Primes and Particles : Mathematics, Mathematical Physics, Physics 9783031497759, 9783031497766

Table of contents :
Preface
Witten on Mathematics and Physics
Acknowledgments
Contents
About the Author
Chapter 1: Introduction
1.1 Our Examples
1.2 The Ising Model in Two Dimensions: An Identity in a Manifold Presentation of Profiles
1.3 Dedekind-Weber
1.4 The Stability of Matter
1.5 Packaging Functions, Riemann Zeta Function
Appendix
A.1 Subheads of Dyson and Lenard´s 1967-1968 Papers on the Stability of Matter
A.2 The ``Numbered´´ Flow of Theorems and Lemmas of the Dyson-Lenard Proof
A.2.1 Theorems
A.2.2 Lemmas
A.3 The Flow of the Dyson-Lenard Proof, as Lemmas Hanging from a Tree of Theorems
A.4 The Structure of Lieb and Thirring´s Argument in ``Bound for the Kinetic Energy of Fermions Which Proves the Stability of ...
A.5 Subheads and Subtopics of C.N. Yang, ``The Spontaneous Magnetization [M] of a Two-Dimensional Ising Model´´ (1952)
Chapter 2: Why Mathematical Physics?
2.1 The Big Ideas
2.2 Ising in Two Dimensions: An Identity in a Manifold Presentation of Profiles
2.3 Ising Susceptibility
2.4 Where´s the Physics?
2.5 Dedekind-Weber and Reciprocity
Chapter 3: Learning from Newton
3.1 Lessons from Newton
3.2 Creativity
3.3 Mathematical Physics
3.4 Influence
3.5 The Apocalypse
Chapter 4: Primes and Particles
4.1 The Thermodynamics and Music of the Numbers
4.2 A Potted History
4.3 Symmetry and Orderliness
4.4 Coherence
4.5 Decomposition
4.6 Hierarchy
4.7 Adding-Up and Linearity
4.8 Divisibility
Chapter 5: So Far and in Prospect
5.1 Kinship and Particles
5.2 Primes and Particles
5.3 Effective Field Theory
5.4 Packaging Functions Connecting Spectra to Surprising Symmetries
5.4.1 Multiple Ways of Computing Packaging Functions, Revealing Other Symmetries in the Spectrum
5.4.2 Algebraic, Arithmetic, Analytic: An Analogy of Analogies: Syzygies
5.5 The Right Particles or Parts
5.5.1 Fermions
Chapter 6: Creation: When Something Appears Out of Nothing
6.1 Points
6.2 Vacua
6.3 Mathematical Sleight of Hand: So to Speak
6.4 Points, Again
Chapter 7: Packaging ``Spectra´´ (as in Partition Functions and L/ζ-Functions) to Reveal Symmetries in Nature and in Numbers
7.1 Geometry and Harmony
7.2 Parts and the Right Parts
7.3 Plenitude
7.4 Manifold Perspectives or Profiles
7.5 Layers
7.6 Fermions
7.7 A Concrete Realization of the Dedekind-Weber Program
7.8 Another Multiplicity
Chapter 8: Legerdemain in Mathematical Physics: Structure, ``Tricks,´´ and Lacunae in Derivations of the Partition Function of...
8.1 Examples
8.2 The Two-Dimensional Ising Model
8.2.1 The Meaning of the Numbers
8.2.2 An Amazing Invention
8.2.3 Employing a Device of the Past
8.2.4 Where Did That Come From?
8.2.5 ``A Useful Identity, Easily Seen´´
8.2.6 Signposting Along the Way
8.2.7 ``Further Details of Simplifications Like This Will Not Be Reported Here´´
8.3 The Stability of Matter
8.3.1 ``Hacking Through A Forest Of Inequalities´´
8.3.2 ``Thomas-Fermi Atoms Do Not Bind´´
8.3.3 ``An Elementary Identity, Fourier Analysts Are Quite Familiar with It. Gruesome Details, Nasty and Ghastly Calculations,...
8.4 Genealogy Reconsidered
Chapter 9: Mathematical Physics
Bibliography
Index

Citation preview

Martin H. Krieger

Primes and Particles Mathematics, Mathematical Physics, Physics

Primes and Particles

Martin H. Krieger

Primes and Particles Mathematics, Mathematical Physics, Physics

Martin H. Krieger University of Southern California Los Angeles, CA, USA

ISBN 978-3-031-49775-9 ISBN 978-3-031-49776-6 https://doi.org/10.1007/978-3-031-49776-6

(eBook)

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

Once More, For David

Preface

Mathematical physics is the discipline of people who try to reach a deep understanding of physical phenomena by following the rigorous style and method of mathematics. Freeman Dyson1

Mathematics and physics have an intimate relationship, albeit on-and-off. Mathematics provides both machinery for calculation and structures for analogy. Primes and Particles describes that relationship in detail in terms of serious mathematics and serious physics. Mathematical physics uses comparatively sophisticated mathematics to understand and analyze physical situations and theories. Mathematical physics differs from theoretical physics in that it demands substantial rigor (as would be the case for mathematics), in part because that rigor reveals interesting physics. (In general, if you have a good physical theory that accounts for experiments, and it allows you to think about the world—yet it is manifestly not so rigorous, you focus on your explanation of the phenomena and not on the lack of rigor. You figure that the mathematics will eventually catch up.) Mathematics is not only machinery for working out a model; it often brings to the fore physical content. Mathematical details, such as the nature of a limiting process, may build in the physics. The mathematics is not prescriptive about the world, in that the actual world is quite contingent, not necessarily following the mathematics you are using. We often find that there are several very different mathematical machineries that apply to the same situation—and we would like to know what it is about that situation that enables it to be so variously described. There would seem to be an identity in a manifold presentation of profiles.2 What is that identity?

1

From Eros to Gaia, pp. 164–165. I take this term from R. Sokolowski, Introduction to Phenomenology (Cambridge: Cambridge University Press, 1999). The mathematician Gian-Carlo Rota wrote about this in terms of mathematical beauty. 2

vii

viii

Preface

Rereading my earlier work now shows how a small number of themes recur and are elaborated. Primes and Particles is the fifth in a series: Marginalism and Discontinuity (1989), Doing Physics (1992, 2003), Constitutions of Matter (1996), and Doing Mathematics (2003, 2015).3 As they are needed, I have repeated material discussed in the earlier books on the Ising Model, the Stability of Matter, and the Dedekind-Weber (1882) understanding of Riemann’s work. Many details of the mathematical moves and their physical significance are in Constitutions of Matter. The Dedekind analogy is discussed in Doing Mathematics. In the second half of the Introduction, there is an exposition of the main features of the examples I shall employ. The reader is encouraged to refer to this exposition as I invoke these examples. I have tried to make the mathematics and the physics I describe more available to the lay person and to physicists and mathematicians outside their area of expertise. When I use the terms the physics or the mathematics I mean to be specific about the actual physical processes or the actual mathematical devices. I am not making a claim about physics or about mathematics, in general.

