form γ µ Dµ E, p, m; λ I ψ % p0
(6.42)
where D0
Dj
eλE cosh λm , sinh λm
1 pj 2eλE cosh λE cosh λm 2 , p sinh λm
(6.43)
(6.44)
I is the identity matrix, and the γ µ are the familiar “γ matrices”. 6.6.5. The “soccerball noproblem” Within the specific setup of toy DSR test theories of the type I considered in the phenomenology section, there is a natural issue to be considered,
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which in the first days of development of such toy test theories KowalskiGlikman and I lightly referred to as “the soccerball problem”. The point is that nonlinear deformations of the energy-momentum relation are certainly phenomenologically acceptable for fundamental particles (if the deformation scale is Planckian the effects are extremely small), but clearly the same modifications of the energy-momentum relation would be unacceptable for bodies with rest energy greater than the Planck energy, such as the moon or a soccerball. This issue remains somehow surrounded by a “mystique” in some DSR circles, but it is probably not very significant. Even within the confines of that specific type of toy DSR test theories there probably is, as stressed already in Ref. 1, an easy solution. In fact, another challenge for those test theories is the introduction of a description of “total momentum” for a body composed of many particles, and the two difficulties might be linked: it does not appear to be unlikely1 that the proper description of total momentum might be such that the nonlinear properties attributed to individual particles are automatically suppressed for a multiparticle body. Another possibility is the one of reformulating the test theory for frequencies/wavelengths rather than for energy/momentum, in which case the “socerball problem” essentially disappears.21 6.6.6. Curvature in energy-momentum space While not adding to the scientific content of toy DSR test theories of the type I considered in the phenomenology section, it is conceptually intriguing and possibly valuable to notice that the nonlinearities that these test theories implement can be viewed as a manifestation of curvature in energymomentum space. Indeed, as primarily stressed by Kowalski-Glikman55 (applying in the DSR arena a line of reasoning previously advocated, from a wider quantum-gravity perspective, by Majid113,114 ), in such test theories the energy-momentum variables can be viewed as the coordinates of a DeSitter-like geometry. Applying this viewpoint has proven valuable to quickly seeing some properties of a given toy DSR test theory of the type I considered in the phenomenology section. One must however keep in mind the physics of the variables one is handling: it would be for example erroneous to assume that diffeomorphism transformations of energy-momentum variables could be handled/viewed in exactly the same way as diffeomorphism transformations of spacetime coordinates.
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I should also stress that, while it might be valuable to view (when possible) in terms of curvature in energy-momentum space a given framework for which compatibility with the DSR principles has already been independently established, the availability of a natural-looking map from the energy-momentum variables to some coordinates over a curved (e.g. deSitter) geometry does not suffice to guarantee the availability of a DSR formulation of the relevant framework. In order to make this remark more concrete let me propose a simple analogy. The propagation of light in a water-pool is (to very good approximation) described by a dispersion relation E c2water p2 c4water m2 (m 0 for photons), which of course allows a map from the energy-momentum variables to some coordinates on a Minkowski geometry, but we know that the scale cwater is not observer independent, and the laws of propagation of light in water do not admit a special-relativistic formulation. 6.6.7. A gravity rainbow? In a sense similar to the usefulness of the observation concerning curvature in energy-momentum space it may also be useful to look at toy DSR test theories of the type I considered in the phenomenology section as theories which, at least to some extent, are characterized by an energy-dependent metric. This was first advocated by Magueijo and Smolin24 and is based on a corresponding description of modified energy-momentum relations: at least for massless particles some of the modified energy-momentum relations that have most attracted attention in the relevant literature, which can always be cast in the form f pµ ; Lp 0, could be rewritten (and be accordingly reinterpreted) in the form f pµ ; Lp pµ g p0 ; Lp µν pν 0. This would invite one to look for other aspects of a given toy DSR theory based on modified energy-momentum relations that admit analogous description in terms of an energy-dependent metric. It would clearly be (if for no other reasons, at least practically/computationally) advantageous to find a collection of features of a DSR theory that could all be codified under the common unifying umbrella of a given energy-dependent metric (a “rainbow metric”). However, in pursuing this objective it is of course necessary to proceed cautiously: if, rather than rephrasing established DSR features of a framework in terms of a rainbow metric, one simply took a formal/technical approach, introducing here and there (wherever feasible) an energy-dependent metric, the end result might well not have DSR-compatible structure . It might not even be a physically meaningful
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framework: there are formulas in which we usually refer to a metric that do not admit generalization (at least not a naive generalization) in terms of an energy-dependent metric simply because there is no obvious characteristic energy scale on which to link the metric dependence. Think for example of formulas involving the energy-momentum of several particles, with large hierarchies between the energies of different particles in the system: on which energy should the metric depend then? And a particularly challenging context might be the one of description of the laws of transformation between inertial observers for the spacetime coordinates: two different inertial observers will assign different quadruplets of spacetime coordinates and different values of energy to the particles in a given ensemble, and whatever role one attributes the energy-dependent metric (which in a DSR framework should be describable as an observer-independent law) should be compatible with the way in which simultaneously energies and spacetime coordinates transform in going from one observer to another. For example, in ordinary special relativity xµ η µν xµ xµ η µν xµ with xµ η µν xµ xµ η µν xµ , but if, say, one was to consider something of the form xµ g µν E xµ then the needed law should probably have the form xµ g µν E xµ xµ g µν E xµ and (for all particles in the system!) should codify/comply with the observer-independent laws for g µν E . And of course, just like a postulate of “curvature in energy-momentum space” would not in itself suffice to qualify the corresponding framework as DSR-compatible, the introduction of some energy-dependent metric does not automatically lead to a DSR theory. For example, the propagation of light in dispersive materials often leads to a dispersion relation that could be formally arranged in the form pµ g p0 ; Lp µν pν 0, thereby introducing an energy-dependent metric, but of course the presence of a dispersioninducing material actually selects a preferred frame, and therefore is incompatible with any relativistic formulation. 6.7. Some Recent Proposals 6.7.1. A phase-space-algebra approach Already in some of the first investigations of the DSR idea a few authors (perhaps most notably Kowalski-Glikman (see, e.g., Ref. 9) had explored the possibility of obtaining a DSR-compatible framework by postulating a noncommutativity of spacetime coordinates that would be merged within a single algebraic structure with the symmetry-transformation generators,
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thereby forming a 14-generator “phase-space algebra”. Recently the development of a new approach essentially based on that perspective was started in Refs. 43–46, with the objective of constructing point-particle models that possess a noncommutative (and non-canonical) simplectic structure and satisfy a modified dispersion relation, also hoping that such a setup might facilitate the description of interactions and the introduction of quantum properties for the point particles. A valuable tool that has been developed44 for this approach is a map connecting the novel phase-space structures to ordinary canonical phase space. And among the features that already emerged from this approach I should at least also mention an alternative derivation of the Dirac equation (both in the case of the maximummomentum toy DSR test theory that I used as example in the phenomenology section, and for the maximum-energy toy DSR test theory that was first considered by Magueijo and Smolin), and a proposal of a Nambu-goto setup which is expected44 to be suitable for a DSR-compatible description of point particles. 6.7.2. Finsler geometry From various perspectives one can look at the structure of the DSR principles as suggesting that spacetime should be described in terms of some “exotic” geometry, and indeed, as mentioned, the majority of DSR studies assume a noncommutative spacetime geometry. There has been recently some exploration59 (also see Ref. 44) from the DSR perspective of another candidate as exotic spacetime geometry: Finsler geometry. This idea is in its very early stage of exploration, and in particular for the Finsler geometries considered in Ref. 59 the presence of 10-generator Poincar´e-like symmetries (a minimum prerequisite to even contemplate a formalism as DSR candidate) has not yet been established. It may be significant that the Finsler line element is not invariant67 under DSR-type transformations. 6.7.3. A 5D perspective Several DSR studies, while aiming for a proposal for 4D physics, finds useful to adopt a formalism or a perspective which is of 5D nature. For those exploring the possibility of obtaining a DSR-compatible picture with theories formulated in κ-Minkowski noncommutative spacetime, an invitation toward a 5D perspective comes from the fact that for 4D κ-Minkowski spacetime it is not unnatural to consider a 5D differential calculus.101,102
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This mainly has its roots in the fact that the κ-Poincar´e Hopf algebra is most naturally introduced87 as a contraction of the q-deSitter Hopf algebra (whose dual spacetime picture is naturally described in terms of a 5D embedding environment). And even some authors who are not advocating the noncommutative spacetime formulation (and are probably unaware of the peculiarities of the 5D differential calculus) have independently argued62,63 that it might be beneficial to set up the construction of DSR theories from a sort of embedding-space 5D picture of spacetime variables and/or energymomentum variables. It is probably fair to say that at present it is still unclear whether the type of intuition generated by these 5D perspectives can be truly valuable, but it is certainly striking that different areas of DSR research, working independently, ended up advocating a 5D perspective. 6.7.4. Everything rainbow Taking off from the “rainbow metric” perspective, which I briefly discussed in the previous section, one can find arguments to advocate also60 that the quantum-mechanics scale and the speed-of-light scale c should acquire an energy dependence. This is too recent a proposal for me to comment robustly on it, but I should stress that the challenges for interpretation of some applications of the “rainbow metric” that I mentioned in the previous section of course also apply to this “everything rainbow” approach, and probably apply in more sever way. There are formulas in physics that refer simultaneously to the metric, to , to c, and to a multitude of energy scales (formulas that apply to a large system, composed of many subsystems, for which one can meaningfully introduce a “subsystem energy” concept), and in such cases it seems that one should find a miraculous recipe in order to constructively obtain an effective metric, effective , and effective c. In partly related work61 it has been argued that rather than concepts such as maximum space momentum, maximum energy and minimum wavelengths (that I had so far mentioned in describing work on toy DSR test theories), one could perhaps consider the development of a toy DSR theory in which the new relativistic scale actually sets a maximum value for energy density. This in an intriguing proposal, which might gain supporters if a satisfactory relativistic formulation is found and explicitly articulated in terms of operatively well-defined entities. The construction of such a relativistic formulation may prove however rather challenging, even more challenging than it has been for the type of toy DSR test theories I considered in the phenomenology section. In fact, the laws of transformation
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of energy density should require from the onset a description of symmetry transformations that acts simultaneously on energy-momentum variables (energy) and spacetime variables (the volume where the relevant energy is “contained” which is of course needed for the energy-density considerations). 6.8. Closing Remarks Doubly-Special Relativity is maturing quickly, as a result of the interest it is attracting from various research groups, each bringing its relevant expertise to the programme. A key factor for the success of this large multi-perspective effort is the adoption of a common language used to give and gain full access to the progress achieved toward the development of the DSR idea. I still think the characterization I originally gave in proposing this DSR idea1 could serve us well for these purposes, so I tried to propose it once again here, emphasizing its advantages especially from the perspective of characterizing results from a physics viewpoint (rather than technical/mathematical). Clearly the key objective at this point should be the one of constructing a full theory (with spacetime observables, energy-momentum observables, frequencies, wavelengths, cross sections...) compatible with the DSR principles. I have mentioned here some partial results which might suggest that this ambitious objective is now not far. In the meantime the use of “toy DSR test theories” (in the sense I have here clarified) can be valuable, both to develop some intuition for what type of effects could be predicted within a DSR framework, and to anchor on some formulas (of however limited scope) the debate on conceptual aspects of DSR research. References 1. G. Amelino-Camelia, gr-qc/0012051, Int. J. Mod. Phys. D 11, 35 (2002); hep-th/0012238, Phys. Lett. B 510, 255 (2001). 2. J. Kowalski-Glikman, hep-th/0102098, Phys. Lett. A 286, 391 (2001). 3. G. Amelino-Camelia, gr-qc/0106004 (in “Karpacz 2001, New developments in fundamental interaction theories” pag. 137-150). 4. N.R. Bruno, G. Amelino-Camelia and J. Kowalski-Glikman, hepth/0107039, Phys. Lett. B 522, 133 (2001). 5. S. Alexander, R. Brandenberger and J. Magueijo, hep-th/0108190. 6. J. Magueijo and L. Smolin, hep-th/0112090, Phys. Rev. Lett. 88, 190403 (2002).
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Chapter 7 Nuances of Neutrinos
Amitava Raychaudhuri Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211 019, India Neutrinos are elementary particles which have captured the attention of the scientific community in the past decade. After a brief survey of the status of this subject, we turn to the India-based Neutrino Observatory (INO) project. Here we give a brief introduction to the goals of INO and report on its status.
Contents 7.1 Introduction . . . . . . . . . . . . . . . . . 7.2 Neutrino Oscillations . . . . . . . . . . . . . 7.2.1 Two flavors . . . . . . . . . . . . . . 7.2.2 Matter effects . . . . . . . . . . . . . 7.2.3 Three flavors . . . . . . . . . . . . . 7.2.4 Open issues . . . . . . . . . . . . . . 7.3 INO . . . . . . . . . . . . . . . . . . . . . . 7.3.1 ICAL: genesis and plans . . . . . . . 7.3.2 The ICAL detector . . . . . . . . . . 7.3.3 Up/Down asymmetry in atmospheric 7.3.4 Simulation, prototype, ... . . . . . . . 7.3.5 Future: Long baseline . . . . . . . . 7.4 Conclusions . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . .
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7.1. Introduction I feel honored, indeed humbled, to have the privilege to speak at this meeting to mark seventy-five years of this august institution. We, the members of the scientific community of today, owe much to visionaries like Professor Prasanta Chandra Mahalanobis who spent considerable effort in building the research institutions of the country and for spreading the culture of 171
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science in India. The Indian Statistical Institute stands today as one of the major players in attracting bright minds to a career in science. The protagonist of this report will be an almost undetectable elementary particle – the neutrino. Some of the things which we know about it now are the following: (1) It is a chargeless elementary particle which barely interacts. (2) There are three neutrinos – νe , νµ , and ντ – partnering the three charged leptons e, µ, and τ . (3) An important result of the last decade is that neutrinos have mass. (4) In addition, the familiar flavor eigenstate neutrinos – νe , νµ , and ντ – are superpositions of neutrino states of definite mass, which are nondegenerate. Such a superposition is permitted in quantum mechanics. (5) This leads to a curious feature termed neutrino oscillations. The characteristic of this oscillation is the wavelength which behaves like λ E ∆m2ij , E being the energy and ∆m2ij m2i m2j being the mass square splitting among the i-th and the j-th mass eigenstate. (6) These results have attracted much attention due to the fact that a nonzero neutrino mass goes beyond the very successful Standard Model of particle interactions and has impact on diverse areas ranging from cosmology to nuclear physics. The neutrino made what must be termed a back-door entry into the realm of physics. Unlike the electron which was experimentally observed, the neutrino was postulated to uphold sacred principles like the conservation of energy and angular momentum. In the early nineteen thirties, Wolfgang Pauli, when he made the proposal of a light, uncharged particle – the neutrino – to address the beta decay conundrum, was well aware that such an entity would be very elusive, quite difficult to detect. Until its detection, it was at best an elegant stop-gap fix to enigmatic features of beta decay experimental observations. In fact, it took the better part of twenty years for the neutrino to be tracked down in a laboratory and lay this uneasy issue to rest, only to open up new nu horizons – a veritable Pandora’s box – which have kept scientists busy since then. The weakness of the neutrino interaction can be gauged from the following fact. The sun radiates a huge number of neutrinos along with the heat and light and as many as 50 trillion solar neutrinos pass through the human body every second. The weakness of the neutrino interaction ensures that they pass right through us without any interaction and no harm
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is done. Obviously, this characteristic of the neutrino also makes it so very difficult to detect. Only with extremely strong neutrino sources and huge detectors can we expect to really pin down this elusive entity. Painstaking experiments over many decades have helped us develop an understanding of the properties of the neutrino. But much still remains to be done. Since the nineteen fifties novel discoveries in neutrino physics have emerged with regular frequency. Some of these are: detection of antineutrinos from reactors, producing controlled beams of neutrinos at accelerators, establishing that the νµ is different from the νe , verification of a key prediction of the Standard Model through the neutrino neutral current, the observation of the third kind of neutrino (the ντ ), etc. The India-based Neutrino Observatory (INO) is a proposal to set up a neutrino detector in India. India has a good track record in such experiments. It may not be out of place to recall that the signature of the νµ in cosmic rays was first noted by an Indian experiment and the world’s first dedicated proton decay experiment was located at the Kolar Gold Fields (KGF) in India. Here, after a short summary of the physics behind neutrino oscillations and the rˆ ole of masses and mixings, I will turn to the INO. I will deal in brief succession with the genesis of this proposal, the Iron Calorimeter (ICAL) detector design, the physics prospects, and the current status. 7.2. Neutrino Oscillations Flavour oscillation is an intriguing observed property of neutrinos. It is experimentally seen that neutrinos of any type (flavor) oscillate in time to states of other flavor. For illustration, if we imagine only two types of neutrinos, say νe and νµ , then this means that if initially we have a νe state then at a later time t it becomes a νµ state and then again back to a νe state and so on. This is much like the oscillation of a simple pendulum where the bob swings from one extremity to another periodically. Here the two neutrino flavors correspond to the two extremities. After one oscillation period the νe is back to the initial state while at exactly half the period it is a pure νµ . At intermediate stages it is a superposition of the νe and νµ states; this simply means that there is some probability of the state appearing as an electron neutrino and a complementary probability of its behaving like a muon neutrino. Thus at an arbitrary time there is an ‘oscillation probability’ Peµ and a ‘survival probability’ Pee which together add up to unity. In quantitative terms the oscillation probability of a νe
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after a time t is
2
2
2
Peµ t 4 cos θ sin θ sin
∆m2 t 4E
(7.1)
where ∆m2 m21 m22 and θ is the ‘mixing angle’. The survival probability Pee t 1 Peµ t. The behavior of neutrino oscillations is depicted in Fig. 1 for maximal mixing, i.e., θ 45 . Eq. (7.1) is a consequence of quantum mechanical time evolution1,2 . We turn to this now.
Fig. 7.1.
Two flavor neutrino oscillation probabilities for maximal mixing.
7.2.1. Two flavors We begin with a discussion of oscillation between two neutrino flavors. The time evolution of a stationary state ψk (in units such that c 1) is: ψk
t ψk exp
i
Ek t,
(7.2)
where Ek is the energy eigenvalue corresponding to ψk . Thus, the stationary state vectors at different times differ simply by an overall phase change. The time evolution of an arbitrary, i.e., non-stationary, state, ψ , is more complicated. For such a state we can write at t 0:
ψ 0 ak ψk , k
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where ak are constants. Using eq. (7.2) one finds:
ψ t ak ψk exp iEk t. k
For neutrinos, the basic assumption is that the familiar electron and muon neutrinos (νe and νµ ) – the flavor eigenstates – are not the mass eigenstates (i.e., the stationary states) ν1 and ν2 , but their superpositions: ψνe ψν1
c
ψν2
s;
ψνµ ψν1
s
ψν2
c,
where c cos θ and s sin θ. For two flavors a single angle, θ, suffices to completely specify one basis in terms of the other. Consider now the state vector of a νe produced at t 0. Thus, in a slightly more compact notation, initially ψ 0 ψνe cψ1 sψ2 . If the stationary states ψ1 and ψ2 correspond to energies E1 and E2 , respectively, then at a later time the state vector will be: ψ
t cψ1 exp
iE1 t
sψ2 exp
iE2 t.
The probability, Peµ t, of the state ψ t (originating as a νe at t appearing as a νµ is ψνµ ψ t2 and can be expressed as: Peµ t c2 s2 exp
iE1 t
exp
2
iE2 t
0)
.
The neutrinos are expected to have small masses, mi , and are in the ultram2 relativistic regime Ei p 2pi ) where p $ mi ) is the magnitude of the neutrino momentum3 . In this situation: ∆m2 πL Peµ t 4c2 s2 sin2 t sin2 2θ sin2 , (7.3) 4p λ where ∆m2
m22 m21 and λ 2.47m
E MeV
eV2 ∆m2
,
(7.4)
is the so-called oscillation length expressed here in terms of the neutrino energy E p. We use L and t interchangeably, since the neutrinos move with essentially the speed of light (c = 1). In the right hand side of eq. (7.3) the first factor is a consequence of the ‘mixing’ while the second factor leads to the ‘oscillatory’ behavior. For vacuum oscillations, the former, dependent on the mixing angle θ, is a constant but in the MikheyevSmirnov-Wolfenstein (MSW) matter effect, discussed later, it changes with the matter density.
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From eq. (7.3) Pee t 1 Peµ t 1 sin2 2θ sin2
πL λ
.
It is seen from the above that Pee t can be less than or equal to unity. The essential ingredients for this are twofold: (1) The neutrinos must be massive and non-degenerate. (2) The mass eigenstates of the neutrinos – ν1 , ν2 – must be different from the flavor eigenstates – νe , νµ . 7.2.2. Matter effects It is clear from eq. (7.3) that the vacuum oscillation probability depends only on the magnitude of ∆m2 . Though this information is indeed valuable, the actual neutrino mass spectrum will be determined only when the mass squared splittings are known in magnitude as well as in sign. This is where interactions with matter have a rˆ ole to play. Of the variants of the basic theme of neutrino flavor change the most prominent is the MSW4 matter induced effect, originally put forward by Wolfenstein. We do not enter into a detailed discussion here. Suffice it to say that interactions affect the inertia or mass of any particle and neutrinos are no exception. Therefore, neutrino masses and mixing depend on the ambient matter density. The resultant effect emerges as a contribution of a fixed sign which adds to ∆m2 . Obviously, depending on the sign of ∆m2 this will increase or decrease the oscillation effect thus opening a window on this important issue. Besides the effective mass squared splitting, matter effects also change the mixing angle θ and in some cases can result in a resonance-like behavior. The MSW effect is particularly important for solar neutrinos during their passage through the sun. For INO, another feature of the MSW effect will be of especial relevance. The preponderance of matter over antimatter in the ambient environment leads to the MSW contribution being different for neutrinos and their antiparticles, the antineutrinos. In fact, these contributions are of opposite sign. Therefore, to determine the sign of the mass splitting one must be able to make separate measurements of oscillations of neutrinos and antineutrinos. A comparison of the these oscillations gives a handle on the sign of the neutrino mass splitting. This is the feature which turns out to
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be valuable for the ICAL detector at INO which has a magnetic field and therefore can set apart neutrino events from the antineutrino ones. We will return to this later. 7.2.3. Three flavors It is now established that there are three light neutrinos. A natural question then is how do experiments constrain three-neutrino mixing? The previous results are readily generalized to the case of three or more flavors. Here, for the purpose of illustration, we focus on three neutrinos. The 3 3 neutrino mass matrix Mν can be diagonalized according to5 U Mν U
diag m1 , m2 , m3 ,
(7.5)
where U is the Maki-Nakagawa-Sakata2 (MNS) unitary matrix which relates flavor (α) and mass (i) eigenstates of neutrinos through να Uαi νi and m1 , m2 , m3 are the eigenvalues. The flavor states α 1,2, and 3 correspond to e, µ, and τ , respectively. The general expression for the probability that an initial να of energy E gets converted to a νβ after traveling a distance L in vacuum is
U Uβj sin2 πL , Pνα νβ δαβ 4 Uαi Uβi (7.6) αj λij j i where, as in eq. (7.4), λij 2.47m E MeV eV2 ∆m2ij ), ∆m2ij mi 2 mj 2 . The full forms of the various survival and transition probabilities depend on the spectrum of ∆m2ij and the MNS mixing matrix U relating the flavor states to the mass eigenstates. A popular parametrization of U is: " $ c12 c13 s12 c13 s13 eiδ U # c23 s12 s23 s13 c12 eiδ c23 c12 s23 s13 s12 eiδ s23 c13 %. (7.7) s23 s12 c23 s13 c12 eiδ s23 c12 c23 s13 s12 eiδ c23 c13 Here, cij , sij CP-phase.
cos θij , sin θij , θ12 , θ23 , θ13 are the mixing angles and δ a
7.2.4. Open issues Within the three-flavor picture, there are six oscillation parameters in the neutrino sector: the two independent mass splittings – ∆21 and ∆31 (the third is ∆23 ∆21 ∆31 ) – and, as seen from eq. (7.7), three mixing angles (θ12 , θ23 , θ13 ) and the CP-phase, δ. Much information exists about
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these from the solar, atmospheric, reactor, and long baseline experiments. We summarize below the current status at the 3σ level. The solar neutrino experiments have pretty much pinned down the (12) sector. It is now known that ∆21 = (7.1 - 8.3) 105 eV2 and 0.26 sin2 θ12 0.40. The atmospheric neutrino experiments, on the other hand, tell us that ∆31 = (2.0 - 2.8) 103 eV2 and 0.34 sin2 θ23 0.67. The third mixing angle has only an upper bound sin2 θ13 0.05, set by reactor antineutrino experiments. It is noteworthy that the mass splitting ∆31 is known only in magnitude while even the sign of ∆21 has been determined. This is because the solar neutrinos during their passage through the sun have significant MSW matter interactions and the effect makes its impact on observations. As noted earlier, this contribution is of a fixed sign. Depending on the sign of the vacuum mass splitting this will enhance or decrease the oscillation effect. This is the key to determine the sign of ∆ij . What then are the open issues for neutrino oscillation physics? Over the last few years there has been a paradigm shift in the nature of research in this area. The current focus is on making increasingly precise measurements of neutrino mass and the mixing matrix. Experiments planned to yield results over the next fifteen to twenty years reflect this change in focus. At this juncture the main goals of neutrino physics research are:
Firstly, a confirmation of an actual oscillation – a dip and a rise – needs to be seen. The experiments so far, by and large, have observed a depletion of the number of neutrinos which can be explained by neutrino oscillations. However, to rule out other alternatives and for an unequivocal evidence one would like to measure the probability (or equivalently the number of events) behaving in an oscillatory fashion with L or E or LE (see eqs. (7.3, 7.4)). The SuperK experiment has presented tantalizing evidence in this direction. Next in the priority list would be a determination of the mixing angle θ13 on which only a limit currently exists. No less important is the determination of the sign of ∆31 . In the absence of this information one cannot fix the ordering of the three neutrino masses, which is an essential ingredient for model building. The CP-phase δ is completely unknown. It will settle the issue whether CP-violation is restricted to only the quark sector or has a leptonic analogue as well.
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The current limits on θ23 are centered around 45o , which corresponds to maximal mixing. Many models of neutrino masses have appeared in the literature which lead to maximal mixing in this sector. Experiments should unequivocally tell whether there is a deviation from maximality. Ideas which may also be explored through future experiments include the possibility of sterile neutrinos (which have no charged lepton partner) and new physics beyond the Standard Model of particle physics.
7.3. INO As noted, several important results in neutrino physics have been uncovered in the past years through sophisticated experimentation. This gives confidence and encouragement to design new experiments to unravel finer details of the neutrino mass spectrum, leptonic CP-violation, etc. The India-based Neutrino Observatory is but a step in this direction. It is a proposal to set up an underground laboratory where experiments requiring a low-background environment can be performed. The first experiment proposed for INO is with a magnetized iron detector – an iron calorimeter (ICAL). Here we discuss only this aspect of the INO programme. INO will start its activity with ICAL. Over time it is expected to develop into a full-fledged underground science laboratory to host experiments that require a low background environment. The laboratory may also house experiments in other disciplines such as geology and biology that can profit from its special environment and infrastructure. The geographical location for any India-based neutrino laboratory is advantageous7 as most of the neutrino detectors are at latitudes above 35o . A near-equator location permits neutrino astronomy searches covering the whole celestial sky, study of solar neutrinos passing through the earth’s core and also neutrino tomography of the earth.
7.3.1. ICAL: genesis and plans Historically, the Indian initiatives in cosmic ray studies and neutrino physics go back to the nineteen fifties. The first ever atmospheric neutrino interaction was observed as early as in 1965 by the KGF experiment. In view of the current resurgence of neutrino physics as outlined above and the past Indian contribution, it was felt that non-accelerator particle physics research ought to be revived through an India-based Neutrino Observatory.6
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Taking into account the physics possibilities, the available expertise, and the cost, it was decided that a modern magnetized Iron Calorimeter with Resistive Plate Chambers (RPC) as the detector elements will be wellsuited for the purpose. This detector will have the capability to achieve many of the physics goals discussed in subsection 7.2.4. In the first phase the ICAL detector at INO will make precision measurements of neutrino properties using atmospheric neutrinos. It is foreseen that the detector will be in a position to deliver results on a timeline competitive with other international efforts in this direction. On a longer term, ICAL could usefully serve as an end-detector for very long baseline neutrino experiments where the source could be a neutrino factory or a beta-beam located in Europe or the U.S. The site for the Observatory has to provide sufficient coverage from all directions to remove cosmic-ray and other backgrounds. A suitable location has been identified (PUSHEP, Latitude: N11.5 , Longitude: E76.6 )) near Masinagudi in the Nilgiris about 250km from Bangalore. The Observatory will be located under a mountain providing an all round rock coverage of more than one km which corresponds to a water equivalent of 2800 metres. The Observatory will be inside a cavern with good access by road tunnels. Geological and environmental studies have progressed and an Engineering Task Force is in the process of preparation of a Detailed Project Report about the civil infrastructure work. An INO Centre will be located within the Mysore University campus about 90km from the site. It will be the base for many of the INO scientists, will house detector and electronics laboratories and will be in charge of training and outreach programmes. The current plan is to start the experiment in about five years. An INO project report was prepared and submitted to the Department of Atomic Energy. This report was sent for evaluation to half-a-dozen international referees of high standing and their responses are all positive. Funding for INO has been approved by the Department of Energy. Release of funds will require some further clearances at different levels of government. Internationally, many new experiments are planned for precision studies of neutrino oscillation parameters. Some, like T2K and Noνa, will use neutrinos produced at accelerators while others, e.g., Daya Bay and Double CHOOZ, will rely on antineutrinos from reactors. It is a very competitive environment and INO could become a prominent member of this group.
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7.3.2. The ICAL detector ICAL is the proposed magnetized iron calorimeter which will be the first neutrino detector at INO. Of course, the uncharged neutrinos will not be seen directly. The detection will be through muons which the νµ , ν¯µ produce in interactions with the target iron in the detector. The RPCs will be triggered when the muons pass through them and successive hits will define a µ µ track. The magnetic field will enable charge identification which is crucial for probing CP-violation, to study earth matter effects on atmospheric neutrinos, as well as being an absolute necessity for the far-end detector of a long baseline experiment based on a neutrino factory. To look for oscillations in atmospheric neutrinos, a detector should have excellent position, angular, and time resolutions. These goals can be achieved by a good tracking calorimeter like ICAL. In such a detector, the neutrino energy for fully-confined events can be estimated from the track length whereas for partially contained events the energy of the escaping muon is obtained from the track curvature in the magnetic field. The path length L covered by the neutrino is estimated from the neutrino direction. A sub-nanosecond time resolution can ensure an almost perfect up-down discrimination of the event.
7.3.2.1. Detector Structure The proposed detector, see Fig. 7.2, will have three modules of lateral sizes 16 m 16 m each and a height of about 12 m. This is composed of 140 layers of iron plates of thickness 6cm each, inter-leavened with active detector elements of thickness 2.5cm each, to be described below. The iron plates can be magnetized up to 1–1.4 Tesla. The total mass of such a detector including the support structure etc, will be approximately 50 kton. The whole detector as described above may be surrounded by an external layer of scintillation counters. This will act as a veto layer and will be used to identify muons entering the detector from outside as well as to identify partially confined events with vertex inside the detector. In addition to the main detector, two smaller detectors of equal area but with fewer layers of active detector elements are proposed to be located on either side of the main detector, at a later stage, to increase the aperture to study neutrinos from astrophysical objects.
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0000000000000000000000000000000 1111111111111111111111111111111 1111111111111111111111111111111 0000000000000000000000000000000 0000000000000000000000000000000 1111111111111111111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 0000000000000000000000000000000 12m 1111111111111111111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 0000000000000000000000000000000 16m 1111111111111111111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 0000000000000000000000000000000 1111111111111111111111111111111
16m
16m
16m
6cm 2.5cm
Sketch of the iron calorimeter detector (left) and of a typical Resistive Plate Chamber (right).
Fig. 7.2.
7.3.2.2. Active Detector Elements The active area of the detector is 48 16 m2 between two successive planes, or a total of the order of 100000 square meters. Resistive Plate Chambers are low cost active detectors with nano-second timing resolution and are a good choice to span such a large area. The RPC, shown in Fig. 7.2 right panel, is a gas-filled detector with two parallel electrodes of 2mm thick float glass. The plates, kept 2mm apart by suitable spacers, contain the gas. For a suitable combination of gas mixture and electrical field the detector can be operated in a spark mode. The high resistivity of the electrodes and the choice of the gas mixture ensure the containment of the spark as well as the recovery time. The high voltage (in the range of 8–10kV) will be applied to the electrodes either by means of graphite coating or resistive adhesive film on the glass plate. 7.3.2.3. Present Status Prototype 1m 1m RPCs have been fabricated at the Tata Institute of Fundamental Research (TIFR) and the Saha Institute of Nuclear Physics (SINP). An advanced gas mixing unit is designed at SINP with many builtin features. A major milestone has been crossed with the efficiency of the RPCs at TIFR surpassing 90% above 8.6 kV. Apart from the efficiency, the
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other important feature of this detector element is the fast timing which is useful for discriminating the up-going muons from the down-going ones. The timing for the prototype RPCs is found to be better or as good as in the case of a scintillator detector. They are being tested for long lifetimes, essential for the experiment. down
2 ∆m31-precision θ
K
CNGS
SK+K2K
T2 ICAL
0
CNGS T2K
0.2
θ
SK+K2K exluded at 3σ
νA NO OS MIN
Relative error at 2σ
0.4
2
sin θ23-precision
ICAL L
MIN
OS
N
O
νA
up
-0.2 SK+K2K current data -0.4
1
2 3 2 -3 2 True value of ∆m31 [10 eV ]
41
2 3 2 -3 2 True value of ∆m31 [10 eV ]
4
Fig. 7.3. Down-going and up-going atmospheric neutrinos (left) and a comparison of the precision in measuring the atmospheric mixing angle and mass-splitting that may be achieved by ICAL compared with the expectations from other experiments (right).
7.3.3. Up/Down asymmetry in atmospheric neutrinos Oscillation is manifested in atmospheric neutrinos through an up-down asymmetry. This is easily illustrated by considering downgoing and upgoing νµ , ν¯µ which arrive at the detector from opposite directions. For upgoing neutrinos with any zenith angle θ, the corresponding downgoing neutrino has an angle π θ (see left panel of Fig. 7.3). The number of neutrinos at any zenith angle depends on the height of the atmosphere in that direction. So, it is natural to expect that the flux of downgoing events in any direction will roughly match with the number of upgoing events from the opposite direction. But the upgoing neutrinos travel over distances comparable to the diameter of the earth before reaching the earth and have scope for oscillating to a different flavor. Oscillations will lead to a depletion of the ratio
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of the number of upgoing events to the number of downgoing events in the opposite direction. One may thus regard the downgoing neutrinos with a zenith angle θ to be the ‘no oscillation’ standard for upgoing neutrinos of neutrino angle π θ. The ‘up/down’ ratio is then a measure of the impact of oscillation, the survival probability Pµµ . For atmospheric neutrinos, the survival probability for νµ (see e.g., eq. (7.3)) is a simple oscillating function of LE. A verification of the oscillation hypothesis requires an identification of the minima and maxima. The most recent analysis of Super-K events shows evidence for the first dip and rise in Pµµ . Although this is suggestive of muon neutrino oscillation, only an observation of the full pattern over more than one period will constitute a complete confirmation. This can be ensured by a detector which extends over a larger range of LE with better E and L resolution (which requires an accurate measurement of the position and direction). ICAL is designed with this in mind. Since the neutrino travel distance is fixed by the zenith angle, it will be bread and butter physics for ICAL to obtain the ‘up/down’ ratio as a function of the neutrino LE. Simulations indicate that ICAL, as designed, will be able to see the first oscillation dip and the subsequent rise in a five year run. The precision in the determination of ∆31 and sin2 θ23 that can be achieved by ICAL in a 250kT-year run is shown in the right panel of Fig. (7.3) where the expectations of the other planned experiments are also given. The magnetic field in ICAL permits charge identification of the muons. Thus neutrino events can be distinguished from antineutrino events. Because of matter effects, the ‘up/down’ ratio for νµ and ν¯µ will be different and this will allow a determination of the sign of ∆31 , i.e., the neutrino mass ordering. Simulations indicate that it would take ten years data taking with a 100kT ICAL to make a statistically significant measurement. To summarise, the main characteristic of the ICAL detector is a clean identification of muons with good energy and time resolution. Also, the presence of the magnetic field will distinguish positive and negative charged particles. 7.3.4. Simulation, prototype, ... Work on the ICAL detector simulation is in progress from various directions. The first goal of the experiment will be to extract precise information about neutrino masses and mixings using atmospheric neutrino events. A GEANT3-based ICAL simulation program has been used to explore the en-
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ergy and angular resolutions, charge identification efficiency, etc. and to identify appropriate cuts for event reconstruction. Atmospheric neutrino charged current (CC) events are generated using the NUANCE package8 for this simulation. Currently, work is in progress on a GEANT4 version of ICAL simulation. The design for the ICAL magnetic field is being pursued at the Bhabha Atomic Research Centre and the Variable Energy Cyclotron Centre (VECC). A 1m3 prototype of ICAL is being set-up at VECC in Kolkata. The iron for this purpose has already been delivered and installed in the test-area. Further work in this direction is in progress.
7.3.5. Future: Long baseline The possibility of neutrino beams from muon storage rings has received a lot of attention in the recent literature. Such facilities provide intense, controlled high luminosity neutrino beams that are almost pure νµ ν¯e or νe ν¯µ depending on the sign of the stored muon. With its charge discrimination capability, ICAL will offer unique opportunities to exploit the physics potential of such sources. We do not go into the details of these possibilities except to list some of them.
Determination of θ13 Sign of ∆31 Probing CP violation in the leptonic sector Detecting large matter effects in νµ ντ oscillations
The location of INO is particularly suited for a very long baseline experiment with a high intensity source located in Europe because the sourcedetector distance is very close to the ‘magic baseline’. At this special distance, many of the parameter degeneracies arising from the unknown CPphase δ drop off and it is possible to cleanly measure θ13 to a high precision and determine the sign of ∆31 . A beta-beam source – which is a pure beam of νe or ν¯e – is particularly well-suited for this9 . ICAL may also present other physics opportunities, e.g., looking for Kolar events, Ultra high energy neutrinos and muons, etc., which could further enrich the physics program.
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7.4. Conclusions In this article, the theoretical basis of neutrino oscillations has been discussed. Neutrino oscillations pin down the mass and mixing in the neutrino sector – a pointer to new physics. The INO project and the ICAL detector which are expected to probe the mass spectrum and neutrino mixing angles to very high precision are elaborated upon. The atmospheric neutrino physics program possible with a magnetized iron tracking calorimeter like ICAL is substantial and germane. One can observe a clear signal of oscillation by observing the full oscillation swing. Also the precision of the parameters, ∆31 and θ23 can be improved to 10 %. Among the physics capabilities are the sensitivity to matter effects and the sign of ∆31 . In addition atmospheric neutrinos can be used to probe CPT invariance. An apology is due for the paucity of references to the original papers. This is ameliorated to some extent by the availability of the latest as well as the earlier references on the internet10 . Acknowledgements I wish to thank the organizers of THEOPHYS-2007, in particular Professors Subir Ghosh and Guruprasad Kar, for arranging such a wonderful meeting and for providing an opportunity to share the ideas about INO with many esteemed scientists. This research was supported by the XI Plan ‘Neutrino’ project at HRI. References 1. B. Pontecorvo, JETP 6, 429 (1958). 2. Z. Maki, M. Nakagawa, S. Sakata, Prog. Theor. Phys. 28, 870 (1962). 3. Here we assume that all neutrinos share the same momentum p. To include the effect of momentum spread one must consider a neutrino wave-packet. In most situations, such an analysis makes no practical difference from the above. See, for example, B. Kayser, Phys. Rev. D24, 110 (1981); C. Giunti, C. W. Kim, W. Lam, ibid. D44, 3635 (1991). 4. M. Mikheyev, A. Smirnov, Sov. J. Nucl. Phys. 42, 913 (1986); L. Wolfenstein, Phys. Rev. D17, 2369 (1978), ibid. D20, 2634 (1979). ν c , neutrinos the l.h.s of Eq. (7.5) 5. For Majorana, i.e., self conjugate ν T reads U Mν U . 6. Details of the INO proposal may be seen at http://www.imsc.res.in/ ino. 7. It is possible to reach down to almost 8o latitude in South India.
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8. D. Casper, hep-ph/0208030. Also http://nuint.ps.uci.edu/nuance. 9. S.K. Agarwalla, A. Raychaudhuri, and A. Samanta, Phys. Lett. B629, 33 (2005), S.K. Agarwalla, S. Choubey and A. Raychaudhuri, arXiv:0711.1459 (to appear in Nucl. Phys. B) and references therein. 10. See, for example, The Neutrino Oscillation Industry Home Page: http://neutrinooscillation.org.
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Chapter 8 Dynamics of Proton Spin: Role of qqq Force
A. N. Mitra 244 Tagore Park, Delhi-110009, India The analytic structure of the qqq wave function, obtained recently in the high momentum regime of QCD, is employed for the formulation of baryonic transition amplitudes via quark loops. A new aspect of this study is the role of a direct (Y -shaped, Mercedes-Benz type) qqq force in generating the qqq wave function The dynamics is that of a Salpeter-like equation (3D support for the kernel) formulated covariantly on the light front, a la Markov-Yukawa Transversality Principle (MYTP) which warrants a 2-way interconnection between the 3D and 4D BetheSalpeter (BSE) forms for 2 as well as 3 fermion quarks. The dynamics of this 3-body force shows up through a characteristic singularity in the hypergeometric differential equation for the 3D wave function φ, corresponding to a negative eigenvalue of the spin operator iσ1 .σ2 σ3 which is an integral part of the qqq force. As a first application of this wave function to the problem of the proton spin anomaly, the two-gluon contribution to the anomaly yields an estimate of the right sign, although somewhat smaller in magnitude.
Contents 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Theoretical Ingredients . . . . . . . . . . . . . . . 8.1.2 Plan of the Paper . . . . . . . . . . . . . . . . . . 8.2 Structure of the Full BS Wave Function Ψ . . . . . . . . 8.2.1 Instant vs LF Representations of Momenta . . . . 8.2.2 From Ψ to Φ via Gordon Reduction . . . . . . . 8.2.3 3D-4D Interlinkage by Green’s Function Method 8.3 Proton Spin Formalism . . . . . . . . . . . . . . . . . . 8.3.1 BS Normalization of qqq Wave Function . . . . . 8.3.2 Spin Matrix Elements in Lowest Order . . . . . . 8.4 Spin Correction from Two-gluon Anomaly . . . . . . . . 8.4.1 Two-gluon Anomaly Operator . . . . . . . . . . . 8.4.2 2-gluon anomaly correction to spin amplitude . . 189
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8.4.3 ‘Kinematical’ Part of the Spin Correction . . . . . . . . . . . . . . . . 203 8.5 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
8.1. Introduction The concept of a fundamental 3-body force (on par with a 2-body force) is hard to realize in physics, leaving aside certain ad hoc representations of higher order effects, for example those of ∆, N resonances in hadron physics. At the deeper quark-gluon level on the other hand, a truly 3-body qqq force shows up as a folding of a ggg vertex (a genuine part of the gluon Lagrangian in QCD) with 3 distinct q¯gq vertices, so as to form a Y -shaped diagram. Indeed a 3-body qqq force of this type, albeit for ‘scalar’ gluons, was first suggested by Ernest Ma,1 when QCD was still in its infancy. [A similar representation is also possible for N N N interaction via ρρρ or σσσ vertices, but was never in fashion in the literature2 ]. We note in passing that a Y -shaped (Mercedes-Benz type) picture3 was once considered in the context of a preon model for quarks and leptons. In the context of QCD as a Yang-Mills field, a ggg vertex has a momentum representation of the form4 Wggg
igs fabc
k1 k2 λ δµν
k2 k3 µ δνλ k3 k1 ν δλµ
(8.1)
where the 4-momenta emanating from the ggg vertex satisfy k1 k2 k3 0, and fabc is the color factor. When this vertex is folded into 3 q¯gq vertices of the respective forms gs u¯ p1 iγµ &λa1 2'u p1 , and two similar terms, the resultant qqq interaction matrix (suppressing the Dirac spinors for the 3 quarks) becomes5 Vqqq
gs4 1 2 3 &2' &3'&λ1λ2 λ3 '&k2 k2 k2 ' iγ . k2 k3 γ .γ 1 2 3 23
(8.2)
where ki pi pi ; λi are the color matrices which get contracted into the corresponding scalar triple products in an obvious notation. [Note that the flavor indices are absent here since the quark gluon interaction is flavor blind]. This interaction will be considered in conjunction with 3 pairs of qq forces within the framework of a Bethe-Salpeter type dynamics to be specified below. Before proceeding further, a possible motivation for the use of a direct qqq force, apart from its intrinsic beauty, comes from the issue of “proton spin” which, after making headlines about two decades ago, has come to the fore once again, thanks to the progress of experimental
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techniques in polarized deep inelastic scattering off polarized protons, and their variations thereof, which allow for an experimental determination of certain key QCD parameters by relating them to certain observable quantities emanating from external probes; (see a recent review6 for references and other details). On the other hand it is also of considerable theoretical interest to determine these very quantities directly from the intrinsic premises of QCD provided one has a “good” qqq wave function to play with. Such a plea would have sounded rather utopian in the early days of QCD when phenomenology was the order of the day. Today however many aspects of QCD are understood well enough to make such studies worthwhile by hindsight, with possible ramifications beyond their educational value. For simplicity we work in the experimentally accessible regime of valence quarks. and make free use of the results of a recent paper5 for many details on the specific effect of the 3-body force (1.2) on the analytical structure of the qqq wave function, while giving more emphasis on the formalism relating to the loop diagrams towards the determination of proton spin with appropriate 4D BS normalization.
8.1.1. Theoretical Ingredients In the valence quark regime, we need to consider a qqq system governed by pairwise qq forces as well as a direct 3-quark force of the type 8.2. A further simplification occurs in the high momentum regime where the effect of confining forces may be neglected, so that only coulombic forces are relevant. As explained in,5 we shall take the dynamics of a qqq hadron in the high momentum regime to be governed by the Salpeter equation7 formulated in a covariant manner, which has the remarkable property of 3D-4D interlinkage (see5 for a detailed picture). A covariant formulation of the Salpeter Equation in turn, is centered around the hadron 4-momentum Pµ in accordance with the Markov-Yukawa Transversality Principle (MYTP),8,9 which is a ‘gauge principle’ in disguise,10 and ensures that the interactions among the constituents be transverse to the direction of Pµ . In the high momentum regime to be considered here, the confining interaction has been ignored for simplicity,5 which leaves the 3D form of the BS dynamics inadequate for mass spectral determination, yet its dynamical effect on the spin-structure of the wave function should be realistic enough for dealing with the hadron spin in the high momentum limit.
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A further ingredient concerns the use of Dirac’s light-front form of dynamics11 which has a bigger {7} stability group than the more conventional instant form whose stability group is only {6}. All this can be covariantly formulated; see5 and12 for details.
8.1.2. Plan of the Paper The plan of the paper, which is based on an interlinked 3D-4D BSE formalism characterized by a Lorentz-covariant 3D support for its kernel a la MYTP,8,9 adapted to the light front (LF)12 is as follows. In Section 2 we summarise the principal results of ref 5 on both the 3D (φ) and the full-fledged 4D(Ψ) forms of the qqq wave function, so as to give a basically self-contained picture omitting the non essential details from.5 Section 3 outlines the construction of the normalized 4D wave function after assessing the possible options on BS normalization for the same. (As a check, some of the conventional results are reproduced). Section 4 is devoted to the principal result of this investigation, viz., the construction of the two-gluon coupling to the axial operator iγµ γ5 and its insertion into the quark lines involved in the two types (self-energy and exchange) of possible baryonic transition diagrams for such coupling. Section 5 concludes with a short discussion of the results obtained vis-a-vis experiment.
8.2. Structure of the Full BS Wave Function Ψ In this Section we collect the principal results of ref 5 on the full structure of the BS wave function in both the 3D (φ) and 4D (Ψ) forms.
8.2.1. Instant vs LF Representations of Momenta We first record the correspondence between the instant and LF forms of 12 for the LF quantities p the dynamics, starting with some definitions p0 p3 defined covariantly as p n.p 2 and p n ˜ .p 2. while the perpendicular components continue to be denoted by p in both notations. For a typical internal momentum qµ , the parallel component P.qPµ P 2 of n, the instant form translates in the LF form as q3µ zPn nµ , where Pn P.˜ 2 z 2 M 2 which shows that zM plays and z n.q n.P . As a check, qˆ2 q the role of the third component of qˆ on LF. Next, we collect some of the
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more important definitions/results of the LF formalism12 q
q qn n; qˆ q zPn n; z
Pn
P.˜ n; P.q
2
P.ˆ q Pn q.n; qˆ
q.nP.n; qn
Pn q.n P.nqn ; qˆ.˜ n P .q
q M z ; P 2
2 2
2
M
q.˜ n;
(8.3)
0;
2
For a qqq baryon, there are two internal momenta, each separately satisfying the relations 8.1. Note that for any 4-vector A, A.n and An correspond to 1 2 times the usual light front quantities A A0 Az respectively. But since a physical amplitude must not depend on the orientation n, a simple device termed Lorentz completion via the collinear trick12 yields a Lorentz-invariant amplitude for a transition process with three external lines P, P , P P P as explained in.5,12 And for ready reference, the precise correspondence between the instant and LF definitions of the ‘parallel (z)’ and ‘time-like (0)’ components of the various 4- momenta for a qqq baryon ( i = 1,2,3):13
piz ; pi0
M pi M pi ; ; P
2P
pˆi
&pi , piz '
(8.4)
The last part of Eq.(8.4) defines a covariant 3-vector on the LF that will frequently appear as arguments of 3D wave function φ for the qqq proton. 8.2.2. From Ψ to Φ via Gordon Reduction The full wave function for three fermion quarks complete with all internal d.o.f.’s, satisfies the following Master equation whose kernel includes both qq and direct qqq forces14 :
Ψ p1 p2 p3
3
SF p1 SF p2 gs2
1
SF
d4 q12 γ 1 γ 2 Dµν k12 Ψ p1 , p2 , p3 2π 4 µ ν
p1 SF p2 SF p3
d4 p3 d4 q12 Vqqq Ψ p1 p2 p3 2π 8
(8.5)
where the definitions for the various momenta, and the phase conventions for the quark propagators are those of,14 while the direct 3-quark interaction Vqqq in the last term is given by 8.2. Here the internal variables must be defined in a pre-assigned basis, say indexed by #3 as13
2ξ3
p1 p2 ;
6η3
2p3 p1 p2 ;
P
p1 p2 p3
(8.6)
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where the time-like and space- like parts of each are given by 8.4, and the corresponding 3-vector defined as pˆi &pi , piz '. (Two identical sets of momentum pairs ξ1 , η1 and ξ2 , η2 are similarly defined, but can be expressed in terms of the set 8.6 via permutation symmetry). The solution of this Master equation 8.5 was then achieved in three steps (A, B, C). Step A consists in defining an auxiliary scalar function Φ related to the actual BS wave function Ψ by14 Ψ Π123 SFi1
pi Φ
p i p 2 p 3 W P
(8.7)
where the quantity W P is independent of the internal momenta but includes the spin-cum-flavour wave functions χ, φ of the 3 quarks involved (see15 for notation and other details) :
W P χ φ χ φ 2
(8.8)
The quantities φ , φ are the standard flavour functions of mixed symmetry16 [not to be confused with the 3D wave function φ !], and χ ,χ are the corresponding relativistic spin functions. The latter may be defined either in terms of the quark # indices as in Eqs 8.2 or 8.5, or sometimes more conveniently in a common Dirac matrix space as15,17
χ ; χ
M
iγ.P
2M
iγ5 ; iˆ γµ
3C 2βγ
1; γ5 γ ˆµ u
P α (8.9)
where the first factor is the βγ-element of a 4 x 4 matrix in the joint spin space of the quark #s 1, 2,17 and the second factor the α element of a 4 x 1 spinor in the spin space of quark # 3; C is a charge conjugation matrix with the properties18 γ ˜µ
C 1 γµ C; γ˜5
C 1 γ5 C;
and γˆµ is the component of γµ orthogonal to Pµ . Finally, the representations of the flavour functions φ , φ satisfy the following relations in the “3” basis19 τ 3φ φ 1; 1 τ 3 φ φ 1; (8.10) 3 Step B now consists in recasting Eq. 8.5 in terms of the scalar quantity Φ a la Eq. 8.7 with a simultaneous use of Gordon reduction on the pairwise kernels V ξˆi ηˆi and the 3-body kernel Vqqq , as described in5 following the original treatment of.20 This has the effect of eliminating the Dirac matrices in favour of the Pauli matrices σµν . [We skip these details which may be found in5 ].
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8.2.3. 3D-4D Interlinkage by Green’s Function Method The next step (Step C) now consists in a reduction of the 4D BSE for the quantity Φ defined above to one for a 3D scalar φ by the standard method of elimination of the time-like variables, and a reconstruction of the 4D quantity Φ, thus establishing a 3D-4D interconnection between these two wave functions. This last is facilitated by the Green’s function approach13 adapted to the LF formalism, as described in.5 Calling the 4D Green’s functions associated with Ψ and Φ by GF and GS respectively, the connection between them, analogously to 8.7, may be written as GF ξη; ξ η W P Π123 SFi1
ξη; ξ η Π123 SFi1
P (8.11) where we have indicated the 4-momentum arguments of the Green’s functions involved, in a common S3 basis ξ, η , and expressed the spinflavour dependence of GF as a matrix product implied by the notation ¯ P . W P W It was shown in5 how the 3D-4D interconnection is first achieved at the level of the ‘scalar’ Green’s functions whose 4D and 3D forms are labelled by GS and gs respectively, and thence to the corresponding wave functions Φ and φ by the method of spectral representations. Finally the connection to the 4D spinor wave function Ψ is established via Eq. 8.9. We skip these steps which are given in sufficient details in.5 The final result for Ψ in terms of φ is Ψ ξ, η Π123 SF pi D123
pi GS
φ
123
ˆ ηˆ ξ,
¯ pi W
1 W P 2πi2
(8.12)
where the structure of D123 is expressed by a double integral over two time-like momenta: 1 D123
P 2 dq12 dp3 4M 2 2iπ 2 ∆1 ∆2 ∆3
(8.13)
and the 3D wave function φ satisfies the equation ˆ ηˆ 2π 3 D123 φ ξ,
p3z d3 ξˆ3 Vqq3 φ ξˆ3 , ηˆ3 2 123 1 d3 ξ d3 η Vqqq φ ξˆ3 , ηˆ3 3 3 2π 3
(8.14)
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The solution of this equation has been obtained in5 in coordinate space, using combinations analogous to 8.6, viz.,
2s3
r1 r2 ;
6t3
2r3 r1 r2
(8.15)
The final result for φ in coordinate space (see5 for details) is φ F a, b3x;
a b 2;
ab
β ; 2
β
0.058
(8.16)
where F is a standard hypergeometric function of its arguments, and has a particularly convenient representation for a b 221 1
F a, b; 3; x
dyy a1
0
1 y a ; 1 xy a
a 2 β 2
(8.17)
where x R2 R02 , R2 s2 t2 , and x 1 corresponds to the point R R0 . This completes our summary of the full structure of the 4D qqq wave function Ψ in terms of the 3D quantity φ a la.5 8.3. Proton Spin Formalism As a first application of this wave function, we shall determine the baryon spin, together with its corrections, in a general enough manner involving loop diagrams. To that end a key ingredient is the baryon normalisation within the Bethe Salpeter formalism, for which the appropriate diagram is Fig 8.1 with the spin operator iγµ γ5 replaced by an appropriate one signifying conservation of charge, mass or probability with corresponding operators e 16 τ 2iγµ , M 2 or 1 respectively. We adopt the last one (probability) in preference to the others in view of its simplicity as well as universal appeal as an even operator.
p1
QQ Q Q Q p2 p3
iγµ γ5 X
Q Q Q Q Q
Fig. 8.1. Schematic baryon spin diagram, with internal quark momenta p1 , p2 , p3 ; basic spin operator iγµ γ5 is inserted in line p1 .
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8.3.1. BS Normalization of qqq Wave Function Consider Fig 8.3 where the 3 internal quark lines (1, 2, 3) are labelled by momenta p1 , p2 and p3 respectively. and the operator iγµ γ5 is temporarily replaced by 1 to signify probability conservation for a BS normalization calculation. This exercise is patterned closely on the lines of,15 albeit in a suitably corrected form in which the matrix elements are not factored into two parts (as done erroneously in15 ), but otherwise maintaining its mixed symmetric [m’ m”) notation for the matrix elements for each separate d.o.f. (spin and flavour). Keeping track of the indices (1, 2, 3), the two spin matrix elements N , N for BS normalization (taken between the functions 8.9) may be written in an obvious notation as, N
N
N N N N N N1;23 1;32 2;31 2;13 3;12 3;21 N N N N N N1;23 1;32 2;31 2;13 3;12 3;21
(8.18)
and should be multiplied by the corresponding flavour matrix elements 8.10 in accordance with the structure of the function W P of 8.8 . The individual terms in Eq 8.18 are related by permutation symmetry, and two typical elements are given by
N1;23
u¯ P Ps SF p1 &1'SF p1 PE
N1;23
γ5 C C 1 γ5 SF p 2 PE SF p3 Ps u P 2 2 (8.19)
u ¯ P Ps γˆρ γ5 SF p1 &1'SF p1 PE
C 1 γρ PE SF p3 γ5 γρ Ps u P 6
γρ C SF p 2 6
where P is the baryon 4-momentum with mass M ( P 2 Ps
1 iγ.sγ5 2;
PE
M
(8.20) M 2 ),
iγ.P 2
and (8.21)
and the normalization condition is ( c.f.,15 ) 2
dτ N
φ 1 φ
N
φ 1 φ dτ N N 2N
(8.22) where the flavour functions φ , φ are defined in 8.10 and dτ is the full measure of the internal integration variables defined by 8.6
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dτ
ˆ ηˆ2 d4 ξd4 η D123 φ ξ,
(8.23)
and the 3D wave function φ and the associated denominator function D123 are as defined in Eq (2.10). Note that the time-like variables ξ0 and η0 of Eq.8.4 do not appear in the factors D123 φ on the rhs of 8.23. We may now use the same pattern for the evaluation of some standard physical quantities which may serve as checks on the self-consistency of this formalism. Thus for the nucleon charge, the probability operator 1 employed for BS normalization above should be replaced by
1 eiγµ 16 τ3 2
(8.24)
and the corresponding matrix elements Q and Q may be written down in the same notation and phase convention as for N and N above, and then divided by the total BS normalizer N for correct overall normalization. The final result for the nucleon charge, after evaluating the flavour matrix elements a la Eq.(8.10) is
2QN
dτ Q 16 τ3 2 Q 16 τ3 6
(8.25)
where Q and Q are given by Eqs (8.19) and (8.20) respectively, except for the replacement of &1' by iγµ , and τ3 has the values 1 for proton/neutron. The momentum integrals are involved, but if terms of order ξ 2 , η 2 M 2 are ignored compared to unity in the integrands concerned, some remarkable simplifications bring out the full flavour of SU 6 symmetry, albeit in a relativistic manner. Thus as a first check on the self-consistency of the formalism, the proton/neutron charges work out as e and 0 respectively. 8.3.2. Spin Matrix Elements in Lowest Order We now employ this formalism for the determination of nucleon spin in lowest order, for which the basic spin operator is iγm uγ5 , (as in Fig 8.1), multiplied by appropriate flavour matrices. It is simplest to speak of the ‘axial charges’ whose proportionality to the spin vector sµ comes out from analogous equations to (2) and (3) of Ref. 6, with the substitution of &1' by 3 iγµ γ5 in Eqs 8.19 - 8.20 above. The flavour dependent axial charges gA , 8 0 gA and gA of Ref. 6 are then reproduced by the multiplication of this
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spin operator with the successive Gell-Mann matrices λ3,8,0 respectively, and taking their matrix elements between the states defined by 8.10. Now 0 the spin anomaly occurs mainly with respect to gA , while the other two parameters remain almost unaffected. In the lowest order, i.e., neglecting terms of order ξ 2 , η 2 M 2 , these quantities may be worked out in the same normalization as defined in Section 8.3.1 above, to yield the values
3 109; g 8 23; g 0 23 A A
gA
(8.26)
Comparison with Eq.(8.26) of Ref. 6 reveals a difference of a factor of 23 between the two results. This is due to the BS normalization employed here, viz., a relativistic one normalizing direct to unit probability which does not distinguish between the proton and the neutron ), instead of to the charge which does, as in Ref. 15. The latter agrees with the standard non-relativistic value cited in Ref. 6, but the former indicates a welcome alternative possibility to ensure better with experiment without relativistic 0 corrections. Further, it is only the last one, gA , that is subject to anomaly corrections arising mainly from two-gluon effects that we consider next. 8.4. Spin Correction from Two-gluon Anomaly
@
Q @ @ @ @
Q Q iγµ γ5 Q QX
Q Q Q iγµ γ5 QQ X
Fig. 8.2. Two gluon operator (crossed box) representing a sum of two distinct diagrams for axial vector coupling
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8.4.1. Two-gluon Anomaly Operator The 2-gluon anomaly operator ∆µνλ appears in Fig 8.2 as a ‘crossed box’ represented by a sum of two triangle diagrams, the second one being merely the effect of exchanging the two gluon lines connected to the triangle loop. In this Section we indicate its evaluation in a general manner in preparation for its insertion in the internal quark lines (Fig 8.3) for obtaining the gluon 0 anomaly corrections to gA . The 2-gluon anomaly operator, with gluon momenta k1 k (entering) and k2 k (leaving) may be expressed in the form igs2 4 T r d qiγ S q k1 iγµ γ5 SF q k2 iγλ SF q ∆µνλ k ν F 2π 4 (8.27) A second one is obtained by the simultaneous interchanges k k and ν λ. The calculation is straightforward and will be mostly skipped except for a quick indication of how to incorporate gauge invariance. While the modern method is that of dimensional regularization, it should be adequate to follow an old-fashioned (simpler) method due to Rosenberg,22 which effectively amounts to subtracting out the non-gauge-invariant terms at the integrand itself, so as to ensure separate conservation of currents at the two vertices ν and λ. After the trace evaluation in 8.26, this procedure leaves a numerator proportional to q in the integrand. This needs at least an extra power of q arising from an expansion of the propagator denominators in powers of q.k q 2 k 2 . In the lowest order in k, the integral over q 2 becomes convergent, and after standard q integration via the Feynman auxiliary variable u, reduces to an integral over u ∆µνλ
2αs π
1 0
duu2 m2q k 2 u
which for small m2q further reduces to a very simple form : ∆µνλ
αs µνλσ kσ ;
m2q
k2
(8.28)
8.4.2. 2-gluon anomaly correction to spin amplitude The operator ∆µνλ is now ready for insertion in the internal quark lines of Fig 8.3 signifying the forward scattering amplitude of the baryon. The insertion can be done in two different ways : self-energy like insertion in line p1 a la Fig 8.3(a); and exchange like insertion connecting two quark lines p1 and p2 , as in Fig 8.3(b). We designate these contributions by Σ , Σ ;
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@
p1
Q Q Q Q Q
QQ Q Q Q p2 p3
a
p1 QQ Q Q Q p2 p3
@
Q Q Q Q Q
b
Fig. 8.3. Two-gluon operator, fig (8.2), inserted in the internal quark lines of the baryon: (a) ‘self-energy’ like insertion in line p1 ; (b) ‘exchange-like’ insertion connecting lines p2 and p3
and V , V respectively, in accordance with the two types of spin matrix elements a la Eq.(8.9). These contributions are further indexed by the subscripts 1; 23, etc since three such diagrams for each type must be added up like in Eq.(8.18). The master expressions for these matrix elements are as follows. Σ1;23
2gs2 3 2π 4
d4 k¯ u P Ps SF p1 ∆µνλ iγν SF p1 k iγλ D2 k
SF p1 PE Σ 1;23
2gs2 3 2π 4
γ5 C SF 2
p2
C 1 γ5 PE SF p3 Ps u P conj (8.29) 2
d4 k¯ u P Ps γˆρ γ5 SF p1 ∆µνλ iγν SF p1 k iγλ D2 k
SF p1 PE
iˆ γρ C SF 6
p 2
C 1 iˆ γρ PE SF p3 Ps u P conj (8.30) 6
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The symbols conj in these equations represent the effects of the crossed diagrams for the 2-gluon anomaly (Fig 8.2). For the exchange type insertions, the corresponding expressions V , V may be written in a similar but slightly simplified form as
V1;23
2gs2 PE γ5 ∆µνλ SF d4 k¯ u P Ps SF p1 4 3 2π 2 γν D k SF p2 γ5 PE SF p3 γλ D k SF
V1;23
p2 k
p3 k Ps u P conj
(8.31)
2gs2 PE γˆρ ∆µνλ SF d4 k¯ u P Ps γˆρ γ5 PE SF p1 3 2π 4 6 γν D k SF p2 γˆρ PE SF p3 γλ D k SF
p3 k γ5 γˆρ u P conj
p 2 k
(8.32)
These quantities, when integrated over dτ , Eq. 8.23, and divided by the normalizer N , Eq.8.22, qualify directly as 2-gluon anomaly corrections 0 (in the same relative normalization) to the spin matrix element gA listed 0 in Eq. 8.26. The result for the fractional correction to gA may be expressed in the form αs 2 0 gA . (8.33) π where the dimensionless quantity θ may be termed the ‘reduced fractional 2-gluon anomaly correction’ . The calculation of θ - a long and elaborate proces - involves two distinct steps: (a) integration over d4 k (b) integration over dτ . While step (a) is necessarily a dynamic correction, step (b) may be further divided into two parts, i) ‘kinematic’ and ii) ‘dynamic’, according as the effects of the internal momenta (ξ, η) are neglected or included respectively. The reason for this break -up is that only the latter involves an interplay of the the 3D wave function φ 2 , appearing via the integration measure dτ , with the internal momenta (ξ, η) which are copiously present in the large number of propagators which make up the integrands of the types (8.29–8.32), while the ‘kinematical’ part almost entirely suppresses this contribution by dropping the effects of these internal momenta from the said propagators. [Note that the hypergeometric form 8.17 of φ which appears through the integral measure dτ , carries the dynamical signature of the ‘spin-part’ of δgA
θ
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the 3-body force !]. In this paper we are able to give only the results of the ‘kinematical’ part, while the calculation of the more difficult ‘dynamical’ part is in progress. To that end, the ‘kinematical’ part is calculable on closely analogous lines to the spin matrix elements in lowest order (see Sect. 8.2), using the normalization of Sect (8.1). The essential steps are very briefly indicated below. 8.4.3. ‘Kinematical’ Part of the Spin Correction First, to incorporate the operator ∆µνλ of 8.27, the following results are useful: γν γλ γσ µνλσ
6γµ γ5 ;
γλ γσ µνλσ
2γν γµ γ5
(8.34)
Next, the (logarithmic) divergence of the k- integration requires the standard process of dimensional regularization,23 with a typical result of the form24 d4 k i
1
du2 1 u 0
k2 Λu 3
k2
2
π γ 1 ln π∆1
(8.35)
where Λu
u∆1 m2g 1 u;
∆1
m2q p21
After the k-integration (step (a)), the dτ integration (step (b)) involves some drastic approximations effectively involving the replacement of the 4-momenta pi of the various propagators by their ‘central’ values. At the end of this exercise, the effect of the factor φ2 in dτ almost ‘decouples’ from that of the various propagators involved in step (b), and the integrations can be performed without much further ado. Omitting these steps, the two contributions θ1 and θ2 from the ‘self-energy’ and ‘exchange’ effects respectively become the following : θ1
0.5;
θ2
1.5
(8.36)
resulting in a total effect ‘kinematical’ contribution θ
2.0
(8.37)
which with α 0.39 in 8.33, amounts to a tiny correction to the spin anomaly, albeit of the right sign.
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8.5. Summary and Conclusion To summarise, we have presented a first application of a new form of dynamics within the framework of QCD in the high momentum limit, viz., the role of a direct qqq force which has been shown5 to produce an additional singularity in the structure φ of the 3D qqq wave function. The application is intended to address the issue of the proton spin anomaly in terms of a two-gluon anomaly effect. To that end, a good part of the paper has been devoted to a fairly general formulation of baryonic transition amplitudes, looked upon as qqq systems in terms of Feynman amplitudes involving appropriate quark loops. The Bethe-Salpeter normalization has been attuned to the total probability which maintains a symmetry between the proton and the neutron, instead of to the total charge which does not. i This relativistic formulation has the advantage that the axial charges gA , (i 0, 8, 3), are already 23 times the corresponding non-relativistic quantities,6 thus obviating major ‘relativistic corrections’6 for them. Thus calibrated, the formalism is applied to the evaluation of two-gluon anomaly 0 corrections [self-energy and exchange] to gA , by inserting the anomaly operator ∆µνλ into the internal quark lines, so as to produce a fractional correction of the general form 8.33, in which the dimensionless quantity θ is a measure of the correction. Unfortunately we have so far been able to calculate only the ‘kinematical’ correction which corresponds to the neglect of the internal momenta (ξ, η) in the integrands of the amplitudes involved. The resulting value of θ is 2.0 which has the right sign, but a rather small magnitude. This still leaves open the possibilities of ‘dynamical’ corrections which involve an interplay of the internal momenta, mostly arising from the various propagators, with the 3D wave function φ whose hypergeometric form 8.16 reflects the dynamics of the 3-body force, namely the negative eigenvalue of the associated spin operator. The ‘correct’ (negative) sign of θ is an encouraging sign for the vast scope for the role of this crucial dynamics yet to be included in its derivation. This calculation is currently in progress. The author is grateful to the organizers of THEOPHYS-07 for an opportunity to present these preliminary results at this Conference. References 1. Ernest Ma, Phys. Rev.D12, 2105 (1975). 2. B. McKellar, private communication ; 1994.
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3. A. N. Mitra, Phys Rev. D28, 1745 (1983). 4. J.C. Taylor, Gauge Theories of Weak Interactions, Cambridge Univ Press, 1978. 5. A.N. Mitra, hep-ph/0704.1103; Ann.Phys.(N.Y.) 2007, in press. 6. S.D. Bass Rev. Mod. Phys. 77, 1257-1302 (2005). 7. E.E. Salpeter, Phys.Rev. 87 (1952) 328. 8. M.A. Markov, Sov. J. Phys. 3 (1940) 452. 9. H. Yukawa, Phys.Rev. 771950) 219. 10. J. Lukierski and M. Oziwicz, Phys. Lett.B69, 339 (1977). 11. P.A.M. Dirac, Rev. Mod. Phys. 21 (1949) 392. 12. A.N. Mitra, Phys. Lett.B463 (1999) 293. 13. A.N. Mitra and B.M. Sodermark, lanl ark hep-ph/0104219. 14. S. Chakrabarty et al., Prog. Part. Nucl. Phys. 22, 43-180 (1989). 15. A.N. Mitra and A. Mittal, Phys.Rev.D29, 1399 (1984). 16. R.P. Feynman et al., Phys.Rev. D3, 2706 (1971) 17. R. Blankenbecler et al., Nucl.Phys.12, 629 (1959) 18. A.S. Davidov Quantum Mechanics, Pergamon Press Oxford, 1965; A.K. Saxena, Textbook of Quantum Mechanics, CBS Publishers New Delhi 2007 19. A.N. Mitra and M.H. Ross, Phys. Rev.158, 1630 (1967) 20. A.N. Mitra, Zeits. f. Phys. Particles & Fields, C8, 25 (1981). 21. E.T. Whittaker and G.N. Watson, A course on Modern Analysis, Camb Univ Press, N.Y., 1952 22. L. Rosenberg, Phys. Rev. 129, 2786 (1963) 23. G. t’ Hooft and M. Veltman, Nucl. Phys.B44, 89 (1972) 24. A.N. Mitra and W.Y. Pauchy Hwang, Eur. Phys. J. C 39, 209 (2005)
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Chapter 9 Whither Nuclear Physics?
Afsar Abbas Centre for Theoretical Physics JMI, Jamia Nagar, New Delhi-110025, India Nuclear Physics has had its ups and downs. However in recent years, bucked up by some new and often puzzling data, it has become a potentially very rich field. We review some of these exciting developments in a few important sectors of nuclear physics. Emphasis shall be on the study of exotic nuclei and the new physics that these nuclei are teaching us.
Nuclear physics, perhaps not unlike any other discipline of scientific enterprise, has had mixed fortunes. Having had its heyday in the midtwentieth century, nuclear physics’ fortunes had sunk rather low towards the last decades of the 20th century. In fact, nuclear physics had been pronounced dead by many an expert. But as in Mark Twain’s case, that announcement was rather “premature”. Being a human, Mark Twain had to pass away ultimately, but in the case of nuclear physics, in recent years, there has been a great upsurge of scientific interest and at present nuclear physics is well and kicking. The reasons which have led to this renaissance in nuclear physics, shall be the focus of this paper. Most of the knowledge of nuclear physics, until recently, was based on radioactive decay studies and by nuclear reactions induced by beams of some 283 species of stable or long lived nuclei one finds on earth. So to say, this has resulted in what may be called the conventional nuclear physics. However, this allows us to study only somewhat limited regimes of nuclear physics. The nucleus, consisting of A number of nucleons, is governed by a large number of degrees of freedom. We have to judiciously choose the ones which are relevant; relevant, both in terms of theoretical challenges as well as in terms of the kind of experiments we are capable of performing today and as to which nuclear degrees of freedom do these experiments allow 207
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us to explore. Given what is being actively pursed today, we may define a three dimensional landscape. The three parametric space within which we may restrict our discussion are: firstly the temperature or excitation energy, secondly the angular momentum space and thirdly the space of the Z .1 neutron-proton ratio NA The first two of the above can be studied by varying the combination of the target and the projectile and by changing the energy of the projectiles. There has been great progress in studying nuclei at higher energies. As we have gone to higher and higher energies, we continue to improve our knowledge and understanding of nuclear physics. The frontier area in this parameter space today is the ongoing attempts to achieve the new phase of quark gluon plasma in heavy ion collisions at laboratories like CERN and RHIC.2–4 Very high angular momenta of 50-80 has been achieved by grazing angle collisions between heavy ions. Hence motion of individual nucleons under these special conditions provide new insights into the nuclear dynamics.5 The search for superheavy nuclei comes under the same category. The studies from the first two parameter space have been very exciting in nuclear physics and have resulted in major improvements in our understanding of the nuclear phenomenon. However the most significant developments which are providing the intriguing possibility of some ‘new physics” in the regime of nuclear physics, have come from the developments arising from the third parametric space, ie those brought in by changing the neutron-proton ratio in nuclei. And we shall concentrate upon these here. Drip line nuclei are those wherein the last neutron or proton is barely bound. As compared to the some 283 stable nuclei, the number of nuclei between the neutron and proton drip lines is close to 7000. This allows for tremendous scope of variation of both the number of protons and neutrons. The recent exciting development of the generation of radioactive beams has made the above studies possible.6 Study of exotic nuclei through these radioactive beams is transforming the whole landscape of nuclear physics, and as perhaps even more significantly, is forcing us to attain and acquire a better understanding on what nuclear physics is all about. The most significant discovery in the sphere of exotic nuclei is certainly that of the neutron halo nuclei.7,8 In conventional nuclei the rms radius of a nucleus is given by 1
R R0 A 3
(9.1)
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where R0 is 1.2 1013 cm. In halo nuclei, it was found that the size is significantly larger than what the above formula gives and which holds well for conventional nuclei. So for example for 11 3 Li8 , the radius was found to be about 3.3 fm which is much larger than what the above formula would give and is close to the radius of a nucleus with A=48 or so. In contrast the radius of 93 Li6 is only about 2.3 fm. In the case of 62 He4 the rms radius determined empirically was 2.57 fm while that of 42 He2 is only 1.46 fm. The 6 two neutron separation energy of 11 3 Li8 and 2 He4 are of very small value 0.3 Mev and 1.0 Mev respectively. Hence one assumes that these large 6 9 4 halo nuclei 11 3 Li8 and 2 He4 consist of a compact core (of 3 Li6 and 2 He2 respectively) and with two extra neutrons orbiting around the core at a large distance away from it.7,8 Hence the two extra neutrons in the above Li and He nuclei are very weakly bound and thus extend to large distances in these so called halo nuclei. In addition it turns out the addition of the two extra neutrons outside 9 6 3 Li6 and outside 2 He4 cores, practically do not modify the electro-magnetic 6 properties of the cores in 11 3 Li8 and 2 He4 . For example the magnetic dipole 9 11 moment of Li and Li are 3.4 and 3.7 nm respectively while the two electric quadrupole moments are -27.4 and -31 mb respectively. Charge changing reaction for 8,9,11 Li on carbon target are nearly the same for all the Li isotopes [8]. This shows that the charge distribution in all these nuclei remains the same. And hence this shows that the two extra neutrons in 11 Li do not disturb the proton distribution of the core. In addition, more significantly, the wave function which works for these halo nuclei is where the core and the halo neutrons decouple so that7,8 Φhalonucleus
Ψcore
&
Ψ2n
(9.2)
17 There are other two neutrons halo nuclei known, eg. 14 4 Be10 and 5 B12 11 etc. In fact there are also single neutron halo nuclei as well, eg 4 Be7 and 19 6 C13 . So it is doubtful if pairing has anything to do with the existence of halo nuclei. In fact, through the determination of the momenta distribution of the halo neutrons, it has been found that they remain significantly far away from each other in the halo.7 The fact that the radii of neutron halo nuclei is so very large and this coupled with the fact that the wave function decouples as given above, is indicative of some “new physics”. This point is consolidated by further studies of the halo nuclei. Below we discuss two more sets of empirical information arising from the study of halo and other exotic nuclei, which
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further strengthen the belief that this indeed points to some “new physics”. This phrase “new physics” just means that we have to go beyond our understanding of nuclear physics based on conventional ideas. In nuclear fusion process, the overlap between the two participating nuclei is important. Hence based on this picture of fusion in conventional nuclear physics, one would expect significant enhancement of the probability of nuclear fusion at low energies if halo nuclei are involved. A precision experiment to detect this effect was performed recently by Raabe et al.9 They studied the reaction of the halo nucleus 62 He4 on 238 92 U146 target at energies near the fusion barrier. Surprisingly, they found no such enhancement which clearly contradicted the logic for halo nuclei based on conventional nuclear physics. This shows that the behavior of halo nuclei is different from the standard nuclei on the basis of which all our understanding of nuclear physics is based upon. Clearly the two extra nucleons in 62 He4 behave differently with respect to the other four nucleons in the core nucleus 42 He2 . This is consistent with the total wave function of 62 He4 being a simple product wave function as given above Φ62 He4
Ψ42 He2
&
Ψ2n
(9.3)
The symmetry inherent in such a product is that one does not antisymmetrize the two neutrons in the halo with respect to the other neutrons in the core. Clearly this is not permitted for a standard single composite nucleus in conventional nuclear physics. But this is consistent with all the other properties of halo nuclei. And this is also consistent with the results of the fusion reaction study discussed above.9 What appears to be happening there, was that the large fission yields that they had obtained, did not result from the fusion with 6 6 2 He4 but from neutron transfer. That is, the last two neutrons in 2 He4 are first transferred to the target. This enriched target then reacts with the left over core of the halo nucleus. This is indeed a new phenomenon indicating the uniqueness of the halo nuclei. Another new puzzling aspect arising from the study of exotic neutron rich nuclei is that of the sudden changing of magic numbers. Magic numbers N,Z = 2,8,20,28,50,82 ... have been the corner stones of conventional nuclear physics. However it has to be admitted that, though phenomenologically these had been incorporated in the shell model in nuclear physics, we really had no basic understanding of how these magic numbers arose.
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Studies of neutron rich nuclei have been indicating clearly that these conventional magic numbers are holy cows no more. Several aberrant magic numbers have been forcing themselves upon physicists. Studies of 12 4 Be8 32 and 12 M g20 showed that N = 8 and 20 were magic no more for these nuclei.10,11 The nucleus 28 8 O20 which should have been particularly stable as per the conventional nuclear shell model, was found to be not even bound.8 New magic numbers like N=14,16 and 32 have been discovered. In fact 12 Clearly all these new magic N=14 in 42 14 Si28 was shown to be magic. numbers are indicative of some “new physics” in nuclear physics.13 One has to ask as to how halo is created, why does it behave so strangely that there is no fusion enhancement due to its large size and also as to what leads to the changing of magic numbers. Here I shall try to provide some answers. Going through the binding energy systematics of neutron rich nuclei one notices that as the number of α’s increases along with the neutrons, each 4 He + 2n pair tends to behave like a cluster of two 31 H2 nuclei. Remember that though 31 H2 is somewhat less strongly bound (ie. 8.48 MeV) it is still very compact (ie. 1.7 fm), almost as compact as 4 He (1.674 fm). In addition it too has a hole at the centre. Hence 3 H is also tennis-ball like nucleus. This splitting tendency of neutron rich nuclei becomes more marked as there are fewer and fewer of 4 He nuclei left intact by the addition of 2n. Hence 7 Li which is 4 He 3 H with 2n becomes 9 Li which can be treated as made up of 3 3 H clusters and should have hole at the centre. Similarly 12 Be consists of 4 31 H2 clusters and 15 B of 5 31 H2 clusters etc. Other evidences like the actual decrease of radius as one goes from 11 Be to 12 Be supports the view that it (ie 12 Be) must be made up of four compact clusters of 3 H. Just as several light N=Z nuclei with A=4n, n=1,2,3,4 ... can be treated as made up of n α clusters, in Table 9.1 we show several neutron rich nuclei which can be treated as made up of n 31 H2 clusters. We can write the binding energy of these nuclei as Eb
8.48n Cm
(9.4)
where n 31 H2 clusters form m bonds and where C is the inter-triton bond energy. We have assumed the same geometric structure of clusters in these nuclei as for α clusters of A = 4n nuclei as given above. All the bond numbers arise due to these configurations. We notice from Table 9.1 that this holds good and and that the inter-triton cluster bond energy is approxi-
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A. Abbas Table 9.1. Inter-triton cluster bond energies of neutron rich nuclei Nucleus 9 Li 12 Be 15 B 18 C 21 N
n 3 4 5 6 7
m 3 6 9 12 15
EB
8.48nM eV 19.90 34.73 45.79 64.78 79.43
C(MeV) 6.63 5.79 5.09 5.40 5.29
mately 5.4 MeV. We notice that this value seems to work for even heavier neutron rich nuclei. For example for 42 Si the inter-triton cluster energy is still 5.4 MeV. Notice that the geometry of these cluster structures of 3 H becomes more complex as the number increases but nevertheless, it holds well. The point is that these neutron rich nuclei, made up of n number of tritons, each of which is tennis-ball like and compact, should itself be compact as well. These too would develop tennis-ball like property. This is because the surface is itself made up of tennis-ball like clusters. Hence as there are no more 4 He clusters to break when more neutrons are added to this ball of triton clusters, these extra neutrons will ricochet on the surface. Hence we expect that one or two neutrons outside these compact clusters would behave like neutron halos. Therefore 11 Li with 9 Li 2n should be two neutron halo nuclei - which it is. So should 14 Be be. It turns out that internal dynamics of 11 Be is such that it is a cluster of α t t (which also has to do with 9 Li having a good 3 α cluster) with one extra neutron halo around it. Next 17 B,19 C,20 C would be neutron halo nuclei and so on. 3 Hence, all light neutron rich nuclei 3Z Z A2Z are made up of Z 1 H2 clusters. 13,14 all these Due to hidden color considerations arising from quark effects, should have holes at the centre. This would lead to tennis-ball like property of these nuclei. One or two (or more) extra neutrons added to these core nuclei would ricochet on the surface of the core nucleus and form halos around it. Practically all known and well-studied neutron halo nuclei fit into this pattern. Also this makes unambiguous predictions about which nuclei should be neutron halo nuclei and for what reason. The proton halo nuclei can also be understood in the same manner. Here another nucleus with a hole at the centre 32 He1 (binding energy 7.7 MeV, size 1.88 fm) would play a significant role. The success of this model here gives us confidence in the new picture proposed.
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Note that ‘n’ and ‘p’ are members of an isospin doublet. These combine together to give a bound triplet state (S=1, T=0), that is deuteron with a binding energy of 2.2 MeV. It has no excited states. The singlet state (S=0, T=1) is unbound by 64 keV. This being isospin partner of (n-n) (S=0, T=1) and (p-p) (S=0,T=1), all these are unbound. Now it turns out that ‘h’ and ‘t’ are also members of a good isospin 1/2. Note that though both ‘n’ and ‘p’ are composites of three quarks these still act as elementary particles as far as low-energy excitations of nuclear physics are concerned. Only at relatively higher energies does the compositeness of ‘n’ and ‘p’ manifest itself. Similarly the binding energies of 3 He and 3 H are 7.72 MeV and 8.48 MeV respectively and also these two have no excited states. Hence for low-energy excitations, of a few MeV, we may consider these as elementary. Their compositeness would be manifested at higher excitation energies. That is we treat ‘h’ and ‘t’ as elementary isospin 1/2 entities here. This is similar to the two nucleon case. Hence, we would expect for (h-t) the triplet (S=1,T=0) to be bound and singlet (S=0,T=1) to be unbound. Also, its isospin partners (h-h) (S=0, T=1) and (t-t) (S=0, T=1) would be unbound too. Herein triton (“t”) 31 H2 helion (“h”) 32 He1 are treated as fundamental representations of this new symmetry group called “nusospin” SU 2A . Even the isospin symmetry between neutron and proton is broken. So we expect that this new symmetry should be broken as well. To see how this new symmetry is likely to manifest itself we look at other symmetry groups which are known to be broken, So in quark model we know that progressively the flavor symmetry groups SU(2), SU(3), SU(4) etc for more number of quark flavors are broken more and more strongly. In fact SU(5) for five quark flavors u,d,s,c and b quarks is very strongly broken. However it still manifests itself in particle physics. Its most clear manifestation is in terms of representations of all the particles built up of any of these 5 quarks. Hence, howsoever badly it may be broken, the physical existence of objects which correspond to the irreducible representation of a particular group is what actually determines the relevance of a particular group in physics. So also in the case of our new symmetry in nuclei SU 2A this may be minimum expectation as well. Hence as already suggested, the particle representation of nuclei of the form 3Z Z A2Z nuclei would be that of Z number of tritons as per the “nusospin” group.. So let us ask as to what this new “nusospin” symmetry SU 2A has to tell us about new magicities. Clearly the fact that 3Z Z A2Z nuclei are made up of Z number of tritons leads to new stability for them.
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We plot S1n as a function of Z for a particular N rather than plotting it as a function of N , as is normally done [11,12]. It appears that what we are plotting brings out the relevance of the (h,t) degree of freedom of the new group nusospin more clearly. Fig 9.1 is S1n for fixed N= 4, 6, 8, 10, 11, 12, 16, 20, 22 and 24 plotted as a function of Z.
Fig. 9.1. One neutron separation energy as a function of proton number Z for different number of fixed neutrons indicated
The reader’s attention is drawn to the extra-ordinary stability manifested by the plotted data for the proton and neutron pairs (Z,N): (6,12), (8,16), (10,20), (11,22) and (12,24). Note that the stability at these pair of numbers is sometimes as prominent as that at the N=Z pair. In fact the Z,N pair (10,20) stands out as the best example of this. Hence it is clear that the separation energy data very clearly shows that there are new magicities present in the neutron rich sector for the pair (Z,N) where N=2Z. For more plots of this kind which show stability of (Z,2Z) nuclei, see.13 What is the significance of this extraordinary stability or magicity for the nuclei 3Z Z A2Z ? We already know that for the even-even Z=N cases it is the significance of α clustering for the ground state of these nuclei which explains this extra stability. Quite clearly the only way we can explain
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the extra magicity for these N=2Z nuclei is by invoking the significance of triton clustering in the ground state of these neutron rich nuclei. Thus 30 3 10 N e20 has significant mixture of the configuration 10 1 H2 in the ground state. It is these tritonic clusters which give the extra stability to these nuclei thereby providing us with these unique new sets of magic numbers. We continue to plot the separation energies a little differently here. We plot S1n and S2n as a function of Z for a particular N . The same with S1p and S2p as a function of N for a particular Z. We do a systematic study of these plots for all the data available in literature at present. We find that these brings out certain very interesting generic features which as we shall find clearly indicate strong evidences of triton (“t”) 31 H2 and helion (“h”) 3 2 He1 clustering in nuclei We study the whole range of data set available. For the sake of brevity, here we show two representative plots. These are one and two neutron separation energy as a function of Z for N = 29 and 30 as plotted in Fig 9.2 and 9.3 below. All the plots of S1n and S2n as a function of Z for a particular N show similar features. The same with S1p and S2p as a function of N for a particular Z. We do a systematic study of these plots for all the data. The features which we shall point out here are there in all the other plots. In fact we shall study here only those features which are generic of all such plots. Certain common features which stand out are as follows (The statements below are made in the context of the plot S1n and S2n as a function of Z for a particular N ): A. For all even-even N=Z nuclei there is always a pronounced larger separation energy required with respect to the lower adjoining nuclei plotted. B. For all odd-odd N=Z nuclei there is always a pronounced larger separation energy required with respect to the lower adjoining nuclei plotted. C. For case A when Z number is changed by one unit, the separation energy hardly changes (sometimes not at all). But when this number is changed by two units, another pronounced peak occurs. D. For case B when Z number is changed by one unit, the separation energy hardly changes (sometimes not at all). But when this number is changed by two units, another pronounced peak occurs. So there are peaks for odd-odd N-Z nuclei (and not for even-even cases). Here, as we are pulling one or two neutrons as a function of Z, the above effects cannot be the result of identical nucleon pairings, What these plots are telling us is as to what happens to last one or two neutron bindings in a nucleus as proton number changes.
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Separation energy (MeV)
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Z
Fig. 9.2.
One and two neutron separation energy as a function of proton number Z for fixed N=29 neutrons
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Separation energy (MeV)
25
34
Z
Fig. 9.3.
One and two neutron separation energy as a function of proton number Z for fixed N=30 neutrons 217 recent
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Clearly the peaks as indicated in A above are due to the fact that the last one or two neutrons must have come from a stable alpha cluster. This is consolidated by the fact that another extra Z does not make a difference to the separation energy. To understand observation B above - note that here the peaks are there for ALL odd-odd N=Z nuclei. This is an amazing fact. These odd-odd nuclei are more stable or “magic” with respect to the adjoining odd-even or even-odd nuclei. Pairing of identical nucleons cannot explain this generic feature. Neither can alpha clustering do so. Obviously it is the formation of triton-helion ‘h-t’ pair which can only explain this extraordinary effect. What is the significance of this extraordinary stability or magicity for all the nuclei 3Z Z A2Z ? Quite clearly the only way we can explain the extra magicity for these N=2Z nuclei is by invoking the significance of triton clustering in the ground state of these neutron rich nuclei. So the nucleus 3Z Z A2Z is made up of Z number of triton clusters, as we showed earlier. To understand this unique feature, the new symmetry “nusospin” symmetry becomes relevant. It is clear that it is tritons which explain the stability of neutron rich nuclei and it it is pair of ‘h-t’ clusters which explain the stability of odd-odd N=Z nuclei in the separation energy as plotted above. Now we can explain the observation C above. Clearly for even-even N=Z nuclei it is one (or more) alpha clusters which explain the data. Hence one extra Z does not affect the separation energy. But two extra Z will tend to make a pair of helions (akin to the two tritons for neutron rich case above). These two ‘h-h’ will make for the extra stability for the adjoining even-even nuclei (and so on). So also can the observation D be understood as the extra 2Z will create an extra helion to attach to the already existing ‘h-t’ pair to make for extra stability of this adjoining odd-odd nuclei. Other peaks in the above plots can be similarly explained as due to alpha or triton and helion clusters. Clearly the new nusospin group provides the (h,t) degrees of freedom to explain the above data. As we have found the nusospin group to be useful in certain physical situations, the relevant enlargened group in nuclear physics should be & SU 2A SU 2I ie a product of the isospin and the nusospin groups. Hence both the (p,n) and (h,t) are relevant degrees of freedom for nuclei. This helps us in resolving a puzzle in the structure of 42 He2 nucleus. It is now well known,17 that contrary to expectations, the ground state of
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4 2 He2
contains very little of deutron-deutron configuration; and the same is actually built upon h-n and t-p configurations.17 It is a puzzle as to how come the first excited state of this even-even nucleus is another 0 with T=0 state (the same as the ground state) at a high value of 20.2 MeV. However this finds a natural explanation in our model of the product group ' SU 2A The wave functions of the ground state and the first SU 2I excited state of 42 He2 in our model is naturally given as Φgs
Ψh
'
Ψn Ψt 2
'
Ψp
(9.5)
' Ψn Ψt Ψp (9.6) 2 ' Clearly this enlargened group SU 2I SU 2A should help us in improving the understanding of nuclear phenomenon. So in summary, the field of nuclear physics today is an active and a very promising branch of physics. The new results are forcing us to rethink some of the fundamentals of conventional nuclear physics. Obviously what has been working (and hence phenomenologically correct) for N=Z and nearby nuclei cannot be wrong. But we have to extend our understanding so as to incorporate the new puzzling data arising mainly from the study of exotic nuclei. The N=Z and nearby cases should be considered as special case of a more general framework which should account for the exotics as well. We have tried to present a picture which tries to tackle some of these issues. Coming up with a comprehensive picture is the exciting challenge of nuclear physics today. Φ20.2
Ψh
'
References 1. A. Richter, Nucl. Phys. A 533 (1993) 417c 2. RHIC Report: Conceptual Design of Relativistic Heavy Ion Collider, BNL Report 52195, Brookhaven National laboratory 3. B. Mueller, “The physics of quark gluon plasma”, (Lecture Notes in Physics Vol 225), Springer Verlag, Berlin 1985 4. A. Abbas, L. Paria and S. Abbas, Eur. Phys. J. C 14 (2000) 695 5. P. J. Twin et al, Phys. Rev. Lett. 57 (1986) 811 6. W. Gelletly, Contemp. Phys. 42 (2001) 285 7. I. Tanihata, J. Phys. G 22 (1996) 157 8. I. Tanihata, Nucl. Phys. A 685 (2001) 811 9. R. Raabe et al, Nature 43 (2004) 823
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10. Z. Dlouhy, D. Baiborodin, J. Mrazek and G. Thiamova, Nucl. Phys. A 722 (2003) 36c 11. M. Thoennessen, T. Baumann, J. Enders, N. H. Frank, P. Heckman, J. P. Seitz and E. Tryggestad, Nucl. Phys. 26 (2003) 61c 12. J. Fridmann et al, Nature 435 (2005) 922 13. A. Abbas, Mod. Phys. Lett. A 20 (2005) 2553 14. A. Abbas, Phys. Lett. B 167 (1986) 150 15. A. Abbas, Mod. Phys. Lett. A 16 (2001) 755 16. G. Audi, A. H. Wapstra and C. Thibault, Nucl. Phys. A 729 (2003) 337 17. K. Langanke, Adv. Nucl. Phys. Vol 21 (1994) 85
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Chapter 10 Generalized Swanson Model and its Pseudo Supersymmetric Partners
A. Sinha and P. Roy Physics & Applied Mathematics Unit Indian Statistical Institute Kolkata - 700 108, India
We study a generalized version of the Swanson model, viz., H A A αA2 βA 2 (α β), replacing the Harmonic oscillator creation and annihilation operators a , a in the original Swanson model by generalized creation and annihilation operators A , A. It is found that the energies are real for a certain range of the parameter values only. We then map the non-Hermitian Hamiltonian H to its Hermitian counterpart h (with identical energies) by means of a similarity transformation ρ. Applying the principles of supersymmetric quantum mechanics we obtain an isospectral partner (H ) of H , with the help of the supersymmetric partner (h ) of h . It is observed that H , are related by pseudo supersymmetry, which is anticipated due to their non Hermitian character. We illustrate our formalism with the help of an explicit example.
Contents 10.1 10.2 10.3 10.4
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Similarity Transformation between H and h . . . . . . . . Pseudo Hermitian Isospectral Partner H of H . . . . . 10.4.1 Pseudo Hermiticity of H . . . . . . . . . . . . . 10.5 Underlying Symmetry between the Partners H . . . . . 10.6 A Model based on Trigonometric Rosen-Morse I Potential 10.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . .
One can find the extended versions of this work in A. Sinha and P. Roy, J. Phys. A : Math. Theor. (2007) 40 10599 A. Sinha and P. Roy, J. Phys. A : Math. Theor. (2008) 41 335306 221
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10.1. Introduction Though study of non Hermitian potentials began almost sixty years ago, nevertheless, interest in such models was revived about a decade ago when Bender and Boettcher1 conjectured that the strict restriction of Hermiticity of the Hamiltonian operator may be replaced by the more physical requirement of space-time symmetry, viz., PT symmetry, and still the system would admit real energies: i.e., Though
H
"
H
H PT
yet
PT H
(10.1)
where P stands for parity and T denotes time reversal operators respectively: PxP
x
PpP
,
T pT
p
,
T i.1T
i.1
(10.2)
In innumerable works (by various scientists) that followed,3,5 it was observed that such Hamiltonians admit real and discrete spectrum when PT symmetry is exact (i.e., when the energy eigenstates are also the eigenstates of PT ), whereas for spontaneously broken PT symmetry the energies occur as complex conjugate pairs. However, it was soon discovered that PT symmetry is neither the necessary nor the sufficient criterion for the spectrum to be real. Subsequent works attributed the reality of the spectrum to its η-pseudo Hermiticity criterion, such that H are linear operators acting in a Hilbert space (generally different from the physical Hilbert space), and satisfying18 : H
ηHη 1
,
i.e.
H η
ηH
(10.3)
where η is a linear, Hermitian, invertible operator, without a unique representation. Furthermore, the pseudo-Hermiticity of H is equivalent to the presence of an antilinear symmetry, and PT symmetry is a very good example of such a symmetry.19 The converse is also true: A quantum system possessing an exact antilinear symmetry is pseudo-Hermitian, and is equivalent to a quantum system described by a Hermitian Hamiltonian h.18,20 Thus if an eigenvalue (Sturm-Liouville) equation or a differential operator H acts in a complex function space V, endowed with a positive definite inner product, such that it is described by the Hilbert space H, then there exists a mapping from the non-Hermitian H to its Hermitian counterpart h, through a similarity transformation ρ;22 i.e., h ρHρ1
(10.4)
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with ρ being the unique positive-definite square root of η: ρ
η
(10.5)
A relation similar to (10.4) holds for observables as well, relating the observable Oh in the Hermitian theory to the corresponding observable in the pseudo-Hermitian theory by O
ρ1 Oh ρ
(10.6)
Since non Hermitian quantum systems have found wide spread interest because of their immense potential for possible applications in a wide range of phenomena,23,24 e.g., nuclear physics, scattering theory (i.e., complex absorbing potentials), field theory, periodic potentials, quantum cosmology, random matrix theory, etc., various attempts have been made to extend the class of such systems for exactly solvable (ES) or quasi-exactly solvable (QES) or conditionally exactly solvable (CES) models, with real, discrete energies, using different approaches — e.g., supersymmetry,25 the related intertwining operator method,28 or the Darboux algorithm.29 We wish to extend this class further by first obtaining a generalized form (H ) of the non Hermitian Swanson model30 H
a a αa2 βa
2
,
α"β
(10.7)
by replacing the Harmonic oscillator annihilation and creation operators a and a d d a x , a x (10.8) dx dx
with generalized annihilation and creation operators A and A A
1 d & 1 α β dx
W
x'
,
A
1 1 α β
d dx
W
x
(10.9) and then obtain its pseudo supersymmetric partner H . The function W x above, called the pseudo superpotential (in analogy with conventional supersymmetry), is given by W x
f0 x f0 x
(10.10)
where f0 x is the ground state wave function of the Schr¨ odinger Hamiltonian H A A. So we would start with the Hamiltonian H
A A αA2 βA
2
,
α"β
(10.11)
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where α, β are real, dimensionless constants, and examine the range of values of the parameters for which the energies are real. Due to this restriction on the parameter values for real energies, the model may be termed as conditionally exactly solvable (CES).24 Our next step would be to generate new non Hermitian Hamiltonians H , which are isospectral to H . We would like to mention here that non Hermiticity may be introduced in two ways — through a scalar term, by replacing x with x i or by taking one of the parameters complex in the expression for the potential V x, or by a momentum-dependent vector term. In this work we have restricted ourselves to the latter case as in this case a similarity transformation of the non Hermitian Hamiltonian (H with solutions ψ ) can be mapped to a Hermitian Schr¨ odinger Hamiltonian (h with solutions φ ), consisting of the standard kinetic term plus a local Hermitian potential term, with possibilities of exact (or quasi-exact) solvability. Additionally, the metric operator can also be obtained in closed form. The same is not true for the scalar case, where neither can the exact form of the metric operator be obtained explicitly, nor can the resulting Hermitian Hamiltonian (which is a is a complicated one) be solved exactly. Having obtained the non Hermitian partner Hamiltonian H of the generalized Swanson Hamiltonian H , our attempt would be to look for some underlying symmetry between H . Due to the non Hermiticity of H , it is anticipated that the partners will be related by pseudo supersymmetry.31 Thus Hermitian supersymmetric partners h are related by a similarity transformation to non Hermitian pseudo supersymmetric partners H . Finally we shall illustrate our formalism with the help of an example. The plan of work is as follows : In Section 2, we give the general formalism for solving a class of non-Hermitian Swanson model with generalized creation and annihilation operators. The similarity transformation ρ, between the Hermitian h and the non-Hermitian H, is established in section 3. In section 4, new non Hermitian Hamiltonians H are generated, which are isospectral to the initial non Hermitian Hamiltonian H . The underlying symmetry between the partners H is studied in section 5. The formalism developed here is actually applied to an explicit example in section 6. Finally, Section 7 is kept for conclusions and discussions. 10.2. Theory As mentioned above, we shall examine a generalization of the Swanson model , viz., 30
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H
A A αA2 βA
2
225
α"β
,
where α and β are constants, dimensionless as well as real. Evidently, H is non-Hermitian for α " β for any real W x. With the help of (10.9), the eigenvalue equation corresponding to (10.11) reduces to ) ( 2 d α β (10.12) W H ψ V x ψ Eψ dx 1 α β Considering a gauge-type transformation of the form23,24 ψ x eµ
φ x
W x dx
with µ
,
α β , α 1 α β
β
"
1
(10.13) reduces (10.12) to the well known Schr¨ odinger form d2 x Eφ x V x φ h φ x dx2 where V x
2 1 4αβ W x 1 α β
1 W 1 α β
x
(10.14)
(10.15)
Using the principles of supersymmetric quantum mechanics,25 h can be written in a factorizable form as a product of a pair of linear differential operators A˜ , A˜ , as h
A˜ A˜
d2 dx2
w2 w
(10.16)
where is the factorization energy, and A˜ , A˜ and w x are given by A˜
d ln ϕ0 x dx (10.17) Here ϕ0 is the ground state eigenfunction of A˜ A˜ with energy E0 . A˜
d dx
w
x ,
d dx
w
x ,
w x
Evidently, if we can identify the term V x in (10.15) above, with an exactly solvable potential, then we can easily find the solutions of h . To this end, for further convenience, V x can be identified with a shape-invariant potential, as using the ideas of supersymmetric quantum mechanics,25 the raising and lowering operator method of harmonic oscillator can be generalized to a whole class of shape invariant potentials, which includes all the
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analytically solvable models. To narrow down the class of potentials further, our strategy would be to write V x in (10.15) in the supersymmetric form w2 x w x as given in (10.16). This identification enables us to find the energies (E) and the eigenfunctions (ψ ) of the eigenvalue equation in (10.12). However, this imposes certain restrictions on the permissible values of α and β. For real energies, supersymmetric considerations require the term containing W 2 x in the expression for V x in (10.15), must be positive. Furthermore, E and ε should have similar behavior. Hence, the parameters α , β must satisfy the following constraints, irrespective of the explicit form of W x : αβ
1
,
4αβ
1
(10.18)
There may be additional restrictions on α , β depending on the particular choice of the model, arising from the normalizability requirement of the wave functions. We shall illustrate this further with the help of an explicit example in a later section. 10.3. Similarity Transformation between H and h In this section we shall determine a similarity transformation, mapping the non-Hermitian H to the Hermitian h .18 If we focus our attention on the gauge transformation ρ eµ W dx relating ψ x and φ x in equation (10.13) and apply the transformation ρ to ψ x in the eigenvalue equation H ψ x Eψ x
(10.19)
then it can be shown that φ x is a solution of the equation h φ Eφ with the same energy E as in (10.19) :
H ρ1 φ x Eρ1 φ x
provided H is mapped to h
or ρ H ρ1 φ x E φ x (10.20) by the similarity transformation
h
ρ H ρ1
Furthermore, this exact form of the similarity operator for this class of models, also gives the wavefunctions in the corresponding Hermitian picture. 10.4. Pseudo Hermitian Isospectral Partner H of H Having mapped a quantum system described by a pseudo Hermitian Hamiltonian H , to its corresponding Hermitian counterpart h , we take refuge
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in the formalism of supersymmetric quantum mechanics (SUSYQM),25 or the equivalent intertwining operator method,28 to find an isospectral partner h of h , given by h
so that h φ x
A˜A˜
d2 dx2
d2 dx2
w2 w
V x φ x
(10.21)
Eφ x
(10.22)
where V x w2 x w x
(10.23)
If we now apply the inverse transformation to the solutions of h , i.e., φ x
ρ ψ x
eµ
ψ x
W x dx
(10.24)
then, using straightforward algebra, equation (10.22) can be written as ) ( 2 d α β W x V x ψ Eψ H ψ dx 1 α β (10.25) Thus, H are of the same form, except for the explicit form of V x. Since h share identical energies, except for the ground state, so should H , with the exception of the ground state. Thus, applying the principles of SUSYQM, we obtain a non Hermitian partner Hamiltonian H of the initial one H , sharing identical energies except for the ground state. 10.4.1. Pseudo Hermiticity of H
If one considers the inverse transformation (10.24), then it is easy to check that both the Hermitian Hamiltonian h and their non Hermitian counterparts H are related by the same similarity transformation H
ρ1 h ρ
(10.26)
Additionally, H are pseudo Hermitian with respect to the same pseudo Hermiticity operator η H
ηH η 1
i.e.