Witten on Mathematics and Physics Edward Witten, the distinguished theoretical and mathematical physicist, presents a view on the relationship of mathematics and physics—somewhat different than I pursue here.4 For Witten, advances in theoretical physics often hint at interesting mathematical problems—strings and geometry, for example. Mathematical developments are often of value to the physicist. Moreover, one cannot know where one’s theoretical endeavors will lead, so string theory eventually pointed to a quantum theory of gravity. Witten’s experience of mathematics and physics gives him 3

My earlier books include: M Marginalism and Discontinuity. New York: Russell Sage Foundation, 1989. P Doing Physics: How Physicists Take Hold of the World. Bloomington: Indiana University Press, 1992 (xx + 168 pp.). 2nd edition, 2012 (xxvi + 218 pp.). C The Constitutions of Matter: Mathematically Modeling the Most Everyday of Physical Phenomena. Chicago: University of Chicago Press, 1996 (xxii + 343 pp.). [MR1425391, 97g:00015] D Doing Mathematics: Convention, Subject, Calculation, Analogy. Singapore: World Scientific, 2003 (xviii + 454 pp.). MR1961400, 2004a:00011]. 2nd edition, 2015 (xxiv + 467 pp.). PP Primes and Particles—this book Location of discussions in these five books, referring to chapters in each case: “The Physics” P6, C8; Adding-Up M2; Centers M3; Dedekind-Weber D5; Economy C5; Elementarity PP3; God D6; Handles & Tools P5, M7; Ising Model D3Appx, C4, C4-6, PP5; Kinship P3; Maxwell D6; Particles & Fields P1; Parts D2; Philosophy of Math D1, D6, P6, C5, M6, PP6-8; Physics & Math C6-7; Stability of Matter D4, C6, PP1, PP5; Statistics/Vacuum D2, C2; The City D6; Thermodynamic Limit C3; Vacuum P4, PP4. 4 My remarks about Witten are derived from various lectures to be found in the web, often on his receiving a major prize.

Preface

ix

confidence in their relationship, albeit one cannot know just what that relationship is until it becomes concrete and specific. Witten is more convinced than is Wigner, more than amazed and struck by “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” For Witten, these apparently autonomous fields are so intertwined, and one might say that there is “The Unreasonable Effectiveness of the Natural Sciences in Mathematics.” Witten’s claim is empirical and experiential, given his career as a theoretical physicist who also made deep and widely appreciated contributions to mathematics. Primes and Particles has little to say to Witten. My claims are rather more modest and concrete. Beverly Hills, CA, USA

Martin H. Krieger

Acknowledgments

Chapter 3, on Newton, derives from a talk I gave at the University of Southern California as part of a project sponsored by the Templeton Foundation. Over the years, Sam Schweber was a cheerleader. By his faithful friendship Gian-Carlo Rota encouraged me. And my undergraduate education at Columbia College and my graduate education at Columbia University enabled me to do my work for almost fifty-five years since then. David Krieger and Jonah Greenstein abided. The Appendix to Chap. 1 is taken from my Doing Mathematics (World Scientific, 2015), pp. 130–131, 133, 150–152, 159–160, by permission of the publisher.

xi

Contents

1

2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Our Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The Ising Model in Two Dimensions: An Identity in a Manifold Presentation of Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Dedekind-Weber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 The Stability of Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Packaging Functions, Riemann Zeta Function . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1 Subheads of Dyson and Lenard’s 1967–1968 Papers on the Stability of Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 The “Numbered” Flow of Theorems and Lemmas of the Dyson-Lenard Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 The Flow of the Dyson-Lenard Proof, as Lemmas Hanging from a Tree of Theorems . . . . . . . . . . . . . . . . . . . . . . . . A.4 The Structure of Lieb and Thirring’s Argument in “Bound for the Kinetic Energy of Fermions Which Proves the Stability of Matter” (1975) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.5 Subheads and Subtopics of C.N. Yang, “The Spontaneous Magnetization [M] of a Two-Dimensional Ising Model” (1952) . . . Why Mathematical Physics? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The Big Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Ising in Two Dimensions: An Identity in a Manifold Presentation of Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Ising Susceptibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Where’s the Physics? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Dedekind-Weber and Reciprocity . . . . . . . . . . . . . . . . . . . . . . .

1 4 5 8 9 13 15 15 16 17

17 19

. .

21 23

. . . .

24 25 26 27

xiii

xiv

Contents

3

Learning from Newton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Lessons from Newton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Creativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Mathematical Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Influence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 The Apocalypse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29 30 31 32 35 37

4

Primes and Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 The Thermodynamics and Music of the Numbers . . . . . . . . . . . . . 4.2 A Potted History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Symmetry and Orderliness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Adding-Up and Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Divisibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41 42 45 47 49 50 52 53 54

5

So Far and in Prospect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Kinship and Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Primes and Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Effective Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Packaging Functions Connecting Spectra to Surprising Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Multiple Ways of Computing Packaging Functions, Revealing Other Symmetries in the Spectrum . . . . . . . . . . 5.4.2 Algebraic, Arithmetic, Analytic: An Analogy of Analogies: Syzygies . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 The Right Particles or Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57 57 57 58

6

Creation: When Something Appears Out of Nothing . . . . . . . . . . . 6.1 Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Vacua . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Mathematical Sleight of Hand: So to Speak . . . . . . . . . . . . . . . . 6.4 Points, Again . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . .

63 66 67 68 68

7

Packaging “Spectra” (as in Partition Functions and L/ζ-Functions) to Reveal Symmetries in Nature and in Numbers . . . . . . . . . . . . . . . 7.1 Geometry and Harmony . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Parts and the Right Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Plenitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Manifold Perspectives or Profiles . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 A Concrete Realization of the Dedekind-Weber Program . . . . . . . . 7.8 Another Multiplicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71 71 72 72 73 73 74 74 75

58 59 59 60 60

Contents

8

Legerdemain in Mathematical Physics: Structure, “Tricks,” and Lacunae in Derivations of the Partition Function of the Two-Dimensional Ising Model and in Proofs of The Stability of Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 The Two-Dimensional Ising Model . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 The Meaning of the Numbers . . . . . . . . . . . . . . . . . . . . . . 8.2.2 An Amazing Invention . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.3 Employing a Device of the Past . . . . . . . . . . . . . . . . . . . . 8.2.4 Where Did That Come From? . . . . . . . . . . . . . . . . . . . . . . 8.2.5 “A Useful Identity, Easily Seen” . . . . . . . . . . . . . . . . . . . . 8.2.6 Signposting Along the Way . . . . . . . . . . . . . . . . . . . . . . . 8.2.7 “Further Details of Simplifications Like This Will Not Be Reported Here” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 The Stability of Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 “Hacking Through A Forest Of Inequalities” . . . . . . . . . . . 8.3.2 “Thomas-Fermi Atoms Do Not Bind” . . . . . . . . . . . . . . . . 8.3.3 “An Elementary Identity, Fourier Analysts Are Quite Familiar with It. Gruesome Details, Nasty and Ghastly Calculations, Modulo Oversimplifications, Applying General Nonsense” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Genealogy Reconsidered . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xv

77 78 79 79 83 86 86 87 88 88 89 90 90

90 91

Mathematical Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97

9

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

About the Author

Martin H. Krieger was trained as an experimental physicist at Columbia University. He is a Fellow of the American Physical Society. Krieger has taught in policy and planning and management at the University of California (Berkeley), Minnesota (Twin-Cities), MIT, University of Southern California, and University of Michigan (Ann Arbor). He has been a fellow at the Center for Advanced Study in the Behavioral Sciences and the National Humanities Center. Krieger is professor emeritus at the University of Southern California. Primes and Particles is his twelfth book. His earlier books include Doing Physics (Indiana, 1992, 2012), Constitutions of Matter (Chicago, 1996), and Doing Mathematics (World Scientific, 2003, 2015).

xvii

Chapter 1

Introduction

I began the Preface saying: Mathematics and physics have an intimate relationship, albeit on-and-off. Mathematics provides both machinery for calculation and structures for analogy. Primes and Particles describes that relationship in detail in terms of serious mathematics and serious physics. I want to provide detailed examples that evidence and describe that relationship in specific rather than generic terms. Philosophy is written in this grand book, which stands continually open before our eyes. (I say the ‘Universe’), but can not be understood without first learning to comprehend the language and know the characters as it is written. It is written in mathematical language, and its characters are triangles, circles and other geometric figures, without which it is impossible to humanly understand a word; without these one is wandering in a dark labyrinth. (Galileo, Il Saggiatore, 1623)1) [often condensed as “Mathematics is the language with which God has written the universe.”]

Note that Dante begins the Divine Comedy (~1320) with: Midway upon the journey of our life I found myself within a forest dark, For the straightforward pathway had been lost.2

It is a commonplace to quote Galileo to the effect that Nature is written in the language of mathematics. It would seem that there are many such dialects, although it may be possible to say roughly the same thing in some of the dialects. It is also true that poetry employs in part the language of everyday speech. The critic’s problem is

1 La filosofia è scritta in questo grandissimo libro, che continuamente ci sta aperto innanzi agli occhi (io dico l’Universo), ma non si può intendere, se prima non il sapere a intender la lingua, e conoscer i caratteri ne quali è scritto. Egli è scritto in lingua matematica, e i caratteri son triangoli, cerchi ed. altre figure geometriche, senza i quali mezzi è impossibile intenderne umanamente parola; senza questi è un aggirarsi vanamente per un oscuro labirinto. 2 Nel mezzo del cammin di nostra vita mi ritrovai per una selva oscura, ché la diritta via era smarrita.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 M. H. Krieger, Primes and Particles, https://doi.org/10.1007/978-3-031-49776-6_1

1

2

1

Introduction

to show how a lingua franca does the work. More generally, “language” is used in various ways, and it might be imagined that mathematics is used in all of those ways. I will begin with an attempt to justify mathematical physics by describing the work it does for the physicist and for the mathematician.3 Throughout, I will use two examples: proofs that everyday matter in the regime of electromagnetic forces is stable and does not collapse; and solutions of a two-dimensional lattice of magnetic spins, interacting simply with their neighbors, as a model of a ferromagnet (the Ising model). I shall be describing work that proved prize winning, that exhibited in an elegant fashion how mathematics makes physics sing. In the next chapter, on Newton, I am concerned with the archetypal mathematical natural philosopher and how his mathematical work on mechanics and gravity is intrinsic to his other work on alchemy and theology. They are endeavors to discern the logic of nature and of God’s Providence. For Newton, to do mathematical physics is to do theology. Moreover, there are remarkable structural similarities in various mathematical accounts of Nature. For example, the nineteenth century account of the numbers, how the prime numbers are structured, is rather similar to the twentieth century account of the structure of the elementary particles. In each case, what is prime or elementary might well be discovered to be composite if we change the environment. Adding in the square root of a prime will make some primes composite: (3 – √2) × (3 + √2) = 9–2 = 7. If we add energy to a collection of what we once thought of as elementary particles, we find that some of them reveal their composite nature: atoms become electrons + nuclei, new particles appear in the collision of known particles. The dictionary provided by mathematics, a dictionary drawn from the work of mathematicians and physicists, allows the physicist to subtly describe the range of what happens at liminal points in the natural world, when there is Nothing (a vacuum) and then the appearance of a phenomenon, Something. It is also remarkable that it is possible to package a set of numbers (say the primes, or the frequencies of the basic tones of a drumhead) into a single function, for example, g(τ) = Σ an einτ packaging the an, or sin x = Σ x – x3/3! + . . . packaging the odd factorials. A packaging function (as we shall see, the zeta function or the physicist’s partition function) may have notable symmetries not at all apparent from the set of numbers. Moreover, when we are mathematically describing and solving a physics problem, the details and difficulties in the mathematics are actually often about the physics, as if a machine’s working would tell us about what it is producing. The tricks and esoteric techniques are in the end physical.