H η
ηH
(10.27)
where ρ and η are inter-related through ρ η .23,32 It is interesting to study the behavior of the wave functions ψ x. Since H are η-pseudo Hermitian, the wave functions should be normalized as
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ψ η ψ .18 With η ρ2 and ψ x ρ1 φ x, the above normalization condition reduces to the conventional normalization of Hermitian quantum systems, viz., φ φ , easily available in standard text books of quantum mechanics for the shape-invariant potentials considered here.25
10.5. Underlying Symmetry between the Partners H To explore the underlying symmetry between the isospectral partners H , we start with their Hermitian counterparts h , which form a pair of supersymmetric partners, with super Hamiltonian h 0 (10.28) h 0 h
and are generated by supercharges 0 A˜ , q 0 0
q
so that
0 0 A˜ 0
(10.29)
* + q ,q
h
(10.30)
Since H ρ1 h ρ, one can define two operators D as D
ρ1 A˜ ρ
D ρ1 A˜ ρ
,
,
d dx
d dx
µW
x w x -
µW
(10.31)
x w x
in terms of which the isospectral Hamiltonians H can be written as H Thus D
D D
H
,
D D
(10.32)
play the role of intertwining operators for H :
D H
H D
,
H D
D H
(10.33)
It must be kept in mind that the functions W x and w x appearing in the explicit form of D are related to each other by the expression 2 1 4αβ 1 w2 x w x W x W x (10.34) 1 α β 1 α β Since the isospectral partner Hamiltonians H are pseudo Hermitian, we expect them to be embedded in the framework of pseudo supersymmetry.31
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Straightforward algebra shows that the operators D are pseudo-adjoint of one another ˜ D (10.35) D η 1 D η η 1 ρA˜ ρ1 η ρ1 Aρ So if we construct a new Hamiltonian from the partners H as H 0 H 0 H
and define two operators Q and Q as 0 D
Q , Q η 1 Q η 0 0
0 0 D 0
(10.36)
0 0 D 0
(10.37)
then it is easy to observe that H
* + Q ,Q
(10.38)
Additionally, &Q, Q'
*
Q , Q
+
0
(10.39)
Thus we obtain the standard pseudo super algebra of non Hermitian supersymmetry.31 Hence, the isospectral partners H are related by pseudo supersymmetry, with the operators Q and Q playing the role of pseudo super charges, the anticommutator of which gives the pseudo super Hamiltonian H. Furthermore, it can be shown from the similarity mapping between H and h that the super charges q , q of conventional supersymmetry are related to the pseudo supercharges Q , Q of pseudo supersymmetry through Q ρ1 q ρ
(10.40)
An explicit example will further illustrate our formalism. 10.6. A Model based on Trigonometric Rosen-Morse I Potential For potentials belonging to the shape invariant class, the function w x consists of two parts, denoted by f x and g x, i.e. w x
λ1 f x
δ1 g x
,
with λ1 , δ1 constants
(10.41)
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If analytic solutions of the highly non-trivial Ricatti equation (10.34) are to be obtained, the function W x used in the construction of A and A in (10.9) is assumed to be of the same form as w x: W x λ2 f x
δ2 g x
,
with λ2 , δ2 constants
(10.42)
Our aim is to write V x in terms of w2 x w x, with W x and w x being inter-related through (10.34). Substituting (10.41) and (10.42) in (10.34), the expression takes the explicit form
*
1 4αβ 2
1 α β
λ22 f 2 δ22 g 2 2λ2 δ2 f g
λ21 f 2 δ12 g 2 2λ1 δ1 f g λ1 f
+
1 λ2 f 1 α β
δ2 g
δ1 g
(10.43) This general expression relates the unknown parameters λ1 , δ1 in terms of the known ones λ2 , δ2 , for all shape invariant potentials where the parameters of the original potential and its partner are related to each other by translation. This enables one to write the partner potential V x, and hence the partner Hamiltonian H , in terms of the parameters of the starting Hamiltonian H . As an explicit example, we consider the trigonometric Rosen-Morse I model25 described by the potential V x a a 1 csc2 x 2b cot x a2
with W x a2 cot x
b2 a2
b2 a2
,
0xπ
,
a2
0 , b2
0
(10.44)
(10.45)
Obviously, a suitable ansatz for w x would be w x
a1
cot x
b1 a1
,
a1
0 , b1
0
(10.46)
Without going into detailed calculations, we just quote the results here: The eigen energies of the positive and the negative sector are related through En with
b2 En a22 22 a2
En 1
,
with
n 0, 1, 2,
(10.47)
1 4αβ b21 2 a1 n , n 0, 1, 2, 1 α β 2 a1 n2 (10.48)
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Table 10.1. Some values of the parameters for the model with W x as in (10.49) β α α β 4αβ a2 b2 µ1 µ2 σ a1 1/4 1/2 3/4 1/2 3/2 1/8 3/2 1/12 12 4 1/4 2/3 11/12 2/3 1 1/2 5 5/2 36 6.52 1/3 1/2 5/6 2/3 1 2 1 2 6 3
231
given b1 1 24 36
The partner potentials are obtained as b22 1 4αβ 1 4αβ 2 2 a1 a1 1 csc x2b2 cot x V x a2 2 a2 1 α β 2 1 α β 2 (10.49) and the (unnormalized) solution of H is calculated to be ψn x e
b1 a1 n
µ1 x sina1 n µ2 x P s ,s y n
(10.50)
where y
i cot x
,
s
a1 n i
b1 a1 n
(10.51)
and µ1
a2 µ a2
α β 1 α β
,
µ2
b2 b2 α β µ a2 a2 1 α β
(10.52)
The solutions of its partner H can be obtained by applying the transformation ψn x
ρ1 φ x n
(10.53)
where φn are the solutions of the supersymmetric partner Hamiltonian h . It is evident from the explicit expressions for ψn x that for its welldefined behavior, α and β must obey additional constraints a1 n µ2
0
,
b1 a1 n
µ1
(10.54)
which , in turn, implies αβ
(10.55)
Choice of parameters To show there are parameter values actually satisfying (10.18), we show a few possible values in Table 10.1. There may be innumerable such combinations of α , β, λ2 and δ2 . We show a few possible values of the parameters a2 , b2 appearing in the Rosen Morse I model, in Table 10.1.
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10.7. Conclusions To conclude, we have studied a class of pseudo-Hermitian Hamiltonians of the form H A A α A2 β A 2 , where α and β are real, dimensionless constants (α " β), and A and A are generalized creation and annihilation operators. Incidentally, Swanson studied a similar model,30 although with harmonic oscillator creation and annihilation operators only. The range of parameter values ensuring the reality of the spectrum, is obtained. This non-Hermitian Hamiltonian H is mapped to its Hermitian counterpart h , with the help of a similarity transformation ρ. This straightforward approach provides us a simple way of determining the similarity transformation ρ, the metric operator η through η ρ2 , as well as the corresponding Hermitian Hamiltonian h . We have also found out a non Hermitian partner of H , denoted by H . It is observed that H form a pair of pseudo super symmetric partners of a pseudo super Hamiltonian H, and share identical energies except for the ground state. Furthermore, the same similarity transformation operator ρ maps the pair of non Hermitian Hamiltonians H to their respective Hermitian counterparts h , through H ρ1 h ρ, and these Hermitian maps form a pair of supersymmetric partners, generated by supercharges q, q . The pseudo super charges Q, Q generating the pseudo super algebra of H are also related to q, q through the similarity transformation : Q ρ1 q ρ. It may be mentioned here that though two Hamiltonians may be related by similarity transformations, yet they can reveal different physical aspects of the dynamical system. In fact, for a particular class of potentials, certain physical properties are expected to emerge more distinctly in the non Hermitian framework. For example, exceptional points, or branch-point singularities of the spectrum and eigenfunctions, are associated with non Hermitian operators.33 However, when one goes from the non Hermitian to the corresponding Hermitian picture, the exceptional points are lost, and consequently the entire information related to such phenomena. Additionally, though the super symmetric partners h of a Hermitian Hamiltonian can always be mapped to non Hermitian ones (say H ) by a similarity transformation, there is absolutely no way to determine whether H are isospectral or not. This is due to the fact that to write the pseudo Hermitian partner Hamiltonian H in terms of the generalized annihilation and creation operators A and A is still an open problem. As a result, while h look similar in appearance (being expressed in terms of the creation and
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annihilation operators A and A), H are not look-alikes. Nevertheless, we have been able to express H in terms of the operators D , thus proving them to be related by pseudo super symmetry. Finally, we have applied our formalism successfully to the shapeinvariant Rosen Morse I model. Incidentally, our present formalism is applicable to all the known classes of shape-invariant models where the parameters of the original potential and its shape-invariant partner are related through translation. The interesting point to note here is that the wave functions of H are automatically normalized following the normalization criterion for pseudo Hermitian systems.18 Acknowledgment AS acknowledges financial support from SERC, DST, Govt. of India. References 1. C. M. Bender & S. Boettcher, Phys. Rev. Lett. 80 (1998) 5243. 2. C. M. Bender and S. Boettcher, J. Phys. A : Math. Gen. 31 (1998) L273. 3. For a variety of papers on this topic see the Special issues on Conference Proceedings on Pseudo-Hermitian Hamiltonians in quantum Physics in link http://gemma.ujf.cas.cz/ znojil/conf/index.html, viz., 4. Czech. J. Phys. 54 (2004), Czech J. Phys. 55 (2005), J. Phys. A : Math. Gen. 39 (2006), Czech J. Phys. 56 (2006), J. Phys. A : Math. Theor. 41 2008. 5. M. Znojil, J. Phys. A : Math. Gen. 33 (2000) 4561. 6. G. L´evai and M. Znojil, J. Phys. A : Math. Gen. 33 (2000) 7165. 7. P. Dorey, C. Dunning and R. Tateo, J. Phys. A : Math. Gen. 34 (2001) 5679. 8. C. M. Bender, S. Boettcher, H. F. Jones, P. N. Meisinger and M. Simsek, Phys. Lett. A 291 (2001) 197. 9. Z. Ahmed, Phys. Lett A 282 (2001) 343, ibid. Phys. Lett. A 287 (2001) 295. 10. B. Bagchi and C. Quesne, Phys. Lett. A 273 (2000) 285, ibid. Phys. Lett. A 300 (2002) 18. 11. C. M. Bender, D. C. Brody and H. F. Jones, Phys. Rev. Lett. 89 (2002) 270401, ibid. Phys. Rev. Lett. 92 (2004) 119902(E). 12. C. M. Bender and B. Tan, J. Phys. A : Math. Gen. 39 (2006) 1945. 13. E. Caliceti, F. Cannata and S. Graffi, J. Phys. A : Math. Gen. 39 (2006) 10019, and references therein. 14. C. Quesne, J. Phys. A : Math. Theor. 40 (2007) F745-F751. 15. F. Cannata, M.V. Ioffe, D.N. Nishnianidze, Phys. Lett. A 369 (2007) 9. 16. C. M. Bender, Rep. Prog. Phys. 70 (2007) 947. 17. Paulo E. G. Assis and A. Fring, J. Phys. A : Math. Theor. 41 (2008) 244001. 18. A. Mostafazadeh, J. Math. Phys. 43 (2002) 205, ibid. 2814, ibid. 3944. 19. L. Solombrino, J. Math. Phys. 43 (2002) 5439.
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F. G. Scholtz, H. B. Geyer and F. J. W. Hahne, Ann. Phys. 213 (1992) 74. R. Kretschmer and L. Szymanowski, Phys. Lett. A 325 (2004) 112. A. Mostafazadeh and A. Batal, J. Phys. A : Math. Gen. 37 (2004) 11645. A. Sinha and P. Roy, J. Phys. A : Math. Theor. 40 (2007) 10599. A. Sinha and P. Roy, J. Phys. A : Math. Theor. 41 (2008) 335306, and references therein. F. Cooper, A. Khare and U. Sukhatme, Supersymmetry in Quantum Mechanics, World Scientific, 2001. B. Bagchi, Supersymmetry in Quantum and Classical Mechanics, Chapman and Hall, 2000. H. Kalka and G. Soff, Supersymmetry, Teuber, 1997. L. Infeld and T. E. Hull, Rev. Mod. Phys. 23 (1951) 21. V. V. Fatveev and M. A. Salle, Darboux Transformations and Solitons, New York, Springer, 1991. M. S. Swanson, J. Math. Phys. 45 (2004) 585. A. Mostafazadeh, Nucl. Phys. B 640 (2002) 419. H. F. Jones, J. Phys. A : Math. Gen. 38 (2005) 1741. W.D. Heiss, Exceptional Points of Non Hermitian Operators, arXiv : quantph / 0304152v1.
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Condensed Matter Phenomena
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Chapter 11 The Relevance of Berry Phase in Quantum Physics
Pratul Bandyopadhyay The relevance of Berry phase in various quantum systems has been outlined here. Specifically its relationship with the chiral anomaly in quantum field theory, with quantum Hall effect and with quantum entanglement associated with various spin systems has been discussed.
Contents 11.1 Introduction . . . . . . . . . . . . . . . . . . 11.2 Berry Phase and Chiral Anomaly . . . . . . 11.3 Berry Phase and Quantum Hall Effect . . . 11.4 Quantum Entanglement and Berry Phase . 11.5 Concurrence and Berry Phase . . . . . . . . 11.6 Concurrence in Various Spin System . . . . 11.7 Entanglement of Two Delocalized Fermions 11.8 Hubbard Model . . . . . . . . . . . . . . . . 11.9 Conclusion . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . .
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237 238 239 240 242 243 246 247 250 250
11.1. Introduction It has been observed by Berry [1] that when a quantum system is described by a cyclic time dependent Hamiltonian then the instantaneous eigenstate acquires a geometric phase apart from the dynamical phase. This phase depends on the nontrivial topology of the parameter space spanned by the cyclic parameter. If Un t is an instantaneous eigenstate of the Hamiltonian H t then the geometric phase is given by
T
γn T i
Un t U n t dt
0
during the cycle T. In an alternative way we can write 237
(11.1)
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γn T
where A R Un R can also be written as
,
Un
A RdR
R being an adiabatic parameter. This
γn T
(11.2)
Bd s
(11.3)
where B R A. A and B are known as Berry connection and curvature respectively. Later it has been shown by other authors that adiabaticity is not an essential criterion. 11.2. Berry Phase and Chiral Anomaly In the realm of quantum field theory it has been observed that the geometric phase is associated with chiral anomaly. When a chiral current interacts with a gauge field we have chiral anomaly in the sense that the divergence of the axial vector current does not vanish. Indeed we have the relation dµ jµ5
1 8π 2
( Fµv Fw
(11.4)
where jµ5 is the axial vector current jµ5 Ψγµ γ5 Ψ, Ψ being the spinorial field. Fµν is the field strength and Fµν is the Hodge dual
F µν
1 2
µνλσ
Fλσ
(11.5)
From this we note that the chiral anomaly is related to the Pontryagin index given by
q
2µ
1 d4 x Fµν Fµν 16π 2 1 dµ jµ5 d4 x 2
(11.6)
Here q is an integer and known as Pontryagin index. µ can take the values of 12, 1, 32 and represents monopole strength. µ 12 corresponds to one magnetic flux quantum. It has been shown that when a particle
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encircles certain magnetic flux quanta in a cyclic evolution, the Berry phase is given by ei 2π µ .2 From our above result, it is observed that when a scalar particle moves around one magnetic line it attains the phase eiπ µ 12 representing a fermion. When a fermion encircles one magnetic line, this may be viewed as a scalar particle encircling two magnetic flux lines so that the phase is ei 2π . When a fermion moves about another fermion and traverses a halfcircle the phase is eiπ . This is the statistical phase which appears when two fermions are adiabatically exchanged. Thus we note that the statistical phase also may be viewed as a manifestation of Berry phase. 11.3. Berry Phase and Quantum Hall Effect The relationship between chiral anomaly and Berry phase plays a significant role in various aspects of condensed matter physics. In 2 + 1 dimension the QHE action is given by the Hopf invariant which arises from the mapping S 3 S 2 and is given by S
1 2
d3 x µνλ Fµν Aλ
(11.7)
If ρ denotes a 4 dimensional index then 1 ρµνλ Fρµ Fνµ (11.8) 2 This connects the Hopf invariant with chiral anomaly. When we consider a 2 dimensional surface as the surface of a 3 dimensional sphere with a magnetic monopole of strength µ at the center, the total number of magnetic flux lines passing through the surface is 2 µ. The Hall states attain the phase ei2πµ . The filling factor ν follows from the Dirac quantization condition n which eµ 12 n, n being an integer. Indeed we have the filling factor ν 2µ may be expressed in the form δ ρ ρµνλ Aµ Fνλ
ν
n 2mn 1
(11.9)
where 2µ which is an odd integer is expressed as 2mn 1, m n being an integer. This is the Jain classification scheme which has been derived in terms of the Berry phase factor µ [3]. The Hall conductivity is given by
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σH
νe2 h
(11.10)
Thus we note that the Barry phase attained by quantum Hall states plays a significant role in determining the integer and fractional quantum Hall conductivity.
11.4. Quantum Entanglement and Berry Phase In a recent paper [4] it has been shown that the Barry phase plays a significant role in the peculiar correlation between two distant objects in a quantum system. In fact when a fermion is viewed as a scalar particle attached to a magnetic flux line [5] entanglement of a spin system can be viewed as to be caused by the deviation of the internal magnetic flux line associated with one particular fermion in presence of the other. This effectively represents the interaction of spin with a rotating magnetic field. To be specific let us consider the entangled state of two spin 1/2 systems (Bell) Ψa
) *
b
* )
(11.11)
where a and b are complex coefficients. A rotating magnetic field with an angular velocity ω around the z-axis under an arbitrary angle θ may be taken as
B t B n θ, t
(11.12)
where n θ, t is a unit vector given by
"
Sin θ Cos wt
$
. Æ . Æ . n θ, t . Sin θ Sin wt Æ Æ % #
(11.13)
Cos θ The interaction Hamiltonian is given by
H
K B. σ 2
(11.14)
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where K gµB , µB being the Bohr magneton and g is the Lande factor. The instantaneous eigenstate of a spin operator in the direction n θ, t in the αZ -basis is given by )n ; t
Cos
θ 2
)z
Sin
θ iwt e 2
*z
*n ; t
Sin
θ 2
)z
Cos
θ iwt e 2
*z
(11.15)
For the time evolution from t 0 to t τ 2π w each eigenstate will pick up a geometric phase apart from the dynamical phase )n ; t *n ; t
0 )n t τ
0 *; t τ
eiφB θ eiν
eiφB θ eiν
)n ; t
*n ; t
0
0 (11.16)
where ΦB is the Berry phase and νI represents the dynamical phase, To study entanglement, let us consider the initial Bell state as the entanglement of the interaction Hamiltonian
Ψ t 0 a
)n *n b *n )n
(11.17)
After a complete rotation the state picks up a geometric phase (apart from the dynamical phase which we omit here).
Ψ t τ aeiφB
)n *n b
eiφB
*n )m
(11.18)
Here ΦB is the Berry phase given by the half of the solid angle swept out by the magnetic flux line [6]. QB
π 1 Cos θ
QB
π 1 Cos θ
(11.19)
θ being the deviation of the magnetic flux line from the z-axis. From the relation ΦB ei2πµ we find φB
µ
1 1 Cos θ 2
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φB
µ
1 1 Cos θ 2
(11.20)
where µ is the measure of the formation of entanglement. For up (down) spin θ 0 π corresponds to disentangled (maximally entangled) state. 11.5. Concurrence and Berry Phase For the Bell state
Ψ a ) * b * )
(11.21)
where a and b are complex coefficients, the measure of the formation of entanglement known as concurrence is given by [7] C
2 a b
(11.22)
In the present formulation a and b are implicit functions of θ. We write
" $ " $ a f θ 1 # % # % 2 b g θ
(11.23)
The angle θ here just corresponds to the deviation of up (down) spin under the influence of the other and represents the angle associated with the Berry phase ΦB given by Equation (10.19). For the maximally entangled state (MES) we have θ π as it represents the maximum deviation of a spin from the z-axis when the spin direction is reversed. For this state we have a b 12 and C 1. Again for the disentangled state θ 0 and we have C 0. These constraints imply f θ θπ g θ θπ 12 and f θ θ0 0 or g θ θ0 0
(11.24)
From these constraint equations and for the positive definite norms 0 a 1 and 0 b 1, we can parameterize
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" 1 # 2
a
$
"
% #
b
f θ
$
"
% #
Cos2 n θ4
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$ %
(11.25)
2
I θ
Sin n θ4
where n is an odd integer. It is noted that according to Equation (10.25) the relation a 2 b 2 1 is satisfied only in the case of θ π implying the MES. So to have the probability interpretation the generalized relation may be defined by incorporating the normalization factor 21 2 in Equation
a b
(10.21). The system under consideration suggests that the range of θ lies between 0 θ π. So in expression (10.25) we take n 1. This relates concurrence with the Berry phase and is given by [4]
c 2 a b Sin2 θ2
φB 1 1 Cosθ 2 2π
(11.26)
11.6. Concurrence in Various Spin System 1. Transverse Ising system The Hamiltonian for the transverse Ising system on one dimensional lattice with N sites is given by
H
M
λ σix σix 1 σiz
M
(11.27)
Here σi is the Pauli matrix at the site i, λ is the inverse strength of the magnetic field and M N21 for odd N. A family of Hamiltonian is introduced by applying a rotation of φ around the Z-axis to each spin
Hφ
1
gφ Hgφ
gφ
exp
iφazi 2
(11.28)
The Berry phase of the ground state accumulated by varying angle φ from 0 to π is described by
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φB
i
g
0
φ
y
dφ
π 1 Cos θk
(11.29)
k 0
k
M, M 1, . . . M.
The ground state G is given by
G Π
k 0
θk Cos 2
0 k 0 k
ie
i2φ
θk Sin 2
1 k *k
(11.30)
Where 0 k and 1 k are the vacuum and single fermionic excitation of the k-th mode respectively. We have Cos θk
1 λCosφk
(11.31)
1 λ2 2λCosφk
where Cos φK 2πk N . In the thermodynamic limit the Berry phase is given by π
φB
i
1
1 λ2 2λCosφ
0
At the critical point λ λc
1 λCos φ
dφ
(11.32)
1, we find φB
π 2
(11.33)
This implies that the concurrence is given by [4]
φB 0.18 (11.34) 2π This is in good agreement with the result obtained in standard method by Osborne and Nielson C 0.1946 at the critical point [8]. C
It is to be mentioned that at other points of λ, the concurrence µ depend on a function of λ which is related to the external field. Indeed, as observed by Carollo and Pachos [9], the geometric phase is the witness of the singular point. The relation between concurrence and Berry phase becomes more pronounced by the fact that near the critical point both concurrence and Berry phase follow the scaling behavior as has been observed by Ostesloh et al. [10] and Zhu [11] respectively.
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2. Heisenberg Antiferromagnetic Chain The Hamiltonian for the linear antiferromagnetic Heisenberg chain is given by H
SiX SYX
Y
Si
SjY
Z
Si
SjZ
(11.35)
i,j
i, j are nearest neighbor sites. The rotational symmetry for S implies
SiX SYX
SiY SjY
SiZ SjZ
0 state
(11.36)
For spin 12 fields SiZ SjZ 14 Cosθij as under the influence of the other spin, the spin at site deviates from the quantization axis by an angle θ. Thus implies that for the S 0 sector, the correlation
3 Cos θij (11.37) 4 The Berry phase obtained by a rotation of the spin around the Z-axis is
Si Sj
φB
π 1 Cos θ
(11.38)
Thus in the linear chain, the accumulated Berry phase may be written as φB
π 1 4 S i S j
(11.39)
i,j
The ground state energy is Eg eg N 3Nnn Γg where eg is the ground state energy per site given by eg ln2 14 , Γg is the nearest neighbor correlation Siz Sjz , Nnn is the number of nearest neighbors pair of spins. In the thermodynamic limit we have N Nnn and from the above relations we find [4] φB
π 1 4eg
(11.40)
This gives the value of the concurrence φB 1 1 4eg 2ln 2 1 0.386 (11.41) 2Π 2 This is identical with that obtained by o’conner and Wooters [12] as well as by Wang and Zanardi [13]. C
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11.7. Entanglement of Two Delocalized Fermions We consider spin entanglement of two electron states in two spacial regions A and B on a latter of the form
Ψ
N
1 C 0 Ψij Ci Cj Ψ C ij i j 2 i,j 1
(11.42)
creates an electron with spin s at site i [14]. Here Ψ Ψ where Ci,S ij ij is the amplitude of probability to find the two electron state with one having spin ) in region A and another with spine * in region B. The whole set of probabilities gives the wave function for the two electron system in the continuous limit. To study the concurrence associated with the entanglement of such a system in terms of the Berry phase acquired by the spin of one electron in the presence of the other electron we consider a rotating of the spin around the z-axis under an angle θ at each site where θ varies from 0 to π Ψ ij
ziθ Ψ ij e
(11.43)
The Berry phase acquired by the spin is given by π
i
φB
Ψ
θΨ
dθ
(11.44)
0
which on the lattice takes the form
φB
2π.2
Ψ ij Ψji
(11.45)
So the expression for the concurrence takes the form
φB 2 Ψij Ψji (11.46) 2π Ramsak et al. [1] have expressed concurrence of such a system in terms of the operators C
SA B z For the state with Stot
SA B
0 we have
Ci Ci
i A B
(11.47)
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S 2 SA B
2
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(11.48)
This can also be formulated in a more familiar form by considering the state in analogy to the Bell state
Ψij
1 Ψij Ψ ji 2
over all pairs i, j such that i A and j is given by
C
(11.49)
B. The expression for concurrence
2 2 Ψ
Ψij ij
(11.50)
i,j
which is equivalent to the expression (10.48). We note that the result is identical with expression (10.46) obtained from the relationship of Berry phase with concurrence. 11.8. Hubbard Model To consider a physical situation of our above formalism we have considered the Hubbard model. Let us consider two interacting electrons in onedimensional lattice with N α. The corresponding Hamiltonian is
H
t
i,j
C l.c. Cis js
Uij nis njs1
(11.51)
i,j,s1 ,s1
where t is the hopping parameter, U represents the on site repulsion and nis is the number of electrons at the site i with spin s. Let one electron with spin ) is initially confined in the region A and the other site spin * in the region B. The initial state is defined by two wave packets the left with momentum k and the right with momentum - q. After collision the electrons move apart with non-spin flip amplitude tkq and spin flex amplitude γkq . For sharp momentum resolution we take k q k0 . To study the entanglement of these two electrons in terms of the Berry phase acquired by their spins we take resort to the well known fact that for strong coupling and at half filling the system with the Hamiltonian reduces to the Heisenberg antiferromagnetic chain and the Hamiltonian is given by
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H
J
Six Sjx Siy Sjy Siz Sjz
with J 4t2 U. In the S 0 sector (S try of the Hamiltonian implies
Six Sjx
Siy Sjy
(11.52)
totalspin) the rotational symme-
Siz Sjz
(11.53)
If θ be the deviation of the spin at the site from the z-axis under the influence of the other spin we can write 1 Cos θ (11.54) 4 We consider collision of the two electrons initially at regions A and B. After the collision the electrons move to the final state in these two regions either with spin flip or non-spin flip configurations. The Berry phase acquired by the up (down) configuration is given by
Siz Sjz
φB φB π 1 Cos θ π 1 Cos θ
(11.55)
After the collision the initial spin position get changed so that for spin flip and non-spin flip cases we have the two phases φB
π 1 Cos θ θΠ
φB
π 1 Cos θ θ0
(11.56)
The generalized expression may be written as φB
π 1 Cos θ
(11.57)
When the spin flip and non-spin flip amplitudes coincide the concurrence is given by φB 1 1 Cos θ θ0,Π 1 ν2π 2 This is identical with the definition of concurrence C
C
2 tkq γkq
1
(11.58)
(11.59)
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for tkq γkq . This corresponds to k0 0, π. However when the spin slip and non-spin flip amplitudes do not coincide i.e. when tkq " γkq the concurrence can be derived from the measure of the angle θ in terms of the momentum k0 . This can be achieved from an analysis of the very relations in the Hubbard model and Heisenberg antiferromagnetic chain in the ground state. In the Hubbard model when no particles meet at a lattice point the many particle energy is given by
E
2t
Cos k0
(11.60)
In the Heisenberg antierromagnetic chain with the correlation given by eqn. (10.54), the energy per site to given by
E
J
3 Cos θ 4
(11.61)
In the Hubbard model the occupation number of each species of spin nis 12. Thus with J 4t2 U the energy of one particle can be related to the energy per site in the antiferromagnetic chain by the relation
t Cos k0
4t2 3 Cos θ U 4
(11.62)
Cos θ
1 Cos k0 3
(11.63)
For t U, we find
So the concurrence for different values of k0 at t terms of the Berry phase through the relation
C
1 1 Cos 2
θ 0,π
1 2
1
1 Cos k0 3
U can be obtained in
(11.64)
k0 0,π
The results are found to be in excellent agreement with those obtained by Ramsak et al. [15] using the relation C 2 tkq γkq . The significant implication of this result is that we can have a generalized relationship between the entanglement of distinguishable spins and that of delocalized electrons.
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11.9. Conclusion We have discussed here the relevance of Berry phase in various fields of quantum physics. In the realm of quantum field theory this phase is related to chiral anomaly. This helps to relate this with the Hopf invariant in a 2 1 dimensional system when the two dimensional surface is taken to be the surface of a three dimensional sphere with a monopole at the center. This implies that integer quantum Hall effect (IQHE) as well as fractional quantum Hall effect (FQHF) can be analyzed in terms of this phase. Indeed we can achieve the Jain classification scheme in FQHE from this analysis. In a quantum system this phase is found to bear the signature of the specific phenomenon of correlations between distinct objects known as entanglement. Indeed the concurrence of two-site entanglement of a spin system is found to be related to this phase acquired by the spins due to a complete rotation of the entangled state. We have evaluated the concurrence in terms of this phase for transverse Ising model as well as Heisenberg anti-ferromagnetic chain and the results are found to be in good agreement with that obtained by standard method. We have also studied the spin entanglement of two delocalized electrons in terms of the Berry phase acquired by their spins and the result is found to be in excellent agrement with that obtained by other method. This helps us to generalize the relationship between the entanglement of distinguishable spins and that of delocalized electrons. References 1. M. V. Berry : Proc. Roy. Soc. (London) A 392, 46 (1984). 2. D. Banerjee and P. Bandyopadhyay : J. Math. Phys. 33 990 (1992). 3. For a review See. B. Basu and P. Bandyopadhyay : Int. J. Mod. Phys. B 12, 2049 (1998). 4. B. Basu and P. Bandyopadhyay : Int. J. Geom. Methods in Modern Physics 4, 707 (2007). 5. P. Bandyopadhyayand and K. Hajra : J. Math. Phys. 28, 711 (1987). 6. R. A. Bertlmann, K. Durstberger, Y. Segaawa, B. C. Hersmayr : Phys. Rev. A 69, 032112 (2004). 7. W. K. Wooters : Phys. Rev. Lett. 80, 2245 (1998). 8. T. J. Osborne and M. A. Nielson : Phys. Res. A 66, 032110 (2002). 9. A. C. M. Carollo and J. K. Pachos : Phys. Rev. Lett. 95, 157203 (2005). 10. 10 A. Osterloh, L. Amico, G. Falco and R. Fazio Nature 416, 608 (2002). 11. S. L. Zhu : Phys. Rev. Lett. 96, 077206 (2006). 12. K. M. O’Connor and W. K. Wooters : Phys. Rev. A 63, 052302 (2001).
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13. X. Wang and P. Zanardi : Phys. Lett. A 39, 1 (2002). 14. B. Basu and P. Bandyopadhyay : J. Phys. A 1, 055301, (2008). 15. A. Ramsak, J. Sega and J. A. Jefferson : Phys Rev. A 74, 010304 (R) (2006).
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Chapter 12 Quantum Hamiltonian Diagonalization and Equations of Motion with Berry Phase Corrections Pierre Gosselin1 , Alain B´erard2 and Herv´e Mohrbach2 1
Institut Fourier, UMR 5582 CNRS-UJF, UFR de Math´ematiques, Universit´e Grenoble I, BP74, 38402 Saint Martin d’H`eres, Cedex, France 2 Laboratoire de Physique Mol´eculaire et des Collisions, ICPMB-FR CNRS 2843, Universit´e Paul Verlaine-Metz, 57078 Metz Cedex 3, France It has been recently found that the equations of motion of several semiclassical systems must take into account anomalous velocity terms arising from Berry phase contributions. Those terms are for instance responsible for the spin Hall effect in semiconductors or the gravitational birefringence of photons propagating in a static gravitational field. Intensive ongoing research on this subject seems to indicate that actually a broad class of quantum systems might have their dynamics affected by Berry phase terms. In this article we review the implication of a new diagonalization method for generic matrix valued Hamiltonians based on a formal expansion in power of . In this approach both the diagonal energy operator and dynamical operators which depend on Berry phase terms and thus form a noncommutative algebra, can be expanded in power series in . Focusing on the semiclassical approximation, we will see that a large class of quantum systems, ranging from relativistic Dirac particles in strong external fields to Bloch electrons in solids have their dynamics radically modified by Berry terms.
Contents 12.1 12.2 12.3
12.4
12.5
Introduction . . . . . . . . . . . . . . . . . . . . . . . Recursive Diagonalization of Quantum Hamiltonian The Semiclassical Approximation . . . . . . . . . . . 12.3.1 The semiclassical energy . . . . . . . . . . . 12.3.2 The equations of motion . . . . . . . . . . . Physical Applications . . . . . . . . . . . . . . . . . 12.4.1 Electron in a magnetic bloch band . . . . . . 12.4.2 Photon in a static gravitational field . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . 253
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12.1. Introduction Since the seminal work of Berry,1 the notion of Berry phase has found several applications in branches of quantum physics such as atomic and molecular physics, optics and gauge theories. Most studies focus on the geometric phase that a wave function acquires when a quantum mechanical system has an adiabatic evolution. Yet, the Berry phase in momentum space has recently found unexpected applications in several fields. For instance, in spintronics, such a term is responsible for a transverse dissipationless spin-current in semiconductors in the presence of electric fields.2 In optics, it has been recently found that a monopole in momentum space causes the gravitational birefringence of photons in a static gravitational fields.3 Both effects are two manifestations of the spin Hall effect which can be interpreted at the semiclassical level as due to the influence of Berry curvatures on the semiclassical equations of motion of spinning particles.4 Similarly, a new set of semiclassical equations with a Berry phase correction was proposed to account for the semiclassical dynamics of electrons in magnetic Bloch bands5 .6 In a more exotic application, intrinsic Berry phase effects in the particle dynamics of the doubly special relativity theory was recently described.7 In the above cited examples, the semiclassical equations of motion with Berry phase corrections (anomalous velocity terms) can be derived in a representation where the Hamiltonian is diagonalized at the semiclassical order. It was indeed shown that the semiclassical diagonalization results in an effective energy operator with Berry phase corrections as well as noncommutative covariant coordinates and momentum operators. These dynamical operators being corrected by Berry terms,8 this leads directly to these new Berry effects. This is another illustration of the fact that the physical content of quantum systems is most often best revealed in the representation where the Hamiltonian is diagonal. The paradigmatic example is provided by the Foldy-Wouthuysen (FW) representation of the Dirac Hamiltonian for relativistic particles interacting with an external electromagnetic field. In this representation the positive and negative energy states are separately represented and the non-relativistic Pauli-Hamiltonian is obtained.9 Actually even if several exact FW transformations have been found for some definite classes of potentials,10–12 the diagonalization of matrix valued Hamiltonian is, in general, a difficult mathematical problem re-
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quiring some approximations, essentially a perturbation expansion in weak fields. To overcome this limitation we have recently proposed a new method based on a formal expansion in powers of the Planck constant 13 which is not restricted to Dirac Hamiltonians but also applicable to a large class of quantum systems. It is worth mentioning that recently a variant of the FW transformation valid for strong fields and based also on an expansion in of the Dirac Hamiltonian was presented.14 The main advantage of the diagonalization procedure of 13 is that it embraces several different physical systems ranging from Bloch electrons in solid to Dirac particles interacting with any type of external fields (for instance in Refs. 3 and 15 electrons and photons in a static gravitational field were considered). In this paper we review the recursive diagonalization procedure of Ref. 13 from which can deduce the expressions of the semiclassical energy and of the dynamical operators. We then consider, as a physical application, an electron in a magnetic Bloch band and the spin hall effect of light in a gravitational field. 12.2. Recursive Diagonalization of Quantum Hamiltonian In this section we consider 3 a quantum mechanical system whose state space 2 is a tensor product L R V with V some internal space. In other words, the Hamiltonian of this system can be written as a matrix H0 P, R of size dim V whose elements are operators depending on a couple of canonical variables P and R, the archetype example being the Dirac Hamiltonian with V C 4 . In13 we found a diagonalization process for this matrix valued quantum Hamiltonian H0 P, R recursively as a series expansion in powers of which gives the quantum corrections to the diagonalized Hamiltonian with respect to the classical situation 0. For example, the first order correction in corresponds to the semiclassical approximation. In this approach we derived the expansion recursively in the following way. The Planck constant is formally promoted to a dynamical parameter α in order to establish a differential equation connecting the two diagonalized Hamiltonians at α and α dα. The integration of this differential equation allows then the recursive determination of the different terms in the expansion of the diagonalized Hamiltonian in powers of α. To start with, consider the diagonalization at the scale α Uα P, R H0 P, R Uα P, R εα P, R if P, R iα and similarly for α dα.
(12.1)
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Uα dα P, R H0 P, R Uα
dα P, R εα dα
P, R if P, R i α dα
Let us develop this last relation to the first order in dα, εα dα P, R Uα H0 Uα dα α Uα H0 Uα Uα H0
(12.2)
α Uα
(12.3)
After rewriting the r.h.s. of this equation in terms of Berry connections
Pl
l AR α iUα ∇Pl Uα and Aα iUα ∇Rl Uα we arrive at the following dif13 ferential equation : d εα P, R dα
α Uα
,
P, R Uα P, R , εα P, R
1 Rl l A ∇Rl εα P, R ∇Rl εα P, R AR α 2 α Pl l AP ∇ ε P, R ∇ ε P, R A Pl α Pl α α α
P R i * Pl l l l εα P, R , AR α Aα εα P, R , Aα Aα 2 R + i P εα P, R , Aα l , Aαl &Asym &∇Pl ∇Rl εα P, R' 2
+ Uα Asym &∇Pl ∇Rl H0 P, R' Uα i Xεα P, R εα P, R X
(12.4) 2
with the notation X
Asym ∇Rl ∇Pl Uα P, R Uα P, R
where the linear operation Asym8 acts on a symmetrical function in P and R in the following way : , 1 1 1 A R B P B P A R B P , A R (12.5) Asym 2 2 2 the functions A R and B P being typically monomials in R and P arising in the series expansions of the physical quantities. As shown in,13 we can separate the energy equation Eq. 12.4 in a diagonal and a non diagonal part such that we are led to the following two equations d εα P, R dα 0
P R.H.S. of Eq. 12.4
(12.6)
P R.H.S. of Eq. 12.4
(12.7)
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These two equations are supplemented by the differential unitarity condition 0
P, RUα P, RU α P, R i R i l X X Aα l , AP α 2 2 α Uα
P, R
αU α
(12.8)
The three equations Eqs. 18.1- 12.8 allow to determine recursively in powers of α the energy of the quantum system in question. Actually, the integration over α of Eq. 18.1 gives εα P, R at order n in α when knowing all quantities at order n 1. By the same token, Eqs. 12.7 and 12.8 (whose meaning is that Uα P, R is unitary at each order in α) involve α Uα P, R, and allow to recover Uα P, R at order n by integration over α. As a consequence, the diagonalization process is perfectly controlled order by order in the series expansion in α.
12.3. The Semiclassical Approximation In this section we consider the Hamiltonian diagonalization at the semiclassical level and the resulting equations of motion. Actually, the semiclassical approximation has recently found new important applications in particle and solid state physics. Notably, the equations of motion reveal a new contribution coming from the Berry curvature. This contribution, called the anomalous velocity, modifies profoundly the dynamics of the particles. For instance, the spin Hall effect of electrons and holes in semiconductors,2 as well as the new discovered optical Hall effect3,4,16,17 can be interpreted in this context. Similarly, the recent experimental discovery of the monopole in momentum can also be elegantly interpreted as the influence of the Berry curvature on the semiclassical dynamics of Bloch electrons.18,19
12.3.1. The semiclassical energy The consideration of Eq. 12.4 alone is sufficient to deduce the semiclassical diagonal Hamiltonian. Indeed, writing εα ε0 αε1 , with ε0 the diagonalized energy at the zero order, Eq. 12.4 is solved by (putting α )
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ε P, R
ε0
P, R l AP 0 ∇Pl ε0
1 Rl l A ∇Rl ε0 P, R ∇Rl ε0 P, R AR 0 2 0 Pl P, R ∇Pl ε0 P, R A0
/ i l l P ε0 P, R , AR AP 0 0 2
0 l l ε0 P, R , AP AR 0 0 (12.9) R P A0 and AP 0
R where Pwe have introduced the notations A0 P A0 . This latter expression can also be written i l l AP ε p, r ε0 p, r P ε0 p, r , AR 0 0 2 P R 2 ε0 p, r , A0 l A0 l O
(12.10)
where we have defined the projected dynamical operators (covariant coordinates and momentum operators) r R AR 0 p P AP 0
(12.11)
with AR 0
i U0 ∇P U0 ,
AP 0
i
U0 ∇R U0 ,
and AP,R ∇R ∇P U0 U0 . 0 The matrix U0 P, R is the diagonalization matrix for H0 when the operators are supposed to be commuting quantities, the diagonalized energy being ε0 P, R . When P and R do not commute, the matrix U0 P, R does not diagonalize H0 anymore. In order to get the corrections to the energy at the semiclassical order due to the noncommutativity of P and R we have to compute ε1 P, R. From the diagonal Hamiltonian, we can now derive the equations of motion for the covariant operators. 12.3.2. The equations of motion Given the Hamiltonian derived in the previous subsection, the equations of motion can now be easily derived. The evolution equations have to be considered, not for the usual position and momentum, but rather for the projected variables r and p . Actually, these latter naturally appear in our
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diagonalization process at the order. Let us remark that their components do not commute any more. Actually 2 rr 2 2 ri , rj i Θij i ∇Pi ARj ∇Pj ARi ARj , ARi 2 pp 2 2 pi , pj i Θij i ∇Ri APj ∇Rj APi APi , APj 2 pr 2 pi , rj iδij i Θij iδij i ∇Ri ARj ∇Pj APi 2 (12.12) APi , ARj the Θij being the so called Berry curvatures. Using now our Hamiltonian yields directly to general equations of motion for r, p : i i i r, P ε p, r , ARl APl ε p, r , APl ARl 9 r r, ε p, r 2 i i i p 9 p, ε p, r p, P ε p, r , ARl APl ε p, r , APl ARl 2 (12.13) The commutators can be computed through the previous commutation rules between r and p. The last term in each equation represents a contribution of “magnetization” type and has the advantage to present this general form whatever the system initially considered. In the context of Bloch electrons in a magnetic field, it gives exactly the magnetization term revealed in5 (see6 and below). For spinning particles in static gravitational fields, this term gives a coupling between the spin and the intrinsic angular momentum with magneto-torsion fields3 .15 12.4. Physical Applications 12.4.1. Electron in a magnetic bloch band This topic was first dealt with in5 in the context of wave packets dynamics. In the context of the Hamiltonian diagonalization it was considered in6 and.8 The purpose is to find the semiclassical diagonal energy operator for an electron in a periodic potential facing an electromagnetic field. To apply our formalism, consider an electron in a crystal lattice perturbed by the presence of an external electromagnetic field. As is usual, we express the total magnetic field as the sum of a constant field B and small nonuniform part .. δB R. The Schrodinger equation reads H0 eφ R Ψ R EΨ R with H0 the magnetic contribution (φ being the electric potential) which
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reads H0
P 2m
2 eA
R eδA R
V
R ,
P i∇
(12.14)
where A R and δA R are the vectors potential of the homogeneous and inhomogeneous magnetic field, respectively, and V R the periodic potential. The large constant part B is chosen such that the magnetic flux through a unit cell is a rational fraction of the flux quantum he. The advantage of such a decomposition is that for δA R 0 the magnetic translation operators T b exp iK.b, with K the generator of translation, are commuting quantities allowing to exactly diagonalize the Hamiltonian and to treat δA R as a small perturbation. The state space of the Bloch electron in the periodic zone scheme20 is spanned by the basis vectors of plane waves n, k k n with n corresponding to a band index and k vary in R3 . The state n can be seen as a canonical base vector n 0...010...0... (with 1 at the nth position) such that U k n un k with un k the periodic part (in space) of the magnetic Bloch waves5.21 In this repoperator is resentation K n, k k n, k and consequently the position R i k, implying the canonical commutation relations Ri , Kj iδij . We first perform the diagonalization of the Hamiltonian in Eq. 12.14 for δA 0 by diagonalizing simultaneously H0 and the magnetic translation operators T. The diagonalization is performed as follows: start with an arbitrary basis of eigenvectors of T. In this basis H0 can be seen as a square matrix with operators entries.H0 is diagonalized through a unitary matrix U K which should depend only on K (since U should leave K invariant, i.e., U KU K) and whose precise expression is not necessary for the derivation of the equations of motion, such that U HU E K eφ U RU , where E K is the diagonal energy matrix made of elements En K with n the band index (i.e. the diagonal representation of H0 ). Now, to add a perturbation δA R as in (6 ), that breaks the translational symmetry, we have to replace K in all expressions by ˜ K e δA R K
(12.15)
and as the flux δB on a plaquette is not a rational multiple of the flux ˜ i since quantum, we cannot diagonalize simultaneously its components K they do not commute anymore. Actually ˜ i, K ˜ j ieεijk δBk R K
(12.16)
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˜ , so that To do the semiclassical diagonalization we replace U K by U K
and AKl the non projected Berry connections are ARi iU ∇K i U ∇Rl δAk RARk . From these we can define the nth intraband position and ˜ n δB rn O ˜ eAn k ˜n K momentum operators rn RAn and k
the projection of the Berry connection on the with An Pn U ∇ U K chosen nth Band.6 It can be readily seen that the matrix elements of An can be written An k i un k ∇k un k (see also Ref. 21 for the derivation of the position operator in the diagonal representation). What is totally new here is the transformation on the momentum operator k˜n which get also a Berry connection correction. Using our general results of section II, the full Hamiltonian Eq. ˜ 12.14 can thus be diagonalized through the transformation U K i ˜ plus a projection on the chosen n-th Band as it is usual ARl , AP l U K 4
in solid state physics (the so called one band approximation) and we obtain the energy operator of the n th band as ˜ HU K ˜ Pn U K k ˜ i E K, U ∇Ki U εijk δB r U ∇Kj U
Pn E k 4 k ijk δB r i
U ∇Kj U E K, U ∇Ki U ε 4 ˜n M K ˜ .δB rn O 2 En k (12.17) ˜n are the same as En K with k ˜n replacing K. where the energy levels En k ˜ Pn ie E K ˜ , A K ˜ A K ˜ can be written The magnetization M K 2 under the usual form21 in the k, n representation ie ijk ε En En Aj nn Ak n n Minn 2 n n We mention that this magnetization (the orbital magnetic moment of Bloch electrons), has been obtained previously in the context of electron wave packets dynamics.5 From the expression of the energy Eq. 12.17 we can deduce the equations of motion (with the band index n now omitted) ˜ k ˜ k ˜ Θ k ˜ r E k
˜ k
eE er δB
r M δB r
(12.18)
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1 with Θij k 1 i Aj k 1 j Ai k 1 the Berry curwhere ri , rj iΘij k 6 vature. As explained in these equations are the same as the one derived in5 from a completely different formalism. 12.4.2. Photon in a static gravitational field We now apply our general approach to the case of a photon propagating in an arbitrary static gravitational field, where g0i 0 for i 1, 2, 3, so that ds2 g00 dx0 2 gij dxi dxj 0. As explained in3 the photon description is obtained by considering first a Dirac massless particle (massless neutrino) and then by replacing the Pauli matrices σ by the spin-1 matrices S. Therefore we start with the Dirac Hamiltonian in static gravitational field which can be written ˆ H
˜ εβγ Γβ σ γ g00 α.P 0 4
i
0β Γ αβ 4 0
(12.19)
˜ given by P˜α hiα R Pi εβγ Γβ σ γ with hiα the static orwith P i 4 the spin connection components thonormal dreibein α 1, 2, 3, Γαβ i and εαβγ σ γ 8i γ α γ β γ β γ α . The coordinate operator is again given by R i p . Note that here we consider the general case where an arbitrary static torsion of space is allowed. It is known22 that for a static ˆ gravitational field (which is the case considered here), the Hamiltonian H ˆ is Hermitian. We now want to diagonalize H through a unitary transfor˜ . Because the components of P ˜ depend both on operators P mation U P and R the diagonalization at order is performed by adapting the method ˆ detailed above to block-diagonal Hamiltonians. To do so, we first write H in a symmetrical way in P and R at first order in . This is easily achieved using the Hermiticity of the Hamiltonian which yields ˆ 1 g00 α.P ˜ .αg00 εβγ Γβ σ γ . ˜ P H 0 2 4 Using the general expression Eq. 12.10 we arrive at the following expression for the diagonal positive (we have projected on the positive energy subspace) energy representation ε˜ : ε˜ ε
λ p.Γ0 4 p
B.σ 2ε
AR p.B . ε r
(12.20)
where we have introduced a field Bγ 12 Pδ T αβδ εαβγ with T αβδ lβ αδ hδk hlα l hkβ hlβ l hkα hlα Γβδ the usual torsion for a static l h Γl
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metric (where only space indices in the summations give non zero contributions). We have also defined in Eq. 12.20 λ Γi r.p λ Γj r.p εc pi g ij g00 pj , 4 p 4 p with the γ-th component of the vector Γi as Γi,γ εβγ Γβ r and the i helicity λ p.σ . Note that the dynamical operators are now p PΣ (12.21) 2ε2 PΣ ˜ p P c2 (12.22) ∇R P 2ε2 Interestingly, this semi-classical Hamiltonian presents formally the same form as the one of a Dirac particle in a true external magnetic field8 .23 The term B.σ is responsible for the Stern-Gerlach effect, and the operator L AR p is the intrinsic angular momentum of semiclassical particles. The same contribution appears also in the context of the semiclassical behavior of Bloch electrons (spinless) in an external magnetic field65 where it corresponds to a magnetization term. Because of this analogy and since T αβδ is directly related to the torsion of space through T αβδ hδk hiα hjβ Tijk we call B a magnetotorsion field. However, this form for the energy presents the default to involve the spin rather than the helicity. Actually one can use the property λp2p σ 2 AR p to rewrite the energy as r
R c2
ε1 ε
λ p.Γ0 4 p
λg00 B.p 2ε p
(12.23)
The semi-classical Hamiltonian Eq. 12.23 contains, in addition to the energy term ε, new contributions due to the Berry connections. Indeed, Eq. 12.23 shows that the helicity couples to the gravitational field through the magnetotorsion field B which is non-zero for a space with torsion. As a consequence, a hypothetical torsion of space may be revealed through the presence of this coupling. Note that, in agrement with,25 this Hamiltonian does not contain the spin-gravity coupling term Σ.∇g00 predicted in.24 From Eqs. 12.21 and 12.22 we deduce the new (non-canonical) commutations rules i j r , r iΘij (12.24) rr i j ij p , p iΘpp (12.25) i j ij ij p , r ig iΘpr (12.26)
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where Θij ζη ζ i Aη j η i Aζ j Aζ i , Aη j where ζ, η.mean either r or p. An explicit computation shows that at leading order Σ.p pγ αβγ i j ε hα hβ 2ε4 4 Σ.p pγ Θij ∇ri pα ∇rj pβ εαβγ pp c 2ε4 4 Σ.p pγ Θij ∇ri pα hjβ εαβγ (12.27) pr c 2ε4 From the additional commutation relations between the helicity and the dynamical operators ri , λ pi , λ 0 we deduce the semiclassical equations of motion Θij rr
c
4
9Θrr 9 r 1 Θpr ∇p ε˜ p p 9
1 Θpr ∇r ε˜ 9 r Θpp
(12.28)
To complete the dynamical description of the photon notice that at the leading order the helicity λ is not changed by the unitary transformation which diagonalizes the Hamiltonian so that it can be written λ p.Σp. After a short computation one can check that the helicity is always conserved d p.Σ (12.29) 0 dt p for an arbitrary static gravitational field independently of the existence of a torsion of space. Eqs. 12.28 are the new semiclassical equations of motion for a photon in a static gravitational field. They describe the ray trajectory of light in the first approximation of geometrical optics (GO). (In GO it is common to work with dimensionless momentum operator p k01 k with k0 ω c instead of the momentum16 ). For zero Berry curvatures we obtain the well known zero order approximation of GO and photons follow the null geodesic. The velocity equation contains the by now well known anomalous contribution p 9 Θrr which is at the origin of the intrinsic spin Hall effect (or Magnus effect) of the photon in an isotropic inhomogeneous medium of refractive index n r.4,16,17,26 Indeed, this term causes an additional displacement of photons of distinct helicity in opposite directions orthogonally to the ray. Consequently, we predict gravitational birefringence since photons with distinct helicities follow different geodesics. In comparison to the usual velocity 9 r ∇p ε˜ c, the anomalous velocity term v is obviously i 1 rj g ij being proportional to the wave length λ. 1 cλ∇ small, its order v
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The momentum equation presents the dual expression 9 r Θpp of the anomalous velocity which is a kind of Lorentz force which being of order does not influence the velocity equation at order . Note that similar equar Θpp were predicted tions of motion with dual contributions p 9Θrr and 9 for the semiclassical dynamics of spinless electrons in crystals subject to small perturbations5 .6
12.5. Conclusion Some recent applications of semiclassical methods to several branches of Physics, such as spintronics or solid state physics have shown the relevance of Berry Phases contributions to the dynamics of a system. However, these progresses called for a rigorous Hamiltonian treatment that would allow for deriving naturally the role of the Berry phase. In this paper we have considered a diagonalization method for a broad class of quantum systems, including the electron in a periodic potential and the Dirac Hamiltonian in a gravitational field. Doing so, we have exhibited a general pattern for this class of systems implying the role of the Berry phases both for the position and the momentum. In such a context, the coordinates and momenta algebra are no longer commutative, and the dynamical equations for these variables directly include the influence of Berry phases through the parameters of noncommutativity (Berry curvatures) and through an abstract magnetization term. Acknowledgment It is a great pleasure for us to thank Prof. S. Ghosh for collaborations and for having given to H. M. the opportunity to give a talk at the Kolkata conference on recent developments in theoretical physics. References M. V. Berry, Proc. R. Soc. A 392 (1984) 45. S. Murakami, N. Nagaosa, S. C. Zhang, Science 301 (2003) 1348 P. Gosselin, A. B´erard, H. Mohrbach, Phys. Rev. D 75 (2007) 084035. A. B´erard, H. Mohrbach, Phys. lett. A 352 (2006) 190. M. C. Chang, Q. Niu, Phys. Rev. Lett 75 (1995) 1348; Phys. Rev. B 53 (1996) 7010; G. Sundaram, Q. Niu, Phys. Rev. B 59 (1999) 14915. 6. P. Gosselin, F. M´enas, A. B´erard, H. Mohrbach, Europhys. Lett. 76 (2006) 651.
1. 2. 3. 4. 5.
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7. P. Gosselin, A. B´erard, H. Mohrbach, S. Ghosh, Phys. Lett. B 660 (2008) 267. 8. P. Gosselin, A. B´erard and H. Mohrbach, Eur. Phys. J. B 58, 137 (2007). 9. L. L. Foldy and S. A. Wouthuysen, Phys. Rev. 78 (1950) 29. 10. E. Eriksen, Phys. Rev. 111, 1011 (1958). 11. A. G. Nikitin, J. Phys. A 31, 3297 (1998). 12. A. J. Silenko, J. Math. Phys. 44, 2952 (2003). 13. P. Gosselin, J. Hanssen and H. Mohrbach, Phys. Rev. D. in press, arXiv:condmat/0611628. 14. A. J. Silenko, arXiv: math-ph: 0710.4218. 15. P. Gosselin, A. B´erard, H. Mohrbach, Phys. Lett. A 368 (2007) 356 16. K. Y. Bliokh, Y. P. Bliokh, Phys. Lett A 333 (2004), 181; Phys. Rev. E 70 (2004) 026605; Phys. Rev. Lett. 96 (2006) 073903. 17. M. Onoda, S. Murakami, N. Nagasoa, Phys. Rev. Lett. 93 (2004) 083901. 18. Z. Fang et al., Science 302, 92 (2003). 19. A. B´erard, H. Mohrbach, Phys. Rev. D 69 (2004) 127701 20. C. Kittel, Quantum Therory of Solids, Wiley, New York,1963.. 21. E. M. Lifshitz, L. P. Pitaevskii, Statistical Physics, vol 9, Pergamon Press, 1981. 22. M. Leclerc, Class.Quant. Grav. 23 (2006) 4013. 23. K. Y. Bliokh, Eur. Lett. 72 (2005) 7. 24. Y. N. Obukhov, Phys. Rev. Lett 86 (2001) 192. 25. A. J. Silenko, O. V. Teryaev, Phys. Rev. D 71 (2005) 064016. 26. C. Duval, Z. Horvath, P. Horvathy, L. Martina, P. Stichel, Phys. Rev. Lett 96 (2006) 099701; D. Xiao, J. Shi, Q. Niu, Phys. Rev. Lett. 96 (2006) 099702.
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Chapter 13 The Hall Conductivity of Spinning Anyons
B. Basu Physics and Applied Mathematics Unit Indian Statistical Institute Kolkata-700108 Anyon excitations have been predicted and observed experimentally in Ga-As semiconductor. We explain the intrinsic spin Hall Effect in Ga-As alloys from generic anyon dynamics in the presence of external electromagnetic field. The free anyon is represented as a spinning particle with an underlying non-commutative configuration space. The interacting anyon model is identified with the Bloch electron in Ga-As alloy (in electromagnetic field) in a semiclassical framework and bulk parameters of the latter are mapped to the parameters of the anyon model, giving rise to new observational possibilities. The Berry curvature plays a major role in our analysis.
Contents 13.1 13.2
Introduction . . . . . . . . . . . . . Introductory Concepts . . . . . . . 13.2.1 Spin Hall effect . . . . . . 13.2.2 Berry phase . . . . . . . . 13.2.3 Anyons . . . . . . . . . . . 13.3 Berry Phase of Anyons . . . . . . . 13.3.1 Motivation . . . . . . . . . 13.3.2 Framework . . . . . . . . 13.3.3 Mathematical formulation 13.4 Berry Curvatures and Equations of 13.5 Physical Significance . . . . . . . . 13.5.1 Hall conductivity . . . . . 13.5.2 Spin Hall conductivity . . 13.6 Conclusion . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . .
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13.1. Introduction The semiclassical dynamics of Bloch electrons in external fields provides a theoretical framework for understanding different properties of crystals. After the discovery of the geometric phase, it is shown that there is a Berry curvature correction to the semiclassical equations of motion. There is a clear indication1 that the geometric phase2 modifies the local dynamics of Bloch electrons and thus affects the transport properties of semiconductors and metals. In recent years, there are successful applications of these ideas in the context of anomalous Hall effect in ferromagnetic semiconductors and metals3–7 as well as in spin transport8 in paramagnetic metals. The consequence of spin-orbit interactions in a crystal is that they lead to a change in the group-velocity of Bloch electrons, an anomalous velocity, that is linear in electric field. This anomalous velocity, which was proposed long ago9 is proportional to the Berry curvature of Bloch electrons in momentum space and can contribute to various linear transport coefficients. It has been demonstrated that the wave packet formalism1,10 capture the physics connected with adiabatic motion and the Berry phase, in particular the Berry curvature correction to the semiclassical equations of motion. The Berry phase appears naturally as part of the wave packet distribution. The Berry curvature correction to the wave packet equations of motion is believed to play an important role in the anomalous Hall effect and spin Hall effect. In the field of spintronics11 understanding the dynamics of spin current is extremely important. The prediction12 and observation13 of an intrinsic spin Hall Effect has evoked a lot of interest since the dissipationless spin Hall current can be an efficient means of injecting spin current in (Ga-As) semiconductors. Furthermore the spin Hall conductivity can be quantized14 and the resulting quantum spin Hall liquid will have exotic features such as fractional statistics indicating the presence of anyons 15 as quasi-particles. In fact, signatures of anyon in Ga-As heterostructures have been reported.16 This discussion poses the motivation of the present talk. We demonstrate, from very general considerations, that anyon dynamics in the presence of external electromagnetic field naturally yields the abovementioned anomalous velocity. The Berry curvature plays a major role in the analysis. We have organized the paper in the following way. In the next section, we briefly sketch the introductory concepts of the spin Hall effect, geometric phase or the Berry phase, and anyons. In section III, we set up our mathematical modeling of anyons within noncommutative framework. to
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construct the derive the Berry phase of anyons. The next section (section IV) is devoted to discuss the semiclassical equations of motion and derive the conductivity of anyons. In section V, we have discussed the physical significance of our formulation. The final section VI gives the conclusion and discussion of our work. 13.2. Introductory Concepts 13.2.1. Spin Hall effect Charged carriers subjected to a magnetic field can be deflected and produce a Hall voltage across a sample.17 Confining the charged carriers to move in two dimensions can give rise to the quantum Hall effect.18 However, charge carriers also possess spin, and recent theoretical work indicates that spins can be affected by a voltage bias and give rise to a spin Hall effect (SHE) for carriers confined to two dimensions. The spin hall effect was predicted long ago by the theoretical physicists,19 but the recent experimental support13 has revived a lot of theoretical research papers in this area. SHE causes spin up and spin down electrons to build up on opposite sides of a sample in the presence of an electric field. The existence of the spin Hall effect shows it is possible to direct spins depending on their orientation within conventional semiconductor circuits in the absence of a magnetic field. The ability to manoeuvre electron spins with an electric field rather than a magnetic field could prove useful for making spintronic devices that manipulate spin rather than charge13 13.2.2. Berry phase In 1984, Prof. Michael Berry showed that a quantum system when adiabatically transported round a closed circuit C in the space of external parameters acquires, besides the familiar dynamical phase, a non integrable phase depending on the geometry of the circuit C. We can express the 1-form Berry phase as
γn C
An R dR
(13.1)
Since its discovery, Berry’s geometric phase has applications in the field of atomic and molecular physics, optics, spin in a magnetic field, AharonovBohm effect and so on. The Berry phase is also discussed in the context of quantum Hall effect and fractional quantum Hall efeect.20
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13.2.3. Anyons In space of three or more dimensions, particles are restricted to being fermions or bosons, according to their statistical behavior. Fermions respect the so-called Fermi-Dirac statistics while bosons respect the BoseEinstein statistics. In the language of quantum physics this is formulated as the behavior of multiparticle states under the exchange of particles. This is in particular for a two-particle state (in Dirac notation): ψ1 ψ2 ψ2 ψ1
(13.2)
(where the first entry in ... is the state of particle 1 and the second entry is the state of particle 2. So for example the left hand side is read as particle 1 is in state ψ1 and particle 2 in state ψ2 ). Here the “+” corresponds to both particles being bosons and the “-” to both particles being fermions (composite states of fermions and bosons are not possible). In two-dimensional systems, however, quasiparticles can be observed which obey statistics ranging continuously between Fermi-Dirac and BoseEinstein statistics, as was first shown by J.M. Leinaas and J.Myrheim.21 In our above example of two particles this looks as: ψ1 ψ2
eiθ ψ2 ψ1
(13.3)
Here, i is the imaginary unit from the calculus of complex numbers and θ is a real number. So in the case θ π we recover the Fermi-Dirac statistics (minus sign) and in the case θ 2π the Bose-Einstein statistics (plus sign). In between we have something different. Later the term anyon was coined16 to describe such particles, since they can have any phase when particles are interchanged. 13.3. Berry Phase of Anyons 13.3.1. Motivation The study of SHE attracted a great deal of attention. The SHE can be either extrinsic (ESHE) or intrinsic (ISHE). The ISHE is believed to be due to spin-orbit interaction. The prediction12 and observation13 of an intrinsic spin Hall Effect has evoked a lot of interest since the dissipationless Spin Hall current can be an efficient means of injecting spin current in (Ga-As) semiconductors. Furthermore, the spin Hall conductivity can be quantized14 and the resulting quantum spin Hall liquid will have exotic features such as fractional statistics indicating the presence of anyons 15 as
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quasi-particles. In fact, signatures of anyon in Ga-As heterostructures have been reported.16 This discussion poses the motivation of the present talk. We demonstrate, from very general considerations, that anyon dynamics in the presence of external electromagnetic field naturally yields the abovementioned anomalous velocity. The Berry curvature plays a major role in the analysis. We consider (free relativistic) anyons as 2 1-dimensional spinning particles and study their dynamics in the presence of external electromagnetic field.22–24 Their behavior is quite different from that of a point charge due to the inherent spin-orbit coupling in the former. We interpret this system as an effective model of semi-classical dynamics1 of anyon excitations in the Ga-As alloy. We demonstrate the clear analogy between our results and physical properties of Ga-As alloy such as anomalous Hall conductivity. In a quantitative way we also show that new results, such as value of anyon spin and its gyromagnetic ratio in a solid, can be predicted. The key role is played here by the non-commutative (NC) geometry25 of the anyon configuration space (or more generally phase space)22–24 and very interestingly Berry curvature emerges here through NC geometry. Let us dwell on this point a little longer. In general, the (multi-component) vectorial structure of the wavefunction of the effective excitation such as Bloch electron or photon, associated with the particle spin, plays a vital role in all the Berry curvature studies.10 Anyons possess arbitrary spin15 and, conforming to the above idea, this makes them a prime candidate in Berry phase study. Indeed, an analogue of the Dirac equation, to describe a free relativistic anyon, has been formulated26 that requires an infinite component wavefunction. However, instead of exploiting the multicomponent anyon wavefunction of,26 we will employ the spinning particle model.22–24 In this model, the anyon wavefunction is a single component scalar and the arbitrary spin is induced by the underlying NC configuration spacetime of anyon. The NC parameter appears as the anyon spin. The non-Abelian nature of the U 1 gauge theory in NC spacetime has already been noticed.25 We find its echo in the Condensed Matter scenario where the Berry gauge field assumes a non-Abelian character. Lastly, we note that the Berry curvature in mixed position-momentum space, (first highlighted in27 ), plays an important role in determining the anomalous velocity.
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13.3.2. Framework We start by studying the dynamics of anyons in an external constant electromagnetic field E1 , E2 , B . We will restrict ourselves to the lowest nontrivial order in the electromagnetic coupling e and consider a generalized modanyon model24 with arbitrary gyromagnetic ratio g. In these anyon 22–24 2 the spin tensor Sµν is not independent, Sµν s µνλ pλ p where els s and pµ are the arbitrary spin parameter and momentum respectively. Hence spin operators can always be replaced by momentum operators and spin effects, e.g. in equations of motion, derived using the NC phase space algebra), are identified through the parameter s. This phenomenon is at the heart of the Berry phase interpretation in Born-Oppenheimer scenario: the dynamics of the “slow” momentum variable is modified by spin, the “fast” variable. There is a natural and consistent relation between the parameters of our model (e.g. spin, mass and gyromagnetic ratio) with those of the bulk system (e.g. magnetic length, filling fraction).28 The interesting connection between NC spacetime and Berry curvature effect was demonstrated recently in,29 in the context of momentum space singularity in Anomalous Hall effect.3 The anyon phase space variables rµ , pµ are the covariant physical degrees of freedom. But, as mentioned above, they obey an NC algebra that reduces to the canonical phase space Poisson brackets for s 0. Because of this rµ , pµ can not be used directly in the Einstein-Brillouin-Keller quantization scheme1 or in the identification of the Berry potentials.10 However, one can “solve” the NC algebra in terms of a canonical (Darboux) set of variables Rµ , Pν igµν , Rµ , Rν Pµ , Pν 0. The set Rµ , Pµ can be used in the Einstein-Brillouin-Keller framework. From the mapping between rµ , pµ and Rµ , Pµ the all important Berry potentials Apµ , Arµ can be simply read off.10 However, to compare with experiments, we have to finally reexpress the results in terms of the physical degrees of freedom rµ , pµ . Explicitly, the Darboux transformation, Berry potential and 10 curvature components Ωrp µν etc. are defined as in, Rµ
p
rµ Aµ
, Pµ
pµ Arµ ; Ωrp µν
p
r
r
p
pµ , Aν Aµ , rν Aµ , Aν .
(13.4)
The other curvature components follow in an obvious way. Notice that the NC phase space requires a more general definition of the curvature Ωrp µν than 10 r p the one prescribed in, apart from the non-abelian A , A commutator term. The Hamiltonian equations of motion are, p µ
ipµ , G
; rµ
irµ , G,
where G is generator of (relativistic) time evolution.
(13.5)
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The Einstein-Brillouin-Keller quantization condition is straightforward in terms of canonical Rµ , Pµ coordinates:
~ P~ .dR
j
ν , 4
(13.6)
where j is an integer and ν the Maslov index. In terms of rµ , pµ , the physical variables, (13.6) read,
~ P~ .dR
p~
~ r .d ~r A ~p A
p ~.d~r
~ r .d~r A
where geometric part of ϕ Berry phase.
~ r .d~r A
~ p .d~ A p,
(13.7)
~ p .d~ A p is generally termed as the
13.3.3. Mathematical formulation After discussing the general setup we now turn to the particular case at hand. The anyon dynamics is governed by the following generator G24 and O e NC phase space brackets:22–24
G
1 2 p m2 2m
ge Sµν F µν 2
1 2 p 2m
ges p~ p
m2
~ E
Bp0
,
(13.8)
rµ , rν
ifµν ie f F f
igµν i F f
; p µ , rν
; pµ , pν
ieFµν , (13.9) 3 gii sµνσ pσ p2 2 , the metric is g00 1 and Fi0 where fµν Ei , Fij ij B. With this algebra (16.3), the Lorentz generator Jµ that transforms rµ , pµ correctly, contains a spin-part,22,23 µν
µν
µνλ rν pλ
Jµ
(13.10)
spµ p,
and is structurally very similar to the angular momentum defined by Murakami et al. in.3 The canonical coordinates Rµ , Pµ to O e are computed in terms of the physical rµ , pµ variables, by exploiting the relations derived in22 and we obtain, e Pµ p µ pµ Arµ , Fµν rν sαν p 2
Rµ
rµ sαµ p˜
es Fρν rρ sαρ p 2
where
αµ p
µνρ pν η ρ λ, λ
αµ pν
p2
sαρ
αν pµ
p2 p0 , ηµ
rµ Apµ (13.11)
1, 0, 0 ,
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p˜µ
e pµ Fµν rν 2
ν
sα p
and the Berry potentials are introduced. Explicit forms of the potentials, to O e, are, Ar0
e s E.r ij Ei pj , Ari 2 λ Ap0
0 , Api
e r0 Ei B ij rj 2
s pi , λ
s esB 1 ij pj . λ 2λ
(13.12)
(13.13)
The Berry phase contribution, the intrinsic term (in the non-relativistic limit), is given by, ϕ e0
s p e 0d A p 2 m
area
of p orbit
(13.14)
13.4. Berry Curvatures and Equations of Motion It is noted that the Berry curvatures induce the NC brackets (16.3): es sp0 ij pi Ej pj Ei p0 B ij iΩpp ij ; p3 p3 pi , pj ieB ij iΩrr (13.15) ij , es pi , rj iδij i
jk Ei pk Bp0 δij iδij iΩrp ij , etc. (13.16) p3 ri , rj
i
It should be mentioned that external Fµν are also included in our definition of the curvature Ωµν . e 0 and es-terms yield the purely intrinsic part and spin-dependent part respectively in the curvature. Clearly, even to O e, the commutator terms Aµ , Aν are non-zero which shows the non-Abelian nature of the curvature. In our formalism, this is due to the underlying NC geometry (16.3,16.7). We derive the equations of motion (see also24 ) by using (13.5,16.4,16.7): ri
pi g es pi , pi 1 ij Ej p0 B p E m 2 mp p2
e m
p0 Ei B ij pj .
(13.17) Notice that in the anomalous (or non-canonical) part of the velocity equation for ri , the g-term comes from the Hamiltonian and the rest is contributed by the coordinate-momentum mixed curvature Ωrp whereas the canonical Lorentz force equation for p i is induced by Ωrr .
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13.5. Physical Significance To discuss the significance and application of our results, we note that in the Berry phase contribution, the intrinsic term (in the non-relativistic limit), ϕ e0
s p e 0d A p 2 m
area
of p orbit
(13.18)
will modify the energy spectra1 and density of states30 of the excitation. Indeed, there are further spin-orbit (es) contributions in ϕ as well, that can be generated from (16.5,16.6). 13.5.1. Hall conductivity We can now find the conductivity in this case. From the equations of motion, in the non-relativistic limit, we get, ri
1
1
g es pi p E 3 2 m m
1
g s
ij p j 2 m2
(13.19)
where the last term is the anomalous velocity component. On the other hand, coming to the Hall effect considerations, we rewrite the velocity equation in (13.19) in the form, ri
g es
ij Ej , 2 m2 (13.20) where the O 1m4 term is dropped. The effective mass is now m and the last term signifies Hall motion since it induces a velocity, transverse to the Ex , 0 we find Electric field. For E 1
1
g esB pi 2 m2 m
x
1
px , y m
g es
ij Ej 2 m2
py m
1
pi m
1
g es Ex . 2 m2
(13.21)
Clearly the Hall conductivity is given by28 σxy
jy Ex
ey Ex
1
g se2 2 m2
e 2 ν.
(13.22)
Here, e can be considered as the effective (or fractional) charge. The parameter ν is the intrinsic Berry phase (13.18) and as expected it also appears in the intrinsic NC coordinate algebra: x1 , x2 ism2 iν and following28 it can be identified as the “magnetic length”. Hence, as we set out to demonstrate, the Hall conductivity σxy is given by the Berry curvature.
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13.5.2. Spin Hall conductivity We can also estimate the spin Hall conductivity as, s σxy
Jy E x
p~ s y Ex m
1
g p ~ e s ν. 2 m
(13.23)
~ appears as the Fermi momentum (see Murakami et al. in12 ). Typically, p From the experimental observations of fractional charge and Hall conductivity, it is possible to obtain values of s and g which are anyon parameters. This is a new observation.31
13.6. Conclusion Presence of fractional statistics has been predicted for quantum spin Hall liquid and anyon excitations have been observed experimentally, both in Ga-As alloys. It seems natural that the effects should be explained from anyon dynamics point of view. In the present work we have shown that the anomalous velocity of Bloch electron in a semiclassical analysis, (resulting in spin Hall effect), emerges naturally when equations of motion for anyons are studied in external electromagnetic field. A non-commutative phase space structure governs the anyon dynamics. In both the frameworks Berry curvature plays a key role. Possibility of measuring the anyon spin and gyromagnetic ratio from experimental observations is a new prediction of our scheme. Acknowledgement The author highly appreciates the discussion with her co-authors Subir Ghosh and S. Dhar. References 1. M.C.Chang and Q.Niu, Phys.Rev.Lett. 75, 1348(1995), G.Sundaram and Q.Niu, Phys.Rev.B 59 14915 (1999); A.Bohm et.al., The Geometric Phase in Quantum Systems, Springer, 2003. 2. M.V.Berry, Proc.Roy.Soc. London A392 45 (1984); A.Shapere and F.Wilczek, Geometric Phases in Physics, World Scientific, 1989. 3. T.Jungwirth, Q.Niu and A.H.Macdonald, Phys.Rev.Lett. 88 207208(2002); 4. M.Onoda and N.Nagaosa, Phys.Rev.Lett. 92 037204 (2004). 5. Y. Yao et. al. Phys. Rev. Lett. 90 206601 (2003). 6. F. D. M. Haldane, Phys. Rev. Lett.93 206602 (2004).
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13. 14. 15. 16.
17. 18. 19. 20. 21. 22. 23. 24. 25.
26. 27. 28. 29. 30.
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B. Basu and P. Bandyopadhyay, Phys. Scripta 75 448 (2007) J. Sinova et. al. Phys. Rev. Lett. 92 126603 (2004). R.Karplus and J.M.Luttinger, Phys.Rev. 95 1154(1954). For a review, see K.Yu.Bliokh and Yu.P.Bliokh, Ann.Phys.(N.Y.)319 13 (2005). S.A.Wolf et.al., Science 294 1488 (2001). S.Murakami, N.Nagaosa and S.-C.Zhang, Science 301 1348 (2003); J.Sinova et.al., Phys.Rev.Lett.92 126603 (2004). For a review, see S.Murakami, condmat/0504353. Y.K.Kato et.al., Science 306 (2004) ; J.Wunderlich et.al., Phys.Rev.Lett.94 047204 (2005). B.A.Bernevig and S.-C.Zhang, cond-mat/0504147. See for example F.Wilczek, Fractional statistics and anyon superconductivity, World Scientific, Singapore, 1990. J. Ihm, Phys. Rev. Lett. 67 251 (1991); F. E. Camino, W. Zhou and V. J. Goldman, Phys. Rev. Lett. 95 246802 (2005); Eun-Ah Kim, Phys. Rev. Lett.97, 216404 (2006); Phys. Rev. Lett. 95, 176402 (2005) and condmat/0604325; Eun-Ah Kim, M. J. Lawler, S. Vishveshwara and E. Fradkin, Phys. Rev. B 74, 155324 (2006); Phys. Rev. Lett. 95, 176402 (2005). E. Hall 2 (1879) 287 K.von Klitzing, G. Dorda and M. Peper 45(6) (1980) 494 M. I. D’yakonov, V. I. Perel, JETP Lett. 13, 467 (1971); M. I. D’yakonov, V. I. Perel, Phys. Lett. A 35, 459 (1971) B. Basu and P. Banydyodhyay, Int. J. Mod. Phys. B 49(1998) 1249 J.M.Leinaas, and J.Myrheim, Nuovo Cimento B37 (1977), 1 C.Chou, V.P.Nair and A.P.Polychronakos, Phys.Lett. B304 105 (1993). S.Ghosh, Phys.Lett.B338 235 (1994); Erratum-ibid. B347 468 (1995); Phys.Rev.D51 5827 (1995); Erratum-ibid. D52 (1995) 4762. C. Duval, P.A.Horvathy, Phys.Lett.B594 402 (2004). N.Seiberg and E.Witten, JHEP 9909 032 (1999). For reviews see for example M.R.Douglas and N.A.Nekrasov, Rev.Mod.Phys.73 977 (2001); R.J.Szabo, Quantum Field Theory on Noncommutative Spaces, hep-th/0109162. M.S.Plyushchay, Phys.Lett. B248 107 (1990); R.Jackiw and V.P.Nair, Phys.Rev D43 1933 (1991). P.Zhang and Q.Niu, cond-mat/0406436. Z.F.Ezawa, Quantum Hall Effects Field Theoretical Approach and Related Topics, (World Scientific) (2000). A.Berard and H.Mohrbach. Phys.Rev. D69 95 127701(2004). Apart from the Anomalous and Spin Hall effects,3,12 a number of phenomena can be studied through Berry curvature effect in a unified way, such as, the modified density of states of electrons in Anomalous Hall Effect (D.Xiao, J.Shi and Q.Niu, Phys.Rev.Lett.95 137204 (2005); C. Duval et.al., Phys.Rev.Lett. 96 099701 (2006), existence of magnetic monopole (in momentum space) (M.Onoda and N.Nagaosa, J.Phys.Soc.Jpn. 71 19(2002); Z.Fang et al., Science 302 92(2003); , Hall Effect of light (V.S.Liberman and B.Ya.Zel’dovich, Phys.Rev.A46 5199 (1992);
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K.Yu. Bliokh and Yu.P.Bliokh, Phys.Lett. A 333 181 (2004); M.Onoda, M.Murakami and N.Nagaosa, Phys.Rev.Lett.93 083901 (2004); C.Duval, Z.Horvath, P.A.Horvathy, Phys.Rev.D74 021701(R) (2006); A.Berard and H.Mohrbach, Phys.Lett.A352190 (2006).), photon-gravitation interactions (P.Gosselin, A.Berard and H.Mohrbach, hep-th/0603227.) etc. 31. S. Dhar, B. Basu and Subir Ghosh, Phys. Lett A 341, 406 (2007)
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Chapter 14 Quantum Annealing, Spin Glasses, Optimization and Analog Quantum Computation Arnab Das1 and Bikas K. Chakrabarti2 1
The Abdus Salam International Centre for Theoretical Physics, Starda Costiera 11, 34014 Trieste, Italy 2
Theoretical Condensed Matter Physics Division and Centre for Applied Mathematics and Computational Science, Saha Institute of Nuclear Physics, 1/AF, Bidhannagar, Kolkata-700064, India and Economic Research Unit, Indian Statistical Institute, 203 B.T. Road, Kolkata-700108, India We review here the recent success in annealing or optimizing the cost functions of complex systems utilizing quantum fluctuations rather than the classical (thermal) fluctuations. These are introduced through studies of mapping of such computationally hard problems to classical spin glass problems, quantum spin glasses and consequent annealing or analog quantum computation.
Contents 14.1 14.2
14.3
14.4
14.5
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . Classical Spin Glasses . . . . . . . . . . . . . . . . . . . . . 14.2.1 Spin glass ordering and Ergodicity breaking . . . . 14.2.2 Replica Theory and RSB in the spin glass models . Optimizations, Thermal Annealing and Spin Glasses . . . . 14.3.1 Combinatorial Optimization Problems . . . . . . . . 14.3.2 Simulated Thermal Annealing . . . . . . . . . . . . 14.3.3 Statistical Mechanics of Optimization Problems Heuristics . . . . . . . . . . . . . . . . . . . . . . . . 14.3.4 Traveling Salesman Problem . . . . . . . . . . . . . Quantum Spin Glasses . . . . . . . . . . . . . . . . . . . . . 14.4.1 Phase Diagram . . . . . . . . . . . . . . . . . . . . . 14.4.2 Replica Symmetry in Quantum Spin Glasses . . . . Optimization Using Quantum Annealing . . . . . . . . . . . 14.5.1 Quantum Monte Carlo Annealing . . . . . . . . . . 279
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
280 282 282 283 286 286 288 290 290 293 294 296 299 302
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14.5.2 Quantum Annealing Using Real-time Adiabatic Evolution . . . 14.5.3 Quantum Adiabatic Search for a Hole in a Golf-Course . . . . . 14.5.4 Experimental Realization of Quantum Annealing . . . . . . . . 14.6 Non-stationary Quantum Annealing of a Kinetically Constrained System 14.7 Quantum Quenching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.8 Convergence of Quantum Annealing Algorithms . . . . . . . . . . . . . . 14.9 Summary and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1 Suzuki-Trotter Formalism . . . . . . . . . . . . . . . . . . . . . . A.2 Quantum quenching of a long range TIM . . . . . . . . . . . . .
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14.1. Introduction Quantum mechanical tunneling through classically localized states in glasses has opened up a new paradigm for solving hard optimization problems through adiabatic tuning of quantum fluctuations. We introduce the idea in details through successive steps.17,19 a) Physics of classical spin glasses have already contributed enormously to our knowledge of the landscape structure of the energy or the thermodynamic potentials and of the unusually slow (glassy) dynamics of many-body systems in presence of frustration and disorders. Mapping of computationally hard problems, like the traveling salesman problems etc., to classical spin glass models also helped understanding their complexity. b) The ground (and some low-lying) state structures of frustrated random systems in presence of quantum fluctuations have also been studied in context of quantum spin glasses. It has been shown that because of the possibility of tunneling through infinitely high but narrow barriers in the classical energy landscape, quantum fluctuations can in fact restore replica symmetry in the systems with broken replica symmetry in the classical limit. This indicates that quantum dynamics can be ‘more ergodic’ (in contrast to classical or thermal ones) and explore the energy landscape much better. Nature of these quantum phase transitions in such systems are also extensively studied. c) The most natural connection between the paradigm of classical spin glasses and hard optimization problems comes through a widely used and well established optimization technique, namely, the simulated annealing. This technique provides a very general framework for constructing heuristics for finding approximate solutions of hard optimization problem employing the strategy of reaching the ground state by slow cooling. The possibility of quantum tunneling through classically impenetrable barriers, as indicated from the studies of quantum spin glasses, naturally suggests an
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elegant and often more effective alternative to simulated annealing. This novel concept, known as quantum annealing, employs quantum fluctuations to anneal a glassy system to its ground state instead of thermal fluctuations (see Fig. 14.1).
Cost/ Energy
Thermal Jump
Quantum C
Tunneling
C’
Configuration
Fig. 14.1. While optimizing the cost function of a computationally hard problem (like the ground state energy of a spin glass or the minimum travel distance for a traveling salesman problem), one has to get out of a shallower local minimum like the configuration C (spin configuration or travel route), to reach a deeper minimum C . This requires jumps or tunneling like fluctuations in the dynamics. Classically one has to jump over the energy/cost barrier separating them, while quantum mechanically one can tunnel through the same. If the barrier is high enough, thermal jump becomes very difficult. However, if the barrier is narrow enough, quantum tunneling often becomes quite easy.
In QA, one introduces a (non-commuting) quantum mechanical tunneling field term in the classical Hamiltonian (mapped cost function) and tunes that eventually to zero. This effectively tunes the Planck’s constant to zero to reach the classically optimized (minimum cost) state. This tuning, when done completely adiabatically, assures reaching the ground state of the classical glass at the end, provided there is no crossing of energy levels with the ground state in the course of evolution, and one has started with the ground state of the initial Hamiltonian. Initially the tunneling field being much higher compared to the interaction term, the ground state (a uniform superposition of all classical configurations) is trivially realizable. Experiment shows that relaxations are often much faster in course of
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such quantum annealing, compared to that during thermal annealing. Simulations also clearly demonstrate that quantum annealing can often reach the ground state of a complex glassy system much faster than its thermal counter part. Thus quantum annealing paves the realization of analog quantum computation, which is an independent and powerful alternative to digital quantum computation, where discrete unitary transformations are implemented through quantum logic gates. 14.2. Classical Spin Glasses 14.2.1. Spin glass ordering and Ergodicity breaking Spin glasses are random magnetic systems which show random freezing of the spins or magnetic moments below a certain noise level (here temperature Tc ). The main features essential for such freezing are the presence of quenched randomness and frustration (which means that there are such competing interactions that all of them could not be satisfied by any spin configuration) in the spin-spin interactions. Presence of random ferromagnetic and antiferromagnetic bonds (interactions) in a spin system can produce such a situation.9,56 The interaction energy of such a system may be represented by a Hamiltonian of the form H
N
Jij Si Sj ,
(14.1)
i j
where Si s denote spins and Jij s the interactions between them. These Jij s here are quenched variables which vary randomly both in sign and magnitude following some distribution ρ Jij . The typical distributions are (zero mean) Gaussian continuous distribution of positive and negative Jij values 2 N Jij ρ Jij A exp , (14.2) 2J 2 and binary ρ Jij pδ Jij
J
1 pδ Jij
J ,
(14.3)
with probability p of having a J bond, and 1 p of having a J bond. Two well studied models are the SK model (introduced in70 ) and the EA model (introduced in23 ). In SK model the interactions are infinite-ranged, while in EA, the interactions are between the nearest neighbors only. For
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both of them, however, ρ Jij is Gaussian; A N 2πJ 2 12 for normalization in SK model. The freezing is characterized by some non-zero value of the thermal average of the magnetization (below Tc ) at each site (local ordering). However, since the interactions are random and competing, the spatial average of single-site magnetization (below Tc ) is zero. Above Tc , both spatial and temporal average of the single-site magnetization of course vanishes. A relevant order parameter for this freezing is therefore 1 Si 2 , N i N
q
(14.4)
(overhead bar denoting the average over disorders and ... denotes the thermal average). Here q " 0 for T Tc , while q 0 for T Tc . As will be seen in the following, the existence of a unique order parameter q will indicate ergodicity and replica symmetry (RS). In the spin glass phase (T Tc ), the whole free energy landscape gets divided (cf. Fig. 14.1) into many valleys (local minima of free energy) separated by free energy barriers, which often become infinitely high in the thermodynamic limit (N ; see.9,56 The valleys then get completely separated from each other, so that if the system initially be within one valley, then it remains confined there for infinitely long time and thus becomes non-ergodic. Hence the observable free energy (and hence local magnetization and all other physical quantities) for such a trapped system is given by the averages over the configurations within that valley only, instead of the one over the whole configuration space. The valleys are thermodynamically stable and are termed as pure states. Confinement in such a random symmetry broken pure state allows the spins to explore only a very restricted (and correlated) part of the configuration space and thus make them “freeze” to have a magnetization that characterizes the state (valley) locally. Consequently, as the ergodicity is lost, RS is broken (RSB) and a distribution P q of order parameter emerges corresponding to the distribution of these thermodynamically separated frozen states. 14.2.2. Replica Theory and RSB in the spin glass models Extensive Monte Carlo studies, together with the analytical solutions for the mean field of SK and EA models, have revealed the nature of spin glass transition (see, e.g.,.9,56 In order to find the thermodynamic behavior of such a system with quenched disorder, one has to calculate free
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energy of a given realization (i.e., for a given set of Jij s) of a large sample. This means, one has to calculate the sample-specific partition function Z Jij T r&exp HT ' and the free energy F Jij T log Z Jij . Finally, one has to average (over disorders) the corresponding free energy: 3N F ij dJij ρ Jij F Jij . Employing the replica trick (log Z limn 0 Z n 1n) and averaging over the Gaussian distribution ρ Jij (14.2) in SK model, one can express (see9 ) the effective partition function as ( ) n n N 4 J2 2 Zn dqµν exp 2 q qµν Siµ Siν , (14.5) T µν i µν µν where n 0. Here, the matrix elements qµν µ " ν can be given a straightforward physical interpretation in terms of their equilibrium values as follows.22 Equilibrium condition is given by δZn δqµν 0, which leads to qµν
N 1 Siµ Siν ; N i
µ " ν.
(14.6)
In order to minimize the corresponding free energy function f q limn 0 Zn n (using saddle point method, say) one needs to know the structure of the matrix q. Since the µ-th and the ν-th replicas represent the same interaction Hamiltonian, an intuitively reasonable ansatz would be to put qµν q for all µ " ν. This is known as the Replica Symmetric (RS) ansatz and the parameter q is nothing but the EA order parameter. But for the SK spin glass, it turns out that the ansatz fails to give even a physically correct result, not to speak of its instability (detδ 2 f qδq2 0) in the whole low temperature T Tc region. The result in fact leads to negative entropy at sufficiently low temperature (S 12π at T 0).9,56 This indicates that there is some serious flaw in the RS assumption. One then assumes different ansatz to incorporate more parameters to capture the structure of the matrix q (Replica Symmetry Breaking or RSB). An elegant scheme due to Parisi;58 see also49 ) suggests that the ergodicity broken state in a random system cannot be characterized by a single value of the overlap order parameter qµν , rather, one should have a distribution function T
P q lim
1
0 n n 1 µν δ qµν q.
n
(14.7)
and the resultant free energy should be minimized functionally with respect to P q .
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This clearly shows that replica symmetry breaking always implies breaking of ergodicity. It is not clear yet that whether in real systems there is a true breaking of ergodicity, in the sense, that the lifetime of such symmetry broken states really diverge in N limit. It is a very difficult issue to address experimentally, since the time scales involved are far larger than that of any conceivable real life experiment, if not really divergent. The RSB results in if the barrier heights diverges with N . On the other hand, for very high but finite barriers would give rise to metastable states with very large but finite life time, and there would be no real symmetry breaking (and hence no true thermodynamic transition). Theoretically, explicit RSB in short range EA spin glasses could not be shown analytically yet. However numerical simulations indicate that EA model has a true spin glass phase for dimensionality 3 and above. Continuing the previous discussion on the free energy landscape of such spin glasses, one can proceed as follows. As the temperature is lowered further, each valley breaks up into further valleys, as a continuous sequence of symmetry breaking occurs as temperature is lowered continuously from Tc .22 Each pure state is characterized by an independent order parameter; one may denote the order parameter for the valley α by qα . The state within one valley contains no information whatsoever about the state in the other valleys, and hence about the structure of the free energy landscape as a whole. For an alternative picture of the RSB or of ergodicity breaking, let us imagine that two identical replicas (having exactly the same set Jij s) of a spin glass sample are allowed to relax thermally below Tc , starting from two different random (paramagnetic) initial states. Then these two replicas (labeled by α and β, say) will settle in two different valleys, each characterized by different values of the order parameter and that correspond to an overlap parameters qαβ having a sample-specific probability distribution function
eFα Fβ T δ q qαβ . PJ q α,β
Here J denotes a particular sample with a given realization of quenched random interactions (Jij s) between the spins. Assuming self-averaging behavior, PJ q may be replaced by its average over disorders in thermodynamic limit, i.e., by 4 dJij ρ Jij PJ q , P q
i j
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where ρ Jij is the distribution function from which Jij s are drawn. P q gives the probability distribution for the two pure states to have an overlap q, assuming that the probability of reaching any pure state α starting from a random (high temperature) state is proportional to the thermodynamic weight exp & Fα ' of the state α. This distribution function can distinguish the spin glass phase clearly from the ordinary paramagnetic or ferromagnetic phase. For an ordinary ferromagnetic phase, below its ordering temperature, P q will be a double δ function with the peaks at q m2 , m T being the average magnetization of the system as a whole at temperature T . But for a spin glass state, P q is a non-trivial continuous distribution between two δ function peaks at qmax T . For any value of q between qmax , there is a finite non-zero value of P q . The entire free-energy landscape could no more be characterized by a single order-parameter, instead we need a whole distribution function of the same. This physical picture of ergodicity breaking is connected to the concept of Replica Symmetry Breaking (RSB) as illustrated in the next section. For J Ising model, where Jij s are distributed according to (14.3), a number of exact results are known along a line, known as Nishimori line, in the phase diagram in p T plane. The line is given by the relation J T 12 ln 1 pp. On this line the ferromagnetic order parameter m equals the spin glass order parameter q. Thus on the line, quenched and annealed averaging becomes equal, and it thus marks a crossover between purely ferromagnetic region (m q) and a randomness-driven region (m q) within the ferromagnetic phase.56 Later it will be shown that in certain quantum spin glasses the ergodicity may be restored, at least partly, because of quantum tunneling60 between pure states. This becomes possible since unlike the classical (thermal) fluctuations, which can only help by scaling the barrier height, quantum tunneling brings about a fair chance to tunnel through even a very high barrier if it is thin enough (see section 14.5). 14.3. Optimizations, Thermal Annealing and Spin Glasses 14.3.1. Combinatorial Optimization Problems Multi-variable optimization problems occur frequently where one has to choose the best bargain from a host of available options. Mathematically one may often cast such a task as a problem of minimizing a given cost function H S1 , S2 , ...SN with respect to N variables S1 , S2 , ...SN (some-
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times subject to some constraints). The task is to find out a set of values for the variables (a configuration) for which the function H &Si ' has the minimum value (cf. Fig. 14.1). In many important optimization problems, the set of feasible configurations from which an optimum one is to be chosen, is a finite set. In such a case we say that the problem is combinatorial in nature. If the variables Si s are discrete and each takes up a finite number of values, then the problem is clearly a combinatorial one. Moreover, certain problems with continuous variables (like linear programming problem) can also be reduced to combinatorial problems.57 Here we focus on this type of optimization problems, and assume that we have to minimize (without any loss of generality) H &Si ' with respect to the discrete state of the variables Si . An optimization problem is said to belong to the class P (P for Polynomial), if it can be solved in polynomial time (i.e., the evaluation time goes like some polynomial in N ) by some deterministic Turing machine or by any computational model polynomially equivalent to it.33 Existence of such a polynomial bound on the evaluation time is somehow interpreted as the “easiness” of the problem. However many important optimization problems seem to fall outside this class. There is another important class of problems which can be solved in polynomial time by nondeterministic machines. This class is the famous NP (Nondeterministic Polynomial) class. P is included completely in the NP class, since a deterministic Turing machine is just a special case of nondeterministic Turing machines. Unlike a deterministic machine, which takes a specific step deterministically at each instant (and hence follow a single computational path), a nondeterministic machine has a host of different ‘allowed’ steps at its disposal at every instant. At each instant it explores all the ‘allowed’ steps and if any of them leads to the goal, the job is considered to be done. Thus it explores in parallel many paths (whose number goes roughly exponentially with time) and see if any one of them reaches the goal. Among the NP problems, there are certain problems (known as NPcomplete problems) which are such that any NP problem can be “reduced” to them using a polynomial algorithm. This roughly means that if one has a routine to solve an NP-complete problem of size N then using that routine one can solve any NP problem at the cost of an extra overhead in time that goes only polynomially with N . The problems in this class are considered to be hard, since so far no one can simulate a general nondeterministic machine by a deterministic Turing machine without an exponential loss in
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execution time. In fact it is largely believed (though not been proved yet) that it is impossible to do so (i.e., P"NP) in principle. However, assuming this to be true, one can show that there are indeed problems in NP class that are neither NP-complete nor P.33 14.3.2. Simulated Thermal Annealing There are some excellent deterministic algorithms for solving certain optimization problems.57,61 These algorithms are however quite small in number and are strictly problem specific. For NP or harder problems, only approximate results can be found using these algorithms in polynomial time. These approximate algorithms are also even more problem specific, in the sense that if one can solve a certain NP-complete problem up to a certain approximation using some polynomial algorithm, then that does not ensure that one can solve all other NP problems using it up to the said approximation in polynomial time. A general approach towards formulating approximate heuristics for solving these problems may be based on stochastic (randomized) iterative improvements. The most preliminary one in this family is the local minimization algorithm. In this algorithm one starts with a random configuration C0 and makes some local changes in the configuration following some prescription (stochastic or deterministic) to generate a new configuration C1 and calculates the corresponding change in the cost. If the cost is lowered by the change, then the new configuration C1 is adopted. Otherwise the old configuration is retained. Then in the next step a new local change is attempted again, and so on. This reduces the cost steadily until a configuration is reached which minimizes the cost locally. This means that no further lowering of cost is possible by changing this configuration using any of the prescribed local moves. The algorithm essentially stops there. But generally in most optimization problems (like in spin glasses), there occur many local minima in the cost-configuration landscape and they are mostly far above the global minimum (see Fig. 14.1). It is likely that the algorithm therefore gets stuck into any of them and ends up with a poor approximation. One can then start afresh with some new initial configuration and end up with another local minimum. After repeating this for several times, each time with a new initial configuration, one may choose the best result from them. But much better idea would be to get out of high local minima while relaxing stochastically. One can introduce some fluctuation or noise in the process so that the movement is not always to-
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wards lower energy configurations, but there is a finite probability to go even to the higher energy configurations (the higher is the final energy, the lower is the probability to move to that), and consequently a chance to get out of shallow local minima. Initially a strong fluctuation is to be adopted (i.e., the probability to go to higher energy configurations is relatively high) and slowly it is reduced until finally it is tuned off completely. In the mean time the system gets a fair scope to explore the landscape more exhaustively and settles into a reasonably deep cost or energy minimum. Kirkpartick et al.46 suggested an elegant way: A fluctuation is implemented by introducing an “artificial” temperature T into the problem such that the transition probability from a configuration Ci to a configuration Cf is given by min &1, exp ∆if T ', where ∆if Ef Ei , Ek denoting the cost of the configuration Ck . A corresponding Monte Carlo dynamics is defined, say, based on detailed balance, and the thermal relaxation of the system is simulated. In course of simulation, the noise factor T is reduced slowly from a very high initial value to zero following some annealing schedule. At the end of the simulation one is expected to end up with a configuration whose cost is a reasonable approximation of the globally minimum one. If temperature is decreased slow enough, say,
T t N log t
(14.8)
where t denotes the cooling time, and N the system size, then the global minimum is attained with certainty in the limit t .34 Even within a finite time and with a faster cooling rate, one can achieve a reasonably good approximation (a crystal with only few defects) in practice. This simulated annealing method is now being used extensively by engineers for devising real-life optimization algorithms. We will refer to it as classical annealing (CA), to distinguish it from quantum annealing (QA) which employs quantum fluctuations. It is important to note that though in this type of stochastic algorithms the system has many different steps with their respective probabilities at its disposal, it finally takes up a single one, say by tossing coins, and thus finally follow a single (stochastically selected) path. Hence it is not equivalent to a non-deterministic machine, where all allowed paths are checked in parallel at every time-step.