3

For detailed references and further discussion of some of what I say here, see Krieger, 1996, 2015, Chapters 3–5. For more recent work, and the Prize-winning papers I refer to at the beginning of this chapter, at the IMU site there are popular accounts of the work, as well as laudatios. For DuminilCopin, see “100 years of the (critical) Ising model in the hypercubic lattice” (at https://www. mathunion.org/fileadmin/IMU/Prizes/Fields/2022/hdc.pdf and https://www.mathunion.org/imuawards/fields-medal/fields-medals-2022. For Lieb, https://www.mathunion.org/imu-awards/carlfriedrich-gauss-prize/carl-friedrich-gauss-prize-2022

1

Introduction

3

My purpose here is to be quite specific about how that language is employed, here in the realm of mathematics used in describing physical systems. In each case, I shall explain both the mathematics and the physics and how the mathematical language does physical work. What will make this study difficult is the actual mathematics and the physics, not at all dumbed down. I have deliberately allowed for much repetition of the two main examples, Ising and the Stability of Matter, in part because the lay reader is unlikely to magically recall their features. At the same time, I have tried to write so that the lay reader can appreciate my points even if some of the technical materials are unfamiliar. I have also repeated the material on the Dedekind-Weber triplet, on packaging functions, and on number theory, since readers may come from diverse fields. Some names recur: Baxter, Dyson, Lieb, Onsager, Riemann, Wilson, and Yang, and a drum, fermions, K0, kinship, the vacuum, and WMTB. Our themes will be: • • • • •

The parallel of primes and particles. Proofs of the stability of matter. The two-dimensional Ising model. The Dedekind-Weber (1882) analogy of arithmetic, algebra, and analysis. A function that packages a set of numbers, that function then has lovely analytic properties and symmetries (so telling us about that set). • The analogy between kinship theory and the rules of interaction of particles. • Just what is mathematical physics. In 1983, the mathematician Alexandre Grothendieck wrote in a letter to G. Faltings: What my experience of mathematical work has taught me again and again, is that the proof always springs from the insight, and not the other way round – and that the insight itself has its source, first and foremost, in a delicate and obstinate feeling of the relevant entities and concepts and their mutual relations. The guiding thread is the inner coherence of the image which gradually emerges from the mist, as well as its consonance with what is known or foreshadowed from other sources – and it guides all the more surely as the “exigence” of coherence is stronger and more delicate.4

Here, we might ask why mathematics is so useful for understanding mathematics. Mathematical physicists seem to take mathematics as real (ersatz Platonists), but that the mathematics might actually work in physical explanation is a matter of what you can actually do. Few believe that the mathematics will tell you the truth about physics, although they surely know of Einstein’s use of differential geometry in his account of general relativity. They are aware that Einstein spent several years looking for the mathematics that would be just right, and lots of mathematics was not helpful. If they rely on mathematics to guide them in their theorizing, they are aware of how readily they might well be led astray if their reliance is not tempered by what works and is illuminating about the natural world.

4

Translation and quotation from Grothendieckcircle.org, letters, June 27, 1983.

4

1 Introduction

When Wigner spoke of the surprising (or “unreasonable”) usefulness of mathematics in physics, in effect he treated mathematics as something divorced from natural science, the mathematics miraculously unreasonably effective therein (Wigner. 1970). A similar argument might be made about everyday language in thinking or in poetry, and that we speak in prose (of course, in actuality, we speak in a sort of prose) is perhaps no more surprising than our legs are useful for walking. These questions need to be made more specific about both the mathematics and the physics if they are not to lead to commonplaces. Mathematical physics may provide useful examples for exploring conventional questions about epistemology, ontology, and reality. Unfortunately, the philosophers’ questions are often far from the actual practice I describe here. If I am fortunate, the examples I provide will prove useful to professional philosophers— although I believe the examples will redirect the philosophers to rather new questions about how the mathematics and the physics are made suitable for each other. It may be useful to sketch briefly how I understand some of the conventional philosophic questions about the relationship between mathematics and physics. Although I am focusing on mathematical physics, I am not claiming that the world is mathematical, as such. Rather, mathematics is employed creatively, much as language is employed in everyday discourse or poetry. Yes, mathematics may be taken to be formal and logical, but in its being used to work on physical problems I am impressed by how the mathematical physicist adapts the physics (and the models employed) and the mathematics so that they might work together. Surely, the mathematics as applied affects what we think of as the physical objects (say particles), since we discover many of those physical objects in describing the physics in mathematical terms. As far as I can tell, from the point of view of the practicing mathematical physicist the use of mathematics is not so different than when we use everyday language to describe our world. There is a literature on how mathematics is employed in physics, and my examples may provide material for those philosophical inquiries. Crucially, we need to pay attention to the actual use and application, rather than mathematics as such.

1.1

Our Examples

I have here presented the main features of the examples I use throughout. The reader is encouraged to check here to acquaint herself or to remind himself of the details of the models. I should note that I have focused on the initial efforts on Ising, in particular. Ising now appears in quite sophisticated mathematical contexts. (Bulletin of the American Mathematical Society, 60 (October 2023) on Vaughn Jones’ work and its influence, for example), and one speaks of “Baxterisation,” as well as in earlier work on Yang-Baxter (the star-triangle transformation).

1.2

The Ising Model in Two Dimensions: An Identity in a Manifold. . .

5

Fig. 1.1 The two-dimensional lattice of spins of the Ising model, where we might say that " = +1 and # = -1, and the interaction energy E is = -J Σover adjacent spins σi × σi + 1

1.2

The Ising Model in Two Dimensions: An Identity in a Manifold Presentation of Profiles

The two-dimensional Ising model of ferromagnetism, a lattice of magnetic spins considered in a temperature bath, T, so fixing the temperature, allows for a wide variety of methods for deriving the statistical mechanics partition function, PF, where PF = exp-F/kBT, kB is Boltzmann’s constant and F is the free energy. PF = Σ exp‐E i =kB T ð = Σ exp‐βE i where β = 1=kB TÞ; • Ei (=Σ –J σiσj) is the total energy of a configuration of the components (here the spins). • where the spins are adjacent to each other, and J > 0 so that the spins prefer to be aligned with each other; since then the energy is lower. • and one is summing over all allowed configurations. If the temperature is above a critical point, if there is such a critical point, the random motions due to thermal energy impede that alignment (Fig. 1.1). One needs to discover the right objects, namely, the composition of each configuration, that may be added up to understand the behavior of the grid. The right parts here could be the interactions of the individual spins, as above, or as we shall see, the rows of spins, or of the polymer-like strings of like-pointing spins, or the boundaries between islands of up-pointing spins, say, and down-pointing spins. Or, we might