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14.3.3. Statistical Mechanics of Optimization Problems and Thermal Heuristics As has been mentioned already, many combinatorial optimization problems may be cast into the problem of finding the ground state of some classical (spin glass like) Hamiltonian H &Si '. One can therefore analyze the problem using statistical mechanics in order to be able to apply physical techniques like simulated annealing. If one naively takes the number of variable N as the size, then the entropy and the energy are often found to scale differently with N and the application of standard thermodynamic arguments become difficult. One needs to scale temperature and some other quantities properly with N in order to be able to talk in terms of the concepts like free energy minimization etc. Moreover, the constraints present in the problems are often very difficult to take into account easily. Two different approaches are adopted generally to deal with the constraints. One is the softer method, where all the configurations are allowed in the sampling, but heavy penalties are associated with those which are to be excluded to satisfy the constraints.49 The other method is to restrict the calculation in the specific subspace of the configuration space which satisfies the constraints. Thus, for example, in Monte Carlo simulation of such a system, only those specific moves are allowed which satisfy the constraints. 14.3.4. Traveling Salesman Problem In traveling salesman problem (TSP), there are N cities placed randomly in a country having a definite metric to calculate the inter-city distances. A salesman has to make a tour to cover every city and finally come back to the starting point. The problem is to find out a tour of minimum length. An instance of the problem is given by a set &dij ; i, j 1, N ', where dij indicates the distance between the i-th and the j-th city, or equivalently, the cost for going from the former to the later. We mainly focus on the results of symmetric case, where dij dji . The problem can be cast into the form where one minimizes an Ising Hamiltonian under some constraints, as shown below. A tour can be represented by an N N matrix T with elements either 0 or 1. In a given tour, if the city j is visited immediately after visiting city i, then Tij 1, or else Tij 0. Generally an additional constraint (which naturally restricts the search to a subspace that contains the minimal path, for many metrics) is imposed that one city has to be visited once and only once in a tour. Any valid tour with the above restriction may be represented by a T matrix whose each row and each column
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has one and only one element equal to 1 and rest all 0s. For symmetric metric, i.e., dij dji a tour and its reverse tour has same length, and it is ˜ , more convenient to work with an undirected tour matrix U 12 T T ˜ where T, the transpose of T, represents the reverse of the tour given by T. Clearly, U must be a symmetric matrix having two and only two distinct entries equal to 1 in every row and every column, no two rows being identical, and so is not any two columns. In terms of Uij s, the length of a tour can be represented by H
N 1 dij Uij . 2 i,j 1
One can rewrite the above Hamiltonian in terms of Ising spins Sij s as HT SP
N 1 1 Sij dij 2 i,j 1 2
(14.9)
where Sij 2Uij 1 are the Ising spins. The Hamiltonian as such appears to be similar to that of a non-interacting Ising spins on an N N lattice, with random fields dij on the lattice points &i, j '. The frustration is introduced by the global constraints on the spin configurations in order to conform with the structure of the matrix U discussed above. The problem is to find the ground state of the Hamiltonian subject to these constraints. 2 There are N 2 Ising spins, which can assume 2N configurations in absence of any constraint, but the constraint here reduces the number of valid configurations to that of the number of distinct tours, which is N !N . The hardness of the problem depends on the nature of the metric dij , or rather on the dimensionality of the space in which the cities are embedded. Mainly two distinct classes are studied, the one with an Euclidean dij s in finite dimension, and the other with random dij s in infinite dimension. In the first case, N cities are uniformly distributed within a hypercube in a d-dimensional Euclidean space. Finding a good approximation for large N is rather easier for this case, since the problem is finite-ranged. Here a d-dimensional neighborhood is defined for each city, and the problem can be solved by dividing the whole hypercube into number of smaller pieces and find the least path within each smaller part and then join them back. The correction to the true least path will be due to the unoptimized connections across the boundaries of the subdivisions. For suitably made division (not too small), this correction will be of the order of the surface-to-volume ratio of each division, and thus will tend to zero in the
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N limit. This method, known as “divide and conquer”, forms a reasonable strategy for solving approximately such finite range optimization problems (including finite range spin glasses) in general. In the second case, dij s are assigned completely randomly, with no Euclidean correlation between them. The problem in this case becomes more like a long range spin glass. A self avoiding walk representation of the problem was made using an m-component vector field, and the replica analysis was done49 for finite temperature, assuming replica symmetry ansatz to hold. The results, when extrapolated to zero temperature, do not disagree much with the numerical results.49 However, the stability of replica symmetric solution has not yet been proved for low temperature region. However numerical results of thermal annealing for instances of size N 60 to N 160 yielded many near optimal tours, and the corresponding overlap analysis shows a sharply peaked distribution, whose width decreases steadily with increase in N . This is somewhat an indication of existence of replica symmetric phase for the system. In applying simulated (thermal) annealing to TSP, one faces the problem of “entropic catastrophe”, which is generic to many combinatorial problem. This is due to the fact that in these problems the number of possible configurations goes typically as N !, which is greater than any exponential in N . In TSP, the number of distinct undirected tour is N !2N . This makes the entropy S (at infinite temperature) a non-extensive quantity, as S N log N ; N , though the energy (path-length) continues to be of the order of N .49 Simulated (thermal) annealing of a Euclidean TSP on a square having length N 12 (so as to render the average nearest neighbor distance independent of N ) has been reported.46 In this choice of length unit, the optimal tour length per step (Ω) becomes independent of N for large N . Thermal annealing rendered Ω 0.95 for N up to 6000 cities. This is in fact much better than what is obtained by so called “greedy heuristics”, (where being at some city in a step, one moves to the nearest city not in the tour in the next step) for which Ω 1.12 on an average. However, an analytical bound on the average (Normalized by N 12 ) value of optimal path-length per city (Ω) for TSP on 2-dimensional Euclidean plane has been calculated (where one calculates the average distance of successive cities within the strips of a given width, and then optimize that with respect to the width), and has been found to be 58 Ω 0.92.7 Careful scaling analysis of the numerical results obtained so far indicates the lower bound to be close to 0.72.14,59
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To summarize, when cast in some energy minimization problem, the combinatorial optimization problems may exhibit glassy behavior during thermal annealing. Even RSB behavior may be observed (like in case of Graph Partitioning Problem, see31 ), since the underlying Hamiltonian need not be short ranged, and the constraints can bring about frustrations into the problem. One can intuitively conclude that thermal annealing or other heuristics would not be able to solve such problems easily to a good approximation within reasonable time. However, in some cases like TSP and certain NP-complete short range spin glasses the RS ansatz seems to hold, at least to a good approximation.4 But these problems still belongs to the hard NP-complete class. This is because existence of RS only tells us about the overall accessibility of the configuration space in thermodynamic sense, but not about the existence of a good gradient in the problem that leads to the deepest valley from any point in the landscape (as it is the case, say, for a simple ferromagnet). Hence it practically says nothing about the time required to solve the worst case instance exactly. Specifically, in some cases, where good solutions are thermodynamically very insignificant in number and there is no monotonic gradient towards them, the entropy might make a classical search exponentially difficult, though the landscape might still remain completely ergodic. Later we will see quantum searches can bring about spectacular improvements in some such cases. 14.4. Quantum Spin Glasses In quantum spin glasses,8,12,41,62,64 the order-disorder transition can (in principle) be driven both by thermal fluctuations as well as by quantum fluctuations. Quantum spin glasses can be of two types: vector spin glasses, where of course quantum fluctuation cannot be tuned, or a classical spin glass perturbed by some tunable quantum fluctuations e.g., as induced by a non commutative transverse field. The amount of quantum fluctuation being tunable, this transverse Ising spin glass (TISG) model is perhaps the simplest model in which the quantum effects in a random system can be and have been studied extensively and systematically.13,45 Here we will focus only on TISG, since the tunability of quantum fluctuation is the key feature required for quantum annealing. The recent spectacular upsurge in the interest in zero temperature quantum spin glass in TISG models have been have been complimented all along by the experimental studies in several systems which have been established
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to be represented accurately by transverse field Ising model (TIM). Recent discovery of the compound material LiHox Y1x F4 with the magnetic Ho ion concentration x 0.1671,76,77 representing accurately a TISG, has led to renewed interest. Here, the strong spin-orbit coupling between the spins and the host crystals restricts the effective “Ising” spins to align either parallel or anti-parallel to the specific crystal axis. An applied magnetic field, transverse to the preferred axis, flips the “Ising” spins. This feature, together with the randomness in the spin-spin interaction makes it a unique TISG system. TISG model described here by the Hamiltonian H
N
i j
Jij Siz Sjz Γ
N
Six ,
(14.10)
i
where Γ denotes the tunneling strength at each site and Jij s are distributed randomly following the distribution ρ Jij given by (14.2). The unique interest in such quantum spin glass system comes from the possibility of tunneling through the (infinitely high) barriers in the potential energy landscape of the classical spin glass models (e.g., SK model) due to the quantum fluctuations induced by the transverse field. In classical systems, the overriding of an infinitely high barrier is infinitely hard for thermal fluctuations at any finite temperature. But quantum fluctuation can make a system tunnel through such a barrier, if its width is infinitesimally small. 14.4.1. Phase Diagram The phase transitions in quantum spin glasses can be driven both by thermal and quantum fluctuations as mentioned before and the equilibrium phase diagrams also indicate how the optimized solution (in SG phase) can be obtained either by tuning of the temperature T or the tunneling field Γ, or by both. We will show later (in context of quantum annealing) that reaching the phase by tuning Γ may often be more advantageous (faster) than that by tuning T . The short-ranged version of this TISG model was first studied by Chakrabarti,12 and the long-ranged version, discussed here, was first studied by Ishii at al.41 Several analytical studies have been made to obtain the phase diagram of the transverse Ising SK model (giving in particular the zero-temperature critical field). The problem of SK glass in transverse field becomes a nontrivial one due to the presence of non-commuting spin oper-
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ators in the Hamiltonian. This leads to a dynamical frequency dependent self-interaction for the spins. 14.4.1.1. Mean Field Estimate One can study an effective-spin Hamiltonian for the SK model in transverse field within the mean field framework (see13 ). The spin glass order parameter in a classical SK model is given by a random ‘mean field’ h r having Gaussian distribution (see9 ) q
drer
2
2 tanh2 hz rT ; hz r J qr hz
(14.11)
where hz denotes the external field (in z direction), the mean-field h r being also in the same direction. In the presence of the transverse field, as in (14.10), h r has components both in z and x directions h r hz rzˆ Γˆ x; h r hz r2 Γ2 , and one replaces the ordering term tanh2 h rT in (14.11) by its component hz rh r2 tanh2 h rT in the z-direction. Putting hz 0 and q 0 one gets the phase boundary equation as (see13 ) Γ Γ tanh . (14.12) J T This gives Γc T 0 J Tc Γ 0 and a phase diagram qualitatively similar to the experimental one shown in Fig.14.2. 14.4.1.2. Monte Carlo Studies Several Monte Carlo studies have been performed for SK spin glass in transverse field applying Suzuki-Trotter formalism (see Appendix A), mapping a d-dimensional quantum Hamiltonian to an effective d 1 dimensional anisotropic classical Hamiltonian; (see also38 ). The partition function gives the effective classical Hamiltonian in M th Trotter approximation as
H
N M
Kij Sik Sjk
i j k
N M
i
KSik Sik 1 ,
k
with Kij
Jij ; MT
K
1 ln coth 2
Γ MT
,
(14.13)
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where Sik denotes the Ising spin defined on the lattice (i, k), i being the position in the in the original SK model and k denoting the position in the additional Trotter dimension. Although the equivalence between classical and the quantum model holds exactly in the limit M , one can always make an optimum choice for M . The equivalent classical Hamiltonian has been studied using standard Monte Carlo technique. The numerical estimates of the phase diagram etc. are reviewed in details in8 and.62 Ray et al.60 took Γ J and their results indeed indicate a sharp lowering of Tc Γ. Such sharp fall of Tc Γ with large Γ is obtained in almost all theoretical studies of the phase diagram of the model (see e.g.,8 and62 ). The Hamiltonian for the E-A spin glass in presence of transverse field is given by (14.10) only the random interactions are restricted among the nearest neighbors and satisfies a Gaussian distribution with zero mean and variance J, as given by Eq. (14.2). With Γ 0, the above model represents the E-A model with order parameter q Siz 2 1 (at T = 0). When the transverse field is introduced, q decreases, and at a critical value of the transverse field the order parameter vanishes. To study this quantum phase transition using quantum Monte Carlo techniques, one must remember that the ground state of a d-dimensional quantum model is equivalent to the free energy of a classical model with one added dimension which is the imaginary time (Trotter) dimension. 14.4.1.3. Experimental Studies of Phase Diagram As discussed earlier in this section, LiHox Y1x F4 with x 0.167 gives a spin glass system, for which the external magnetic field transverse to the preferred axis scales like the square root of the tunneling field Γ in (14.10). With increasing transverse field, glass transition temperature decrease monotonically, as shown in Fig. 14.2. 14.4.2. Replica Symmetry in Quantum Spin Glasses The question of existence of replica-symmetric ground state in quantum spin glasses has been studied extensively in recent years. Replica symmetry restoration is a quantum phenomena arising due to the quantum tunneling between the classically ‘trapped’ states separated by infinitely high (but infinitesimally narrow) barriers in the free energy surface. To investigate this aspect of quantum glasses, one has to study the overlap distribution function P q given by (14.7). In quantum glass problem one can study
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Fig. 14.2. Phase diagram of LiHo0.167 Y0.833 F4 according to the dynamical measurements (filled circles) and of the nonlinear susceptibility measurements (open circles). Filled squares indicate the freezing boundary obtained from AC susceptibility measurements (taken from78 ).
this overlap distribution PN q ; and if the replica symmetric ground states exists, the above function must tend to a delta function in thermodynamic limit (N ). In para-phase, the the distribution will approach a delta function at q 0 for the infinite system. Ray et al.60 performed Monte Carlo simulations, mapping the ddimensional transverse SK spin glass Hamiltonian to an equivalent (d 1)dimensional classical Hamiltonian and addressed the question of stability of the replica symmetric solution, with the choice of order parameter distribution function given by
1 P q δ q
N M 1 α β S S N M i1 k1 ik ik
,
where, as mentioned earlier, superscripts α and β refer to the two identical samples (replicas) evolved through different Monte Carlo dynamics. It may be noted that a similar definition for q (involving overlaps in identical Trotter indices) was used by Guo et al.37 Goldschmidt and Lai35 performed Monte Carlo studies with larger system-size (N 100) and studied the or-
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der parameter distribution function
N 1 α β 5 P q δ q S S , N i1 ik ik where the overlap is taken between different (arbitrarily chosen) Trotter indices k and k ; k " k . Their studies indicate that P5 q does not depend upon the choice of k and k (Trotter symmetry). One can also define qαβ N M α β in similar way qαβ 1N M i kk Sik Sik . There are striking differences between the results Goldschmidt and Lai obtained with the results of Ray et al. For Γ Γc , P1 q is found to have an oscillatory dependence on q with a frequency linear in N (which is probably due to the formation of standing waves for identical Trotter overlaps). However, with increase in N , the amplitude of oscillation decreases and the magnitude of P1 q 0 decreases, indicating that P1 q might go over to a delta function in thermodynamic limit. The envelope of this distribution function appears to have an increasing P1 q 0 value as the system-size is increased. Ray et al.60 argued that the whole spin glass phase is replica symmetric due to quantum tunneling between the classical trap states. Goldschmidt and Lai on the other hand, do not find any oscillatory behaviour in P5 q . In contrary, they get a RSB in the whole spin glass phase from the nature of P5 q , which in this case, has a tail down to q 0 even as N increases. According to them their results are different from60 because of different choices of the overlap function. Goldschmidt and Lai have also obtained replica symmetry breaking solution at first step RSB. Using static approximation,75 found stable replica symmetric solution in a small region close to the spin glass freezing temperature near the phase boundary. But as mentioned earlier, in the region close to the critical line, quantum fluctuations are subdued by the thermal fluctuations. Thus the restoration of replica symmetry breaking, which is essentially a quantum effect, perhaps cannot be prominent there. All these numerical studies are for equivalent classical Hamiltonian, obtained by applying the Suzuki-Trotter formalism (see Appendix A) to the original quantum Hamiltonian, where the interactions are anisotropic in the spatial and Trotter direction. One cannot easily extrapolate the finite temperature results in zero temperature limit. The results of exact diagonalization of finite systems (N 10) at T 0 itself do not indicate any qualitative difference in the behaviour of the (configuration average) mass gap ∆ and the internal energy Eg from that of a ferromagnetic transverse
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Ising case, indicating the possibility that the system might become “ergodic”. On the other hand, the zero temperature distribution for the order parameter does not appear to go to delta function with increasing N as is clearly found for the corresponding ferromagnet (random long range interaction without competition). In this case the order parameter distribution P q is simply the number of ground state configurations having the order parameter value as q. This perhaps indicate broken ergodicity for small values of Γ. The order parameter distribution also shows oscillations similar to that obtained by Ray et al.60 Kim and Kim45 investigated the SK model in transverse field using imaginary time replica formalism, under static approximation. They have shown that the replica-symmetric quantum spin glass phase is stable in most of the area of the spin glass phase, as have been argued by Ray et al. in contrary to the results of Lai et al. and Thirumalai et al. (see e.g.,.13 ) Though it seems that replica symmetric solution for SK-model in transverse field is not stable at low temperature region, the effect of quantum tunneling through the energy barriers is clearly reflected in the fact that the glass transition temperature Tc steadily decreases and goes to zero at a finite value of Γ. Moreover, the level of RSB could not be indicated (whether it is a full scale RSB), and first stage RSB solution has been considered to be a good enough approximation. Hence it might be the case that quantum effect makes a large part of the landscape ergodic (though not the entire one) and hence restores replica symmetry at higher stages.
14.5. Optimization Using Quantum Annealing In the previous sections we have seen how thermal fluctuations can be utilized to frame fast heuristics to find an approximate ground state of a glassy system, or equivalently, a near-optimal solution to combinatorial problem, whose cost-configuration landscape has glassy behaviour due to occurrence of many local minima. There are two aspects of an optimization problem which might render thermal annealing to be a very ineffective one. First, in glassy landscape, there may exist very high cost/energy barriers around local minima which does not correspond to a reasonably low cost. In case of infinite range problems, these barriers might be proportional to the system-size N , and thus diverge in thermodynamic limit. Thus there might occur many unsatisfactory local minima, any of which can trap the system for very long time (which actually diverges in thermodynamic limit
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for infinite range systems) in course of annealing. The second problem is the entropy itself. Since thermal annealing deals with one configuration at a time, if the solutions are rare and there is no gradient in the landscape that by and large leads to the solution. In that case it is clearly an exponential or higher problem (depending how the size of the configuration space scales with the system-size N ), and no better than a random search algorithm. This is the case for golf-course type potential-energy landscape. One can imagine that quantum mechanics might have some solutions to both these problems, at least, to some extent. This is because quantum mechanics can introduce (classically unlikely) tunneling paths even through very high barriers if they are narrow enough.60 This can solve ergodicity problem to some extent as discussed earlier. Even in places where ergodicity breaking does not take place in true sense, once the energy landscape contains high enough barriers (specially for infinite ranged quenched interactions), quantum tunneling provides much faster relaxation to the ground state;19,27,47,65 see also71 and.67 In addition, quantum mechanical wave function can delocalize over the whole configuration space (i.e., over the potential energy landscape) if the kinetic energy term is high enough. Thus it can actually “see” the whole landscape simultaneously at some stage of annealing. These two aspects can be expected to improve the search process when employed properly. In fact such improvements can indeed be achieved in certain situations, though quantum mechanics is not a panacea for all such diseases, and certainly has its own inherent limitations. What intrigues is the fact that the limitations due to quantum nature of an algorithm are inherently different from those faced by its classical counter part, and thus it is not yet clear in general which wins when. Here we discuss some results regarding the quantum heuristics and some of their general aspects understood so far. For more detailed review of the subject we refer to recent reviews in18 and.66 The basic idea of QA is to appoint a tunable quantum fluctuation into the problem instead of a thermal one (29,43 In order to do that, one needs to introduce an artificial quantum kinetic term Γ tHkin into the problem, which does not commute with the classical Hamiltonian HC representing the cost function. The coefficient Γ is the tuning parameter which controls the quantum fluctuations. The total Hamiltonian is thus given by Htot
HC
Γ tHkin
(14.14)
The ground state of Htot is a superposition of the eigenstates of HC . For a
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classical Ising Hamiltonian of the form (14.1), the corresponding total quan tum Hamiltonian might have the form (14.10), where HC ij Siz Sjz and Hkin i Six . Initially Γ is kept very high so that the Hkin dominates and the ground state is trivially a uniform superposition of all the classical configurations. One starts with that uniform superposition as the initial state, and slowly decreases Γ following some annealing schedule, eventually to zero. If the process of decreasing is slow enough, adiabatic theorem of quantum mechanics assures that the system will always remain at the instantaneous ground state of the evolving Hamiltonian Htot . When Γ is finally tuned to zero, Htot will coincide with the original classical Hamiltonian HC and the system will be found at the ground state of it, as desired. The special class of QA algorithms, where strictly quasi-stationary or adiabatic evolutions are employed are also known as Quantum Adiabatic Evolution (QAE) algorithms.27 Two important questions are how to choose an appropriate Hkin and how slow the system should evolve so as to assure adiabaticity. According to adiabatic theorem of quantum mechanics, for a non-degenerate spectrum with a non-zero gap between the ground state and 1st excited state, the adiabatic evolution is assured if the evolution time τ satisfies the following conditionτ where
Htot max
∆2min
Htot max
∆2min
,
6 6 86 67 6 6 dHtot 6 6 6 φ1 s 6 max 66 φ0 s 66 6 6 0 s 1 ds 2 min ∆ s , s tτ
0 s 1
(14.15)
(14.16)
φ0 s and φ1 s being respectively the instantaneous ground state and the first excited state of the total Hamiltonian Htot , and ∆ s the instantaneous gap between the ground state and the first excited state energies (see26 ). It is impossible however to follow the evolution of a full wave function in a classical computer using polynomial resource in general, since it requires tracking of the amplitudes of all the basis vectors (all possible classical configurations), whose number grows exponentially with system-size. Such an adiabatic evolution may be realized within polynomial resources only if one can employ a quantum mechanical system itself to mimic the dynamics. However, one may employ quantum Monte Carlo methods to
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simulate some dynamics (may not be the real-time quantum dynamics) to sample the ground state (or a mixed state at low enough temperature) for a given set of parameter values of the Hamiltonian. Annealing is done by reducing the strength Γ of the quantum kinetic term in the Hamiltonian from a very high value to zero following some annealing schedule in course of simulation. In case of such Monte Carlo annealing algorithm, there is no general bound on success time τ like the one provided by the adiabatic theorem for true Schr¨ odinger evolution annealing. Here we will separately discuss the results of real-time QAE and Monte Carlo QA. Apart from these quasi-stationary quantum annealing strategies, where system always stays close to some stationary state (or low-temperature equilibrium state), there may be cases where quantum scatterings (with tunable amplitudes) are employed to anneal the system.20 14.5.1. Quantum Monte Carlo Annealing In quantum Monte Carlo annealing, one may employ either a finite (but low) temperature algorithm, or a zero temperature algorithm. Most of the quantum Monte Carlo annealing are done using a finite temperature Monte Carlo, namely, Path Integral Monte Carlo (PIMC), since it is much straightforward to implement18,66 compared to the zero temperature Monte Carlo methods. Among the zero temperature Monte Carlo used for annealing are a zero temperature transfer matrix Monte Carlo and Green’s function Monte Carlo. However, these algorithms suffer severely from different drawbacks, which renders them much slower than PIMC algorithms in practice. The Green’s function Monte Carlo effectively simulates the real-time evolution of the wave function during annealing. But to perform sensibly, it often requires a guidance that depends on an a priori knowledge of the wave function. Without this guidance it may fail miserably.66 But such an a priori knowledge is very unlikely to be available in case of random optimization problems, and hence its scope so far seems very restricted. The Zero Temperature Transfer Matrix Monte Carlo, on the other hand, samples the ground state of the instantaneous Hamiltonian (specified by the given value of the parameters at that instant) using a projective method, where the Hamiltonian matrix itself (a suitable linear combination of the Hamiltonian, actually) is viewed as the transfer matrix of a finitetemperature classical system of one higher dimension.18 But the sparsity of the Hamiltonian matrix for systems with local kinetic terms, leaves the
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classical system highly constrained and thus very difficult to simulate efficiently for large system size. Mainly Path integral Monte Carlo has been mostly used for QA so far. The basic idea of path integral Monte Carlo rests on Suzuki-Trotter formalism (see Appendix A), which maps the partition function of a d-dimensional quantum Hamiltonian H to that of an effective classical Hamiltonian Hef f in (d 1)-dimension (see Appendix A). Quantum annealing of a Hamiltonian Htot using PIMC consists of mapping Htot to its equivalent classical and simulate it at some fixed low temperature so that thermal fluctuations are low. The quantum fluctuations are tuned from some very high initial value to zero finally through tuning of Γ in course of simulation. Clearly, the simulation dynamics is not the true Schr¨ odinger evolution of the system, and also cannot simultaneously see the whole configuration space as does a delocalized wave function. It feels the landscape locally and make moves just like a classical system. The first attempt of quantum annealing using PIMC was done by Kadowaki and Nishimori43 for solving TSP and an extensive use of the technique to explore multitude of problems has been done by the group of Santoro and Tosatti.66 Here we will discuss some PIMC quantum annealing results briefly for different systems below. (i) 2D EA Spin Glass Quantum annealing of an EA spin glass in 2-dimension (square lattice) using transverse field (see Eq. (14.10)) for large lattice size (up to 80 80) using PIMC turns out to be much more efficient compared to thermal annealing (CA) of the same system in finding the approximate ground state.47,65 The quantity which is measured is the residual energy res τ E τ E0 , E0 being the true ground state energy of the system and E τ being the final energy of the system after reducing the transverse field strength Γ linearly within time τ from some suitably large initial value to zero. Here τ is the fictitious time given by the number of Monte Carlo steps. Classically, for a large class of frustrated disordered system it can be shown using very general arguments that the residual energy decreases following some power law in the logarithm of the annealing time τ , namely,
res log τ ζ , with ζ 2.39 In PIMC one effectively deals with a classical frustrated system but with a rather uncommon anisotropy and perfectly correlated randomness along the extra dimension. However, the PIMC annealing results show that the quantum effect (taken into account through Suzuki-Trotter mapping) does not change the basic relaxation behaviour of
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res τ . But still a dramatic improvement in evaluation time is achieved since it turns out that the value of the exponent ζ can be much higher (ζ 6) for QA than the Huse-Fisher bound of ζ 2 for classical annealing. This is in fact a tremendous amount of gain in computational time if one thinks in term of the changes in τ required to improve res equally by some appreciable factor in respective cases of classical and quantum annealing. An interesting asymptotic comparison for results of QA and CA for an 80 80 lattice shows that to reach a certain value of res , the PIMC-QA would take one day of CPU time (for the computer used) whereas CA would take about 30 yrs.47,65 The result would not be much different in this case if the real time Schr¨ odinger evolution would have followed in this case, as has been argued using Landau-Zener cascade tunneling picture.47,65 Landau-Zener tunneling theory gives an estimate about the probability of tunneling non adiabatically from a lower to a higher level when the system encounters an avoided level crossing between the levels during time evolution. Let the gap between two energy levels a t and b t of a time-dependent Hamiltonian vary linearly with time (gap ∆ αt) and encounter an avoided level crossing. Here the levels are energy levels of the classical part of the Hamiltonian (say, the potential energy levels of an Ising model). Let the system is so evolved that the characteristic time it spends while passing through the crossing region is τ . Let there be a quantum tunneling field Γ that induces transitions between the levels. Then if the system evolved being at the lower branch a before encountering the avoided level crossing, the probability that it tunnels to the higher branch b while passing through the crossing decreases with the time τ as P τ exp τ τΓ , where τΓ αΓ 2π∆2min , ∆min being the minimum value of the gap attained at the avoided level crossing. for spin glasslike systems with non-zero gap, treating multiple level-crossings, each with small ∆, as a cascade of independent Landau-Zener tunneling, one may argue that residual energy goes as res log τ ζQ , where ζQ is essentially greater than the bound ζ 2 for thermal annealing, and might be as high as 6.47,65 (ii) Random TSP Quantum annealing of TSP with random metric in infinite dimension using PIMC was also found to be more efficient than CA in finding an approximately minimal tour.48 An N -city tour in a random TSP problem can be represented by a configuration of N 2 constrained Ising spins, and the
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Excess length after annealing [%]
tour length by the Hamiltonian (14.9). Now to do the annealing, one needs to introduce a set of moves (spin flip operations) that satisfies the constraints. Classically one very important class of moves is the 2-opt moves, which starting from a valid tour, can generate all possible tours without generating any invalid link. Let a valid tour contains two links i j and k l. A 2-opt move may consist of removing those two links and establishing the following links: i k and j l (here i denotes i-th city). For classical annealing of the Hamiltonian can be done by restricting the Monte Carlo moves within 2-opt family only. However, for the quantum case, one needs to design a special transverse field (non-commuting spin-flip term) to abide by the constraints. It can be realized by a spin-flip term of the form flips down (1 1) S k,i S l,j S j,i S l,k , where the operator Si,j z z and Sji when they are in 1 state, and similarly for the Ising spins Sij
the flip-up operators Si,j s. However, avoiding the Trotter break-up with these complicated kinetic simple kinetic term of the form terms, a relatively S H.C. is used for the quantum to classical Hkin Γ t i,j i,j mapping, but the Monte Carlo moves were kept restricted within the 2opt family to avoid invalid tours. The results were tested on an instance of printed circuit board with N 1002. PIMC-QA was seen to do better than the CA and also much better than standard Lin-Kernighan algorithm.
3 SA QA, PT=100, P=30 QA, τ multiplied by P
2
1 0.9 0.8 0.7 0.6 0.5 0.4 1 10
2
10
3
10
10
4
10
5
Monte Carlo τ (in units of MC Steps)
10
6
Fig. 14.3. Average residual excess length found after CA and QA for a total time τ 1002 instance pr1002 of the TSPLIB.The dashed horizontal (in MC steps), for the N line represents the best out of 1000 runs of the Lin-Kernighan algorithm. QA is clearly faster than CA (taken from6 ).
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PIMC-QA works much better than CA and also Lin-Kernighan algorithm in finding the least path for the printed circuit board version of TSP, as shown in Fig. 14.3. The relaxation behaviour of residual path-length for TSP (see Fig. 14.3) is found to be quite similar to that of the 2-d EA glass discuss earlier (see6 ). This indicates that random TSP also has spin glass-like potential energy landscape (PEL), as has already been hinted by the replica analysis study of the problem discussed in earlier section. In that case Landau-Zener tunneling picture would also be applicable for TSP, and little betterment can be expected following real-time Schr¨ odinger dynamics instead of Monte Carlo methods then. There are of course cases where PIMC-QA did much worse than CA. In case of random 3-SAT problem using linear schedule for decreasing Γ, PIMC-QA gives much worse result than CA.6 Both CA and PIMC-QA are worse than ad-hoc local search Heuristics like Walksat. In case of application of PIMC-QA in image restoration, on the other hand, the performance is exactly same as that of CA.40 Even for a particle in a simple double-well potential, it seems that with naive Monte Carlo moves, PIMC-QA can produce results which are much worse than that one could obtain from real-time Schr¨ odinger evolution of the system. There is in fact no general prescription to choose the right moves that will do the job, unless one has precise idea about the PEL of the problem. The kinetic term to be introduced the problem is somewhat arbitrary, and lot might depend on the choice. It is seen that relativistic kinetic term in fact can do a better job than a non-relativistic one in the case of a particle in a double-well potential (6 ). PIMC-QA also suffers from the difficulty in calculating the SuzukiTrotter equivalent of the quantum Hamiltonian with arbitrary kinetic term designed to satisfy the constraints of the problem. The constraints may be taken care of while making Monte Carlo moves, but that may not always produce expected results. Finally, presence of finite temperature in the problem does not allow one to focus exclusively on the role of quantum fluctuation in the problem. Above all, like any other Monte Carlo method, PIMC-QA is going to do worse if the number of good approximations are significantly low in the configuration space, and there is no overall gradient in the landscape to guide towards them. (iii) Random Field Ising Model: How Choice of Kinetic Term Improves Annealing Results
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QA algorithms enjoy an extra flexibility which a thermal annealing algorithm cannot. A QA algorithm can have a host for choice of its kinetic energy term from which it can pick up a suitable one. A good choice of kinetic term can in fact bring about a lot of betterment. This point is illustrated nicely by Morita and Nishimori52 for QA of random field Ising model, by introducing a ferro-magnetic transverse field interaction, in addition to the conventional single-spin-flip transverse field term (as present in the Hamiltonian (14.10)). The Hamiltonian of the random-field Ising model with standard singlespin-flip transverse term is given by H t HC
1
HKin ,
(14.17)
where HC
J
N
ij
Siz Sjz
N
hi Siz ,
(14.18)
i 1
hi being the random field assuming values 1 or 1 with equal probabilities, and ij denotes sum over the nearest neighbour on a 2-dimensional square lattice, and n
1 Six . HKin Γ t
i 1
The result of QA in such a system in not satisfactory when J is much higher compared to hi ’s.68 If a ferromagnetic transverse term of the form
2
Hkin
Γ t
N
ij
Six Sjx
(14.19)
is added to the Hamiltonian (14.17), the result of QA is seen to improve considerably.52 This happens (as indicated by exact diagonalization results on small system) because the ferromagnetic transverse field term effectively increases the gap and thus decreases the characteristic timescale for the system. This is an example how one can utilize the flexibility in choosing the kinetic term in QA to formulate faster algorithms. 14.5.2. Quantum Annealing Using Real-time Adiabatic Evolution Following up a real-time Schr¨ odinger evolution is not easy in a classical computer, but might be a tractable one in quantum computer. Thus quan-
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tum annealing scheme provides a ground for continuous time analog quantum computation. First analog algorithm in this line was due to Farhi et al. for solving Grover’s search problem.25 Grover,36 showed that quantum mechanical search can reduce an O N classical search time to O N time in finding out a marked item form an unstructured data-base.36 In the analog version, the problem was to find a given marked state among N orthonormal states by quantum evolution. The algorithm was formulated in the following way. There were N mutually orthogonal normalized states, the i-th state being denoted by i. Among all of them, only one, say, the w-th one, has energy E " 0, and rest all have zero energy. Thus the state w is “marked” energetically and the system can distinguish it thus. Now the question is how fast the system can evolve under certain Hamiltonian (our oracle) so that stating from an equal superposition of the N states one reaches w. It was shown that evolving the system under a time-independent Hamiltonian of the form Htot
E ww E ss,
(14.20)
where N 1 i, N i1
s
no betterment over Grover’s N speedup is possible. Later the problem was recast in form of a spatial search,15 where there is a d-dimensional lattice and the basis state i is the of being localized at the i-th lattice site. Similarly as before, on-site potential energy E is zero everywhere except at w, where it is 1. The job is to reach the marked one starting from the equal superposition of all the is. The kinetic term is formulated through Laplacian of the lattice, which effectively introduces uniform hopping to all nearest neighbours from any given lattice site, and is kept constant. The model is in essence Anderson model with only a single-site disorder of strength O 1. Grover’s speed up can be achieved for d 4 with such a Hamiltonian, and no further betterment is possible. The algorithms succeeds only near the critical value of the kinetic term. An adiabatic quantum evolution algorithm for Grover search was formulated in terms of an orthonormal complete set of l-bit Ising-like basis vectors, where the potential energy of a single basis vector (among the 2l ones) is 1, and for rest of all, it is 0. The kinetic term is just the sum of all single bit-flip terms (as the transverse-field term in Eq. 14.1). It connects each basis vector to all those who can be reached by a single bit flip from it. The kinetic term is
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reduced from a high value to zero following a linear schedule. A detailed analysis in light of adiabatic theorem showed that one can at most retrieve Grover’s speedup in that case. This is so because the minimum value of the gap goes as ∆min 2.2l2 , and the spin-flip kinetic term being local, the numerator Htot of the adiabatic factor (see Eq. (14.15)) is at best O l.26 Adiabatic QA following real-time Schr¨ odinger evolution for satisfiability problems (for smaller size though) also gives strikingly different result compared to that obtained using PIMC-QA. Adiabatic QA of an exact cover problem (known to be an NP-complete problem) is studied for small systems, where a quadratic system-size dependence is observed.27 In this problem also the basis vectors are the complete set of 2l orthonormal l-bit basis vectors, denoted by &z1 +z2 ...zl ', each of zi is either 0 or 1. The problem consists of a cost function HC which is the sum of many 3-bit clause functions hCl zi , zj , zk , each acting on arbitrarily chosen bits zi , zj and zk . The clause function is such that hCl zi , zj , zk 0 if the clause Cl is satisfied by the states zi , zj and zk of the three bits, or else hCl zi , zj , zk 1. The cost Hamiltonian is given by HC Cl hC l. Thus if a basis state &zi ' dissatisfy p clauses, then HC &zi ' p&zi '. The question is whether there exist a basis vector that satisfies all the clauses for a given HC . There may be many basis vectors satisfying the clause. All of them will be the ground state of HC with zero eigenvalue. If the ground state has a non-zero (must be a positive integer then) eigenvalue, then it represents the basis with lowest number of violated clauses, the number being given by the eigenvalue itself. The total Hamiltonian is given by t t Hkin HC , (14.21) Htot t 1 τ τ where Hkin is again the sum of single bit flip operators. The initial state at t 0 is taken to be the ground state of Hkin , which is an equal superposition of all basis vectors. The system is then evolved according to time dependent Schr¨ odinger equation up to t τ . The value of tau required to achieve a pre-assigned success probability are noted for different system sizes. The result shows a smooth quadratic system size dependence for l 20.27 The result is quite stimulating, but does not really assure a quadratic behaviour in asymptotic (l ) limit. A quadratic relaxation behaviour ( res 1τ ζ , ζ 2) were reported 72,73 for real-time adiabatic QA (employing exact method for small sysin tems and DMRG technique for larger systems) of a one-dimensional tight-
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bonding model with random site energies and also for a random field Ising model on 2-d square lattice. In the first the case kinetic term was due to hopping between nearest neighbours, while in the second case it was simply the sum of single-spin flip operators as given in Eq. (14.10). Wave function annealing results using similar DMRG technique for spin glass on ladder has been reported in.63 It has been demonstrated that for finite ranged systems, where interaction energy can be written as a sum of interaction energies involving few variables, quantum adiabatic annealing and thermal annealing may not differ much in efficiency. However, for problems with spiky (very high but narrow) barriers in the PEL (which must include infinite range terms in the Hamiltonian, since in finite range systems, barrier heights can grow at best linearly with barrier width), quantum annealing in fact does much better than CA28 as has been argued by Ray et al.60 14.5.3. Quantum Adiabatic Search for a Hole in a GolfCourse So far it has been argued that the advantage of quantum annealing hinges mainly on the fact that quantum tunneling can penetrate through very high but narrow barriers, which are very hard to jump over thermally and thus make the cost-configuration landscapes more accessible to local moves. This, in fact, is a key feature that works even in case of quantum Monte Carlo methods like PIMC-QA, where one finally samples only a very small section of the configuration space. However, there is another remarkable advantage that quantum mechanics provides is the ability to “sense” the whole configuration space simultaneously through a delocalized wave function. This sensing is largely handicapped by presence of random disorder in the system, because the wave function tend to localize in many places, often being unable to to pick up the deepest well very distinctly. But this feature may be utilized in searching a golf-course like PEL which is essentially a flat landscape with very deep and narrow wells occurring very rarely. If there is only one deep well, the problem is just the spatial version of the Grover’s problem. As we have seen, ifthe depth χ of the well is O 1 (independent of N ), then no more than O N speed-up can be achieved. This can be interpreted as the inability of an exceedingly large system to sense a given small wound. So one might want to make the depth of the well large enough so that it cannot be scaled away as the system-size is increased. We first consider the case where the well-depth goes as χ αN ,
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where α is some constant. For a well depth of such order for a spatial search at infinite dimension (kinetic term is infinite ranged) one gets a stunning result: The evolution time that guarantees any given probability of success becomes independent of N ;16,17,19 see Fig. 14.4. However, if χ scales like χ N γ , with γ 1, then the speed-up is lost again - a consequence of quantum mechanical non-adiabaticity.
τmin
8 4 2 1
P(|w>)
(a)
1 1
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(b)
1e-04 1e-08
0.01
1e+06
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1
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100
Fig. 14.4. Panel (a) in the figure shows numerical verification of the N -independence 0.33. Initially, the of the minimum time τmin to achieve success probability P w system is delocalized equally over all sites and evolves with time according to time dependent Schr¨ odinger equation with the Hamiltonian (14.22). As expected from exact analytical result for the adiabaticity condition, it is seen that τmin becomes independent of N . Panel (b) shows the variation of final probability P w of finding the state w with annealing time τ for different final value of Γ, for N 106 .
We formulate the problem in the following way. As in the case of spatial search discussed above, we denote the state of being localized at the i-th cite by i. All the cites except w has zero on-site potential energy, where w has an on site potential well of depth χ t. The system is embedded in infinite dimension and the thus there is a uniform tunneling term Γ between any two sites. To do quantum annealing, we evolve χ t from zero to its
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final value χ0 keeping Γ fixed at some moderate constant value. The total Hamiltonian is given by Htot
χww Γ
ij ,
(14.22)
i,j;i j
where χ χ0 1 tτ and Γ independent of time (χ0 , Γ 0). The timedependent eigen problem can be solved exactly for the above Hamiltonian. The spectrum consists of a ground state E0 t and 1st excited state E1 t (not the stationary states, but the eigenstates of Htot t with in the order of increasing eigen values) with energies 1 N 2 Γ χ N Γ χ2 4χΓ E0 2 and 1 E1 N 2Γ χ N Γ χ2 4χΓ , 2 respectively. The instantaneous gap is thus given by ∆ t 2 N Γ χ 4χΓ. The second excited state is two-fold degenerate, with eigenvalue Γ, and the time evolving Hamiltonian never mixes the first two eigenstate with any of the second excited states. This can be easily argued noting that a state of the form E2 12 i j i, j " w is an eigenstate of Htot t with eigenvalue Γ, and E2 E0 E2 E1 0 for all t. For all allowed combinations of i and j we get N 2 such linearly-independent eigenstates. Form these N 2 linearly-independent eigenstates we can construct N 2 mutually orthogonal eigenstates, each of which will obviously satisfy the above non-mixing condition. Thus we have to take care of only two lowest lying states and the gap between them. It can be shown easily that if χ0 αc N , where αc is some constant, then in the N limit, the adiabatic factor
Htot max
∆2min t
αc , 8Γ2
(14.23)
which means that the annealing time τ required to reach the state w adiabatically (with a probability close to unity), is independent of N . We also confirm it numerically by finding out τmin , which is the minimal τ required for obtaining a success probability P w 0.33 for different N through many decades. Here we have chosen a moderate constant Γ, and have evolved the well depth χ with time. The evolution is computed solving time-dependent Schr¨ odinger equation numerically and τmin is figured out
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up to an accuracy of 104 employing a bisection scheme (i.e., running for higher (τhigh ) and lower values(τlow ) of τ necessary for reaching the target probability, and then decreasing the difference between τhigh and τlow by bisecting it. The results (Fig. 14.4a) clearly show that P w tends to becomes independent of N as N becomes larger and larger. This is completely in accordance with the analytical result (see Eq. (14.23)). The relaxation behavior for large N for a given annealing time τ of course depends on the value of Γ (see Fig. 14.4b). If Γ is too small, the system takes a longer time to feel the changes in the landscape, and hence the adiabatic relaxation requires longer time (the adiabatic factor becomes bigger; see Eq. (14.23)). On the other hand, if Γ is too large, the ground state itself is pretty delocalized, and hence the final state, though more closer to the ground state, has again a small overlap with the target state w. For Γ 0.5, the result is best. For higher and lower values of Γ, the results are worse, as shown in Fig. 14.4(b). seen to be linear with the annealing time τ . 14.5.4. Experimental Realization of Quantum Annealing Brooke et al.10 showed experimentally (see also1 ) that the relaxation behaviour in reaching deep inside the glass phase in Fig. 14.5 depends on the path chosen. A TISG realized by a sample of LiHo0.44 Y0.56 F4 in laboratory transverse field (Γ) is taken from a high temperature paramagnetic to a low temperature glassy phase following two separate paths in the Γ T plane (see Fig. 14.5). Along the classical path of cooling (CA), the transverse field is kept zero through out, and is switched on only after reaching the final temperature. The quantum cooling (QA), on the other hand, is done in presence of a high transverse field, which is lowered only on reaching the final temperature. As the sample is cooled, spectroscopy of the sample at different temperatures (both during CA and QA) are done to reveal the nature of the distribution of spin relaxation time scales. QA is seen to produce states whose relaxation is up to 30 times faster than those produced by the CA at low temperature (see1 ). This clearly indicates that quntum tunneling is much more effective in exploring the configuration space in the glassy phase than thermal jumps (as indicated in Fig. 14.1). An experimental realization of quantum adiabatic annealing for 3-bit instances of MAXCUT problem using NMR technique has been reported.74 Here the the smoothly varying time dependent Hamiltonian is realized by
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Fig. 14.5. Experimental realization of QA and CA in LiHo0.44 Y0.56 F4 is illustrated on its phase diagram on the temperature T and transverse field Γ (measured by the magnitude of the external laboratory field in kOe) plane. The material behaves like a conventional ferromagnet in the region labeled FM, and shows slow relaxation in the glassy domain wall state labeled G. The two paths of relaxations, from an initial point A to a final point C on the phase diagram are shown by arrow-headed lines. Along the classical path A B C (dashed arrow) the transverse field is not applied until the end, so that the relaxations observed are purely thermal. Whereas, along the quantum path A D C (continuous arrow), there is a segment where the temperature is small enough, and the transverse field is high, so that the fluctuations are mainly quantum mechanical. Relaxations observed along the quantum path are often found to be much faster than those observed along the classical path at low enough temperature. Taken from.1
the technique of quantum simulation (see55 ), where a smooth time evolution is achieved approximately (Trotter approximation) through the application of a series of discrete unitary operations. Existence of an optimal run time of the algorithm has been indicated from the results. 14.6. Non-stationary Quantum Annealing of a Kinetically Constrained System Adiabatic theorem of quantum annealing assures convergence of a quantum algorithm when one starts with the ground state of the initial (trivial) ground state of the Hamiltonian and evolve slow enough so that the system is always in the ground state of the instantaneous Hamiltonian. However, the benefit of tunneling may be extended even in cases where one does not
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precisely know the eigenstate of the initial Hamiltonian (say for a given unknown potential energy landscape) and hence unable to start with it. One might rather start with a wave-packet (a superposition of many eigenstates of the Hamiltonian) that explores the potential energy landscape. Quantum tunneling will still allow it to move more easily through the potential energy landscape than a classical particle if the landscape has many high but narrow barriers. Here a semi-classical treatment of such a nonstationary annealing is discussed in context of a kinetically constrained system (KCS).20 We demonstrate here the effectiveness of quantum annealing in the context of a certain generalized Kinetically Constrained Systems (KCS).30 KCS’s are simple model systems having trivial ground state structures and static properties, but a complex relaxation behaviour due to some explicit constraints introduced in the dynamics of the system. These systems are very important in understanding how much of the slow and complex relaxation behaviour of a glass can be attributed to its constrained dynamics alone, leaving aside any complexity of its energy landscape structure. In KCS’s one can view the constraints to be represented by infinitely high energy barriers appearing dynamically. We discuss quantum annealing in the context of a kinetically constrained system, which can be represented by a generalized version of East model;42 a one dimensional KCS. We also compare the results with that of thermal annealing done in the same system. The original East model is basically a one-dimensional chain of non-interacting classical Ising (‘up-down’) spins in a longitudinal field h, say, in downward direction. The ground state of such a system is trivially given by all spins down. A kinetic constraint is introduced in the model by putting the restriction that the i-th spin cannot flip if the i 1-th spin is down. Such a kinetic constraint essentially changes the topology of the configuration space, since the shortest path between any two configurations differing by one or more forbidden flips, is increased in a complicated manner owing to the blockage of the ‘straight’ path consisting of direct flips of the dissimilar spins. Further, the constraint becomes more limiting as more spins turn down, as happens in the late approach to equilibrium. As a result, the relaxation processes have to follow more complex and lengthier paths, giving rise to exponentially large 2 timescale ( e1T ;42 ). In the model20 a chain of asymmetric double-wells (each with infinite boundary walls), each having a particle localized within them. The asymmetry is due to an energy difference of 2h between the two wells of a double
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well. The particle in one of the two (asymmetric) well can change its location to the other well stochastically, either due to the thermal fluctuation or due to quantum fluctuation present in the system. The generalized kinetic constraint is introduced by assuming that if the particle in the (i 1)-th double-well resides in the lower one of the two wells, then there appears a barrier of height χ and width a between the two wells of the i-th doublewell. In such a situation the particle in the i-th double-well has to cross the barrier in order to change its location from one well to the other. On the other hand, if the particle of the i 1-th is in its upper well, there is no such barrier to cross for the particle to go from one well to the other. Following the approximate mapping done in case of symmetric double-well,13 this model can be approximately represented by a generalized version of East model, where each Ising spin is in a local longitudinal field h in downward direction. The spin at the i-th site sees a barrier of height χ and width a between its two energy states when the i 1-th spin is down, while no such barrier occurs for the i-th spin when the i 1-th spin in up This kinetic constraint is the same in both cases irrespective of whether the dynamics is classical or quantum. When dynamics of the particle is due to quantum fluctuations, the tunneling probabilities come from the following semi-classical picture of scattering of a particle in a double-well with infinitely remote outer boundaries. If a particle is put in one of the wells of such a double-well with some kinetic energy (actually the expectation value) Γ, then it will eventually be scattered by the separator (a barrier or step) between the two wells. In such a scattering, there is a finite probability P that the particle manages to go to the other well. We calculate P from the simple picture of scatterings of a particle by one dimensional potentials as prescribed below. In thermal case we take simple Boltzmann probabilities for crossing the same barriers. The minimum of the energy of the Ising chain (equivalent to the potential energy of the chain of the double-wells) trivially corresponds to the state with all the spins down, i.e., aligned along the longitudinal field h (where all the particles are in their respective lower wells). The minimum of the energy of the Ising chain (eqivalent to the potential energy of the chain of the double wells) trivially corresponds to the state with all the spins down, i.e., aligned along the longitudinal field h (where all the particles are in their respective lower wells). However, if one starts with a random configuration and kinetic energy Γ is not sufficient for tunnelling to the upper well, then the system, more or less, will exhibit the zero temperature (energy minimization) relaxation behaviour of the classi-
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cal East model, and will extremely slowly approach the ground state (i.e., the minimum of the potential energy). For sufficiently high Γ, the system occasionally tunnels through the infinite barriers corresponding to the constraints and thus can take up some of the relaxation paths forbidden classically. However, at any nonzero Γ, the ground state (lowest potential energy state) will be mixed with higher potential energy eigenstates. To reach the ground state in quantum case, we start with a very large initial value of Γ and then reduce it following an exponential schedule given by Γ Γ0 exp tτQ . Here t denotes the time, and τQ sets the effective time scale of annealing. At zero temperature the slow spin flip dynamics occurs only due to the tunneling (kinetic energy) term Γ, and hence the system ceases to have any relaxation dynamics in the limit Γ 0. To simulate the time evolution of the particles strongly localized within the double well in terms of spin-flip dynamics, we have crudely approximated the motion of a particle as that due to successive scatterings by the walls and the internal (kinetically appearing) barrier of the well just like a classical particle, only the scattering probabilities are quantum mechanical. Using the probabilities from an elemantary scatering picture54 we have the following flipping probabilities P for the i-th spin in different possible situations used in our Monte Carlo simulation: I. If the i 1-th spin is up and the i-th spin is also up, then P 1. II. If the i 1-th spin is up and the i-th spin is down, then (a) P 0 for Γ 2h, and (b)P min&1, 4Γ Γ 2h12 Γ Γ 2h2 ', for Γ 2h. is down and the i-th spin is up then P III. If the (i 1)-th spin 12 min&1, 4Γ Γ 2h Γ Γ 2h2 g 2 '. is up then (a) P 0 IV. If the (i 1)-th spin is down and the i-th spin 12 Γ Γ 2h2 g 2 ' for Γ 2h, and (b)P min&1, 4Γ Γ 2h for Γ 2h (h and Γ denoting the magnitudes only). Here g χ2 a, χ and a being respectively the height and width of the barrier representing the kinetic constraint. The above expressions for P are actually the transmission coefficients in respective cases of one-dimensional scattering across asymmetric barrier or step (according to the form of the potential encountered in passing from one well to the other, see e.g.,54 ). Similarly, in thermal case, we start with a high initial temperature T0 and reduce it eventually following an exponentially decreasing temperature schedule given by T T0 exp tτC ; τC being the time constant for the thermal annealing schedule. Here, when i 1-th spin is down, the flipping probability for the i-th spin ( exp χT ). Otherwise, it flips with
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probability P 1 if it were in the up state, and with Boltzmann probability P exp hT if it were in the down state. In the simulation,20 N Ising spins (Si 1, i 1, ..., N ) were taken on a linear chain with periodic boundary condition. The initial spin con figuration is taken to be random such that magnetization m 1N i Si is practically negligible (mi 0). We then start with a tunneling field Γ0 and follow the zero temperature (semi-classical) Monte Carlo scheme as mentioned above, using the spin flip probabilities P ’s appropriate for the four cases I-IV. Each complete run over the entire lattice is taken as one time unit and as time progresses, Γ is decreased from its initial value Γ0 according to Γ Γ0 etτQ .
1
1
m
0.8
0.6
Classical Annealing
0.75
6
with τC = 10
Quantum Annealing with τQ=1800
0.5
0.25
m
0
0
t
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30000
0.2
0
1e+06
2e+06
3e+06
4e+06
t
5e+06
6e+06
7e+06
8e+06
Fig. 14.6. Comparison between classical and quantum annealing for a chain of 5 104 spins (for the same initial disordered configuration with mi 103 ). We show the 2 6 1.8 10 (for quantum) and τC 10 (for classical) with h 1; a results for τQ lower τC would not produce substantial annealing. Starting from the same initial values T0 100, (and g 100 in the quantum case) we observe that classical annealing Γ0 requires about 107 steps, whereas quantum annealing takes about 104 steps for achieving 0.92. the same final order mf
The results of thermal and quantum annealing are compared in Fig. 14.6 for the same order of initial value and time constant for Γ and T (barrier height χ is taken to be 1000 in both the cases while g was taken to be 100 in the quantum annealing case, or equivalently the barrier width a is taken to be of the order of 0.1). It is observed that to achieve a similar degree of annealing (attaining a certain final magnetization mf ), starting
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from the same disordered configuration, one typically requires much smaller τQ compared to τC ; typically, τC 103 τQ for equivalent annealing (for similar optimal values of final order mf 0.92). For annealing with final order mf 1, we find τC 104 τQ . This comparison of course depends on the barrier characteristics (value of g). 14.7. Quantum Quenching Instead of annealing, one can quench such a systems by tuning the transverse field from Γi Γc to Γf Γc and follow the relaxation dynamics after that. Here Γc is the critical value of the transverse field for the system. This can help preparing the the state of the system with substantial order in a very short time. Recently, quantum quenching dynamics in different systems, in particular for quenches across a quantum critical point, has been studied extensively.11,20 The (exact) results for the after-quench dynamics of an infinite range Ising model in transverse field (,21 see Appendix B) is shown in Fig. 14.7. The results indicate that for a purely quantum system (small S), one might reach a non-stationary oscillatory state with a substantial value for the long-time average (over the largest timescale) of the order Siz 2 t if the quenching is done from Γi Γc to Γf Γc 2. Thus one gets a state (though non-stationary) which has considerable order, in just one step quenching. Interesting results are coming up due to a recent upsurge in the activity in this field69 following the line of Kibble and Zurek44,79 on the defect distribution (i.e., the error in reaching the ground state) for finite time quenching (fast annealing).
14.8. Convergence of Quantum Annealing Algorithms Here we briefly summarize some important recent results derived by Morita and Nishimori51,53 on the convergence of QA algorithms for TIM systems. The results are valid for both the Quantum Monte Carlo and the real-time Schr¨ odinger evolution versions of QA. The Hamiltonian here is the same as given in (14.10) with a time dependence in the transverse field Γ Γ t. No assumption regarding either on the nature of distribution of Jij or the spatial dimensionality is required. In order to perform QA at temperature T using PIMC, one constructs the Suzuki-Trotter equivalent (see Appendix A) d 1 dimensional classical
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z 2 as a function of Fig. 14.7. Plot of the long-time average (see Appendix B) O Stot Γf J for different S. In the plot the solid (blue), dotted (black), dash-dotted (green) and 50, S 100, S 200 the dashed (red) lines represents respectively the results for S and S 500. O peaks around Γf J 0.25 and the peak value decreases with increasing S. For all plots we have chosen Γi J 2.
system of the d dimensional quantum system, and the resulting (classical) system is simulated using a suitable inhomogeneous Markov Chain. The transverse field Γ t is tuned from a high value to zero in course of simulation. It can be shown that at the end of the simulation the final distribution converges to the ground state of the classical part of the Hamiltonian irrespective of the initial distribution (strong ergodicity) if 1 1 , (14.24) Γ t M T tanh t 22RL where M is the number of Trotter replica in the d 1 dimensional SuzukiTrotter equivalent system. Here R and L are constants depending on the system size N , the spin-flip dynamics appointed for the simulation, the temperature T , etc. For large t the bound reduces to Γ t M T
1 . t 22RL
(14.25)
It is remarkable that in contrast to the inverse logarithmic decay of temperature required for convergence of CA (see Eq. 14.8), QA requires only a power-law decay of the transverse field. In this sense, quantum annealing is
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much faster compared to the classical annealing in general for TIM. Similar result can be derived for more general form of Hamiltonian.51 However, the advantage gained here does not change the complexity class of an NPcomplete problem, since RL is of the order of N , and hence the convergence time is thus still exponential in N . For real-time Schr¨ odinger dynamics at T 0, the bound for the decay of the transverse field is again of the form Γ t ξt 2N 1 , 1
(14.26)
where ξ is exponentially small in N for large N . Though the dynamics of PIMC-QA and real-time Schr¨ odinger evolution are completely different, the power-law bound on the annealing schedule is strikingly similar. 14.9. Summary and Discussions Unlike the gate based quantum computers (see, e.g.,24,32,55 the annealing of a physical system towards the optimal state (encoded in the ground state of the final Hamiltonian) in the classical limit naturally achieves analog quantum computation. As discussed here, quantum mechanical tunneling through classically localized states in glasses has opened up this new paradigm for analog quantum computation of hard optimization problems through adiabatic tuning of quantum fluctuations. We reviewed here the recent success in annealing, or optimizing the cost functions of complex systems utilizing quantum fluctuations, rather than the thermal fluctuations (see66 for a more technical review). As mentioned already, following the early indication in60 and the pioneering demonstrations, theoretically by3,27,29,43 and,65 and experimentally by Brooke et al.,10 the quantum annealing technique has now emerged as a successful technique for optimization of complex cost functions. The literature, exploring its success and also its limitations, is also considerably developed at present. These are introduced here through discussions on the mapping of such hard problems to classical spin glass problems, discussions on quantum spin glasses, and consequent annealing. Physics of classical spin glasses (see Sec. II) offers us the knowledge of the landscape structure of the energy or the thermodynamic potentials and its landscape structure (RS and RSB features). Mapping of computationally hard problem like the traveling salesman etc. problems to their corresponding classical spin glass models also helped understanding their complexity (Sec. III).
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Presence of quantum fluctuations in quantum spin glasses induce (see Sec. IV) the possibility of tunneling through infinitely high but narrow barriers in the classical energy landscape. This shows that quantum dynamics can be more ergodic (in contrast to classical or thermal ones) and explore the energy landscape much better. The possibility of such quantum tunneling through classically impenetrable barriers in such quantum glasses leads to quantum annealing, a device for constructing general heuristics to find approximate solutions of hard optimization problems. While simulated annealing employs the strategy of slow cooling, physical or simulational, to find the ground state of glassy systems, quantum annealing employs quantum fluctuations (see Sec. V). As mentioned before, this effectively tunes the Planck’s constant to zero to reach the classically optimized (minimum cost) state. This tuning, when done completely adiabatically, assures the system to be found in the ground state of the classical glass at the end (provided there is no crossing of energy levels with the ground state in the course of evolution, and one has started initially with the ground state of the Hamiltonian). This has already been realized experimentalist (see Sec. V(c)), where the ground state of a glassy sample is reached faster by tuning the external field (inducing changes in the tunneling field) rather than tuning the temperature. In this way analog quantum computation is realized through a novel route. Recently an equivalence between the adiabatic QA and standard gate-based quantum computation has also been establised (Aharonov et al. 2007, Mizel et al. 2007). Acknowledgments We are grateful to A. Chakrabarti, P. Ray, D. Sen, K. Sengupta and R. B. Stinchcombe for their collaborations at different stages of this work. We acknowledge G. Aeppli, A. Dutta, H. Nishimori, G. Santoro, P. Sen and E. Tosatti for useful discussions. References 1. Aeppli G. and Rosenbaum T. F. (2005), in Das and Chakrabarti (2005), pp 159-169 2. Aharonov D., van Dam W., Kempe J., Landau Z., Lloyd S. and Regev O. (2007) SIAM Journal of Computing, 37 166 3. Amara P., Hsu D., and Straub J. E., (1993) J. Phys. Chem. 97 6715 4. Barahona F. (1982), J. Phys. A 15 3241
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Appendix A.1. Suzuki-Trotter Formalism Here we illustrate this formalism by applying it to a TIM having a Hamiltonian H Γ
N
Six
i 1
i,j
Jij Siz Sjz
Hkin HC
(14.27)
The canonical partition function of H reads Z
T reHkin HC T .
Now we apply the Trotter formula M
exp A1 A2 lim exp A1 M exp A2 M M
,
even when A1 , A2 " 0. On application of this, Z reads
Z lim si exp Hkin M T M
exp
i M
HC M T
si .
(14.28)
Here si represent the i-th spin configuration of the whole system, and the above summation runs over all such possible configurations denoted by i. Now we introduce M number of identity operators 2
N
I
i
si,k si,k ,
k
1, 2, ...M.
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in between the product of M exponentials in Z, and have Z
lim T r
M
exp
M 4
S1,k ...SN,k exp
MT
S1,k ...SN,k exp
k 1
S1,k 1 ...SN,k 1 ,
and periodic boundary condition would imply SN 1,p M 4
MT
k 1
HC
Hkin
1 Jij Siz Sjz M T i,j
S1,p . Now,
S1,k 1 ...SN,k 1
N M
Jij Si,k Sj,k , exp MT i,j 1 k1 where Si,k
1
are the eigenvalues of S z operator. Also,
Γ x S1,k ...SN,k exp S S1,k 1 ...SN,k 1 MT i i k1 N2M 1 2Γ sinh 2 MT N M 1 Γ ln coth exp Si,k Si,k 1 , 2 M T i1 k1 M 4
giving the effective classical Hamiltonian (14.13), equivalent to the quantum one in (14.27). It may be noted from above equation that M should be at the order of 1T ( 1) for a meaningful comparison of the interaction in the Trotter direction with that in the original Hamiltonian. For T 0, M , and the Hamiltonian represents a system of spins in a (d+1)-dimensional lattice, because of the appearance of one extra label k for each spin variable. Thus corresponding to each single quantum spin variable Si in the original Hamiltonian we have an array of M number of classical replica spins Sik . This new (time-like) dimension along which these classical spins are spaced is known as Trotter dimension.
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A.2. Quantum quenching of a long range TIM Let us consider a system of spin- 12 objects governed by the Hamiltonian H
N J z z S S N ij i j
Γ
N
Six .
(14.29)
i
It can easily be rewritten as J 2 x H S z ΓStot , (14.30) N tot z z x x z 2 where Stot i Si , Stot i Si and a constant J 2N i Si J 8 from H in (14.30). The above Hamiltonian can again be cast into the simplified form H h.Stot ,
(14.31)
where h Jmˆ z Γˆ x, giving the mean field (exact in this long-range limit) equation h.ˆ Jm z h z tanh , (14.32) m Stot T 2 Γ2 J 2 h at T 0 and zˆ, x ˆ denote unit vectors along z and x directions respectively. This gives m 0 for Γ Γc and m " 0 for Γ Γc J 2. Since the model is infinite ranged one, the mean field approximation becomes exact in terms of its polar components as S and one can readily express S S sin θ cos φ, sin θ sin φ, cos θ, S being the total angular momentum. One dS h. For can immediately utilize the classical equation of motion dT S an exact analysis of this dynamics, considering the equation for the z and x components, we get from above J dφ dθ Γ sin θ and cos θ Γ cot θ cos φ (14.33) dt dt 2 Here we have S N 2. If the system is now quenched from above its quantum critical point Γ Γc , finally to a Γf Γc , then one can write, equating the energies of the states with and without any order respectively, J cos2 θ Γf sin θ cos φ. 4 Using this, one gets from Eq. (14.33) Γ2f sin2 θ Γf J 4 cos2 θ2 dθ f θ . dt sin θ Γf
(14.34)
(14.35)
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This has zeros (turning points) at θ1 sin1 1 4Γf J and θ2 π 2. z 2 Stot cos2 θ N D, where N can therefore obtain
One
θ2 θ2 2 θ1 dθ cos θ f θ 4 8Γf J 2Γf J and D θ1 dθ f θ , giving a behaviour shown in Fig. 14.7.
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Chapter 15 Liouville Gravity from Einstein Gravity
D. Grumiller and R. Jackiw Center for Theoretical Physics, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139, USA We show that Liouville gravity arises as the limit of pure Einstein gravity in 2 ε dimensions as ε goes to zero, provided Newton’s constant scales with ε. Our procedure – spherical reduction, dualization, limit, dualizing back – passes several consistency tests: geometric properties, interactions with matter and the Bekenstein-Hawking entropy are as expected from Einstein gravity.
Contents 15.1 15.2 15.3
Introduction . . . . . . . . . . . . . . . Constructing the Limiting Action . . . Consistency Checks . . . . . . . . . . . 15.3.1 Geometric properties . . . . . 15.3.2 Does matter curve geometry? 15.3.3 Entropy and coupling constant 15.3.4 Miscellaneous further checks . 15.4 Boundary Terms and Summary . . . . References . . . . . . . . . . . . . . . . . . .
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15.1. Introduction Gravity in or near two dimensions has many manifestations. Gravity in 2 ε dimensions serves as a toy model for quantum gravity and is known to be asymptotically safe.1 The limit of the Einstein-Hilbert action, lim I2 ε
0
ε
lim
1
0 κ2 ε
ε
d2 ε x g R ,
(15.1)
essentially yields the Euler characteristic, scaled by the gravitational coupling constant κ2 ε 16πG2 ε , where G2 ε is Newton’s constant. Taking 331
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into account 1-loop effects makes the limit ε 0 non-trivial, because the relation between bare and renormalized coupling involves a term 1ε.1–4 Also gravity in two dimensions serves as a toy model for quantum gravity and black hole evaporation. The first formulation of 2-dimensional gravity is due to Jackiw and Teitelboim (Bunster).5,6 Since then many similar models have been introduced, for instance the CGHS model7 or the Witten black hole.8 All of them are special cases of general dilaton gravity 1 I2dg d2 x g φR U φ ∇φ2 V φ . (15.2) κ ˆ Here φ is a scalar field, the dilaton. For a review cf. e.g.9 A comprehensive list of potentials U and V can be found in.10 A related 2-dimensional gravity model is Liouville gravity, b 1 d2 x g Q ΦR ∇Φ2 4πµe2bΦ , (15.3) IL 4π
where Q b b1 and b, µ are constant. The fields Φ and g, as well as the action 15.3, may have various interpretations. If metric and scalar field are dynamical fields then the Liouville model 15.3 is a special case of dilaton gravity 15.2 with constant U and exponential V . This is the case of relevance for our present work.a An interesting limit of 15.3 is obtained as b tends to zero. By virtue of the redefinition Φ Qφ the action 15.3 in the limit of b 0 can be brought into the form 1 d2 x g φR ∇φ2 λe2φ , (15.4) IL κ ˆ where λ 4πµQ2 and κ ˆ 4π Q2 have to be rescaled in such a way that they remain finite in the limit. The action 15.4 is recognized as a special case of 15.2 (UL 1, VL λe2φ ) and coincides with the Liouville theory studied e.g. in13 for its Weyl transformation properties. It is the purpose of this paper to consider the limit in 15.1, but rather than keeping Newton’s constant fixed we scale it such that κ2 ε ε. Because the limiting action 15.1 is effectively vanishing as far as bulk properties are concerned, this rescaling of κ2 ε leads to an indeterminancy of the form 00, which is capable to yield an interesting bulk action. However, to make sense of such a limit we need something like a l’Hospital rule. In order to establish such a rule we are guided by the following observation: Einstein gravity in D dimensions exhibits D D 32 graviton modes, yielding at a We shall recall another interpretation arising in the context of conformal field theory and string theory11,12 in the body of this paper.
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D 2 a negative number of graviton modes, which is difficult to interpret. It would be more convenient if, as D is varied to 2, the traversed number of degrees of freedom were positive and did not change continuously. We can achieve this by restricting to the s-wave sector of Einstein gravity, because it exhibits precisely zero propagating physical degrees of freedom, regardless of the dimension. Spherical symmetry implies the existence of D 2 D 12 Killing vectors, which certainly is a strong restriction for large D. However, as D 2 is approached the number of Killing vectors required for spherical symmetry drops to zero, so our Ansatz of restricting to the s-wave sector does not lead to any symmetry constraints on the limiting geometry. This makes it plausible that our procedure captures all essential features. The convenient trick of restricting to the s-wave sector is not sufficient to establish a meaningful limit ε 0, but it allows to exploit properties unique to 2-dimensional dilaton gravity 15.2. In particular, we shall employ a certain duality14 that renders the limit well-defined. Because the duality is involutive we shall dualize back after taking the limit and obtain in this way a non-trivial limit of 15.1, which turns out to be the Liouville action 15.4. Our work is organized as follows. In Section 15.2 we establish the l’Hospital rule as outlined above and obtain the limiting action. We perform several consistency checks in Section 15.3: we demonstrate that the geometric properties are reasonable, that our limiting action is consistent with the standard folklore that “matter tells geometry how to curve”, and that the Bekenstein-Hawking entropy is consistent with the scaling behavior of Newton’s constant. We consider boundary terms and summarize in Section 15.4. 15.2. Constructing the Limiting Action We start with the Einstein-Hilbert action in D dimensionsb 1 ID dD x g D RD , κD M
(15.5)
and make a spherically symmetric Ansatz for the line-element, ds2 b We
D dxµ dxν
gµν
α
gαβ dx
dxβ
1 2D2 dΩ2SD2 , φ λ
(15.6)
omit boundary terms for the time being, because they can be constructed unambiguously once the limit of the bulk action is known. We shall add them in Section 15.4.
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where &µ, ν ' : &1, . . . , D', &α, β ' : &1, 2', dΩ2SD2 is the line-element of the round D 2-sphere and λ is a parameter with dimension of inverse length squared, which renders the scalar field φ dimensionless. The latter – often called “dilaton” – and the 2-dimensional metric gαβ both depend on the two coordinates xα only. We parameterize the dimension by D 2 ε, with the intention to take ε 0 in the end, but for the time being ε need not be small. Inserting the Ansatz 15.6 into 15.5 and integrating over the angular part obtains a 2-dimensional dilaton gravity action 1 1 ε 2 ε 2 12ε d x g φR ∇φ λε 1 εφ , (15.7) I2dg κ M εφ with a gravitational coupling constant given by κ
κ2 ε λε2 ASε
κ2 ε λε2
Γ
1 ε 2 2
2π 2 2 1
ε
,
(15.8)
where the surface area ASε comes from integration over the ε-dimensional unit sphere. The appearance of the dimensionful constant λ in the action and the coupling constant is a consequence of the Ansatz 15.6. It will survive the limit and arises because in 2 ε dimensions Newton’s constant is dimensionful. So far we have not achieved much: the limit ε 0 of 15.7 still either is undefined or trivial, even after suitable rescalings of φ, κ and λ. This is so, because any rescaling that makes the kinetic term in 15.7 finite automatically scales the φR-term to zero. Even a dilaton-dependent Weyl rescaling does not help: the conformal factor becomes singular in the limit, so calculating quantities for conformally related models and taking the limit ε 0 there is possible, but leads to singular quantities in the original formulation and therefore is pointless. However, we note that the solutions for the metric to the equations of motion following from 15.7, 1 dr2 , (15.9) ds2 λ 2a r1ε dτ 2 λ 2a r1ε possess a well-defined limit. The constant of motion a essentially is the ADM mass. We have used Euclidean signature and diagonal gauge to represent 15.9, but of course our statements are gauge independent. Since the limit ε 0 of 15.9 is accessible there is a chance that another action – leading to the same solutions 15.9 – permits a meaningful limit. With this in mind we exploit now a duality discovered in.14 The original action 15.7 leads to the same 2-parameter family of line-elements 15.9 as
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solutions of the classical equations of motions as the dual action 1 ε ˜ 2a 1 εφ˜ε . I˜2dg d2 x g φR κ ˜ M
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(15.10)
In the original formulation 15.7 λ is a dimensionful parameter in the action and a emerges as constant of motion. In the dual formulation 15.10 the respective roles are reversed. The dual dilaton field is related to the original ˜ is arbitrary. one by φ˜ φ1ε . The dual coupling constant κ It is straightforward to take the limit ε 0 of 15.10. 1 ε ˜ 2a I˜C : lim I˜2dg d2 x g φR (15.11) ε 0 κ ˜ M The action I˜C coincides with the geometric part of the CGHS action.7 Thus, we have succeeded to obtain a well-defined non-trivial (dual) limit. Because the duality is involutive,14 we now dualize I˜C and obtain in this way the result we are seeking. The action dual to I˜C is given by 1 d2 x g φR ∇φ2 λe2φ . (15.12) IL κ ˆ M The coupling constant κ ˆ is arbitrary, and we shall exhibit its relation to κ in Section 15.3. The action 15.12 is recognized as the Liouville action 15.4. Therefore, following the l’Hospital rule established in this Section, the limiting action 15.1 is the Liouville action IL . This is our main result. We discuss now some features of 15.12. The range of the dilaton φ naturally is , . This, however, is not the case for spherically reduced gravity 15.7 where φ must be non-negative. We shall see the geometric reason for this extension of the range in the next Section. An interesting property of 15.12 is the invariance of the scalar field equation under local Weyl rescalings, gµν
e2σ gµν ,
φ φσ.
(15.13)
The action 15.12 also is invariant (up to boundary terms and rescalings of λ) independently under global Weyl rescalings and constant shifts of the dilaton, gµν
e2σ0 gµν ,
φ φ σ1 .
(15.14)
With the redefinitions ϕ 2φ and m2 2λ we can represent the action 15.12 as 1 1 d2 x g ϕR ∇ϕ2 m2 eϕ . (15.15) IL 2ˆ κ M 2
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Up to notational differences this coincides with the Liouville action considered e.g. in,13 where further properties of 15.12 are discussed. For m 0 our action 15.15 coincides with the one proposed by Mann and Ross15 as the D 2 limit of General Relativity (cf. also16 ). Their construction employs a Weyl transformation in 2 ε dimensions ( : g µν ∇µ ∇ν ) ˜ eΨ R 1 ε Ψ ε 1 ε ∇Ψ2 , (15.16) R g˜µν eΨ gµν , 4 an ad-hoc subtraction between original and transformed action 1 ˜ , IMR : d2 ε x g R g ˜R (15.17) κ2 ε M and the same rescaling of the gravitational coupling constant employed in the present work, ε 1 d2 ε x g ΨR ∇Ψ2 O ε (15.18) IMR 2κ2 ε M 2 1 1 d2 x g ΨR ∇Ψ2 as ε 0 . (15.19) 2ˆ κ M 2 Thus their ε 0 limiting action 15.19 differs from our 15.12 in that the Liouville exponential is missing. As mentioned in the introduction the metric g and scalar field φ in 15.12 are dynamical fields. For sake of completeness and to avoid confusion we recall here another interpretation of 15.12. In the approach of 17 the emergence of the Liouville action basically comes about as follows. Starting point is the path integral for bosonic strings with flat target-space metric, Z
DgDX e 8π 1
d2 x
g gµν !µ X a !ν X b ηab µ0 ,
(15.20)
where the measure contains the ghost- and gauge-fixing part. The kinetic term for the target-space coordinates is classically invariant under Weyl rescalings of the world-sheet metric g e2σ g, but the measure is not, DX
DX eIL ,
(15.21)
where IL is the Liouville action 15.12, with φ replaced by σ (the coupling κ ˆ depends on the dimension d of the target-space). Analogous considerations apply to the ghost measure, so that for critical strings, d 26, the Liouville contribution to the action generated by 15.21 cancels. For noncritical strings, d " 26, the Liouville contribution survives and is crucial to restore conformal invariance at the quantum level.c We emphasize that c Actually,
the full story is more complicated, involves a conjecture and eventually leads to 15.3, which generalizes 15.12; cf.12 for a review.
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in the approach mentioned in this paragraph the metric (gauge-fixed to conformal gauge) is non-dynamical, while the field φ plays the role of the conformal factor.d This is quite different from the interpretation of 15.12 in the present work, where metric and scalar field are dynamical. 15.3. Consistency Checks 15.3.1. Geometric properties The classical solutions of the equations of motion descending from the Liouville action 15.12 are given by 1 2 dr , φ ln r (15.22) ds2 λ 2a r dτ 2 λ 2a r This is compatible with 15.9 in the ε 0 limit. It should be noted that the coordinates in 15.22 and 15.9 have somewhat unusual physical dimensions: τ has dimension of length squared and r is dimensionless. Exactly like 15.9 the line-element 15.22 exhibits a Killing vector τ . The Killing norm is normalized to λ at r 0, which may be considered as an “asymptotic boundary” because φ tends to . This brings us back to the issue of the range of φ. It is now clear why the limiting geometry 15.22 implies that φ can be also negative: if we restricted φ to positive values, we would impose a cutoff r0 1 on the radial coordinate r. But this cut-off would be artificial, as neither geometry nor dilaton field exhibit any pathological behavior there. Only by allowing φ , is it possible to achieve r 0, . There is a subtlety regarding singularities: even for arbitrarily small ε the line-elements 15.9 have a curvature singularity at r 0. The limiting solution 15.22, however, does not exhibit any curvature singularity. Instead, it is the dilaton field that becomes singular as r 0 is approached. We assume from now on that λ and a are positive. Then for Lorentzian signature there is a Killing horizon at φ φh , where λ . (15.23) φh ln 2a This is consistent with 15.9 which for Lorentzian signature also exhibits a Killing horizon for positive λ and a. The constant of motion a here plays the d Alternatively,
it is also possible to interpret 15.3, supplemented by a kinetic term for the target-space coordinates, as a sigma model, i.e., a critical string theory. Then Φ is the dilaton, g is the world-sheet metric, the target-space metric is flat and the term proportional to µ comes from a non-trivial tachyon background, cf. e.g.12 Another possibility was studied in:18 the metric is dynamical and a specific Liouville action of type 15.3 arises directly from the Polyakov action upon first integrating out the targetspace coordinates, and then ’integrating in’ an auxiliary scalar field Φ.
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role of a Rindler acceleration, except that its dimension is one over length squared rather than one over length. The associated Unruh temperature can be derived in various standard ways, e.g. from surface gravity. The result is TU
a 12 λ . 2π
(15.24)
The appearance of λ12 guarantees that the Unruh temperature TU has the correct dimension of one over length. The same result can also be obtained from calculating the Hawking temperature associated with 15.9 for positive ε, and taking the limit ε 0 in the end. We have thus demonstrated that the limiting solutions 15.22 are consistent with the family of solutions 15.9. 15.3.2. Does matter curve geometry? According to standard folklore “matter tells geometry how to curve”. In particular, in the absence of matter spacetime should be Ricci-flat. Rµν
,
0
Tµν
0
(15.25)
This is certainly the case for Einstein gravity. Consistently, it should be true also for the limiting action 15.12. There is a complication, however. We have to consider the dilaton as part of the geometry, because the result 15.12 arises as the limit 15.1 of a purely geometric action. Therefore, by “matter” we always refer to some additional (physical) degrees of freedom, like scalar or Fermi fields, which we denote by ψi . The total bulk action is given by Itot g, φ, ψi IL g, φ Imat g, φ, ψi ,
(15.26)
and the energy momentum tensor is constructed from Imat in the usual way, Tµν
2
g
δImat . δg µν
(15.27)
Its trace is denoted by T : T µ µ . If the matter action depends on the dilaton we also need the definition 1 Tˆ
g
δImat . δφ
(15.28)
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We investigate now whether we can reproduce 15.25 for generic dilaton gravity 15.2, supplemented by some matter action analog to 15.26-15.28. The equations of motion are R
φU
2U ∇µ φ∇ν φ gµν U ∇φ2 gµν V
∇φ2 2U
φ
2∇µ ∇ν φ 2gµν
φV
κ ˆ Tˆ , (15.29)
φ κˆTµν .
(15.30) The trace of the second equation simplifies to 2V
2
φ κˆT ,
(15.31)
which allows to express the first equation as Rκ ˆ Tˆ κ ˆU T
2U V φ U
∇φ2
φV
.
(15.32)
In two dimensions a spacetime is Ricci-flat if and only if the Ricci scalar vanishes. Therefore, in the absence of matter (Tˆ T 0) spacetime is Ricci-flat if and only if 2U V
φU
∇φ2
φV
0.
(15.33)
The condition 15.33 holds only for a very specific class of dilaton gravity models. It is gratifying that the Liouville action 15.12 [UL 1, VL λe2φ ] belongs to this class. Consistently, also its dual, the CGHS action 15.11 [UC φ 0, VC φ const.], belongs to this class. If matter is not coupled to the dilaton then Tˆ 0 and 15.32 for Liouville gravity can be represented as ˆT . R κ
(15.34)
This is as close an analog of Einstein’s equations as possible in two dimensions.6 The simplest example of an energy-momentum tensor is ˆ , which just amounts to the addition of a cosmological Tµν gµν Λκ constant to IL : 1 d2 x g φR ∇φ2 λe2φ Λ (15.35) ILΛ κ ˆ M The equations of motion yield R 2Λ .
(15.36)
Such spacetimes are maximally symmetric, i.e., they exhibit three Killing vectors. We have thus demonstrated that the limiting action 15.12 is consistent with the Einsteinian relation 15.25. Matter indeed “tells geometry how to curve”.
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15.3.3. Entropy and coupling constant The Bekenstein-Hawking entropy is determined by the dilaton evaluated at the horizon.19 For Liouville gravity 15.12 we obtain S
λ 4π ln κ ˆ 2a
S0
4π ln a . κ ˆ
(15.37)
Here S0 is independent from a, the only constant of motion. On the other hand, the Bekenstein-Hawking entropy for the Schwarzschild black hole in 2 ε dimensions is given by 4π S˜ M εε1 κ2 ε
2πε S˜0 ln M κ
O
ε2 ,
(15.38)
where S˜0 is independent from M , the only constant of motion, and κ is defined in 15.8. Note that the physical units of M are irrelevant here because any change of mass units only shifts S˜0 . The expressions 15.37 and 15.38 are essentially equivalent upon relating the respective constants of motion by a M . To achieve quantitative agreement we must identify εˆ κ 2κ lim κ2 ε .
0
ε
(15.39)
For any finite choice of κ ˆ the relation 15.39 is perfectly consistent with our starting point, the assumption that κ2 ε in 15.1 scales with ε. The factor 2 in the middle equation 15.39 comes from the 0-sphere (which consists just of two points). We have thus demonstrated that the limit 15.1 with κ2 ε ε is consistent with the behavior of entropy 15.37. 15.3.4. Miscellaneous further checks In our paper on the duality14 we considered in detail a 2-parameter family of actions that included spherically reduced gravity from any dimension D. We found there that the dual model is conformally related to spherically ˜ 2D 3 D 2. For D 2 reduced gravity from a dual dimension D ˜ from above we obtain D . Thus, for consistency our Liouville gravity action 15.12 should be dual to a model that is conformally related to spher˜ dimensions. But we know already that ically reduced gravity from D 15.12 is dual to 15.11, so it remains to be shown that 15.11 is conformally ˜ dimensions. Taking the related to spherically reduced gravity from D limit ε in 15.7 and rescaling λ appropriately yields an action of type
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15.2 with UW 1φ and VW φ. This is recognized as the “Witten black hole” action,8 and it is indeed conformally related to the CGHS action 15.11. 15.4. Boundary Terms and Summary We extend now our results as to include boundary terms, i.e., we consider the limit 1 2 ε 1 ε d x g R 2 d x γ K . ΓL : lim Γ2 ε lim ε 0 ε 0 κ2 ε M !M (15.40) Again κ2 ε is supposed to scale proportional to ε; M is a Riemannian manifold in 2 ε dimensions with metric gµν and R is the Ricci scalar; M is the boundary of M with induced metric γµν and K is the extrinsic curvature. Rather than attempting to perform similar steps as above for the boundary action we take a shortcut. Since we know already that the correct bulk action is given by 15.12 we simply supplement the latter by the appropriate boundary terms. They comprise the dilaton gravity analog of the Gibbons-Hawking-York boundary term and a Hamilton-Jacobi counterterm.20 The full limiting action is given by ΓL
1 κ ˆ
d2 x g φR ∇φ2 λe2φ
M
2 κ ˆ
dx
!M
γXK
2 κ ˆ
dx
!M
γ
λ eφ . (15.41)
Our procedure is summarized in the following diagram:
e The
e
only step not discussed so far is the one called “oxidation”, the inverse procedure of “reduction”. Since reduction means integrating out the ε-sphere, oxidation after taking 0 implies “integrating in” the 0-sphere. But this just amounts to rescaling the limit ε the coupling constant κ ˆ by a factor of 2 and therefore is trivial.
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The arrows decorated with a question mark refer to our inability to construct directly a meaningful limit ε 0. Therefore, we started with the upper left corner, proceeded with the steps indicated above the arrows to the upper right corner, then took the limit to the lower right corner, and finally inverted all steps (as indicated below the arrows) to arrive at the lower left corner. Now it is evident how the various approaches to gravity near two dimensions [Eq. 15.1] or in two dimensions [Eqs. 15.2-15.4] are connected. Thus we conclude that the closest analog to the Einstein-Hilbert action in two dimensions is Liouville gravity15.41. Acknowledgments DG thanks Robert Mann for discussions. This work is supported in part by funds provided by the U.S. Department of Energy (DOE) under the cooperative research agreement DEFG0205ER41360. DG has been supported by the Marie Curie Fellowship MCOIF 021421 of the European Commission under the Sixth EU Framework Programme for Research and Technological Development (FP6). References 1. S. Weinberg in General Relativity, an Einstein Centenary Survey, S. Hawking and W. Israel, eds. Cambridge University Press, 1979. 2. R. Gastmans, R. Kallosh, and C. Truffin, “Quantum gravity near twodimensions,” Nucl. Phys. B133 (1978) 417. 3. S. M. Christensen and M. J. Duff, “Quantum gravity in two + epsilon dimensions,” Phys. Lett. B79 (1978) 213. 4. H. Kawai, Y. Kitazawa, and M. Ninomiya, “Scaling exponents in quantum gravity near two-dimensions,” Nucl. Phys. B393 (1993) 280–300, http:// www.arXiv.org/abs/hep-th/9206081hep-th/9206081. 5. R. Jackiw, “Liouville field theory: A two-dimensional model for gravity?,” in Quantum Theory Of Gravity, S. Christensen, ed., pp. 403–420. Adam Hilger, Bristol, 1984. 6. C. Teitelboim, “The Hamiltonian structure of two-dimensional space-time and its relation with the conformal anomaly,” in Quantum Theory Of Gravity, S. Christensen, ed., pp. 327–344. Adam Hilger, Bristol, 1984. 7. C. G. Callan, Jr., S. B. Giddings, J. A. Harvey, and A. Strominger, “Evanescent black holes,” Phys. Rev. D45 (1992) 1005–1009, http://www.arXiv. org/abs/hep-th/9111056hep-th/9111056. 8. E. Witten, “On string theory and black holes,” Phys. Rev. D44 (1991) 314– 324.
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9. D. Grumiller, W. Kummer, and D. V. Vassilevich, “Dilaton gravity in two dimensions,” Phys. Rept. 369 (2002) 327–429, http://arXiv.org/abs/ hep-th/0204253hep-th/0204253. 10. D. Grumiller and R. Meyer, “Ramifications of lineland,” Turk. J. Phys. 30 (2006) 349–378, http://www.arXiv.org/abs/\breakhep-th/ 0604049hep-th/0604049. 11. P. Ginsparg and G. W. Moore, “Lectures on 2-d gravity and 2-d string theory,” http://arXiv.org/abs/hep-th/9304011hep-th/9304011. 12. Y. Nakayama, “Liouville field theory: A decade after the revolution,” Int. J. Mod. Phys. A19 (2004) 2771–2930, http://www.arXiv.org/abs/hep-th/ 0402009hep-th/0402009. 13. R. Jackiw, “Weyl symmetry and the Liouville theory,” Theor. Math. Phys. 148 (2006) 941–947, http://www.arXiv.org/abs/hep-th/ 0511065hep-th/0511065. 14. D. Grumiller and R. Jackiw, “Duality in 2-dimensional dilaton gravity,” Phys. Lett. B642 (2006) 530–534, http://www.arXiv.org/abs/hep-th/ 0609197hep-th/0609197. 2 limit of general relativity,” 15. R. B. Mann and S. F. Ross, “The D Class. Quant. Grav. 10 (1993) 345–351, http://www.arXiv.org/abs/gr-qc/ 9208004gr-qc/9208004. 16. J. P. S. Lemos and P. M. Sa, “The Two-dimensional analog of general relativity,” Class. Quant. Grav. 11 (1994) L11, gr-qc/9310041. 17. J. Distler and H. Kawai, “Conformal field theory and 2-d quantum gravity or Who’s afraid of Joseph Liouville?,” Nucl. Phys. B321 (1989) 509. 18. L. Bergamin, D. Grumiller, W. Kummer, and D. V. Vassilevich, “Classical and quantum integrability of 2D dilaton gravities in Euclidean space,” Class. Quant. Grav. 22 (2005) 1361–1382, http://www.arXiv.org/abs/hep-th/ 0412007hep-th/0412007. 19. J. Gegenberg, G. Kunstatter, and D. Louis-Martinez, “Observables for twodimensional black holes,” Phys. Rev. D51 (1995) 1781–1786, http://www. arXiv.org/abs/gr-qc/9408015gr-qc/9408015. 20. D. Grumiller and R. McNees, “Thermodynamics of black holes in two (and higher) dimensions,” JHEP 04 (2007) 074, http://www.arXiv.org/abs/ hep-th/0703230hep-th/0703230.
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Chapter 16 Exact Static Solutions of a Generalized Discrete φ4
Avinash Khare Institute of Physics, Bhubaneswar, Orissa 751005, India In this article, I compare and contrast various features of the discrete and the corresponding continuum field theory models in 1 1 dimensions. Most of the material in this lecture is based on the work done with Dmitriev, Saxena and Kevrekidis. I point out some of the key issues involved, like loss of Lorentz and translation invariance in the discrete models and its various consequences. In particular, I address the central issue of eventual trapping and pinning of coherent structures in the discrete models and then point out few of the attempts that are being made to overcome this problem. The whole discussion is illustrated through the examples of the celebrated λφ4 field theory as well as nonlinear Schr¨ odinger equation. I also emphasize several common features of a very general discrete λφ4 field theory and a very general discrete Nonlinear Schr¨ odinger equation. I also point out several open questions vis a vis these discrete models. Finally I obtain translationally invariant, static as well as time-dependent, exact solutions of these models including kink, pulse, periodic and short-period solutions.
Contents 16.1 16.2 16.3
16.4
16.5
Why Discrete Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . What are Discrete Models? . . . . . . . . . . . . . . . . . . . . . . . . . . . The Discrete λφ4 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3.1 Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3.2 Two Point Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Generalized DNLS Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.4.1 Conserved Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . 16.4.2 Two-Point Maps for Stationary Solutions . . . . . . . . . . . . . . . Translationally Invariant JEF, Hyperbolic, Trigonometric and Short-Period Static Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.5.1 JEF and Hyperbolic Solutions . . . . . . . . . . . . . . . . . . . . . 16.5.2 Trigonometric Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 345
346 348 349 350 351 352 353 354 354 354 357
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16.5.3 Short-period Solutions . . . . . . . . . . . . . . . . Translationally Invariant Time-Dependent JEF, Hyperbolic, and Short-Period Solutions . . . . . . . . . . . . . . . . . . 16.6.1 Time-Dependent JEF and Hyperbolic Solutions . . 16.6.2 Time-Dependent Trigonometric Solutions . . . . . . 16.6.3 Time-Dependent Short-Period Solutions . . . . . . . 16.7 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . 16.8 Some Open Problems . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.6
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. . . . . . . . . Trigonometric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
358 359 359 360 360 361 363 363
16.1. Why Discrete Models In recent years there has been a growing interest in the analysis of discrete nonlinear models since they play a very important role in many physical applications. For example, the question of mobility of solitonic excitations in discrete media is a key issue in many physical contexts; in particular, the mobility of dislocations, a kind of topological solitons, is of importance in the physics of plastic deformation of metals and other crystalline bodies.1 Similar questions arise in optics for light pulses moving in optical waveguides or in photorefractive crystal lattices (see e.g.,2 for a relevant recent discussion) and in atomic physics for Bose-Einstein condensates moving through optical lattice potentials (see e.g.,3 for a recent review). These issues may prove critical in aspects related to the guidance and manipulation of coherent, nonlinear wavepackets in solid-state, atomic and optical physics applications. One of the important property of the relativistic continuum domain walls (topological defects) is their mobility, i.e. if one obtains a stationary solution, then Lorentz invariance automatically ensures the existence of the corresponding moving solution. Unfortunately, this is no more true in the discrete models and even if one obtains a stationary solution, it is not obvious if there exists a corresponding moving solution. In fact, the discretization even breaks translational invariance, hence in general, in discrete models, stationary kinks can only be centered at countable number of points, usually on a site and midway between two adjacent sites. Only one of the two configurations is stable, which one being highly model dependent. The energy difference between the two configurations (one on a site and the other midway between the two sites) is a kind of potential energy barrier, called Peierls-Nabarro (PN) barrier. This barrier leads to eventual trapping and pinning of coherent structures. Thus one of the central issue in the discrete field theory models is how to avoid this PN barrier. In this context, the translationally invariant
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(TI) discrete models4 have received considerable attention since they admit static solutions that can be placed anywhere with respect to the lattice. Such discretizations have been constructed and investigated for the Kleinodinger equation.16–21 For the Gordon field4–15 and for the nonlinear Schr¨ 5,8 this can be interpreted as the absence of the Hamiltonian TI lattices, Peierls-Nabarro (PN) potential.1 For the non-Hamiltonian lattices, the height of the Peierls-Nabarro barrier is path-dependent but there exists a continuous path along which the work required for a quasi-static shift of the solution along the lattice is zero.12 In general, one can state that coherent structures in the TI models are not trapped by the lattice and they can be accelerated by even a weak external field. This particular property makes the TI discrete models potentially interesting for physical applications and one such physically meaningful model has been recently reported.13 It may be noted that the TI discrete models can support even moving solutions, but only for selected propagation velocities.19 In some cases, the exact static or even moving solutions to the TI models can be expressed explicitly in terms of the Jacobi elliptic functions (JEF). Even in the cases when JEF solutions are impossible, static solutions to a TI model can always be obtained iteratively from a nonlinear map (first integral), solving at each step an algebraic equation. Even as a study in field theory, it is important to study discrete models, since, in case there is fundamental length that would imply that the space is not continuous but has a lattice structure. Normally, in lattice field theory models one is usually interested in the limit when the lattice spacing goes to zero. However, in view of the importance of discrete field theory models, it is also worth considering the results in the discrete models themselves, and not just in the continuum limit. In this article, I shall take this approach and discuss discrete field theory models in their own right specially because of their relevance in several physical applications. In this article, I compare and contrast several features of the continuum and the corresponding discrete field theory models in 1 1 dimensions. As an illustration, I have chosen two examples, one from the relativistic field theory and the other from the nonrelativistic field theory. In particular, I compare and contrast discrete and continuum λφ4 field theory as well as discrete and continuum Nonlinear schr¨ odinger equation. Further, I point out remarkable similarities between a general discrete λφ4 field theory model and a general DNLS model. I also obtain translationally invariant, static
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as well as time-dependent, exact solutions of these models including kink, pulse, periodic and short-period solutions. Some of the issues which I would like to raise and partially answer are: (i) Is there an analogue of Noether’s theorem using which one can find conserved quantities in a given discrete field theory model? (ii) It is worth pointing out here that the Lorentz invariance as well as the translational invariance of the continuum λφ4 field theory, are no more the symmetries of the corresponding discrete model. The question is, if there an analogue of the translational invariance in the discrete case? (iii) One of the useful trick in the continuum λφ4 field theory (and in fact in several models with degenerate minima), is to use Bogomolnyi idea and instead of second order field equation, write down the corresponding first order field equation. The obvious question is if there is a discrete analogue of the Bogomolnyi trick? In particular, instead of the three-point discrete field equation, can one write down a two-point map? (iv) In the continuum case, there is a well known analytic procedure using which one can ensure if a given solution is stable or unstable. Is there a similar procedure in the discrete case? (v) Is there a relation between static TI solutions and the corresponding moving solutions? (vi) In the continuum case, Lorentz invariance ensures that the velocity of propagation cannot exceed the critical velocity. But since translational invariance is lost in the discrete case, one obvious question is whether there is an upper limit for the velocity of propagation? 16.2. What are Discrete Models? In this talk, by discrete field theory models I mean scalar field theory models in which one replaces the second derivative term d2 φdx2 by the corresponding difference operator, i.e. d2 φ dx2
φn 1 φn1 2φn , h2
(16.1)
with h 0 being the lattice spacing. As a result, instead of the continuum (λφ4 ) field equation d2 φ dt2
d2 φ dx2
λφ
1 φ2 ,
(16.2)
in 1 1 dimension, one gets the difference equation d2 φn dt2
φn 1 φn1 2φn h2
λφn f
φn1 , φn , φn 1 ,
(16.3)
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where the only constraint on the function f φn1 , φn , φn 1 is that in the continuum limit it must reduce to λφ3 . Another discrete equation that we shall consider here is the discrete nonlinear Schr¨ odinger (DNLS) equation which follows from the nonlinear Schr¨ odinger (NLS) equation i
du dt
d2 u dx2
2
2u
u 0,
(16.4)
after making the substitution as given by Eq. (16.1). Here u is a complex scalar field and as is well known, for the upper () sign we get the bright soliton (i.e. pulse) while for the lower ( ) sign, one has a dark solition (i.e. kink). On using Eq. (16.1), the corresponding DNLS equation is i
dun dt
un 1 un1 2un f
un1 , un , un 1
0.
(16.5)
Note that following the practice followed in the literature, we have scaled away h2 from the denominator of the second term. Here the function f un1 , un , un 1 must be such that in the continuum limit it reduces to 2u2u, depending on whether we are discussing the attractive or the repulsive case. 16.3. The Discrete λφ4 Model A prototype class of discrete models, relevant to a variety of applications are the so called discrete φ4 models which feature a cubic nonlinearity. It is easily seen that the most general discrete φ4 model with cubic nonlinearity which is invariant under the interchange of φn 1 and φn1 , and which goes over to the continuum φ4 field Eq. (16.2) is given by22 φn
φn 1 φn 1 2φn λφn A1 φ3n A22 φ2n φn 1 φn 1 A3 A5 2 2 φn φ2 n 1 φn 1 A4 φn φn 1 φn 1 2 φn 1 φn 1 φn 1 φn 1 A6 3 2 φ3 n 1 φn 1 ,
h12
(16.6)
with the model parameters satisfying the constraint 6
Ak
λ
(16.7)
k 1
so as to ensure the continuum limit. In Eq. (16.6), φn t is the unknown function defined on the lattice xn hn with the lattice spacing h 0 and overdot means derivative with respect to time t.
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16.3.1. Conservation Laws Unlike Noether’s theorem in the continuum case, we are not aware of any systematic procedure to obtain conserved quantities in the discrete case. However, one can easily verify that momentum P1 defined by P1
φ n φn 1 φn1 ,
(16.8)
n
is conserved in case A1
A3 2 A4
λδ A5
A6
2λγ , A2
λ 1 4γ 4δ ,
(16.9)
Here γ and δ are arbitrary numbers. Thus one has a two-parameter family of non-Hamiltonian TI model which conserves momentum P1 .4 Note that in the continuum limit P1 goes over to the standard momentum operator in the continuum case. On the other hand, if only A4 " 0, then it is easily verified that the corresponding non-Hamiltonian model conserves the following nonstandard momentum operator P2
φ n φn 2 φn2 .
(16.10)
n
If instead A1
A4
4α1 λ ,
A5
A2
0,
6α2 λ,
A3
4α3 λ,
with α1 2α2 α3
A6 1 , 4
2α2 λ, (16.11)
then the field Eq. (16.6) can be obtained from the Hamiltonian
φ 2 n H 2 n
φn φn1 2 2h2
α1 φn α2 φn φn1 4
λ 4
λ 2 φ 2 n
φ2n φ2n1 α3 φ2n φ2n1
(16.12)
and hence energy is conserved in this model. This Hamiltonian model has two free parameters. In the special case when α1 α2 118 and α2 112, this model goes over to the Speight-Ward model5 which is known to support TI kinks.
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16.3.2. Two Point Map One of the useful trick in the continuum case is the so called Bogomolnyi trick, using which, in the static or the stationary problem, instead of the second order field equation, one writes the corresponding first integral which is easily solved compared to the corresponding second order field equation. For example, for the continuum φ4 case, it is well known that the static form of Eq. (16.2) has the first integral 2 λ 1 φ2 C 0, (16.13) φ2x 2 with the integration constant C. The obvious question is, if in the discrete case, one can write a two-point map from which the field Eq. (16.6) can be deduced? Unfortunately, unlike the continuum case, no general procedure is known so far in the discrete case. However, in case one can obtain exact solutions in the form of Jacobi elliptic functions sn, cn, dn, then one can obtain particular two-point map corresponding to each of the solution. On the other hand, there are few cases where the two-point map is already known. For example, in case the parameters Ai satisfy Eq. (16.9) (note this is precisely the model in which momentum P1 as defined by Eq. (16.8) is conserved), then for any value of γ and δ the two point map is given by U φn 1 , φn φn φn1 2 Λφn φn 1 Λγ φ4n φ4n 1 2 1 2 2 2 Λδφn φn 1 φn φn 1 Λ 2γ 2δ φn φn 1 2 CΛ 0, (16.14) 2 with Λ defined by Λ λh2 ,
(16.15)
from which any static solution to Eq. (16.6) can be constructed iteratively, starting from any admissible value of φ0 and solving at each step the algebraic problem. This is so because Eq. (16.6) is nothing but φn
U φn , φn 1 U φn1 , φn . φn 1 φn1
(16.16)
Equation (16.14) is the discretized first integral (DFI),12 i.e., in the continuum limit h 0 it reduces to Eq. (16.13). Note that the two-point map, Eq. (16.14) is quartic in both φn 1 and φn and thus, it cannot be reduced to the integrable nonlinear map reported in23 where the corresponding first integral is quadratic in φn 1 and φn .
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On the other hand, in case only A2 and A4 are nonzero, then as has been shown recently, the two-point map is given by W φn , φn 1 2 Λ 2 Λ Y φ2n 0 .
(16.17)
Since the vanishing of the third bracket of Eq. (16.18) is a trivial possibility, hence for Λ " 2, effectively the map is given by Y CY φ2n φ2n 1 2Zφn φn 1 0, W φn , φn 1 φ2n φ2n 1 2 Λ 2 Λ (16.18) which is precisely of the Quispel form and hence integrable.23 Here Z
2 Λ2 Ch4 A24 , Y 2 2 Λ Ch4 A2 A4
h2 A4 A2 Z .
(16.19)
From this map any static solution to Eq. (16.6) can be constructed iteratively, starting from any admissible value of φ0 and solving at each step the algebraic problem. Speight and Ward instead set C 0 in the static first order continuum field Eq. (16.13) and discretized it as 1 1 6Λ φn φn 1 2 2 2 . (16.20) φn φn φn 1 φn 1 0 , H H 6 Λ 2 3 2 Note that this two-point map is defined in case 0 Λ 6. Since this map is valid only at C 0, hence not surprisingly, Speight and Ward were only able to show the existence of TI kinks. It may be noted that in this model, energy is conserved. It is worth noting that as soon as one obtains a two-point map, it automatically implies that the model is TI. This is because, one can construct solutions iteratively from this map, for any admissible values of φ0 . 16.4. Generalized DNLS Model Another class of discrete models, relevant to a variety of applications are the so called DNLS models. It is easily seen that the most general discrete DNLS model with cubic nonlinearity and having symplectic structure and which is invariant under the interchange of un 1 and un1 , is given by Eq. (16.5) with f being18 f
α1 un 2 un α2 un2 un 1 un1 α3 u2n u ¯n 1 u¯n1
α4 un un 1 un1 α5 un 2
2
u ¯ n 1 un 1 u ¯ n 1 un 1 2 2 α6 u ¯n un 1 un 1 α7 u¯n un 1 un 1 α8 un 1 2 un 1 un 1 2 un 1 ¯n 1 u2n 1 u ¯n 1 u2n 1 α10 un 1 2 un 1 un 1 2 un 1 , (16.21) α9 u
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where u¯ represents complex conjugate and the real valued parameters (α1 , ..., α10 ) satisfy the continuity constraint α1 α7 2 α2 α3 α4 α5 α6 α8 α9 α10 2 .
(16.22)
Following the convention which is followed in the literature, unlike the discrete φ4 case, in the DNLS case, one does not explicitly write h, but instead is scaled away. 16.4.1. Conserved Quantities In all the DNLS models, un is a complex field and the field Eq. (16.5) with f given by Eq. (16.21) is invariant under a global gauge transformation. As a result, norm n un 2 is conserved in this model. While no systematic procedure is known so far to find other conserved quantities, it is easy to check that in case α8
α9 α10 , α1
α4 α6
2α5 α7 ,
(16.23)
while α2 , α3 are arbitrary, then the generalized DNLS Eq. (16.5) with f given by Eq. (16.21) conserves the momentum P 1 defined by
P 1 i un 1 u ¯n u ¯n 1 un . (16.24) n
On the other hand, in case only α5 and/or α7 are nonzero while all other αi 0, then the generalized DNLS Eq. (16.5) with f given by Eq. (16.4) conserves the momentum P 2 defined by
P 2 i un 2 u ¯n u ¯n 2 un . (16.25) n
Finally, in case 2α3
2α8
α2 , α5
α7
α9
α10
0
(16.26)
while α1 , α2 and α3 are arbitrary, the DNLS Eq. (16.5) can be obtained from the Hamiltonian H given by
α1 α6 2 2 2 4 2 2 un un 1 un α4 un un 1 u ¯n un 1 u¯2n 1 u2n H 2 2 n α2 2 2 un un 1 u ¯n 1 un u ¯n un 1 , (16.27) 2 by using the equation of motion iu n
un , H P B
,
(16.28)
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where the Poisson bracket is defined by
dU dV U, V P B dun d¯ un n
dU dV . d¯ un dun
(16.29)
It is worth noticing the close similarity between the results in the DNLS and the discrete φ4 cases as discussed above. 16.4.2. Two-Point Maps for Stationary Solutions With the ansatz un t fn eiωt we obtain from the DNLS Eqs. (16.5) and (16.21), the following difference equation for the amplitudes fn1 2 ω fn fn 1 α1 fn3 α2 α3 fn2 fn1 fn 1 2 2 α4 α6 fn fn1 fn 1 2α5 α7 fn1 fn fn 1 3 3 α8 fn1 fn 1 α9 α10 fn1 fn 1 fn1 fn 1 0. (16.30) It is amusing to notice that the difference Eq. (16.30) is identical to the corresponding static difference Eq. (16.1) for the discrete φ4 case, provided we make the following identification A1
α1 ,
A4
A2
2
α2 α3 , A3
2α5 α7 , A5
2
2
α4 α6 ,
α9 α10 , A6
2α8 ,
Λ ω. (16.31)
It is then straightforward to obtain the two-point maps in this case, simply by comparing with the φ4 results. 16.5. Translationally Invariant JEF, Hyperbolic, Trigonometric and Short-Period Static Solutions We shall first discuss the JEF (Jacobi elliptic function) solutions as well as the hyperbolic solutions which follow from the JEF solutions and later on we shall discuss the sine as well as short-period solutions which exist in almost all the models. 16.5.1. JEF and Hyperbolic Solutions It is easily shown22 that the JEF solutions (i.e. sn, cn, dn), both nonstaggered and staggered, as well as the corresponding hyperbolic kink and pulse solutions can be obtained for the discrete model of Eq. (16.6) in case
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A1 A6 0 in the following seven cases: (i) only A2 nonzero; (ii) only A4 nonzero; (iii) only A2 and A4 nonzero; (iv) only A3 and A5 nonzero; (v) A2 , A3 , and A5 nonzero; (vi) A3 , A4 , and A5 nonzero; (vii) A2 , A3 , A4 , and A5 nonzero. As an illustration, we discuss here nonstaggered as well as staggered sn, cn, dn as well as the kink (tanh) and pulse (sech) solutions in case only A4 " 0. One of the key ingredient in this derivation are the identities for JEF.24 For example, in case only A2 , A4 " 0, it is easily shown that φn
Asnhβ n x0 , m ,
(16.32)
is an exact solution provided 2m h2 A2
2A4 ns hβ, mns 2hβ, m A2 ns2 hβ, m ,
(16.33)
2 Λm 2 A4 ns hβ, m A2 cs hβ, mds hβ, m . (16.34) A2 h2 Here 0 m 1 is the modulus of JEF, A, β are the parameters of the solution and x0 is the arbitrary initial position. Further, ns x, m 1sn x, m, cs x, m cn x, msn x, m and ds x, m dn x, msn x, m. In the limit of m 1 this solution reduces to the kink solution
φn
A tanhβ hn hx0 ,
(16.35)
and the relations (16.33) and (16.34) take the simpler form A2
1 , h2 A4
2
Λ , A2 tanh2 hβ
On the other hand, in case only A2 , A4
φn
A
n
1
"
λ A4 .
(16.36)
0, the staggered solution
snhβ n x0 , m ,
(16.37)
exists provided 2m h2 A2
2A4 ns
hβ, mns 2hβ, m A2 ns2 hβ, m ,
2 Λm 2 A4 ns hβ, m A2 cs hβ, mds hβ, m . A2 h2 In the limit of m 1 the corresponding staggered kink solution
φn
A
n
1
tanhβ hn hx0 ,
(16.38)
(16.39)
(16.40)
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exists provided A2
4 Λ 4 Λ Λ 2h2 A4 2 , tanh . hβ 2 h2 A4 Λ 2 h2 A4 Λ
Another solution in case only A2 , A4 φn
"
(16.41)
0, is
Acnhβ n x0 , m ,
(16.42)
provided 2m h2 A2
hβ, mds 2hβ, m A2 ds2 hβ, m ,
2A4 ds
(16.43)
2 Λm 2 A4 ds hβ, m A2 cs hβ, mns hβ, m . A2 h2 Yet another solution, in case only A2 , A4 " 0 is φn
Adnhβ n x0 , m ,
(16.44)
(16.45)
provided 2 h2 A2
2A4 cs
hβ, mcs 2hβ, m A2 cs2 hβ, m ,
(16.46)
2 Λ 2 A4 cs hβ, m A2 ds hβ, mns hβ, m . (16.47) A2 h2 In the limit of m 1 both these solutions go over to the pulse solution φn
Asechβ hn hx0 ,
(16.48)
and the relations (16.43), (16.44) as well as (16.46) and (16.47) take the simpler form Λ 2cosh hβ 1 0 , A2
Λ 2 Λ 4
2 Λ 2 h2 A4
On the other hand, in case only A2 , A4 φn
A
n
1
"
.
0, the staggered cn solution
cnhβ n x0 , m ,
exists provided 2m 2 2A4 ds hβ, mds 2hβ, m A2 ds hβ, m , h2 A2 2 Λm 2 A4 ds hβ, m A2 cs hβ, mns hβ, m . A2 h2 Yet another solution, in case only A2 , A4 " 0 is
φn
A
n
1
(16.49)
dnhβ n x0 , m ,
(16.50)
(16.51) (16.52)
(16.53)
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provided 2 h2 A2
2A4 cs hβ, mcs 2hβ, m A2 cs2 hβ, m ,
(16.54)
2 Λ 2 A4 cs hβ, m A2 ds hβ, mns hβ, m . (16.55) A2 h2 In the limit of m 1 both these solutions go over to the staggered pulse solution
φn
A
n
1
sechβ hn hx0 ,
(16.56)
and the relations (16.51), (16.52) as well as (16.54) and (16.55) take the simpler form Λ 2 Λ 4 . (16.57) Λ 2cosh hβ 1 0 , A2 2 2 Λ h2 A4 Thus, the staggered pulse solution exists only if Λ 4. In the special case when only A2 or only A4 is nonzero, JEF solutions can be easily obtained from here. Similarly, solutions in the remaining four cases can also be easily worked out. In all the seven cases, one finds that, the amplitude A is always equal to one, for the kink solution. Further, in all the seven cases, while the pulse solution exists only if Λ 0, the staggered pulse solution exists only if Λ 4. 16.5.2. Trigonometric Solutions Unlike the JEF and the hyperbolic solutions, the static TI trigonometric solutions with an arbitrary shift along the chain x0 exist even when all the six parameters Ai are nonzero. In particular, solution of the form φn
A sinhβ n x0 ,
(16.58)
exists provided Λ 2 2 cos hβ h2 A2 sin2 hβ A3 A4 3A6 A5 cos hβ , (16.59) 2hβ A2 A5 cos hβ A6 cos hβ 4 cos2 hβ 3 0 , (16.60) Similarly, the corresponding staggered solution can be written down. In fact looking at the field Eq. (16.6), it is clear that if φn is a nonstaggered solution, then then the staggered solution 1n φn is also a solution to Eq. (16.6) with the coefficients of A1 , A3 and A4 having opposite signs and with 2 Λ replaced by Λ 2.