6

1

Introduction

count the ways of paving a square grid with connections between adjacent spins (what in another context is called a Pflastersatz). Another approach focuses on the symmetries of the lattice and never tries to add up the interactions at all. For example, one notices that a pair of magnets that are aligned when the grid is at a high temperature is analogous to a pair of magnets that are anti-aligned at a very low temperature. A functional equation is derived and solved to obtain the PF. Both ways of computing the partition function come to the same result, as we might hope. We might discover that the first way builds in those symmetries, although they are not explicitly included. In 1941, Montroll, Kramers and Wannier, and Lassetre and Howe (and Ashkin and Lamb (1943) and others) showed how to count up the configurations and their energies algebraically through a matrix that in effect translated the interactions down the grid. They could indicate that there should be a critical temperature and suggest what it was. Using this algebraic approach, the trace of that matrix in effect adding up the spin–spin interactions, Lars Onsager (1944) rigorously derived the partition function and proved that such a system had a sharp phase transition from disorder to order (the spins “all” pointing in the same direction) as the temperature is lowered. Onsager has to diagonalize that matrix system, in order to get the partition function. Onsager’s original solution demanded the development of suitable algebraic devices, involving as it does a quaternion algebra, apparent group representations and group characters, an analogy to quantum mechanics, and physical/mathematical symmetries of the system, as well as elliptic functions (cousins of the trigonometric functions) to parametrize the system. As for the symmetries, the “transfer matrices” (which sum the spin–spin interactions down the lattice), whose trace is the partition function, commute if they have the same normalized temperature, k. The symmetry between high and low temperatures, now called “duality,” is between k and 1/k. In effect, as we mentioned above, a little disorder in a highly ordered lattice is much like a little order in a highly disordered lattice. Onsager’s original paper was seen as complex and difficult to follow, and much of the subsequent work, still mathematical and rigorous, redoes the derivation in ways that are more familiar to physicists. Kaufman redid Onsager’s algebraic work, but now using spinors, familiar to physicists from the Dirac equation. Other derivations are much briefer and quite ingenious. If we carefully examine Onsager’s “ugly” initial paper, we discover that it is filled with ideas discovered in the subsequent work by others. (This may be my retrospective imagining, but I think not.) The algebraic solutions of Onsager and of Kaufman were not intuitive for many physicists. In a subsequent derivation Schultz et al. (1964), starting off as did Onsager, showed algebraically how the Ising model is in effect a field of free fermions (spin ½ particles with antisymmetric wavefunctions, like electrons or muons), using the apparatus of quantum field theory. They say: . . . [There is] the considerable formidability of the method [employed by Onsager, 1944, to solve the two-dimensional Ising Model]. The simplification by Bruria Kaufman using the theory of spinor representations has diminished, but not removed, the reputation of the

1.2

The Ising Model in Two Dimensions: An Identity in a Manifold. . .

7

Onsager approach for incomprehensibility, while the subsequent application of this method by Yang to calculate the spontaneous magnetization has, if anything, restored this reputation. . . . [the present paper presents] the algebraic approach in a way that is both very simple and intimately connected with the problem of a soluble many-fermion system. Except for one or two crucial steps, the approach is straightforward and requires no more than a knowledge of the elementary properties of spin ½ and the second quantization formalism for fermions. . . . The two-dimensional Ising model, rather than being entirely different from the trivially soluble many-body problems, reduces in some ways to one of them, being just the diagonalization of a quadratic form.5

They employed a transformation (due to Jordan and Wigner) that converted mixed fermion-boson spin operators (σij) used to express the transfer matrix into fermion operators, Cm. Further along in their derivation, with the Cm’s now Fourier transformed to ηq’s, they find themselves with expressions such as η{q × ηq. The paired nature of the quadratic forms . . . are extremely reminiscent of the pairHamiltonian and the energy states introduced by Bardeen, Cooper, and Schrieffer in the theory of superconductivity, particularly in Anderson’s formulation . . . We can simplify the results for Vq [the transfer matrix, which is written in terms such as η{q × ηq] in precisely the same way Bogulubov and Valatin simplified the BCS theory, if we introduce the transformation: ξq= (cos φq) ηq + (sin φq) η–q{ .

They found, instead of Onsager’s hyperbolic triangle, particles and (Cooper) pairs of particles. They could explain Onsager’s integrand for computing the partition function as that of particle free-energies rather than as the side of a hyperbolic triangle. The particle energies εq (= Onsager’s γ(ω)) are those of the ξq and ξ–q and the (Cooper) pairs, designated by b+ in SML are η–q{ ηq{. That is, not only is their method more familiar, but they interpret their results in contemporary terms. Rodney Baxter, 1982, made ingenious use of the symmetries of the Ising lattice, functional equations, and properties of elliptic functions in the complex plane (used here to make square roots single valued, “uniformized”) to exactly solve many lattice models. Again, there appears legerdemain, a rather different employment of those elliptic functions Onsager used, and convenient but unproven assumptions of analytic continuation. Baxter’s Exactly Solved Models in Statistical Mechanics (1982), and his papers are filled with adept use of the symmetries of the transfer matrix in terms of horizontal and vertical couplings K and L, a variable k that is both a measure of temperature and the modulus of the elliptic functions (as in Onsager as well), functional equations, and assumptions about analyticity and analytic continuation. In a paper with Enting, he uses all of these, especially the star–triangle relation (of a hexagonal to a triangular lattice and their partition functions), plus a remarkable geometric reconception of the lattice itself, to derive functional equations for the partition function (Baxter and Enting, 1978). His approach allowed for generalization to many other models, and like Onsager he provides an Appendix on elliptic functions.

5

Schultz, Mattis, Lieb, p. 856.

8

1 Introduction

Baxter showed how to use the systems’ symmetries and scaling properties to derive the partition function directly as a solution to a functional equation. Namely, ΛðK, LÞ x ΛðL þ 1=2 π i, ‐KÞ = ð2i sinh 2 LÞn þ ð - 2i sinh 2 KÞn r where Λ is the eigenvalue of the transfer matrix, n is the number of rows, and r is +1 or - 1. Much as in Onsager, there is great skill involved in solving the equation, skills very different than Onsager’s. M. Kac and J. C. Ward (1952), wanting to show how Onsager’s or Kaufman’s algebra counted up closed polygons of like-spins on the lattice (as expected earlier by van der Waerden, 1941), developed an ad hoc matrix whose determinant counted the polymers or self-avoiding random walks, the partition function in effect the Feynman sum-of-histories account (keeping in mind this is a classical system). The trick here was to define the weights of the links of the polymer that when multiplied would then reproduce what was found in the algebraic solutions. Eventually, in later work, the determinant is replaced by a relative of the determinant, the Pfaffian, of a related matrix. In effect, we have a multiplicity of perspectives—algebraic, symmetric, and combinatorial, and graph-theoretic—on an object, those perspectives in toto revealing the identity of the object, each perspective telling us a distinctive feature of that identity that was not manifest from the other perspectives—although since they obtained the same partition function, those features are buried in each method.