A1 A4 A3 cos
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16.5.3. Short-period Solutions It turns out that apart from the JEF, hyperbolic and trigonometric, there are also several short period and even aperiodic solutions of Eq. (16.6). We have obtained fifty one such solutions [three solutions of period two, three of period three, seven of period four, nine of period five, eleven each of period and seven, two each of period eight, nine and ten and one solution of period eleven. Corresponding to most of these solutions, we have also obtained solutions with arbitrarily large period as well as aperiodic solutions. In order to obtain these solutions, it is useful to look at the symmetries of Eq. (16.6). In particular, notice that Eq. (16.6) is invariant under φn1 φn 1 and φn 1 φn1 . Further, Eq. (16.6) is also invariant under φn1 , φn , φn 1 φn1 , φn , φn 1 . A consequence of these two symmetries is that if φn1 , φn , φn 1 is a solution to Eq. (16.6) under certain constraints, then φn1 , φn , φn 1 , φn 1 , φn , φn1 and φn 1 , φn , φn1 are also solutions of Eq. (16.6) provided the same constraints are satisfied. As an illustration, it is easily verified that φn ..., a, a, , ... is an exact solution with period three to Eq. (16.6) provided Ai satisfy a2 h2 A1 A3 A4 Λ 2 , a2 h2 A1 A3 A4 A2 A5 A6 Λ 4 ,
(16.61)
One can in fact generalize this solution and show that even φn ..., a, a, a, a p times, a, ... is also an exact solution to Eq. (16.6) provided Eq. (16.61) is satisfied. One can also show that an aperiodic solution with any number of “a” and “-a” kept at random but with the constraint that at most two “a” or two “-a” are always together, is an exact solution to Eq. (16.6) provided Eq. (16.61) is satisfied. One of the interesting solution that we have obtained is of period four, i.e. φn ..., a, b, a, b, ... is an exact solution to Eq. (16.6) provided A1 A4
A3 ,
a2 b2 h2 A1
Λ 2 , a2
"
b2 , A1
"
0.
(16.62)
Thus one has a one parameter family of solutions. This is a TI solution since it can also be put in the form π (16.63) φn A cos n x0 , 2 In the special case of A3 A4 , A1 0, and Λ 2, one in fact, has a two parameter family of solutions since now “a” and “b” are both free parameters.
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16.6. Translationally Invariant Time-Dependent JEF, Hyperbolic, Trigonometric and Short-Period Solutions We shall first discuss the time-dependent JEF (Jacobi elliptic function) solutions as well as the hyperbolic solutions which follow from the JEF solutions and later on we shall discuss the sine and the short-period solutions which exist in almost all the models. It turns out that all the models which admit static TI solutions also admit time-dependent solutions with arbitrary x0 . Further, we show that in many cases, unlike the continuum case, the propagation velocity can be arbitrarily large. 16.6.1. Time-Dependent JEF and Hyperbolic Solutions It is easily shown25 that time-dependent JEF solutions (i.e. sn, cn, dn), both nonstaggered and staggered, as well as the corresponding hyperbolic kink and pulse solutions can be obtained for the discrete model of Eq. (16.6) in all the seven cases in which the static solutions hold provided A6 0 as before but now A1 " 0. Note that in the static case A1 0. As an illustration, we discuss here nonstaggered sn as well as the kink (tanh) solutions in case only A2 , A4 " 0.25 For example, in case only A2 , A4 " 0, it is easily shown that φn
Asnβ hn hx0 vt, m ,
(16.64)
is an exact solution provided 2 h2 A2
2A4 ns hβ, mns 2hβ, m A2 ns2 hβ, m ,
(16.65)
β 2 v2 2 Λ 2 A4 ns hβ, m A2 cs hβ, mds hβ, m 1 m . (16.66) 2 2 A h A2 Here ns x, m 1sn x, m, cs x, m cn x, msn x, m and ds x, m dn x, msn x, m. In the limit of m 1 this solution reduces to the moving kink solution φn
A tanhβ hn hx0 vt ,
(16.67)
and the relations (16.58) take the simpler form A2
1 , A1
β
2 2
v , h2 A4
2
Λ h2 A1 . tanh2 hβ
(16.68)
It is clear from here that A1 0 and further, for a given h2 and λ, v can be arbitrarily large. Solutions for the special cases when A2 0 or A4 0 can
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be immediately obtained from here. Similarly, solutions for the remaining four cases can be easily worked out. One finds that the amplitude A is always equal to 1 for the kink solution in all the seven cases. 16.6.2. Time-Dependent Trigonometric Solutions The model (16.6) also supports moving staggered as well as unstaggered trigonometric solutions. For example it supports nonstaggered solution of the form φn
A sinβ hn hx0 vt ,
(16.69)
provided Λ 2 2 cos hβ v 2
h2 A2 sin2 hβ A3 A4 3A6 A5 cos hβ , (16.70)
2hβ A2 A5 cos hβ A6 cos hβ 4 cos2 hβ 3 0 , (16.71) It is worth noting that such a solution is valid in most of the special models discussed in the literature, including the Hamiltonian model discussed above.
A1 A4 A3 cos
16.6.3. Time-Dependent Short-Period Solutions Unfortunately, while we have obtained fifty one static short-period solutions, to date we have been able to obtain only six time-dependent shortperiod solutions. These are φn
..., a, a, ...φ t , φn
..., a, a, 0, ...φ t ,
φn
..., a, b, a, b, ...φ t , φn
φn
..., a, 0, a, 0, ...φ t , φn
..., a, a, a, a, ...φ t , ..., a, a, 0, a, a, 0, ...φ t , (16.72)
where φ t could be either sn vtt0 , m, cn vtt0 , m or dn vtt0 , m. This also includes the solutions in terms of sin vt t0 , tanh vt t0 , sech vt t0 . As an illustration, one of the solution is φn
..., a, b, a, b, ...sn vt t0 , m ,
(16.73)
which is an exact solution to Eq. (16.6) provided A1 A4
A3 , A1 a2 b2 2mv 2 , h2 v 2 1 m 2 Λ . (16.74)
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In the limit m 0 we then have the solution φn
..., a, b, a, b, ... sin vt t0 ,
(16.75)
0 , h2 v 2
(16.76)
provided A4
A3 , A1
2 Λ.
On the other hand, in the limit m 1, we have the solution φn
..., a, b, a, b, ... tanh vt t0 ,
(16.77)
provided A1 A4
A3 , A1 a2 b2 2v 2 , 2h2 v 2
2 Λ.
(16.78)
Proceeding in the same way, one can also obtain moving as well as stationary JEF and hence hyperbolic solutions, trigonometric and short period solutions for the generalized DNLS equation discussed in this paper. 16.7. Stability Analysis Let us first discuss the stability of static solutions. Let φ0n be a static solution to Eq. (16.6). To study the dynamics in the vicinity of this solution we substitute the ansatz φn t φ0n εn t into Eq. (16.6), and obtain the following linearized equation εn
Kn,n1 εn1 Kn,n εn Kn,n 1 εn 1 ,
(16.79)
with A2 0 2 φ A3 φ0n1 φ0n A4 φ0n φ0n 1 2 n A5 0 3A6 0 φn 1 2φ0n1 φ0n 1 φn1 2 , 2 2 2 Kn,n λ 2 3A1 φ0n 2 A2 φ0n φ0n1 φ0n 1 h A3 0 φn1 2 φ0n 1 2 A4 φ0n1 φ0n 1 , 2 1 A2 0 2 Kn,n 1 2 φ A3 φ0n φ0n 1 A4 φ0n1 φ0n h 2 n A5 0 3A6 0 φn1 φ0n1 2φ0n 1 φn 1 2 . 2 2 Looking for solutions of Eq. (16.79) of the form εn t Un exp come to the eigen-value problem Kn,n1
1 h2
K U ω
2
U,
(16.80) iωt
we
(16.81)
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where vector U contains Un and the nonzero coefficients of matrix K are given by Eq. (16.80). If φ0n is a TI static solution then it can be shifted along the chain by an arbitrary x0 , φ0n φ n x0 . The eigenvector corresponding to the zerofrequency translational Goldstone mode, UG , has components φn , where prime means derivative of φ with respect to its argument. It is easy to check that UG &φn ' is an exact solution to Eq. (16.79) with ω 2 0. This is because Eq. (16.79) with εn replaced by φn , coincides with the derivative of static version of Eq. (16.6) with respect to x0 . We thus have proved that any TI static solution has the zero-frequency translational mode UG &φn '. The next important issue is to see if the zero mode is the lowest mode 0 or not. For example consider the pulse solution φn Asechhβ n x0 . Thus the corresponding zero mode is given by Un
Asechhβ
n x0 tanhhβ n x0 ,
(16.82)
which is antisymmetric, hence there must exist a symmetric mode with ω 2 0, so that the pulse solution is an unstable solution. Similar argu0 ment, unfortunately does not work for the discrete kink solution φn A tanhhβ n x0 since the corresponding zero mode Un
Asech2 hβ n x0 ,
(16.83)
is symmetric one. In the continuum case, it is a stable mode if its nodeless state. I am not aware of analogue of that argument exists in the discrete case or not, hence strictly speaking, one cannot say if indeed it is a stable mode or not. I suspect that a similar argument should also hold good for the discrete case and hence indeed static discrete kink is a stable solution. Numerically, we have checked that this is true in several cases. We may also add here that but for kink, the JEF as well as the pulse solutions are unstable solutions. Using the above formalism, one can also check the stability of the shortperiod solutions.22 Unlike the static case, the stability of the moving solutions is a tricky issue which has to be done numerically. One finds that unlike the static kink solutions, moving kink solutions are not stable in the whole range of parameters of their existence, but for each given propagation velocity, one finds a range of parameters where the moving kink is stable (in the sense that even the perturbed kink solution, with reasonably large perturbation amplitude, in course of time tends to the exact solution.
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16.8. Some Open Problems Before closing, we spell out some of the open problems. (1) While one has obtained JEF solutions as well as a two-point map in case A2 and/or A4 are nonzero, in the other four cases, while one has obtained JEF solutions, one has not been able to obtain one single unified two-point map. (2) For the general Model (16.9), while a general two-point map as well as conserved momentum operator are known, to date no analytic solution is known which is characterized by the two parameters C and φ0 . Can one find few analytic solutions in this model? (3) There is a belief that all TI models must have some conserved quantity. Unfortunately, out of all the seven models where JEF solutions with arbitrary X0 are known, we so far know only two cases (i.e. when only A2 or only A4 are nonzero) when conserved momentum operator is known. It would be nice if one can find such an operator in the other five cases. One of the open problem is to obtain, a conserved momentum operator in case both A2 and A4 is nonzero. Note that in case only A2 is nonzero then P1 as defined by Eq. (16.8) is conserved while if only A4 is nonzero, then P2 as defined by Eq. (16.10) is conserved. (4) Is it possible to analytically argue about the stability of the discrete kink solution? (5) We have obtained a huge class of short period solutions. It would be nice if one can find some application of some of these solutions. References 1. F.R.N. Nabarro, Theory of Crystal Dislocations (Clarendon Press, Oxford, 1967). 2. T.R.O. Melvin, A.R. Champneys, P.G. Kevrekidis, and J. Cuevas, Phys. Rev. Lett. 97, 124101 (2006). 3. O. Morsch and M. Oberthaler, Rev. Mod. Phys. 78, 179 (2006). 4. P. G. Kevrekidis, Physica D 183, 68 (2003). 5. J. M. Speight and R. S. Ward, Nonlinearity 7, 475 (1994); J. M. Speight, Nonlinearity 10, 1615 (1997); J. M. Speight, Nonlinearity 12, 1373 (1999). 6. C. M. Bender and A. Tovbis, J. Math. Phys. 38, 3700 (1997). 7. S. V. Dmitriev, P. G. Kevrekidis, and N. Yoshikawa, J. Phys. A 38, 1 (2005). 8. F. Cooper, A. Khare, B. Mihaila, and A. Saxena, Phys. Rev. E 72, 36605 (2005).
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9. I. V. Barashenkov, O. F. Oxtoby, and D. E. Pelinovsky, Phys. Rev. E 72, 35602R (2005). 10. S. V. Dmitriev, P. G. Kevrekidis, and N. Yoshikawa, J. Phys. A 39, 7217 (2006). 11. O.F. Oxtoby, D.E. Pelinovsky and I.V. Barashenkov, Nonlinearity 19, 217 (2006). 12. S. V. Dmitriev, P. G. Kevrekidis, N. Yoshikawa, and D. J. Frantzeskakis, Phys. Rev. E 74, 046609 (2006). 13. J. M. Speight and Y. Zolotaryuk, Nonlinearity 19, 1365 (2006). 14. S. V. Dmitriev, P. G. Kevrekidis, A. Khare, and A. Saxena, J. Math. Phys. 40, 6267 (2007). 15. I. Roy, S. V. Dmitriev, P. G. Kevrekidis, and A. Saxena, Phys. Rev. E 76, 026601 (2007). 16. S.V. Dmitriev, P.G. Kevrekidis, A.A. Sukhorukov, and N. Yoshikawa, S. Takeno, Phys. Lett. A 356, 324 (2006). 17. A. Khare, K.O. Rasmussen, M.R. Samuelsen, A. Saxena, J. Phys. A 38, 807 (2005); A. Khare, K.O. Rasmussen, M. Salerno, M.R. Samuelsen, and A. Saxena, Phys. Rev. E 74, 016607 (2006). 18. D.E. Pelinovsky, Nonlinearity 19, 2695 (2006). 19. S. V. Dmitriev, P. G. Kevrekidis, N. Yoshikawa, and D. Frantzeskakis, J. Phys. A 40, 1727 (2007). 20. P.G. Kevrekidis, S.V. Dmitriev, and A.A. Sukhorukov, Math. Comput. Simulat. 74, 343 (2007). 21. A. Khare, S. V. Dmitriev, and A. Saxena, J. Phys. A 40, 11301 (2007). 22. A. khare, S.V. Dmitriev and A. Saxena, arXiv nlin/07101460. 23. G.R.W. Quispel, J.A.G. Roberts, and C.J. Thompson, Physica D 34, 183 (1989). 24. A. Khare, A. Lakshminarayan and U.P. Sukhatme, Pramana (J. Phys.) 62, 1201 (2004); math-ph0306028. 25. S.V. Dmitriev, A. Khare, P.G. Kevrekidis and A. Saxena, Under Preparation.
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Chapter 17 A Model for Flow Reversal in Two-Dimensional Convection Krishna Kumar1 , Supriyo Paul2 , Pinaki Pal3 and Mahendra K Verma4 1 2
Department of Physics and Meteorology, Indian Institute of Technology, Haranguer 721 302, India
3
Kabi Sukanta Mahavidyalaya, Angus, Hooghly 712 221, West Bengal, India 4
Department of Physics, Indian Institute of Technology, Kanpur 208 016, India
A low-dimensional model for flow reversal in two-dimensional RayleighB´enard convection in a thin layer of Boussinesq fluid confined between stress-free and conducting flat boundaries is presented. For higher values of Rayleigh numbers the set of straight rolls shifts in direction normal to its axis. The amount of the shift is random and occurs irregularly in time. This leads to change in the flow patterns. For the shift equal to half the wavelength, the flow direction reverses globally. The result is also compared with that of direct numerical simulation. The histogram of flow reversal time obtained from the model shows power law.
Contents 17.1 Introduction . . . . . . 17.2 Hydrodynamic System 17.3 Results . . . . . . . . 17.4 Conclusion . . . . . . References . . . . . . . . . .
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17.1. Introduction The Rayleigh-B´enard (RB) convection is a well known system for studying pattern-forming instability.1 The RB system consists of a thin layer of Boussinesq fluid confined between large and thermally conducting horizon365
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tal plates and subjected to uniform adverse temperature gradient. When the temperature gradient is increased above a critical value, convection sets in. A set of stationary straight rolls appear just above the onset.2 They show different types of convective patterns.3 At larger values of the Rayleigh number, which is the measure of the adverse temperature gradient applied, convection cells may appear as hexagon4 and asymmetric squares.5 In low Prandtl number fluids, like liquid metals, oscillatory instability6 leads to propagation of traveling waves along the rolls. Recent simulation8 for two-dimensional(2D) convection showed chaotic traveling waves in a direction normal to the roll axis, which may lead to flow reversal or jitters in the flow. This phenomenon is known in binary fluids,7 but not in ordinary fluids. It is however difficult to extract the essential mechanism responsible for this behavior from the DNS. We present here a simple low-dimensional model to capture the possible mechanism of jitters and flow reversal in RB convection. The model shows interaction between real and imaginary parts of the 2D mode W101 through the generation of higher order modes at higher Rayleigh number. The real and imaginary parts of the W101 mode in the direct numerical simulation (DNS) showed an interaction between the real and imaginary parts of any velocity mode. Whenever one of the real and the imaginary parts of the 2D mode changes sign and acquires significantly increase in its magnitude, the system of 2D rolls travel in a direction normal to the the roll axis. This essentially captures the the phenomenon observed in 2D simulation.
17.2. Hydrodynamic System We consider a thin extended fluid layer of fluid of thickness d, kinematic viscosity ν, and thermal diffusivity κ confined between two thermally conducting horizontal boundaries, and heated underneath. We assume the free-slip conditions on the velocity field, which is idealized and makes the construction of the model easier. All the length scales and time scales are made dimensionless by the fluid thickness d and the the thermal diffusion time d2 κ. The temperature filed is made nondimensional by the temperature difference ∆T across the fluid layer. This leads to dimensionless form of the hydrodynamic equations describing the convection in Boussinesq fluid as:
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∇v
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P ∇4 v3 RP ∇2H θ
eˆ3 .∇ ω.∇v v.∇ω ,
(17.1)
0,
(17.2)
tθ
∇ θ v3 v.∇θ, 2
(17.3)
where v x, y, z, t v1 , v2 , v3 is the velocity field, θ x, y, z, t the deviation in temperature field from the steady conduction profile, and ω ω1 , ω2 , ω3 ∇ v the vorticity field in the fluid. The Rayleigh number R is defined as R α ∆T gd3 νκ, where α is the coefficient of thermal expansion of the fluid, g the acceleration due to gravity, and ∆T the temperature difference across the fluid layer. P ν κ, is the Prandtl number. The unit vector eˆ3 is directed vertically upward. The stress-free conducting flat boundary surfaces imply v3 zz v3 θ 0 at z 0, 1. The symbol ∇2H xx yy stands for the horizontal Laplacian. We construct a dynamical system using the standard Galerkin procedure. The spatial dependence of the vertical velocity and the temperature field are expanded in Fourier series, which is compatible with the stress-free flat conducting boundaries and periodic boundary conditions in the horizontal plane. We include minimum modes that can show traveling rolls. The vertical velocity v3 and the temperature field θ are expanded as
v3
θ
W101
t cos kx W¯101 t sin kx sin πz
W202
t cos 2kx W¯202 t sin 2kx sin 2πz
W301
t cos 3kx W¯301 t sin 3kx sin πz
W103
t cos kx W¯103 t sin kx sin 3πz,
θ101
t cos v.∇θkx θ¯101 t sin kx sin πz
θ202
t cos 2kx θ¯202 t sin 2kx sin 2πz
θ301
t cos 3kx θ¯301 t sin 3kx sin πz
θ103
t cos kx θ¯103 t sin kx sin 3πz
θ002
t sin 2πz θ004 t sin 4πz.
(17.4)
(17.5)
The solenoidal property of the velocity and the vorticity fields determine their horizontal components. Projecting the hydrodynamical equations 17.1-17.3 on these modes, we get a eighteen-mode dynamical system, which is explicitly given in the Appendix. We have chosen the real coefficients of for both sine and cosine terms, which is equivalent to choosing
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complex coefficients with exponential terms. The competition between real and imaginary parts of the Fourier modes in DNS would be like competition between the modes representing two sets of parrallel rolls with phase difference of π 4. When the mean value of both the modes do not vary much, the rolls system remains similar without showing any lateral movement. This is a minimum-mode model showing travelling straight rolls. Any reduction in the number of modes removes the interaction between two set of rolls, and the travelling rolls in a direction normal to their axis are not observed. 300
ℜ(W101) ℑ(W101)
W101
150 0 −150 −300
0.2
ℜ(θ
)
101
ℑ(θ101)
θ
101
0.1 0 −0.1 −0.2 0
Fig. 17.1.
20
40
t
60
80
Chaotic time series for modes W101 and θ101 for r
100
229 and P 6.8.
17.3. Results We numerically integrate this dynamical system to investigate the unsteady solutions. We set k kc π 2. For each value of Prandtl number P we vary the reduced Rayleigh number r RRc , where Rc 27π 4 4, in small steps and integrate the system with randomly chosen initial conditions for long periods to get rid of transients. First we present the results of P 6.8. The system shows fixed points for 1 r 192, oscillatory behaviour in time with the movement of the rolls for 192 r 228 and chaotic solutions for r 228. Figure 17.1 shows the time evolution of the large 2D modes W101 & W¯101 (top) and θ101 & θ¯101 (bottom) for r 229. The time series show chaotic behaviour for these modes. The two modes represent two sets of similar straight rolls shifted by λ4, where λ is the wavelength of the
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spatially periodic rolls. The interplay between the modes W101 and W¯101 plays an important part in the lateral movement of straight rolls. The rolls do not move, if both the modes W101 and W¯101 remain in their mean position. As soon as one of the these modes changes sign, and the other mode either changes sign or shows some large variation in its value the rolls start to moving. The time evaluation of the temperature fluctuation also show similar behaviour. t = 145
t = 15
1
1 1 0.9
0 0
0 0
2√2 t = 16.1
2√2
1
0.8 0.7
t = 147.6 1
z
0.6 0.5 0 0
0.4
0 0
2√2
2√2 0.3 t = 148.3
t = 17
1
1
0.2 0.1 0
0 0
0 0
2√2
2√2
x
Fig. 17.2. Traveling rolls in a direction normal to the roll axis and the consequent flow reversal as observed in the DNS (left column) and the model (right column). The DNS 6.8 and r 830 and the model is showing the result for P 6.8 is performed with P 229. The arrows show velocity field and the variation in the grey level shows and r variation in the convective temperature. Darker shade is hotter. Three sets are for different instances.
Figure 17.2 shows the velocity vector embedded on the temperature perturbation due to convection obtained from the DNS (left column) and the model (right column) for different three different instances of time. Three snapshots clearly show the movement of rolls. The direction in which the rolls would move is totally random. If the rolls move by a distance λ2, where λ is the wavelength of straight periodic rolls, a global reversal of convective flow velocity is observed. The temperature field obtained from
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DNS shows the generation of plumes due to the heat flux. The flow reversal are observed in the turbulence regimes of convection.10 This experiment on flow reversal10 observed an exponent of 1 for the probability. We observe flow reversal due to competition between two sets of similar rolls separated by a phase difference of π 2. We also find power law with exponent 1.4. 6
10
−1.4
Ni ~ ∆ti 5
10
4
Ni
10
3
10
2
10
1
10
0
10 −2 10
−1
10
0
10 ∆ti
1
10
2
10
Fig. 17.3. Power law behaviour of the histogram of flow reversal interval ∆ti . Ni 1 ∆ti1.4 .
The time interval between any two flow reversal is random. Figure 17.3 shows the histogram of the ∆t, the time interval between two flow reversals, on log-log scale. The histogram obtained by the model shows a power law dependence. The flow reseal time is computed by observing the velocity field at different points on the lower boundary (z 0). The interval in which this horizontal velocity component changes sign gives the ∆t. The time intervals have many scales. We make the ith time-interval bin with mean value ∆ti . The frequency of ∆t with the value ∆ti is denoted by Ni . Ni scales with ∆ti as ∆tiα . We get α 1.4 (within a few percents) for over three decades in ∆t. 17.4. Conclusion We have presented a low dimensional model showing the possibility of laterally travelling straight rolls in 2D thermal convection. It is due to the nonlinear interaction between two sets of rolls. The rolls shifts irregularly
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showing temporal chaos. Whenever, the shift in the roll system is equal to half the wavelength, we see globally flow reversal. For a shift less than half the wavelength, we observe jitters in the convective flow. The instability leading to travelling waves, similar to ones observed in binary mixtures, is a also a possibility of flow reversal. The histogram of reversal time interval shows a power-law as observed in experiments on flow reversal. The model captures the essential mechanism of the travelling rolls instability and flow reversal as observed in 2D simulation. The model
11 V1 V2 Y1 Y2 V2 V1 3 1 U1 U2 Y1 Y2 U2 U1 3 11 U1 U2 Y σ 4Y P V1 V2 U2 U1 6 7 V1 V2 X1 X2 V2 V1 6 4 U1 U2 X1 X2 U2 U1 3 11 9 U σU σS 3 11 4 Y1 Y2 V2 V1 Y2 Y1 11 9 Y2 Y1 X2 X1 Y2 Y1 11
X
σ
V
X
T
19 3 σV σW 3 19 2 U1 U2 Y2 Y1 U2 U1 19 9 Y1 Y2 X1 X2 Y2 Y1 19
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W1 S1 W2 S2 Y2 S2 W2 W1 S1 U1 V1 U2 V2 2P1 2P2 U2 V2 V1 U1 V1 X1 V1 Z1 2Z2 V2 X2 V2 U1 V1 V2 U2 T2 P 4P rY T1 U2 V2 U1 V1 U1 U2 W1 S1 2W2 X1 2W1 U2 U1 W2 S2 W2 S2 S1 S2 X2 2V1 2V2 W1 S1 S2 S1 Y1 2Z2 Y2 T T rX Y1
2W1 T1 2W2 T2 3 T2 2W2 X1 2V1 2P1 Y2 2W1 T1 X2 2V2 2V2 X2 U1 2P2 3Z1 X1 2V1 U2 W 19 W rV Y1 T1 2S1 2S2 T2 3 T2 2S2 X1 2U1 2P1 Y2 T1 2S1 2U2 X2 X2 2U2 2P2 Z1 2Z2 X X1 2U1 11 S S 3rU Y1
Z1
8 Z1 T1 X1 V1 T2 X2 V2 3 X1 W1 X2 W2 3 U1 S1 U2 S2 32 Z2 Z2 2 V1 T1 V2 T2 2 X1 W1 X2 W2 3 4 Y1 P1 Y2 P2
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where, X X1 , X2 t 3π2 W101 , W¯101 t , 1 t Y Y1 , Y2 t 3π 2 W202 , W¯202 , U U1 , U2 t 9π W301 , W¯301 t , V V1 , V2 t 3π2 W103 , W¯103 t , rπ θ , θ¯ t , T T1 , T2 t 2 101 101 rπ t P P1 , P2 2 θ202 , θ¯202 t , rπ S S1 , S2 t θ , θ¯ t , 2 301 301 rπ W W1 , W2 t θ , θ¯ t , 2 103 103 Z
Z1 , Z2 t
rπ
θ002 , θ004 t and r
4R
27π 4 .
References 1. M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993) 2. A. Schl¨ uter, D. Lortz, and F. Busse, J. Fluid Mech. 23, 129 (1965) 3. E. Bodenschatz, W. Pesch, and G. Ahlers, Annu. Rev. Fluid Mech. 32, 709 (2000) 4. M. Assenheimer and V. Steinberg, Phys. Rev. Lett. 76, 756 (1996) 5. A. Das, U. Ghosal, and K. Kumar, Phys. Rev. E 62, R3051 (2000). 6. F. H. Busse, J. Fluid Mech. bf 52, 97 (1972) 7. R. W. Walden, P. Kolodner, A. Passner, and C. M. Surko, Phys. Rev. Lett. 55, 496 (1985) 8. M. K. Verma, K. Kumar, S. Paul, and D. Carati, arXiv:0704.3795v1 (2007) 9. F. H. Busse and R. M. Clever, Phys. Rev. Lett. 81, 341 (1998) 10. K. R. Sreenivasan, A. Bershadskii, and J. J. Niemela, Phys. Rev. E 65, 056306 (2002)
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Chapter 18 Euclidean Networks and Dimensionality
Parongama Sen Department of Physics, University of Calcutta, 92 Acharya Prafulla Chandra Road, Kolkata 700009, India. Euclidean networks are those in which nodes have well defined position coordinates and the linking probabilities are dependent on these coordinates. A review of the studies concerning the effective dimensionality of a Euclidean network is made and a few unanswered questions are discussed.
During the last few years there has been a lot of activity in the study of networks once it was realised that networks of diverse nature exhibit some common features in their underlying structure; one of the most important of which is the small world property. Networks, which are nothing but graphs, comprise of nodes and links and by the small world property it is meant that the average shortest path (i.e., number of steps) connecting any two nodes scales as log N where N is the number of nodes in the network. The idea of small world property in networks was initiated by the experiments of Milgram et al. in the 1960’s1 in which a few people in Kansas and Nebraska were asked to send letters by hand delivery to specific addresses in Cambridge and Boston. Letters had to be hand-delivered only through persons known on first-name basis. It was found that on an average six people were required for a successful delivery. The number six is “small” and is approximately equal to log N , where N is the total number of people in the USA. Random graphs are also known to have the property of shortest paths scaling as log N , but it is difficult to perceive a social network as a random network. More than thirty years after Milgram’s experiment, the WattsStrogatz (WS) model2 was proposed to simulate the small-world feature of real networks where it was pointed out that the property distinguishing
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random graphs and real world networks is the strong clustering tendency possessed by the latter. Precisely, clustering implies the tendency to form triangles, i.e., if nodes A and B share a link and so do A and C, the probability that B and C would also be linked is a measure of the clustering tendency. In the WS model, the nodes are placed on a ring and each node has connection to a finite number of nearest neighbours initially forming what is called a regular network. The nearest neighbour links are then rewired with a probability p to form random long range links. It was shown that even as p 0 (in the thermodynamic limit), the network behaves as a small world and it was thus claimed that the small world property can be achieved in a regular network by simply incorporating a few long range links. In a slight variation of the WS model, an addition type network was considered in which random long range bonds are added with a probability p, keeping the nearest neighbour links undisturbed. Phase transition to a small world network was again observed for p 0.3,4 Several properties other than the small world effect and clustering were observed in real world networks later. An important one among them is the scale-free feature seen in many networks. The degree of a node is the number of links possessed by it and the scale-free feature indicates that the degree distribution has a power law decay; i.e., there exist some nodes in the networks with very large degree. The Barab´ asi-Albert (BA)5 model was proposed for such networks in which it was assumed that a network evolves from an initial small size and grows as nodes join one by one. Here a new node forms links with an older node with a preferential attachment probability proportional to the older node’s degree. The small world property was also achieved in this model. In this article, we focus on a class of networks in which the nodes are embedded in a Euclidean space. Examples of such networks can be the Internet where the nodes are the servers, hubs, terminals etc., power grids, transport networks and so on. Forgetting the idea of small world networks for the time being, it may be noted that we are in fact very much familiar with Euclidean networks if we think of the regular lattices as networks. The simplest is of course a linear chain on which nodes are placed at regular intervals and are linked to the nearest neighbours. This is nothing but the model of a monatomic lattice chain in which the nearest neighbour atoms have an interaction. One can think of more complicated structures by going to higher dimensions e.g., square, triangular, hexagonal lattices in two dimensions, simple cubic lattice in three dimensions etc.
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Let us consider hyper-cubic lattices in d dimensions where the number of nodes is N Ld such that the average shortest distance scales as L or N 1d . Hence these networks are not small worlds (with the exception of d ) but regular networks. The WS network showed that essentially the dimensionality of a regular lattice or network is increased (infinitely, so to say) by introducing the long range links. However, it must be noted that in this model the probability of long range links is essentially distance independent. What happens in real Euclidean network? There have been some observations to show that the probability of connection between two nodes in a Euclidean network depends nontrivially on the distance between them and is not simply distance independent as assumed in the WS network. Yook et al. observed that nodes of the router level network maps of North America are distributed on a fractal set and the link length distribution is inversely proportional to the link lengths.6 Transport networks also show distinct distance dependence in the link length distribution.7 Even the scientists’ collaboration network has links with strong distance dependence.8 In order to model Euclidean networks with also scale free feature, some models have been proposed earlier.9 In10 for example, growing Euclidean networks were considered where the distance dependence of links in the form of a power law was incorporated in the preferential attachment scheme of the network. This gave rise to phase transitions from scale free to non-scale free behaviour in the network. However, if we are interested in finding out how the presence of long distance links affects the dimensionality of the network, it is best to consider a Euclidean network with no other feature and which can be made identical to the WS model (which has the behaviour of a small world network) in some limit. Such a model is precisely a linear chain or ring with nodes placed at regular intervals and where the linking probability of two nodes placed at distance l 1 is P l lδ .
(18.1)
Obviously δ 0 gives the WS network and for large δ the network is a regular one dimensional network. In order to ensure a fully connected network, we assume that nearest neighbour links are always present. One can define an adjacency matrix for a network, the ijth element of which is equal to one if nodes i and j are connected and zero otherwise. In one dimension, where the nodes are placed on a ring, the typical networks
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site j
δ=0
100 90 80 70 60 50 40 30 20 10 0
0 10 20 30 40 50 60 70 80 90100 site i
site j
δ = 1.0
100 90 80 70 60 50 40 30 20 10 0
0 10 20 30 40 50 60 70 80 90100 site i
site j
δ = 2.0
Fig. 18.1. Structures of the agrams (right panel) for δ 100 (from17 ). The size N neighbours always exist and increased.
100 90 80 70 60 50 40 30 20 10 0
0 10 20 30 40 50 60 70 80 90100 site i
networks (left panel) and the corresponding adjacency di1.0, 2.0 (top to bottom) in the model for a system of adjacency diagrams show that connections to the nearest connections to far away neighbours become fewer as δ is
0,
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generated for different values of δ and the corresponding adjacency matrix are shown graphically in Fig. 18.1. A variation of P l in the above form was first studied by Kleinberg11 on a two dimensional plane with the aim to find out how navigation in the network depends on the parameter δ. Later, a number of other properties of networks (both static and dynamic) with such a probabilistic attachment have been studied.9,12–18 As already mentioned, the known limits in this model are δ 0 and δ where the effective dimension of the network is d and d 1 respectively. Immediately one asks the questions: (i) What is the effective dimension for values of δ in the intermediate range? (ii) Is the effective dimension continuously varying? Naively, one can look at the scaling of the average link lengths and conclude that there exist three regions with different behaviour as δ is varied. The average link length can be computed from the distribution (18.1) to show that it has the following behaviour:15 l
for δ
O 1
1, l
for 1 δ
O N 2δ
2 and l
O N
for δ 2. From this it appears that the network is a small world for δ 1, finite dimensional for 1 δ 2 and regular for δ 2. That the behaviour of the network is regular for δ 2 is more or less confirmed with all the studies agreeing about this. It is also accepted that for δ 1, it is a small world. However, for the region 1 δ 2, the conclusions made in different studies vary to a great extent. According to the findings of 14 and,17 the network behaves as a finite dimensional lattice here. Some earlier studies12,13 however concluded that there is a small world effect right up to δ 2. The clustering coefficients analysed in15 again suggested the existence of the three regions. But in a recent study,18 it was again argued that the shortest path scales as log N for all δ 2, i.e, it is a small world network up to δ 2. The clustering properties studied here also supported this picture. Assuming that the network does behave as a finite dimensional network for 1 δ 2, the question that immediately arises is what is the effective
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dimensionality in this region? Calculating the shortest paths may be one way, and in14 as well as17 precisely this was done. While it was concluded in14 that the effective dimension d lies between four and two in this region on the basis of the scaling of shortest paths, a different approach in17 indicated that it decreases continuously from four to one. There are more than one approach to study the properties of a network. One, as mentioned, is simply the study of shortest paths as a function of the number of nodes. It may also be possible to draw inference about the geometrical properties of the network by studying critical phenomena on it. For example, percolation is a geometrical phase transition and when one studies the Ising model on a percolating lattice, the signature of the percolation critical point is very much there as a finite temperature phase transition is found to exist only beyond it. Similarly, in WS type networks, it was found that a finite temperature phase transition in the Ising model occurs for p 0.19,20 Critical exponents are known to be dependent only on the spatial dimension of the system and the symmetry of the order parameter. Thus one may find out the effective dimension indirectly by calculating the exponents and extracting the dimension from it. In numerical studies, critical exponents can be estimated using the finite size scaling (FSS) analysis. An extraordinary feature emerged when FSS analysis was done for the Ising model on the WS network. In FSS, one generally uses a scaling argument T Tc L1ν where Tc is the critical temperature and ν the critical exponent connected to the correlation function. Here Ld N , but since d and L are unknown one uses instead N 1νd in the scaling argument. While the other exponents showed agreement with mean field theory, it was observed that νd 2 would give an appropriate collapse of the data points in case of the Ising model on small world networks (WS addition type).23 Thus one notices two intriguing facts at δ 0: (a) There is finite size scaling and collapse of data points even though the criticality is of mean field nature. This is not possible in conventional mean field theory. (b) The effective dimension is exactly equal to four (as ν 12 in the mean field theory), the upper critical dimension of the Ising model. It may be recalled that the scaling of shortest paths indicate an infinite dimensionality. Using finite size scaling analysis, exponents of the Ising model were also extracted for the Euclidean network where the linking probability is given by (18.1) for all δ.17,18 For δ 1, FSS analysis gave results identical to
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the δ 0 case, i.e., the effective dimension is always four and independent of δ. However, for 1 δ 2, while in,18 the results again showed mean field nature of the exponents (giving d 4), in,17 the effective dimension was seen to decrease continuously from four to one. For estimating ν and d separately, epsilon expansion was used here. It may be noted here that the lower critical dimension of Ising model is one and therefore a finite temperature phase transition can exist when the effective dimension is above one. Thus according to,14 the effective dimensionality is discontinuous at δ 2, where it changes from two to one abruptly, while in,17 no such discontinuity is observed. On the other hand, according to18 and also another earlier study,13 the small world region exists up to δ 2 again indicating a discontinuity in the effective dimension. The dimensionality at δ 1 is continuous according to all the studies even when it appears to be a transitional point. When the geometrical shortest path scales as N θ , the effective dimensionality can be defined as def f 1θ. Defined in this way, for a small region where δ 1, the effective dimensionality of the network in both14 and17 was found to be greater than four. However, the behaviour of the Ising model is not mean field like here. This is again another unexpected result as the effective dimensionality defined from the shortest paths does not correspond to that of the critical behaviour of the Ising model. On the other hand, assuming the results of 18 to be correct no such anomalous behaviour is present but then it is not clear why the results of 14 are wrong where much larger system sizes were considered. In fact it was argued here that for sizes less than 104 , one may conclude that the network is a small world up to δ 2. Also, several independent studies14–17 suggest that there are two transitional points at δ 1 and δ 2 which is not indicated in13 or.18 In this short review, we have posed more questions than tried to answer. First, we discussed that in the mean field regime, Ising models on networks have some unique features as far as finite size scaling is concerned and the reason is still unknown. Second, the dimensionality of a Euclidean network in which long distance links exist with a probability depending parametrically on the distance between nodes seems not to be known precisely yet. A few studies suggest that the geometrical dimension may not determine the critical behaviour of cooperative phenomena like the Ising model. The continuity of the effective dimension as a function of the parameter governing the linking probability is also under question. An analytical study
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could perhaps settle these questions. Dynamical phenomena can also be studied on this network to see whether it can shed some new light to the unsettled issues. The author acknowledges collaborations in the present topic with Bikas K. Chakrabarti, Arnab Chatterjee, Turbasu Biswas and Kinjal Banerjee, and financial support from CSIR grant no. 03(1029)/05/EMRII. References 1. S. Milgram, Psychology Today 1, 60 (1967); J. Travers and S. Milgram, Sociometry 32, 425 (1969). 2. D. J. Watts and S. H. Strogatz, Nature 393, 440 (1998); D. J. Watts, Small Worlds, Princeton Univ. Press, Princeton (1999). 3. M. E. J. Newman and D. J. Watts, Phys. Lett. A 263, 341 (1999). 4. R. Albert and A.-L. Barab´ asi, Rev. Mod. Phys. 74, 47 (2002). 5. A.-L. Barab´ asi and R. Albert, Science 286, 509 (1999). 6. S.-H. Yook, H. Jeong and A.-L. Barab´ asi, Proc. Natl. Acad. Sc. 99, 13382 (2002). 7. M. T. Gastner and M. E. J. Newman, Eur. Phys. J. B 49, 247 (2006). 8. A. K. Chandra, K. Basu Hajra, P. K. Das and P. Sen, Int J. Mod. Phys. C 18, 1157 (2007). J. S. Katz, Scientometrics, 31, 31 (1994). 9. P. Sen, Physica Scripta T 106, 55 (2003). 10. S. S. Manna and P. Sen, Phys. Rev. E 66, 066114 (2002); P. Sen and S. S. Manna, Phys. Rev. E 68, 026104 (2003); S. S. Manna, G. Mukherjee and P. Sen, Phys. Rev. E 69, 017102 (2004). 11. J. Kleinberg, Nature 406, 845 (2000). 12. S. Jespersen and A. Blumen, Phys. Rev. E 62, 6270 (2000). 13. P. Sen and B. K. Chakrabarti, J. Phys A 34, 7749 (2001). 14. C. F. Moukarzel and M. A. de Menezes, Phys. Rev. E 65, 056709 (2002). 15. P. Sen, K. Banerjee and T. Biswas, Phys. Rev. E 66, 037102 (2002). 16. H. Zhu and Z-X. Huang, Phys. Rev. E 70, 036117 (2004). 17. A. Chatterjee and P. Sen, Phys. Rev E, 74, 036109 (2006). 18. YuF. Chang, L. Sun and X. Cai, Phys. Rev E 76, 021101 (2007). 19. A. Barrat and M. Weigt, Eur. Phys. J. B 13, 547 (2000). 20. M. Gitterman, J. Phys A 33, 8373 (2000). 21. T. Nikoletopoulos, A. C. C. Coolen, I. Prez Castillo, N. S. Skantzos, J. P. L. Hatchett and B. Wemmenhove, J. Phys. A 37, 6455 (2004). 22. C. P. Herrero, Phys. Rev. E 65, 066110 (2002). 23. H. Hong, B. J. Kim, and M. Y. Choi, Phys. Rev. E 66, 018101 (2002).
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Quantum Information
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Chapter 19 Equal Superposition Transformations and Quantum Random Walks Preeti Parashar Physics and Applied Mathematics Unit Indian Statistical Institute 203 B T Road, Kolkata 700 108, India The largest ensemble of qubits which satisfy the general transformation of equal superposition is obtained by different methods, namely, linearity, no-superluminal signalling and non-increase of entanglement under LOCC. We also consider the associated quantum random walk and show that all unitary balanced coins give the same asymmetric spatial probability distribution. It is further illustrated that unbalanced coins, upon appropriate superposition, lead to new unbiased walks which have no classical analogues.
Contents 19.1 Introduction . . . . . . . . . . . . . . . . . . 19.2 The Equal Superposition Ensemble . . . . . 19.3 No-Superluminal Signalling . . . . . . . . . 19.4 Non-Increase of Entanglement under LOCC 19.5 Quantum Random Walks . . . . . . . . . . 19.6 Conclusions . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . .
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19.1. Introduction There has been considerable interest in the recent past to prove the nonexistence of certain quantum unitary operations for arbitrary and unknown qubits. Some of the important ones are: the no-clonning theorem,1 the nodeleting principle,2 no-flipping operator3 and the no-Hadamard operator.4 These no-go theorems have been re-established by other physical fundamental principles, like the no-signalling condition and no-increase of entanglement under local operations and classical communication (LOCC).5–10 It 385
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is then natural to ask that if these operations do not work universally (i.e., for all qubits), then for what classes of quantum states it would be possible to perform a particular task by a single unitary operator. For example, the set of qubits which can be flipped exactly by the quantum NOT operator, lie on a great circle of the Bloch sphere.11 Likewise, the largest ensemble of states which can be rotated by the Hadamard gate was obtained.12 The Hadamard gate creates a superposition of qubit state and its orthogonal complement with equal amplitudes. In the present work, we consider the most general transformation where the superposition is with amplitudes which are equal upto a phase. In other words, the state and its orthogonal superimpose with equal probabilities but not necessarily with exactly the same amplitudes. First, we obtain the largest class of quantum states which can be superposed via this transformation. Second, it is shown, by using the no-signalling condition and non-increase of entanglement under LOCC, that this transformation does not hold for an arbitrary qubit. The Hadamard transformation is known to be intimately connected to quantum random walks.13 It has been used as a ‘coin flip’ transformation (balanced coin) to study the dynamics of such walks.14–17 In the same spirit, we consider the quantum random walk associated with our general transformation and study the probability distribution of the position of a particle. It is found that the entire family of such walks gives the same asymmetric distribution. We have also considered a unitary transformation with unequal amplitudes, serving as an unbalanced coin. It is shown that, after a suitable superposition, both types of coins lead to symmetric (unbiased) walks. However, in the case of unbalanced coin, we obtain new walks that have no classical analogues. The paper is organized as follows: In Sec. 19.2 we present the equal superposition ensemble. Secs. 19.3 and 19.4 pertain to the proving of the non-existence of equal superposition transformation for an arbitrary qubit. If the state and its orthogonal could be superposed, then it must belong to the ensemble presented in Sec. 19.2. This is achieved by imposing the nonviolation of two basic physical principles. Consider two spatially separated parties. Then i it is impossible to send superluminal signals between them, ii it is impossible to increase entanglement between them by LOCC. Sec. 19.5 is devoted to the study of the associated quantum random walks. We end the paper with some conclusions in Sec. 19.6.
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19.2. The Equal Superposition Ensemble The computational basis (CB) states &0, 1' of a qubit can be superposed most generally via the transformation U 0 α0 β 1, U 1 γ 0 δ 1,
(19.1)
α, β, γ, δ being arbitrary non-zero complex numbers. We are, however, interested in equal superposition (upto a phase) of the basis vectors. So let β
eiθ α, δ
eiφ γ.
(19.2)
The transformed states are required to be normalized and orthogonal to each other. This imposes the following constraints αα
γγ
12,
φ θ π.
(19.3)
Eq.(19.1) then becomes U 0 α0 eiθ α1, U 1 γ 0 eiθ γ 1,
(19.4)
with the unitary matrix given by α γ . U iθ e α eiθ γ
(19.5)
This gives an infinite family of ‘equally weighted’ transformations since θ can take any value between 0 and 2π, and α, γ are - numbers satisfying the constraint (19.3). One can get rid of the overall factor by setting α 1 γ in Eq.(19.4). However, the states then become unnormalized. To restore normalization one could simply fix α 1 2 γ. With this choice, the states in (19.4) reduce to the specific form of states lying on the equatorial great circle. So for the sake of generality, we shall refrain from assigning any particular value to these parameters. Now, we address the following question: Which other orthogonal pair of qubit states &ψ , ψ' would transform under U in a similar manner as &0, 1'? More precisely, we wish to find as to which class of qubits would satisfy U ψ αψ eiθ αψ , U ψ γ ψ eiθ γ ψ, αα
γγ
12. (19.6) For this purpose, we start with an arbitrary qubit state ψ and its orthogonal complement ψ as a superposition of the CB states ψ
a0 b1,
ψ
b 0 a 1 ,
(19.7)
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where the non-zero complex numbers obey the normalization condition aa bb 1. Substituting the above states in the first expression of Eq.(19.6) gives U ψ αa eiθ αb 0 αb eiθ αa 1.
(19.8)
U acts linearly on ψ yielding U ψ aU 0 bU 1
αa γb0 eiθ αa eiθ γb1.
(19.9)
Equating the coefficients in (19.8) and (19.9) gives α b eiθ b , γ
a a
eiθ
γ iθ b eiθ b b e α
(19.10)
Thus, we can state our main result: The general equal superposition transformation (19.6) holds for all qubit pairs &ψ , ψ' which satisfy the constraint (19.10). Writing the complex state parameters as a x iy, b u iv, where x, y, u, v are all real, a state from this ensemble reads as ψ &
1 iθ γ e u iv iy '0 u iv 1, 2 α
(19.11)
while its orthogonal would be ψ
γ u iv iy '1 α γ 1 γ eiθ u iv 0 & eiθ u iv iy '1 (19.12) α 2 α u iv 0 &
1 iθ e 2
with the inner product rules explicitly given by ψ1 ψ2 ψ1 ψ 2
1 γ γ γ 6 eiθ eiθ eiθ u1 iv1 u2 iv2 y1 y2 4 α α α i iθ γ e u1 iv1 y2 u2 iv2 y1 ψ 1 ψ 2 , 2 α γ ieiθ u1 iv1 y2 u2 iv2 y1 ψ1 ψ2 . (19.13) α
Our result provides a very convenient unified framework to deduce any desired class of equally superposable quantum states. If the CB states obey a particular transformation (out of the infinite family (19.4)), then in a single shot we can obtain the entire ensemble of qubits which would satisfy the same transformation. To demonstrate its usefulness, we present below, two known examples as special cases of our result.
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1. Hadamard ensemble : Choose α (U UH ) UH 0
1 2
0 1,
where UH
12 , γ
12 , θ
389
0. Then
1 2
(19.14)
1 1 1 . 2 1 1
(19.15)
UH 1
0 1,
This is the well known Hadamard gate with its corresponding transforma tion. Notice that UH 2 I since UH σx σz 2 where σx , σz are Pauli matrices. However, in general U 2 " I. Further, substituting the above choice of the parameters, the constraint (19.10) gives b b , i.e., b is real, and a a 2b, i.e., Re a b. In terms of the real parameters x, y, u, v, the above deductions yield v 0 and u x. Therefore, the qubit states become restricted to ψ
x iy 0 x1,
ψ
x0 x iy 1, 2x2 y 2
1. (19.16)
Hence, we have obtained a special class of states which transform under the action of the Hadamard matrix UH via the transformation 1 1 (19.17) UH ψ ψ ψ, UH ψ ψ ψ. 2 2 In other words, this proves the existence of the Hadamard gate (19.15) for any qubit chosen from the ensemble (19.16). 2. Invariant ensemble : Choose α 12 , γ i2 , θ π2 . Then (U UI ) UI 0 where
1 2
0 i1,
U I 1
1 2
i0 1,
1 1 i . 2 i1
UI
(19.18)
(19.19)
An interesting property of this transformation is that it goes into itself, i.e., UI 0 UI 1 under the interchange 0 1. For this reason we but not shall refer to it as being ‘invariant’. The matrix UI is symmetric hermitian and UI2 iσx (i.e., the NOT gate) since UI I iσx 2. Now, in order to find as to which qubit states would satisfy 1 1 (19.20) UI ψ ψ iψ, UI ψ iψ ψ 2 2
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we substitute the above values of α, γ, and θ in (19.10). This yields b b , i.e., b is real, and a a 0, i.e., Re a 0, implying that a is purely imaginary. Again assuming a x iy and b u iv, these constraints give v 0 and x 0. Therefore, the qubit states become restricted to ψ
iy 0 u1,
ψ
u0 iy 1,
y 2 u2
1.
(19.21)
The above two ensembles were obtained in12 by treating each one separately. Here we have shown that they can be deduced from a single general ensemble of equally superposed qubits. The family of transformations which remain invaraint under the interchange of 0 and 1 is a subset of the general family (19.4), and every member is essentially of the type (19.18). To see this let us consider the general transformation (19.4). For this to be invariant we must have α eiθ γ and γ eiθ α which implies that θ π 2, 3π 2. Substituting γ iα in (19.4) we obtain the general form of the invariant transformation (U UI ) UI 0 α0 i1,
UI 1 αi0 1,
where UI
α
αα
1 i . i 1
12,
(19.22)
(19.23)
Since α is an overall phase factor, it can be readily verified that every member of (19.22) would lead to exactly the same ensemble (19.21). Thus, (19.18) can be regarded as a representative of the invariant family (19.22). In what follows, we shall establish our main result in the context of two other physical principles, namely; the no-superluminal signalling condition and the non-increase of entanglement under LOCC. 19.3. No-Superluminal Signalling Let us consider the CB states transforming via Eq.(19.4), and a qubit state transforming under the same unitary matrix U via the first expression in (19.6). We first show that if ψ is completely arbitrary, then this would imply superluminal signalling. For this purpose, assume that Alice possesses a 3d qutrit while Bob has a 2d qubit and both share the following entangled state:
ψ
1 0A 0B 1A ψ B 2A 1B . 3
φAB
(19.24)
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The density matrix of the combined system is defined as ρAB φAB φ. Alice’s reduced density matrix can be obtained by tracing out Bob’s part ρA
trB ρAB
1 00 11 22 3 a10 a 01 b12 b 21 . (19.25)
Now Bob applies the above mentioned unitary transformation on his qubit states &0, 1, ψ ' in Eq.(19.24). But, he does not communicate any information to Alice regarding his operation. The shared state then changes to I
U φAB φ AB
1 α 00 eiθ α 01 α 1ψ eiθ α 1ψ 3 γ 20 eiθ γ 21 . (19.26)
After this operation, Alice’s new reduced density matrix becomes ρA
1 00 11 22 3 1 1 a eiθ b eiθ b a 10 a eiθ b eiθ b a01 2 2 αγ a eiθ b eiθ b a 12 α γ a eiθ b eiθ b a21 .