1.3

Dedekind-Weber

Riemann (1859) developed a geometric way of understanding algebraic functions of one variable. Algebraic functions are defined as the root of a polynomial equation, so f(x)2 – x = 0, defines f(x) as √x. Subsequently, doubt was cast on Riemann’s analysis, since it would seem to have assumed continuity. (In 1913, Herman Weyl rehabilitated Riemann’s analysis with a rigorous theory of Riemann surfaces.) In 1882, Richard Dedekind and Heinrich Weber responded to that doubt by showing how Riemann’s analysis might be proven algebraically, an algebraic understanding of algebraic functions. They showed how (1) Riemann’s geometric and analysis (continuity) account of algebraic functions of one variable might be proven (2) algebraically using as a model what was then known of (3) algebraic numbers. (Here “algebraic number” means a number defined by the root of a polynomial with integer or polynomial coefficients: x2–2 = 0, so x can be + or – √2.) Dedekind and Weber’s work connected analytic, function theoretic or algebraic, and arithmetic approaches to the same subject, in effect a three-column chart of correspondences. Features in each column had corresponding features (it was hoped) in the other columns. André Weil (1940) described such a chart and its development and called it a Rosetta stone,

1.4

The Stability of Matter

9

referring to the trilingual inscribed stone that earlier allowed Champollion to decipher Egyptian hieroglyphics. The variety of Ising model derivations over the last 80 years may be grouped and classified as follows: analytic and symmetry guided, algebraic, and arithmetic and combinatorial. Here, the mathematical analogy would seem to be mirrored by the physical classification, yet each was developed quite independently. If we could connect the mathematical analogy and the physical classification in a formal way or even a physical way, we would get a better feel for the physics and have as well a nice model of the mathematical analogy.

1.4

The Stability of Matter

Now, it is obvious to us that if we put two stones together, the energy of the system is roughly the sum of the energies of the stones minus any surface interactions (which are manifestly quite small). Put more graphically, putting two stones together does not cause them to become “critical,” as in nuclear fission, perhaps collapse into each other and then release a great deal of energy. “The stability of matter” is a problem in mathematical physics. Using nonrelativistic quantum mechanics, can you prove that matter composed of electrons and nuclei, held together by Coulomb forces, is stable: It will not collapse, and, equivalently, the energy of a system is bounded from below by the number of particles in it times a proportionality constant. Ideally, one would want to show that the energy density of the system is roughly proportional to the density of particles, for then thermodynamics makes sense. (The separation of pressure and Volume, for example.) Accounting for this everyday phenomenon is the problem called “the stability of matter,” and again it is Onsager, in 1939, who made a very rough demonstration that matter is stable. After much preliminary work by others, Dyson and Lenard in 1967 proved the stability of matter, using nonrelativistic quantum mechanics and assuming just Coulomb electrical interactions among the nuclei and electrons. The proof is rigorous, long, and winding (said by Dyson to be “hacking through a forest of inequalities,” but I think this is unfair), although the relevant proportionality constant is 1014 Rydbergs (1 Ry = 13.6 eV), when it should be on the order of 1. Lieb says: The proof was one of the most complicated proofs to appear in mathematical physics. For many years it was regarded with awe, like Onsager’s solution of the two-dimensional Ising model in 1944.6

Dyson and Lenard describe the argument:

6

Lieb and Seiringer, p.130.

10

1

Introduction

In Theorem 9 of Paper I [section 9, stability subject to a smooth charge density] we have a result that almost solves the stability problem . . . . [But] we want to prove that the fermions remain stable in an arbitrary distribution of positive classical point charges, . . . [W]e succeeded in sharpening Theorem 9 only for one negative particle at a time. We are consequently driven to an elaborate and unphysical scheme of chopping up our space into cells, each containing one negative particle (cf. Paper II, Sec. 4). We then prove a sharpened form of Theorem 9 for each cell separately, with its one negative and an arbitrary number of positive particles. Finally, we reassemble the fragments and show that stability in the individual cells implies stability for the whole space (cf. [II] Sec. 5).7. . . In our whole proof of stability, Lemma 9 is the innermost core . . . set[ting] a bound to the interaction energy between negative and positive charges, the bound depending only on the kinetic energy of the negatives and on the potential energy of the positives. . . . a [single] particle of mass m in the periodic Coulomb potential has a ground state binding energy less than 16 Ry[dbergs], . . . . . . The logical structure of the proofs of the two inequalities [roughly, relating averages of density times potential to the average of the gradient of the density times the average of the gradient of the potential times the average density itself] Theorem 11 and Lemma 9, is the same, and the logic of their use in the proofs of Theorem 12 and Theorem 5 [stability for a single particle, stability for a particle in a cube] is also the same. Only the details are more complicated for the case of the cube, and the numerical coefficients are correspondingly less precise.8

For reference, in the Appendix there is an outline of the Dyson-Lenard proof, taken from my Doing Mathematics: in terms of the subheads of the paper, in terms of the theorems and lemmas, and as lemmas hanging from a tree of theorems.9 Namely, the latter—see Fig. 1.2. Now, quantum mechanics shows that a hydrogen atom will not collapse (although the proof is more subtle than is usually supposed10). The problem for a larger atom, or for bulk matter, when you have 1023 atoms, is rather different. Yet, it was 40 years after the advent of quantum mechanics, 40 years after the physics was settled, that Dyson and Lenard provided their proof of the stability of matter. As written up, the proof was rather nicely organized. It appeared to be an elaborate ad hoc construction, again, a tree of theorems from which hung various lemmas, albeit sensible, eventually dividing up space into small volumes. Dyson and Lenard showed that if at least one of the particle types, say the electrons, were not fermions, and all were bosons, matter would not be stable. Fermions are unique particles once we keep in mind all their properties (Pauli Exclusion Principle, chemistry, and atoms), while bosons (such as photons) can have many with the same properties (hence, lasers). The deep point here is that our world, as we know it, depends on electrons being fermions, those unique particles. No fermions, no stability of matter. All this from a mathematical proof! Of course,

Lenard and Dyson, “Stability of Matter II,” p. 699. Lenard and Dyson, “Stability of Matter II,” p. 711. 9 See the Appendix to this chapter. 10 You have to take into account density, and a Sobolev inequality is needed. E. H. Lieb, “The Stability of Matter: From Atoms to Stars,” Bulletin (New Series) of the American Mathematical Society 22 (1990): 1–49. 7 8

1.4

The Stability of Matter

11

Theorem (L=lemma) 1 [N 3 lower bound] 2 ←L1 [a refinement, N 2] 3 ←L2 + L3 (←7, 6) + L4 [improved to N 5/3] [Toy problem: reduction to a one-particlein-all-of-space problem]↓ {1-particle}12,11,10 ←9 ←4 ←8a, 8, L5 [N bound, all particles fermions] **5 ←13 ← (L6, L7, L8, L9) [e− fermions] ↑[good cubes; configurational domains with uniform nearestneighbor separation]

NOTE: Theorem 5 is The Stability of Matter.

Fig. 1.2 The structure of the Dyson-Lenard proof. Note: Theorem 5 is The Stability of Matter.

you need to define “stability of matter” in mathematical terms, so that the definition appeals to our intuitions, and the same for bulk matter. And, you need to do the proof, as well as show that electrons as bosons would not work. By the way, once gravity comes into the picture in a serious way, as in a star, the story changes and you can have collapse. And what happens “next” is one of the great achievements of our time. About a decade later, Elliott Lieb and Walter Thirring provided a much-improved proof where the basic mechanism for stability was much clearer, namely that the relevant model of an atom (Thomas-Fermi) did not lead to binding among atoms. Moreover, that bounding proportionality constant went from about 1014 Rydbergs (1 Ry = 13.6 electron-volt; 1014 Rydbergs = 106 proton masses) per particle in Dyson and Lenard, to about thirty and less in Lieb and Thirring and in subsequent work. For reference, in the Appendix to this chapter there is an outline of the paper:11 The outline begins with: Basic insight: The energy of the Thomas-Fermi model atoms in bulk, with suitably modified constants, can be shown to be a lower bound for the actual N-electron Hamiltonian, HN. And, since Thomas-Fermi atoms do not bind to each other, their (Thomas-Fermi) N-electron Hamiltonian is proportional to N. Hence, we have a lower bound proportional to N, the stability of matter.

11

See the Appendix in this chapter.

12

1

Introduction

Teller’s argument for No-Binding is, roughly: adding some charge to a Thomas-Fermi nucleus will raise the potential, and so building up a molecule is harder than building up two well-separated atoms.

Now, along the way, we need to bind both the kinetic (A) and the potential energies of HN: And a bit later: (B) Bounding the N × N term by an N term: The electron-electron repulsion’s contribution to the potential energy, an N 2 term, needs to be bounded by a term proportional to N. The Thomas-Fermi approximation for the repulsion can be converted to a lower bound for HN’s repulsion. So the total repulsion term is now proportional to N. (The “trick” here is that at a crucial point one interchanges the electron’s and the nucleus’ coordinates.)

1. The N-particle energy, EN, is a sum of the kinetic energy (T ), the attractive energy of electrons and nuclei (A) and the repulsive energy of the electrons with each other (Re) and the nuclei with each other (Rn): EN = T‐A þ Re þ Rn: 2. By [the kinetic energy argument] above: K0 ×

ρ5=3 ≤ T:

3. We make use of No-Binding for Thomas-Fermi atoms, and the interchange of electrons with nucleus coordinates trick, to get Re proportional to N. Lieb and Thirring’s (1975) proof uses much more physics, that Thomas-Fermi model of the atom, and their proportionality constant was about 30. Lieb and Thirring realized that their proportionality constant was still too large given their semiclassical expectations, and over the years Lieb and collaborators have worked diligently to shrink that constant. Charles Fefferman, a mathematician, has provided another such proof (1983) in the middle of a paper on a mathematical “uncertainty principle” to be employed in analyzing and solving partial differential equations, albeit he never deals there with the exact value of the proportionality constant. Fefferman and collaborators (1983–1997) wanted to show not only that matter is stable but also that matter is made up of atoms and molecules and that a gas of those atoms is not just a plasma of electrons and protons, and along the way they worked to provide rigorous estimates of the ground state energy of a large atom. They need to provide formal definitions of the physicists’ notions, say of “an atom,” definitions that were faithful to the physics and would be useful in their rigorous mathematical work. They developed systematic methods of approximating the solutions of the relevant partial differential equations, methods that have wider applicability than just for this problem. Moreover, they provide rigorous justifications for approximations that physicists

1.5

Packaging Functions, Riemann Zeta Function

13

developed ad hoc. A recurrent theme here is how to fill up phase space (of position and momentum) with balls of the right sizes. In Lieb’s earlier work with J. L. Lebowitz, 1972, the spatial task was to fit balls into balls (the so-called “Swiss Cheese Theorem”). More generally, it is a matter of fitting balls into boxes. What physicists take to be Heisenberg’s Uncertainty Principle concerning boxes in phase space, Δx Δp ~ h/2π, or mathematicians speaking about Fourier analysis, becomes elaborated and sophisticated, realizing that some oddly shaped boxes in phase space, whatever their volume, will not accommodate any balls. More recently, Lieb and Seiringer provide a very short proof that is more about mathematics and various inequalities, again using Onsager’s insight that in the end what counts is the interaction of an electron with the nucleus nearest to it—and the constant is 5.6 Rydbergs/atom for H or, in another proof, it is 7.29 Rydbergs. (The semiclassical estimate suggests it should be 4.9 Ry.) All of this work, in both examples, is exact and can be made rigorous and precise. The ugly initial proof eventually yielded slimmer and simpler proofs, typically allowed by deeper insights about the physics and the symmetries of the system and sometimes by more sophisticated or more effective mathematics—say, using some nice inequalities from analysis.

1.5

Packaging Functions, Riemann Zeta Function

“Packaging” means that the function takes a set of numbers and creates something presumably unified that transcends them all. The partition function of statistical mechanics: Σ exp¯βE i where β = 1=kB T, packaging the Boltzmann weights (exp – βΕi) of the possible configurations of a system leads to the thermodynamic Helmholtz free energy (F), namely, PF = exp – βF. The quantum mechanical amplitude for a process is a sum of the amplitudes for all conceivable intermediate processes, a sum of Feynman graphs. More generally, L-functions, the partition functions for particular number systems, very schematically are: Σn χ(n)/ns or in terms of primes = Πp[(1– χ( p)/ps)-1], where χ( p) might be a measure of the number of solutions to an equation modulo p, the character of a Galois group, or the coefficients of a Fourier expansion. The Riemann zeta function is a packaging function for the primes, ζ(s) = Σ(1/ n–s), where n ranges over the integers. For it is not hard to show that ζ(s) = Π(1– pi–s)-1 for all primes pi. To prove this, note that 1/(1-a) = 1 + a + a2 + a3 + . . ., so (1– pi–s)-1 = 1+ pi–s + pi–2s + pi–3s + . . ., and recall the unique factorization of the integers in terms of primes. Zeta has well-defined loci of zeroes, at s = ½ + itn, for a range of t’s > 0, a functional equation (ζ(s) is proportional to ζ(1–s)), and a correspondence with an object, the theta function