(19.27)
Comparing the coefficients of each term in (19.25) and (19.27), it is evident that ρA " ρA for arbitrary choices of the parameters a and b. So, in principle, Alice can distinguish between ρA and ρA , although Bob has not revealed anything to her about his operation. This implies that, with the help of entanglement, superluminal communication has taken place. But faster-than-light communication is forbidden by special theory of relativity. Hence, we conclude that the equally weighted transformation does not exist for an arbitrary qubit. In other words, it is an unphysical operation. If, however, we impose that the no-signalling constraint should not be violated, then ρA and ρA should be equal because the action of U is a trace preserving local operation performed only on Bob’s side. Comparing coefficients of the term 10 in (19.25) and (19.27) we recover the condition a a eiθ b eiθ b . From 12 we have αγ a eiθ b eiθ b a 1 b which yields 2αγ eiθ b b. Substituting γ 2γ , we get the other iθ α constraint b e γ b . Thus, the no-signalling condition gives exactly the same class of states that was obtained initially from linearity.
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19.4. Non-Increase of Entanglement under LOCC Here we shall first show the non-existence of the unitary operation (19.6) for an arbitrary ψ by considering the fact that local operations and classical communication cannot increase the entanglement between two spatially separated quantum systems. It turns out that, ρA and ρA above, have equal eigenvalues 0, 13, 23. This means that there is no change in entanglement before and after the unitary operation. So we consider a different shared resource which has been used in9,10 for studying flipping and Hadamard operations, ΦAB
1
0A 1 b b
1A
0B1 1B2 1B1 0B2
2
(19.28)
0B1 ψ B2 ψ B1 0B2 ,
2
where the first qubit is with Alice while the other two are at Bob’s side. Repeating the protocol, we obtain Alice’s reduced density operator as ρA
1 00 b b11 b10 b 01 . 1 b b
(19.29)
The amount of entanglement given by the von Neumann entropy is zero since the eigenvalues of ρA are 0 and 1. This means that the resource state (19.28) is a product state in the A:B cut. Now Bob applies the trace preserving general transformation on the last particle B2 in Eq.(19.28), which results in the state
Φ AB
1 2N
γ 000 eiθ γ 001 α010 eiθ α011
α10ψ eiθ α10ψ α1ψ0 eiθ α1ψ1 (19.30) ,
where N 2 14 & a a 2 eiθ b eiθ b a a '. Since a and b are arbitrary, so in general, the above state is entangled in the A:B cut. This implies that entanglement has been created by local operation. However, we know that entanglement cannot be increased by local operations even if classical communication is allowed. Therefore, the above contradiction leads us to conclude that the unitary operator (19.5) cannot perform the same task for an arbitrary qubit, as it does for the CB states 0 and 1. We now derive the conditions under which the entanglement in the state would remain zero even after the application of U . For this purpose we have to compare the von Neumann entropy of the respective density matrices on
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Alice’s side. So after Bob’s operation ρA
1 00 N N
111 D 10 D 01 ,
(19.31)
where D 12 &αγ a a eiθ b eiθ b b'. The eigenvalue equation of the above matrix gives two roots, namely, N 2 4 N 1 DD 1 , (19.32) λ 2 2N In order to maintain the same amount of entanglement in the system before and after the unitary operation, we should equate these two roots λ of ρA to the eigenvalues 0 and 1 of ρA (for the entropies to be equal). This furnishes the constraint DD N 1. Substituting the expressions for N, D and D , and rearranging the terms, this condition acquires the form 1 iθ iθ aa e b e b a a 4 1 iθ iθ e b e b γα b αγ b 4 2
eiθ γα b2 eiθ αγ b2 bb
4 a a 2 eiθ b eiθ b a a .
(19.33)
1 1 , γ 2γ on the L.H.S. and adding and subtracting 2aa Using α 2α on the R.H.S., the above relation is recast as iθ b eiθ b 1 a a aa e 4 iθ b eiθ b 1 γ b α b e 2 α γ 2 iθ2 γ b eiθ2 α b e α γ
4bb a a eiθ b eiθ b a a
(19.34)
which can be written more compactly as
a a eiθ b eiθ b 34 a a 14 eiθ b eiθ b 12
eiθ2
γ b α
e
iθ 2
α b γ
2
γ b α
αγ b
.
(19.35)
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For convenience let us denote the two terms on L.H.S. by A and B. Then 2 γ α iθ2 b eiθ2 b AB e . (19.36) α γ In the above, R.H.S. is either a real positive definite quantity or zero. For the L.H.S. to be positive, there, however, exist two possibilities: (i) A 0, B 0: If we suppose that both terms in A are positive (they are already real), then a a eiθ b eiθ b C, where C is a real positive constant. Thus B 0 if αγ b αγ b eiθ b eiθ b 32 C, which certainly is possible. Similarly if we suppose that both terms in A are negative, then A 0 implies eiθ b eiθ b a a C. Thus B 0 if αγ b αγ b a a C2 . (ii) A 0, B 0: In a similar manner, restrictions can be obtained for this case. When R.H.S. is identically zero, then Eq.(19.36) would be satisfied uniquely if A 0, B 0. This gives a a eiθ b eiθ b and b eiθ αγ b , which are exactly the constraints that we have earlier obtained by linearity and no-signalling. It can be easily checked that the other cases A 0, B " 0 and A " 0, B 0 cannot exist due to the constraint fixed by R.H.S. being zero. The above analysis demonstrates the possibility of existence of more solutions from the principle of non-increase of entanglement under LOCC. In the case of Hadamard operation, we had obtained a unique solution10 from linearity, no-signalling and non-increase of entanglement under LOCC. Here we get a larger set of states with zero entanglement, from the last method. However, we must remember that we are looking for orthogonal pairs of states &ψ , ψ' which transform under the unitary operation defined by (19.6). In the above, we have considered only ψ . Therefore, we must now carry out a similar analysis with ψ. More precisely, we take the set of qubit states &0, 1, ψ' and the shared state as 0B1 1B2 1B1 0B2 1 0A ΨAB 1A 1 a a 2 0B1 ψ B2 ψ B1 0B2 . (19.37) 2 Then Alice’s reduced matrix reads as ρA
1 00 a a11 a 10 a01 . 1 a a
(19.38)
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Bob now applies U on the states changing the shared state to
Ψ AB
1 2N
&0, 1, ψ'
γ 000 e
iθ
395
of his last qubit, thereby
γ 001 α010 eiθ α011
γ 10ψ eiθ γ 10ψ α1ψ0 eiθ α1ψ1, (19.39)
where N 2 12 &γα b aa eiθ b eiθ b αγ b aa eiθ beiθ b '. The corresponding reduced density matrix at Alice’s end becomes ρA
1 00 N N
111 D 10 D 01 ,
(19.40)
where D & 14 a a eiθ b eiθ b a2 '. Like the previous case, this matrix has the following two eigenvalues N 2 4 N 1 DD 1 . (19.41) λ 2 2N Equating λ to the eigenvalues 0 and 1 of ρA gives the constraint DD N 1 which can be expanded as
a a eiθ b eiθ b 38 a a 18 eiθ b eiθ b 12
eiθ2
γ b α
eiθ2
α b γ
2
γ b α
αγ b
.
(19.42)
Interestingly, this is a new restriction on the expression on R.H.S. This has to be consistent with the earlier restriction (19.35). Therefore equating (19.42) with (19.35) renders a a eiθ b eiθ b . Substituting this in either (19.35) or (19.42) yields b eiθ αγ b . Thus we finally obtain a unique solution which is exactly the constraint (19.10) that defines our equal superposition ensemble. We remark that such a situation was not encountered in the case of the Hadamard operation.10 The reason is that the Hadamard transformation on ψ is not independent since it can be obtained from the Hadamard transformation on the states &0, 1, ψ ' by using the special property of the Hadamard operator, namely, UH 2 I. However, in the present scenario (and in general), U ψ cannot be deduced from &U 0, U 1, U ψ '. So it is necessary to take U ψ into consideration, although whether this would provide some new restriction or not depends on the particular situation.
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For example, if we proceed with ψ, then linearity and no-signalling give nothing new but the same constraint (19.10) which was obtained from ψ . However, in the framework of non-increase of entanglement under LOCC, this indeed yields a different condition (19.42), thereby forcing the set of solutions to a single unique solution. In view of the above, we are now in a position to make a stronger statement regarding our main result: Any pair of qubit states &ψ , ψ' can be equally superposed via the unitary operation (19.6) if and only if they satisfy the constraint (19.10). 19.5. Quantum Random Walks In the previous sections, we have obtained by different methods the class of qubit states which transform under the action of the unitary matrix U in a manner similar to Eq. (19.4). As an application of this transformation (19.4), we are now going to study the quantum random walk associated with it. A particularly nice detailed survey of quantum walks has been given by Kempe,18 while19 is a short review devoted to their applications to algorithms. The Hadamard matrix UH has been widely used as a balanced coin (translation to the left or to the right with equal probability) to study the properties of a discrete-time quantum random walk (QRW).15–17 For example, the probability of finding the particle at a particular site after T steps of the walk have been investigated in detail. The Hadamard coin gives an asymmetric probability distribution for the QRW on a 1d line. This is because the Hadamard coin treats the two CB states differently; it multiplies the phase by 1 only in the case of 1. It has also been pointed out18,19 that if the Hadamard coin is replaced with the more symmetric coin UI , then the probability distribution becomes symmetric. However, our analysis shows that this is not the case, even though UI treats both 0 and 1 in a symmetrical way. This also motivates us to investigate the discrete-time QRW from a more general point of view. We shall study the behaviour of the walk by taking the general unitary matrix U given by Eq.(19.5) as our balanced coin. Subsequently, we shall comment on some interesting features that these walks share. Consider a particle localized at position z on a 1d line. The Hilbert space HP is spanned by basis states z , where z is an integer. This position Hilbert space is augmented by a coin space HC spanned by the two CB states 0 and 1. To avoid confusion with the position states, we now introduce a change of notation, and instead denote the CB states as ) and *. The total state of the particle lies in the Hilbert space H HC HP .
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The first step of the random walk is a rotation in the coin space. We follow a procedure similar to what was adopted for the Hadamard walk.16,18 In our general scenario, the matrix U given by (19.5) serves as the coin, with the following action (cf. Eq.(19.4)) U ) α ) eiθ α *,
U * γ ) eiθ γ *.
(19.43)
The rotation is followed by translation with the application of the unitary operator
(19.44) S )) z 1z ** z 1z z
z
in the position space HP . Note that S is a ‘conditional’ translation operator since it moves the particle by one unit to the right if the coin state is ), and to the left if it is * S ) z ) z 1,
S * z * z 1.
(19.45)
The particle is subjected to these two alternating unitary transformations. Therefore, the QRW of T steps is defined as the transformation AT , where A acts on the total Hilbert space H and is given by AS U
I
(19.46)
To start with, let the particle be in the ) coin state and located at the position 0. Thus the total initial state is denoted by φ ) 0. Let us now evolve the walk, for a few steps, under successive action of the operator A: φ
α ) 1 eiθ α * 1
α2 ) 2 eiθ α2 * αγ ) 0 e2iθ αγ * 2
α3 ) 3 eiθ α3 * 2α2 γ ) 1 e2iθ αγ 2 ) 1
e3iθ αγ 2 * 3
α4 ) 4 eiθ α4 * 3α3 γ ) 2
e2iθ α3 γ * α2 γ 2 ) 0
e3iθ α2 γ 2 * αγ 3 ) 2 e4iθ αγ 3 * 4
(19.47)
After T iterations, the particle is in an entangled state, say φT . The probability of finding the particle at a particular site z is given by Pz
2
) z φT * z φT
2
.
(19.48)
Let us analyze, step by step, the spatial probability distribution of the walk.
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After T 1: If we measure the position space after the first step, then the particle can be found at the site 1 with probability αα and at the site 1 with the same probability. Since we already know that αα 12 (normalization), so the particle moves with equal probability, one step to the right and one to the left of its original position. The walk is therefore, unbaised, just like the usual Hadamard walk. After T 2: The probabilities of finding the particle at positions 2, 2 and 0 are respectively, P2
α2 α2
αα 2
14,
P2 αα γγ 14, P0 α2 α2 αα γγ 12.
(19.49)
This step is also similar to the case of classical walk since the P s are symmetrically distributed. After T 3: The distribution is P3
α3 α3
αα 3
18,
P3 αα γ 2 γ 2 18, P1 αα 3 4 αα 2 γγ 58, P1 αα γγ 2 18.
(19.50)
After the third step, the quantum walk begins to deviate from its classical counterpart. Although P3 P3 , note that P1 " P1 . So the walk starts to be asymmetric, drifting towards the right since the site 1 has greater probalility. After T 4: Similarly, upon measuring the position space after four iterations, we get the following asymmetric distribution P4
116, P4
116, P2
58, P2
18, P0
18.
(19.51)
Again, this differs from the symmetric classical probability distribution P4 116, P4 116, P2 14, P2 14, P0 38. Proceeding in a similar way one can check the veracity of the foregoing conclusions by considering more number of iterations. Clearly, the parameter θ which appears in the phase factor does not contribute to the probabilities. Also since αα γγ , so for the purpose of probability distribution, only one of the parameters may be regarded as independent. It is observed that the spatial probability distribution of the QRW corresponding to the general matrix U is asymmetrical and coincides exactly with that of the already known Hadamard walk. This means that every
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unitary transformation in which the qubit CB states are equally weighted, leads to the same probability distribution if the particle is taken in the same initial state. Therefore, we infer that even the symmetric coin UI induces an asymmetrical walk. In fact, it can be argued easily as to why a symmmetric probability distribution for the initial state ) 0 (or * 0) is impossible. Let us refer to the distribution (19.50) after three iterations. If we want to make it symmetric, we must have P1 P1 . This implies that αα αα 2 4αα γγ γγ 2 0.
(19.52)
This equation cannot be satisfied since we know that αα term in the bracket equals 1. So L.H.S. can never be zero.
12, and the
The direction of drift in the walk depends on the initial coin state and the bias is the result of quantum interference. So phases play a very crucial role in inducing asymmetry. This bias can, however, be taken care of if we again allow interference, so that the effect of the earlier superposition is negated. Thus, in order to make the walk symmetric or unbiased, we must take a superposition of ) and * as the initial coin state. However, we shall not assume apriori that ) and * are superimposed with equal probability. For our general approach, we shall rather superimpose them with arbitrary amplitudes and obtain restrictions under which we can get a symmetric distribution. So we start the walk in the state
φ
xx yy
x ) y * 0,
1
(19.53)
(x and y are, in general, complex numbers) and let it evolve under the repeated action of the operator A, as was done earlier. After T 1, the state becomes φ1
xα yγ ) 1 eiθ xα yγ * 1 A ) 1 eiθ B * 1
We now demand that the particle should be found at sites 1 and equal probability. This gives the constraint AA
BB
12
(19.54) 1
with
(19.55)
which can be recast in terms of the transformation parameters as xy αγ x yα γ
0.
Clearly, this holds only if xy αγ is purely imaginary.
(19.56)
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After T φ2
2, the state of the particle becomes
αA ) 2 eiθ αA * γB ) 0 e2iθ γB * 1 (19.57)
and the probabilities are P2
αα AA
14,
P2 γγ BB 14, P0 αα AA γγ BB 12. After T
(19.58)
3, the state evolves into φ3
α2 A ) 3 eiθ α2 A * 2xα2 γ ) 1
e2iθ γ 2 B ) 2yγ 2 α * 1
e3iθ γ 2 B * 3
(19.59)
The probabilities for the odd sites are P3
P1
P1
αα 2 AA
18,
P3
γγ 2 BB
1 8
αα 2 AA 4xx αα 2 γγ 18 xx 2 γγ 2 BB 4yy γγ 2 αα 18 yy 2
(19.60)
For symmetry, P1 P1 which, in turn, implies that xx yy . But from normalization, we have xx yy 1. This restricts the value to xx yy 12. So the two amplitudes are equal, upto a phase factor, leading to an equal superposition of ) and *. We have thus found that in order to make the quantum walk associated with the matrix U symmetric, it is necessary to take an ‘equally’ superposed coin state in such a way that xy αγ is purely imaginary. We present a new example to illustrate this situation. i 3π Example: Choose α γ 1
2 and θ 2 . The transformation (19.43) becomes 1i 1i ) i *, Uh * ) i * (19.61) Uh ) 2 2
where Uh
1i 2
1 1 . i i
(19.62)
This has the features of a ‘hybrid’ between the Hadamard and the Invariant transformations discussed earlier. As expected, this gives an asymmetric
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walk. However, if we take the coin state in a superposition with amplii , then the condition that xy αγ is purely tudes x 12 and y 2 imaginary is satisfied. Thus upon evolving the walk with the initial state 12 ) i * 0, we do get the symmetric probability distribution which coincides with the classical one. Unbalanced coin: We have seen above that unitary balanced coins lead to a symmetric walk after appropriate superposition. Now we shall show that even unbalanced (biased) coins can yield an unbiased walk under similar restrictions. Consider the unequal superposition transformation given in12 U * q ) p *,
U ) p ) q *, where, in general, pp
"
pp qq
1, (19.63)
qq and the unitary matrix U is given by p q . (19.64) U q p
This obviously induces an asymmetric walk, and the bias depends on the explicit values of p, q. To symmetrize this walk we follow the earlier procedure of starting with a superposed coin state
Φ
where r, s are non-zero Φ1
rr
r ) s * 0, -
numbers. After T
ss
1,
(19.65)
1, the state becomes
rp sq ) 1 rq sp * 1 E ) 1 F * 1.
For a symmetric probability distribution, we must impose Pz every step. So after the first step we should have EE
FF
12,
(19.66)
Pz at (19.67)
which can be rewritten as 2 rs pq r sp q rr ss qq pp . After T Φ2
(19.68)
2, the particle is in the entangled state
pE ) 2 qE * q F ) 0 p F * 2 (19.69)
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with the probability distribution 1 pp , 2 1 P2 pp F F pp , 2 P0 qq EE qq F F P2
After T
pp EE
qq ,
(19.70)
3, the state of the particle reads
Φ 3
p2 E ) 3 pqE * 1 qq E pq F ) 1
qE qq F * 1 2 p q F ) 1 p F * 3,
p
(19.71)
and the associated probabilities are P3
p2 p2 EE
pp 2 EE
1 pp 2 , 2
P3
pp 2 F F
1 qq 1 pp qq p qEF pq E F , 2 1 P1 qq 1 pp qq p qEF pq E F . 2 Imposing P1 P1 implies that P1
1 pp 2 , 2
p qEF pq E F
(19.72)
0,
(19.73)
which together with (19.68) leads to the following constraint: rs pq r sp q
0,
rr
ss .
(19.74)
Thus, even for an unequal superposition transformation (19.63), the coin state must be in equal superposition (upto a phase) of ) and * in order to obtain a symmetric walk.
Example: Let p 23 , q 12 , r 12 and s i2 . It can be checked that the constraint (19.74) holds for this choice. The associated probabilities are T
1 : P1
T
2 : P2
T
3 : P3
P1
12,
P2
38,
P3
932,
P0
P1
14,
P1
732.
(19.75)
Hence we get a new symmetric distribution which differs from the classical one right in the second step (T 2) itself. This example demonstrates new possiblities in the quantum world which have no classical analoques. The distribution depends only on the values of p and q, after the initial
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constraint (19.74) is satisfied. For p q 12 , we recover the Hadamard walk. So this walk can be thought of as a ‘generalized’ Hadamard walk for unequal amplitudes p and q. 19.6. Conclusions Here we have established that it is not possible to create a superposition with equal probabilities, of an arbitrary qubit state and its orthogonal. The class of states for which this can be achieved is presented. In addition, by using the principles of no-superluminal signalling and non-increase of entanglement under LOCC we have shown that this is the only set of qubits which would satisfy the equally weighted transformation. In other words, a qubit state and its complement can be equally superposed if and only if they belong to the aforementioned ensemble. Through this work, one more impossible operation, namely, no-equalsuperposition has been added to the list of no-go theorems in quantum information. This can be considered as a generalized no-Hadamard principle. It has also been observed how the conditions of no-signalling and no-increase of entanglement puts a restriction on the allowed operations in quantum mechanics. The equal superposition ensemble is shown to be consistent with these two fundamental principles. It would be interesting to obtain an optimal Hadamard operation which would work for all qubits but with some optimal fidelity. The quantum random walk associated with this general unitary equal superposition transformation has been investigated from the point of view of probability distribution of a particle. We have found that the entire family leads to the same asymmetric distribution. This implies that even the symmetric transformation (19.18) gives an asymmetric walk. Our result demonstrates that although phases are crucial in inducing asymmetry, it does not matter in which way they are arranged. The cancellation of phases leading to the asymmetry occurs at exactly the same place (in T 3) for any balanced coin. It may be mentioned that apart from the CB vectors, any state from our ensemble specified by (19.10), can be used as a coin state to study the evolution of the walk. The measurement on the coin register would then have to be carried out in the &ψ , ψ' basis. We have also obtained conditions under which balanced and unbalanced coins would yield unbiased walks. To illustrate this, a few examples have been presented. We find that in general, any quantum walk can be sym-
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metrized by taking an appropriate superposition of the CB states. This is equivalent to applying an additional coin before the start of the walk. For example, the Hadamard walk can be made symmetric by applying the coin UI on the initial state ) 0 (or * 0). This is then followed by the usual alternate sequence of UH and S. On the other hand, if we consider a random walk induced by the coin UI , then, interestingly, the Hadamard coin UH serves as an additional coin to make the UI walk symmetric. This also works for walks obtained from unbalanced coins. An ‘appropriately balanced’ additional coin would render such a walk unbiased (with non-classical distribution). References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.
W.K. Wootters and W.H. Zurek, Nature 299, 802 (1982). A.K. Pati and S. Braunstein, Nature 404, 164 (2000). V. Buzek, M. Hillery and R.F. Werner, Phys. Rev. A 60, R2626 (1999). A.K. Pati, Phys. Rev. A 66, 062319 (2002). N. Gisin, Phys. Lett. A 242, 1 (1998). L. Hardy and D.D. Song, Phys. Lett. A 259, 331 (1999). A.K. Pati, Phys. Lett. A 270, 103 (2000). A.K. Pati and S. Braunstein, Phys. Lett. A 315, 208 (2003). I. Chattopadhyay, S.K. Choudhary, G. Kar, S. Kunkri and D. Sarkar, Phys. Lett. A351, 384 (2006). P. Parashar, Int. J. Quant. Info. 5, 845 (2007). A.K. Pati, Phys. Rev. A 63, 014302 (2001). A. Maitra and P. Parashar, Int. J. Quant. Info. 4, 653 (2006). Y. Aharonov, L. Davidovich and N. Zagury, Phys. Rev. A 48, 1687 (1993). D. Meyers, J. Statistical Phys., 85, 551 (1996). D. Aharonov, A. Ambainis, J. Kempe and U. Vazirani, Quantum walks on graphs, Proceedings of STOC’01, pp 50 (2001). A. Ambainis, E. Bach, A. Nayak, A. Vishwanath and J. Watrous, Onedimensional quantum walks, Proceedings of STOC’01, pp. 37 (2001). A. Nayak and A. Vishwanath, Quantum walk on the line, quant-ph/0010117, (2000). J. Kempe, Contemporary Physics, 44, 307 (2003). A. Ambainis, Quantum walks and their algorithmic applications, quantph/0403120, (2004).
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Chapter 20 Cloning Entanglement Locally
Sujit K. Choudhary and Ramij Rahaman Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India The (im) possibility of exact cloning of orthogonal but equally entangled quantum states under local operations and classical communication is discussed. The amount of entanglement necessary in blank copy is obtained for various cases.
Contents 20.1 20.2 20.3
Introduction . . . . . . . . . . . . . . . . . . . . . . Relative Entropy and Negativity of Entanglement . Cloning Bell States . . . . . . . . . . . . . . . . . . 20.3.1 Local Cloning of Two Bell States . . . . . 20.3.2 Local Cloning of Four Bell States . . . . . 20.3.3 Local Cloning of Three Bell States . . . . . 20.4 Cloning Arbitrary Entangled States of Two Qubits 20.5 Cloning Entangled States of Three Qubits . . . . . 20.5.1 Cloning of GHZ States . . . . . . . . . . . 20.5.2 Cloning of W-States . . . . . . . . . . . . . 20.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . .
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20.1. Introduction Classical states can always be cloned and distinguished but this is not true, in general, for quantum mechanical states. Only orthogonal quantum states can be cloned and distinguished with certainty and that too when the operations for such purposes are performed on the entire system. But a common scenario in quantum information processing is where a multipartite entangled state is distributed among a number of spatially separated parties. 405
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Each of these parties are able to perform only local operations (quantum) on the subsystem they possess and can send only classical information to each other. This is known as LOCC (Local operation and classical communication). Previous works show that under LOCC, even the orthogonal entangled states are not distinguishable in general, but an unrestricted supply of free entanglement makes ‘local cloning’a of these states trivially possibleb . Thus study of local cloning of entangled orthogonal states becomes interesting when the supply of free entanglement is restricted. The study is also important because of fact that local cloning is very closely connected with many important information processing tasks, such as channel copying, entanglement distillation, error correction and quantum key distribution.4 Furthermore, this study is also helpful in understanding the nonlocality of a set.14 The concept of entanglement cloning under LOCC was first considered by Ghosh et al.1 where it was shown that for LOCC cloning of two (four)orthogonal Bell states, 1-ebit (2-ebit) of entanglement is neccessary and sufficient. Since then many works have been done in this direction.2–4 But most of these works deal mainly with maximally entangled states of two qubits. We will first consider here the local cloning of arbitrary but equally entangled orthogonal states of two qubits, the results then will be exploited to explore the (im) possibility of some multipartite entangled states. The organization of this article is as follows: In section-2, we have introduced some measures of entanglement which have got frequent uses throughout the article. Section-3 deals with local cloning of Bell statesc . In section-4, cloning of arbitrary entangled states of two qubits has been discussed. Section-5 deals with cloning of GHZ and W states which are the two extreme representatives of the inequivalent kinds of genuine three qubit entangled states.15
20.2. Relative Entropy and Negativity of Entanglement We introduce below the two measures of entanglement, which LOCC cannot increase. a cloning
under LOCC example, any arbitrary set of four orthogonal states of two qubits can be cloned with the help of 3 ebits, Any set of two orthogonal states need only 2 ebits.1 c by local cloning,or cloning, henceforth, we will mean cloning under LOCC with the help of an entangled blank copy. b For
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(a) The relative entropy of entanglement The relative entropy of entanglement for a bipartite quantum state ρ is defined by:5 ER ρ min
σD H
S ρ +σ
Here D is the set of all separable states on the Hilbert space H on which ρ is defined and S ρ+σ (the relative entropy of ρ to σ) is given by S ρ+σ tr ρ log2 ρ tr ρ log2 σ . Let ρ1 H 1 and ρ2 H 2 be two quantum states and let ER ρ1 S ρ1 +σ1 , ER ρ2 S ρ2 +σ2 ; i.e. σ1 H1 and σ2 H2 are the two separable states which minimize the relative entropies of ρ1 and ρ2 respectively. Let σ be the separable state belonging to the Hilbert space H1 H2 which minimizes the relative entropy of ρ1 ρ2 . Then: ER ρ1 ρ2 S ρ1 ρ2 +σ1 σ2 equality holds when σ1 σ2 It was known6
(20.1)
σ.
S ρ1 ρ2 +σ1 σ2 S ρ1 +σ1 S ρ2 +σ2
(20.2)
ER ρ1 ρ2 S ρ1 +σ1 S ρ2 +σ2
(20.3)
ER ρ1 ρ2 ER ρ1 ER ρ2
(20.4)
hence
i.e.
(b) The Negativity The negtivity of a bipartite quantum state ρ, N ρ, N ρ is given by12 N ρ +ρTB + 1
(20.5)
where ρTB is the partial transpose with respect to system B and +...+ denotes the trace norm which is defined as,
T +ρ B + tr ρTB ρTB (20.6) Like relative entropy of entanglement, the negativity of a bipartite quantum state too cannot increase under LOCC.11
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20.3. Cloning Bell States 20.3.1. Local Cloning of Two Bell States Given below are the four Bell states : 1 1 2πijn2 e j j . m, n, m 0, 1. 2 j 0
Bmn
(20.7)
where one qubit is held by Alice and the other is held by Bob. Any two Bell states can be cloned d with the help of one free ebit.1 To see this let us first suppose that Alice and Bob are supplied with either of the two states– B00 or B10 . If in addition they share a known Bell state (say B00 )as the blank copy, then by applying C-NOToperations locally (where for both parties, the qubit of the unknown Bell state is the source and the qubit of the blank state is the target), they will share two copies of the unknown Bell state. The same result can be obtained for any two Bell states given in (20.7) by first local unitarily transforming the two Bell states to the states B00 , B10 , then cloning of these later two states locally and finally local "2 and B "2 to the respective two copies of the unitarily transforming B00 10 initially given set of two Bell states. In fact it can be easily seen that with less than one ebit of free(i.e., distillable) entanglement such a cloning is not possible. To show this let us take the blank state ρ with distillable entanglement less than 1 bit. Had the cloning of two Bell states (suppose B00 and B10 )been possible with ρ as blank copy, the initially shared separable state ρsep 12 P B00 P B10 of Alice and Bob would have been transformed as: ρsep ρ %
1 P B00 B00 P B10 B10 2
The state in the right hand side has one ebit of distillable entanglement in the Alice vs. Bob cut7 whereas the initial entanglement (entanglement in the left hand side) is equal to entanglement of the blank copy which is less than one ebit. As there can be no increase in entanglement under LOCC, hence we conclude that with less than one ebit of free entanglement, one cannot clone the two Bell states by LOCC. Thus we see that one ebit is the necessary and sufficient amount with the help of which an unknown Bell state given from a set of two known Bell states can be cloned locally. d By
cloning two Bell states, we mean cloning of an unknown Bell state given from a set of two known Bell states, and so forth by the cloning of three and four Bell states
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20.3.2. Local Cloning of Four Bell States In order to obtain the necessary amount of entanglement in the blank copy for local cloning of four Bell states, Ghosh et al1 considered a situation where the two far apart parties Alice and Bob share two copies of one of the four Bell states with equal probabilities. Local exact cloning of four Bell states means that whenever this state is supplied together with a blank copy ρ to an LOCC operation, the result of such an operation would be an equal mixture of three copies of the four Bell states which has two ebits of distillable entanglement9 and hence ρ must have at least two ebits of distillable entanglement because of the fact that the equal mxture of two copies of four Bell states is separable and LOCC cannot increase entanglement. We are showing below that two ebits is also the sufficient amount for the local cloning of four Bell states. Apart from sharing an unknown Bell state (given from a set of four Bell states), let Alice and Bob share two copies of a known Bell state B00 (say) as ancilla. For having the clones of Bell states, let they (Alice and Bob) perform two bilateral XOR or BOXR operations, first one by making the second pair (in state B00 ) source and first pair (in unknown state Bmn ) target and second one by making first pair source and third pair target. The final result of the full operation is given by B00 B00 B00 B00 B00 B00 B01 B00 B00 B01 B01 B00 B10 B00 B00 B10 B00 B10 B11 B00 B00 B11 B01 B10
Now as the two orthogonal states can always be distinguished locally10 hence after the bilateral XOR operation Alice and Bob distinguish between B00 B10 for the last pair by LOCC. If it is B00 they do nothing and if B10 Bob applies flip operation on his qubit. This completes cloning. Here interestingly apart from getting the clone, one obtains one bit of information e (by knowing whether the state to be cloned belongs to the set &B00 , B01 or &B10 , B11 '). e For
cloning four Bell states with the help of two free ebits, reference 4 describes a more complicated protocol whereby one can get three copies of the initial Bell state, but interestingly no such information gain is there.
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20.3.3. Local Cloning of Three Bell States Regarding three Bell states, Owari and Hayashi4 have shown that any three Bell states cannot be cloned if only one ebit free entanglement is supplied as a resource. We, in this section, by entanglement considerations, not only prove the same but also provide the necessary amount of entanglement for such a cloning. Interestingly, this amount comes out as log2 3 ebits. f If cloning of three Bell states (e.g. B00 , B01 , B10 ) is possible with a known entangled state (say B ) as blank copy (resource), then the following state "2 "2 "2 "2 "2 "2 1 3 B00 B00 B01 B01 B10 B10 along with the blank state B given as the input to the cloner will provide the output as:
1 "2 "2 "2 "2 "2 "2 B00 B00 B01 B01 B10 B10 B B 3 1 " 3 " 3 " 3 " 3 " 3 " 3 % ρout B00 B00 B01 B01 B10 B10 3 We now compare the relative entropies of entanglement of ρin and ρout . From inequality (20.4), we have 1 " 2 " 2 " 2 " 2 " 2 " 2 B00 B00 B01 B01 B10 B10 ER ρin ER 3 ER B B 1 " "2 B "2 B "2 B "2 B "2 2 log 3,7 As ER 3 B002 B00 2 01 01 10 10 hence:
ρin
ER ρin 2 log2 3 ER B B At least 2 ebits of entanglement can be distilled from ρout 8 and the distillable entanglement is bounded above by ER , hence ER ρout 2. But relative entropy of entanglement cannot increase under LOCC, and in the output we have at least 2 ebit of relative entropy of entanglement, hence, in order to make cloning possible, log2 3 ebit is necessary in the blank state. Any two qubit state (even a two qubit maximally entangled state) cannot provide this necessary amount of entanglement. f Compare it with the amounts necessary to clone the two and the four Bell states which respectively are equal to log2 2 and log2 4 ebits.
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20.4. Cloning Arbitrary Entangled States of Two Qubits Any two equally entangled orthogonal states can lie either in same plane: (I) Ψ1
a00
b11
Ψ2 b00
a11
or in different planes: (II) Ψ1
a00
b11
Ψ3
a01
b10
where a,b are real and unequal and a2 b2
1.
In both the cases, if one provide two entangled states, each having same entanglement as in the original one, cloning will be trivially possible. Here we investigate the nontrivial case when a single entangled qubit state is supplied as blank copy. Case(I) Suppose there exists a cloning machine which can clone Ψ1 and Ψ2 when a pure entangled qubit state Φ c00 d11; c2 d2 1) is supplied to it as blank copy. Let us supply an equal mixture of Ψ1 and Ψ2 together with the blank state Φ to it; i.e.. the state input to the cloner is: 1 1 P Ψ1 P Ψ2 P Φ (20.8) ρin 2 2 The output of the cloner: 1 1 P Ψ1 Ψ1 P Ψ2 Ψ2 (20.9) 2 2 For proving impossibility of such a cloner, we make use of the fact that Negativity, of a bipartite quantum state ρ, N ρ cannot increase under LOCC.11 The negativity of the input state ρin is ρout
N ρin 2cd
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whereas, the negativity of the output is N ρout 4a2 b2 4 a2 b2 a2 b2 2 The above cloning will not be possible as long as, cd 2a2 b2 2 a2 b2 a2 b2 2
(20.10)
The above inequality has some interesting features, but the most significant feature is that even a maximally entangled state of two qubits cannot help as blank copy for a large number of pairs of nonmaximally entangled states belonging to this class(see fig1). Numerical calculations show that this is the case for 0.230 a 0.973 (except for a 12 . This is surprising as recently Kay and Ericsson13 have given a protocol by which all the pairs of states lying in different planes(II) can be cloned with the help of one free ebit. Other important features are: (a) For a b c d 12 the above inequality becomes an equality. This is consistent with an earlier finding1 that two maximally entangled bipartite state can be cloned with one free ebit. (b) Inequality (20.10) holds even for c a " d b(see fig. 20.1). This in turn implies that same amount of entanglement ( as in the state to be
1.8 1.6 1.4 1.2
N(ρ
)
out
1
Negativity
0.8 0.6
N(ρin)
0.4 0.2 0
0
0.2
0.4
0.6
0.8
1
x
Fig. 20.1. Broken line: Plot of N ρin verses x, where x=c2 1 d2 . Solid line: Plot 1 b2 . Please note that the Negativity of the output of N ρout verses x, where x=a2 is more than that of the input except for maximally(i.e. x 12 ) entangled ones.
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cloned ) cannot help as blank copy, for any pair of nonmaximally entangled states. Case (II) This time we suppose that our cloning machine can clone Ψ1 and Ψ3 if a pure entangled state Φ c00 d11; c2 d2 1) is used as blank copy. Let the state supplied to this machine be: 1 P Ψ1 P Ψ3 P Φ 2 We then have output of the cloner as: ρin
ρout
1 1 P Ψ1 Ψ1 P Ψ3 Ψ3 2 2
Putting for Ψ1 Ψ3 and φ in the expression for ρin and ρout and making use of equations (20.5) and (20.6), we get: N ρin 2cd 1 N ρout 2
2 a6 b 2 a2 b 6
From nonincrease of negativity under LOCC it follows that as long as cd 2 a6 b2 a2 b6 (20.11) the above cloning is not possible. (a) a b c d 12 turns this inequality into an equality. This again is consistent with.1 (b) If we put c a " d b in the above inequality, i.e if we use same amount of entanglement (as in original states) then too cloning remains impossible as can be seen from fig 20.2. (c) Here too the inequality (20.11) shows that the necessary entanglement in the blank copy is always greater than the entanglement of the original states unless they are maximally entangled. As an example, for a 0.3, (i.e. entanglement of the state to be cloned 0.8813), as long as c 0.42, (i.e. entanglement of blank copy 0.9815), cloning is not possible. 20.5. Cloning Entangled States of Three Qubits In this section we will consider local cloning of genuinely entangled states of three qubits. W and GHZ states are the two extreme representatives
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1
N(ρ
0.8
N(ρin)=N(ρout) where x=1/2
0.6
Negativity
)
out
N(ρin)
0.4 0.2 0
0
0.2
x
0.4
0.6
0.8
1
Fig. 20.2. Broken line: Plot of N ρin verses x, where x=c2 1 d2 . Solid line: Plot 1 b2 . Please note that the Negativity of the output of N ρout verses x, where x=a2 is more than that of the input except for maximally(i.e. x 12 ) entangled ones.
of the inequivalent kinds of genuine three qubit entangled states.15 Our result shows that whereas any two GHZ states from the canonical set of eight orthogonal GHZ states can be cloned locally with the help of a GHZ state as the blank copy, no two W-states, taken from the complete set of orthogonal W-states, can be cloned with the help of any three-qubit entangled state. We also find the condition under which a set of three orthogonal GHZ states cannot be cloned with the help of any three-qubit entangled state. 20.5.1. Cloning of GHZ States The full orthogonal canonical set of tripartite GHZ states can be written as (upto a global phase): 1 p Ψp,i,j ABC 0 i j 1 1 i j , (20.12) 2 where p, i, j 0, 1 and a bar over a bit value indicates its logical negation. Consider any pair from (20.12). Let one state of this pair is shared among three parties; Alice, Bob and Charlie. They share another known GHZ state as blank copy. It can be easily shown that control NOT (CNOT) operation (C ij i j i mod 2) by each of the parties will make cloning possible. Take for example the following pair: 1
1 2
Ψ000 ABC 000 111
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1 2
Ψ010 ABC 010 101,
without loss of generality, one can consider the blank state as Ψ000 . Alice, Bob and Charlie each apply CNOT operation on their respective qubits by taking the unknown original state’s qubit as source and blank state’s qubit as target to achieve the required cloning. But for the following pair: 2
1 2
Ψ000 ABC 000 111
1 2 Alice, Bob and Charlie each will apply the CNOT by taking the blank state’s qubit (Ψ000 ) as source and original state’s qubit as target. Interestingly, any pair of the above mentioned GHZ state can be written in either of the above two forms by appropriate basis transformation. Existence of the three GHZ states that cannot be cloned by LOCC: We would like to mention a necessary condition for cloning of a 3 qubit entangled state under LOCC with the help of a 3 qubit state as blank copy which is required for our investigation. A necessary condition for cloning of a 3 qubit state under the usual LOCC (where all the three qubits of the state is operated separately) would be: “The states should remain copiable when the two qubits are operated jointly at one place whereas the third undergo a separate local operation at a different place and there can be classical communication between these two places ”. Consider three states from the set (20.12). The first two qubits of these states are put together in lab-A whereas the remaining third qubit in a different lab (lab-B). These three states are equivalent to the three Bell states in the above mentioned bipartite cut if and only if, all of them have same i and two among them have same j but different p. Now as even 1 free ebit is not sufficient to clone three Bell states and as the maximum bipartite entanglement that a 3-qubit state can have is 1 ebit, so we conclude that these GHZ states with any 3 qubit ancilla state cannot be cloned by LOCC. Any set of three states which are not in the above form in any bipartite cut Ψ100 ABC 000 111
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can always be cloned by LOCC using a known GHZ state as ancilla, where every party uses CNOT. 20.5.2. Cloning of W-States A full set of tripartite W states is given as 1 W1 123 001 100 111 3 1 011 101 110 W2 123 3 1 W3 123 001 100 010 3 1 011 101 000 W4 123 3 1 W5 123 001 010 111 3 1 011 000 110 W6 123 3 1 W7 123 100 111 010 3 1 W8 123 101 110 000 3 We will show below that ‘No set of orthogonal W-states can be cloned by LOCC with the help of any three qubit state as blank copy’. One needs an entangled blank state to clone an entangled state or else, entanglement of the entire system will increase under LOCC which is impossible. So, let us try to clone the W-states with the help of a known et al.15 genuine tripartite entangled state as the blank copy. Recently, Dur have shown that any genuine tripartite entangled state can have entanglement either of the W-kind or of GHZ-kind. So our blank copy can be either of W-class or of GHZ-class.
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(i) Blank copy having GHZ-kind of entanglement. In this case we will show that even a known W state cannot be cloned by LOCC. The minimum number of product terms for a given state cannot be altered by LOCC.15 But such a cloning would imply that the minimal no. of product term for a given state is increased from 6(minimum no. of product term in the input to the cloner) to 9 (corresponding no. in the output), by LOCC which is impossible. (ii) Blank copy having W-kind of entanglement. We first consider a W-class state which is not a W-state as our blank copy. It can be shown that(see the appendix) states belonging to W-class, unless it is a W state, has at least one bipartite cut for which the entanglement, the ‘Bipartite Entanglement’, E 31 log2 13 23 log2 23 . Let us now consider a situation where those two qubits of the blank copy are kept together in lab-A for which the entanglement in that bi-partite cut of the blank copy (W-kind of state in this case) is less than that of corresponding W-state (the state proposed to be cloned). Corresponding qubits of the state to be cloned are also put in Lab-A. Another lab (lab-B) contains the remaining third qubits of these states. As LOCC cannot increase entanglement hence the W-state is not copiable under LOCC between these labs. But as mentioned earlier this is necessary for local cloning of any 3-qubit state, hence we conclude that a W-kind of state (unless it is a W-state) is not helpful in LOCC-cloning of W-states. (iii)W-state as Blank copy For proving the impossibility of cloning of any set of W-states with the W-state as blank copy, we will show that cloning any pair of W states from the above mentioned W-basis by LOCC is not possible. There are twenty eight such pairs. Consider one pair– Wm 123 and Wn 123 . mn n denotes the subspace generated by the support of ρm Let Eij ij andρij , m n where ρij T rk &Wm 123 Wm ' (similar is ρij ). Here i, j, k 1, 2, 3 and i " j " k. A close inspection will reveal that these pairs fall broadly into three categories: mn 2 for at least one value of k are : (A) pairs for which dim Eij (a) W1 , W2 , W3 , W6 for k 2 , (b) W1 , W4 , W6 , W7 for k 1, (c) W2 , W7 , W3 , W4 for k 3. mn 3 for at least one value of k are : (B) pairs for which dim Eij (a) W1 , W6 , W1 , W8 , W5 , W6 , W5 , W8 for k 3, (b) W2 , W3 , W2 , W5 , W3 , W8 for k 1, (c) W4 , W5 , W4 , W7 , W7 , W8 for k 2 are such pairs.
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(C) pairs which don’t fall under above categories W1 , W3 , W1 , W5 , W1 , W7 , W2 , W4 , W2 , W6 , W2 , W8 , W3 , W5 , W3 , W7 , W4 , W6 , W4 , W8 , W5 , W7 , W6 , W8 are such pairs. We consider the above three types of pairs separately. A-Type pairs: mn The k th qubits of the pair for which dim Eij 2 is put in lab-B, whereas th th the i and the j together in lab-A. Under this arrangement, any given pair of this type, for a proper choice of basis, reduces to the form:
1 0A 0B 3
Wm 123
Wn 123
1 0A 1B 3
2 1A 1B 3 2 1A 0B 3
(The subscripts A and B indicates the laboratories occupying the qubits.) We assume now the existence of a cloner which, by LOCC between the labs, can clone a pair Wm 123 and Wn 123 when a known W-state (suppose W1 ) is supplied to it as blank copy. If we supply to the cloner an equal mixture of Wn 123 and Wm 123 together with the blank copy W1 ,i.e. if the input state to this LOCC-cloner is:
ρin
1 P Wn 123 W1 2
1 P Wm 123 W1 , 2
1 P Wn 123 Wm 123 2
the output of the cloner will be ρout
1 P Wn 123 Wm 123 2
Here P stands for projector. For proving impossibility of LOCC-cloning of these states, we make use of the fact that Negativity, of a bipartite quantum state ρ, N ρ cannot increase under LOCC.11 Numerical calculations for negativities give N ρin 0.942809; N ρout 0.993808. As N ρin N ρout , hence we conclude that the states belonging to this pair cannot be cloned under the LOCC between the two labs. and hence by the usual LOCC.
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B-Type pairs: mn 3 is put in Once again the k th qubits of the pair for which dim Eij th th lab-B, whereas the i and the j together in lab-A. For ij vs. k cut and for a proper choice of basis g , a given pair of this type can be written either as : (I) 1 2 Wm I 0A 0B 1A 1B 3 3 2 1 Wn I 1A 0B 2A 1B 3 3 or as: (II)
Wm II
1 0A 0B 3
2 1A 1B 3
1 2 Wn II 0A 1B 2A 0B 3 3 For showing impossibility of local cloning of these pairs, this time we suppose the existence of a cloner which, by LOCC between the labs, can clone a pair Wm I and Wn I when a known W-state (suppose W1 ) is supplied to it as blank copy. i.e. the state supplied to the cloner is: ρin
1 P Wm I 2
W1
1 P Wn I 2
W1 ,
We then should have the output of the cloner as: 1 1 P Wm I Wm I P Wn I Wn I 2 2 (P stands for projector as usual). Calculations for negativities of the input and the output of the cloner give us: N ρin 1.89097; N ρout 2.14597. So, these W-states cannot be cloned by LOCC. ρout
g (i)Take
for example the pair (W1 , W6 ). These states in the ‘12 vs. 3’ cut reduce to 11 , 2 for 01 , 0 form(I)for the substitution: 0 A for 10 12 , 1 A for 00 12 12 A B 2 for 0 3 and 1 B for 1 3 . (ii)Take another pair (W1 , W8 ). This for the ‘12 vs. 3’ cut can be written as (II) under 11 , 2 for 00 11 , 0 for 0 the substitution: 0 A for 10 12 , 1 A for 00 12 12 3 A B 2 2 and 1 B for 1 3 .
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Negativity calculations for type (II)pairs gives: N ρin N ρout 2.49298. where ρin
ρout
1 P Wm II 2
1 P Wm II 2
W1
1 P Wn II 2
W1
Wm II
1 P Wn II 2
2.23802;
Wn II
and P as usual stands for the projector. As N ρin N ρout , hence states belonging to this pair too cannot be cloned. C-Type pairs: Every pair of this set has an important feature that there is one value of mn n 4 and ρm k for which dim Eij ij , ρij " 0. For showing impossibility of cloning, we put those two qubits together in lab-A for which the reduced density matrices of the corresponding W-pairs are noncommuting. The states of a given pair under this arrangement reduce to the following representative form:
Wm
and
Wn
1 0A 0B 3
2 0 A 0B 3
for proper choice of basis h , where: 01A 0 1 A 0, 01 A 0 1A 00 A 1 . 2
2 1A 1B 3
1 1 A 1B 3
0, 00 A
11 A and
Analysis similar to the previous one shows that negativities of the input(equal mixture of Wm and Wn together with a known W state) and output of the assumed cloner (equal mixture of Wm and Wn )are N ρin 2.23802 and N ρout 2.55185 respectively; again denying the existence of such a cloner. h For
example the reduced density matrices of W1 and W3 are noncommuting when the third qubits are traced out. So we keep the first and the second qubits in lab-A and the third in lab-B for LOCC between these labs. 11 , 0 for 01 10 Under the substitution: 0 A for 10 12 , 1 A for 00 12 12 , 1 A for A 2 2 00 12 , 0 B for 0 3 and 1 B for 1 3 ; W1 and W3 reduce to the said form.
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20.6. Conclusion In conclusion, we have discussed here the problem of LOCC cloning of entangled states. To clone three Bell states, one need at least log2 3 ebit in the blank state. So any two qubit state (pure or mixed) cannot serve this purpose. We have also shown the blank state needed should have more free entanglement than the original ones, for cloning any pair of nonmaximal but equally entangled orthogonal states. The necessary amount of entanglement in the blanck state for such cloning to be possible is given by inequalities (20.10) and (20.11). Interestingly this necessary amount is more than 1 ebit for certain set of nonmaximal but equally entangled states contrary to certain other sets for which 1 ebit can serve as blank copy. Our result also establishes one stark difference between two kinds of symmetric three partite genuine entanglement, namely the W-type and the GHZ-type entanglement even for a pair of entangled states (where the LOCC distinguishability is blunt).
Acknowledgments R.R. acknowledges the support by CSIR, Government of India, New Delhi. The authors also acknowledge G. Kar, S. Kunkri and A. Roy for valuable suggestions. References S. Ghosh, G. Kar and A. Roy, Phys. Rev. A 69, 052312 (2004). F. Anselmi, A. Chefles and M. Plenio, New J. Phys. 6, 164 (2004). M. Owari and M. Hayashi, eprint quant-ph/0411143. M. Owari and M. Hayashi, Phys. Rev. A 74, 032108 (2006). V. Vedral and M. B. Plenio, Phys. Rev. A 57, 1619 (1998); V. Vedral, M. B. Plenio, M. A. Rippin and P. L. Knight, Phys. Rev. Lett. 78, 2275 (1997). 6. J. Eisert, eprint quant-ph/0610253 and refernces therein. 7. S. Ghosh, G. Kar, A. Roy, A. Sen(De) and U. Sen , Phys. Rev. Lett. 87, 277902 (2001). j i j i , and 8. The control-not operation C is defined as C i the bilateral control-not operation (BXOR) defined on bipartite system as, j A2 s B2 i A1 r B1 j i A2 s r B2 . Denote B is B i A1 r B1 B m, n as the BXOR operation performed on the mth pair (source) and
3 the nth pair (target), the following operation will give, B 1, 3 B 2, 3 Bmn
2
2
2 9 1 Bm,0 B 3m,n . If this operation is applied on ρout , one get 3 B00 B00 1. 2. 3. 4. 5.
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B00 B00 B01 B01 B10 2 B10 2 B10 B10 . If Alice and Bob do the measurement in 0 , 1 basis on the third copy and communicate, the
9. 10. 11. 12. 13. 14. 15.
results will be either correlated or anticorrelated. When they they are correlated the first two copies are in B00 , and in other case they are in state B10 , therefore distilling two ebits in this process. D. Yang and Y.-X. Chen, Phys. Rev. A 69, 024302 (2004). J. walgate, A. J. Short, L. Hardy and V. Vedral Phys. Rev. Lett.85, 4972 (2000). G. Vidal and R. F. Werner Phys. Rev. A 65, 032314 (2002). K. Zyczkowski, P. Horodecki, A. Sanpera, and M. Lewenstein, Phys. Rev. A 58, 883 (1998). A. Kay and M. Ericsson, Phys. Rev. A 73, 012343 (2006). C. H. Bennett, D. P. DiVincenzo, C. A. Fuchs, T. Mor, E. Rains, P. W. Shor, J. A. Smolin, and W. K. Wootters, Phys. Rev. A 59, 1070 (1999). G. Vidal, and J. I. Cirac, Phys. Rev. A 62, 062314 (2000). W. Dur,
Appendix A generic W-class state15 shared among three parties is: ΨW
a001
b010
c100
d000
where a, b, c 0, and d 1 a b c 0. If possible, let in all the three bipartite cuts, the entanglement of the above W-class state is greater than or equal to 13 log2 13 23 log2 23 . The entanglement in 1 vs 2-3 cut E1:23 is : 1 1 2c2 4cd 1 1 2c2 4cd log2 2 2 1 1 2c2 4cd 1 1 2c2 4cd log2 2 2 Now E1:23
1
3
log2
1 2 3 3
log2
2 3 / ,
1 1 2c2 4cd 1 1 2 2 ; i.e. c 3 2 3 3 3 Similarly, for other cuts, the previous assumption will lead to 1 3
b
1 2 and 3 3
a
2 3
(20.13)
(20.14)
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Both the inequalities (2) and (3), can not hold simultaneously, unless d=0 and a=b=c(i.e. a W-state).QED.