14

1

Introduction

θ(τ) = Σ exp (i π n2 τ + 2 π i n z), where θ(0,–1/τ) is proportional to θ(0,τ), the proportionality factor (√–iτ) is called the modulus. θ retains its form when the independent variable is changed, much as sin(x + a) has the same form as sin x. As for the correspondence, ζ(s) is a Fourier-like Mellin transform of θ(τ), that is ζ(s) ~ M[θ(τ)] = 0!1 τs–1 θ(τ) dτ.12 Packaging functions may exhibit symmetry, in effect making the states of the system a collective whole, the primes a coherent set. The packaging function, in effect an arithmetic counting, might exhibit automorphy, a generalization of the sine function: almost periodicity—all in the complex plane.13 Again, what is remarkable, that integrity, is that the packaging function may sometimes exhibit a symmetry, such as PF(T ) being proportional to PF(1/T) for the partition function of statistical mechanics, or L(s) being proportional to L(1–s), as is the case for the zeta function. That symmetry is sometimes called automorphy, while for L-functions one speaks of a functional equation. Namely, when you change the independent variable (here T or s) the function has a well-defined transformation property, one that retains its shape, so to speak. The value of a packaging function lies in its presumable unification, its integrity, and its behavior independent of how we formed it. Statistical mechanics and quantum mechanics tell us why a partition function might lead to integrity. But why weighted enumeration might lead to automorphy or a functional equation calls for a theory. Put differently, a packaging function may appear to be not well defined over much of the complex s-plane. Yet, because of its putative connections to packaging functions defined by automorphy, we might assure ourselves that not only is the arithmetic or counting packaging function defined by analytic continuation in the complex s-plane, but it also has a functional equation and an expansion in terms of a product, “an Euler product.” So, for example, the regularity of the coefficients of the Fourier transform of that automorphic form, such as those of θ, would seem to be related to the very nice behavior of ζ, defined by enumeration.

12

Note there would seem to be just one zero for the partition function, at the critical temperature, Tc, a real zero, approached by giving T a complex part, iδ, and letting δ go to zero. And the interesting conformal behavior of the Ising lattice at the critical point (angle preserving, scaling, and rotational invariance) would seem to have no correspondence for zeta and the primes. 13 It has been suggested that the arithmetic-automorphic connection is a duality, the hightemperature avatar being a q-series, the low-temperature avatar being an elliptic curve.

Appendix

15

Appendix A.1 Subheads of Dyson and Lenard’s 1967–1968 Papers on the Stability of Matter14 I: 1. 2. 3. 4. 5. 6.

Introduction Statement of Results [Thms. 1, 2, 3, 4, *5* (the stability of matter)] Proofs of Theorems 1 and 2 [N3 and N2 lower bounds, Lem. 1] A Theorem of Electrostatics [Thms. 6 and 7, N × N → N interactions] Proof of Theorem 3 [N5/3 lower bound, using Lems. 2, 3, and 4] Proof of Lemmas 2, 3, and 4 [about functions and their derivatives; the geometry of points; arithmetic of numbers] 7. Proof of Theorem 4 [N bound, all particles are fermions, Lem. 5, using Thm. 8] 8. Proof of Theorem 8 [space occupied by fermions] 9. Smooth Background Charge [Theorem 4, improved to Theorem 9, is almost but not quite Theorem 5 (point nuclear charges), the stability of matter]

II: 1. Introduction [Stability of Matter requires lots more work than Theorems 4 and 9; and now a very large proportionality constant] 2. The Plan of Attack [one particle at a time in a chopped-up space; toy calculation of only one negative particle in space, Thms. 10, 11 (cf. Lem. 9), and 12 (cf. Thm. 5)] 3. Preliminary Simplifications [Now to Theorem 5, the main result, making use of almost all that has gone before] 4. Configurational Domains with Uniform Nearest-Neighbor Separation [chopping up space into good cubes, each containing one negative particle and an arbitrary number of positives, Lem. 6] 5. Reduction of the N-body Problem to a One-Particle Problem [Theorem 9 for each cell, and then reassemble the fragments, using Lems. 7, 8, 9, Thm. 13] 6. Solution of the One-Particle Problem 7. Proof of Lemmas 7, 8, and 9 8. Informal remarks on Lemma 9

14

These are in Krieger, Doing Mathematics.

16

1

Introduction

A.2 The “Numbered” Flow of Theorems and Lemmas of the Dyson-Lenard Proof A.2.1 Theorems I. Argument of Part I 1. Fisher-Ruelle, -N 3 ≤ Emin 2. Refined -N 2 ≤ Emin 3. - N 5/3 ≤ Emin ← 7, L4, L3, L2 4. All particles fermions, -N ≤ Emin ← 8a, L4 *5. e- fermions ← L6, 13, 8a =Stability of Matter 6. Electrostatics—lower bound of Coulomb energy for an arbitrary system of point charges 7. Potential energy of the system bounded by each particle attracted by its nearestneighbor (N × N → N ) 8. Space occupied by particles, given antisymmetry←L5 8a. Space occupied by just two particles, given antisymmetry Argument of Part II: Single Particle *9. Smooth background charge ←4 Toy Calculation: 10. Single negative particle, lower energy bound in a field 11. Sobolev-type inequalities 12. Stability for a single negative particle ←L2 13. Reduction to one-particle case ←L6, 6, L7 (which is similar to L2), L8, L9 [This leads to Theorem 5] A.2.2 Lemmas Argument of Part I: L1 particle in a Yukawa potential need not bind →2 L2 inequality connecting wavefunction, its gradient, and its potential L2a L3 geometrical fact about a finite set of points within a sphere L4 numerical or arithmetic fact L5 uncertainty relation of points within a sphere and functions of them Argument of Part II: L6 partition of space into disjoint suitable cubes L7 L2 for a cube ← L2a L8 comparing energies and potentials for a cube vs. a sphere ←L2

Appendix

17

*L9 bound interaction energy using kinetic energy of negatives and potential energy of positives

A.3 The Flow of the Dyson-Lenard Proof, as Lemmas Hanging from a Tree of Theorems Theorem (L=lemma) 1 [N 3 lower bound] 2 ←L1 [a refinement, N 2] 3 ←L2 + L3 (←7, 6) + L4 [improved to N 5/3] [Toy problem: reduction to a one-particlein-all-of-space problem]# {1-particle}12,11,10 ←9 ←4 ←8a, 8, L5 [N bound, all particles’ fermions] 5 **5 ←13 ←(L6, L7, L8, L9) [e- fermions] "[good cubes; configurational domains with uniform nearest-neighbor separation] NOTE: Theorem 5 is The Stability of Matter.

A.4 The Structure of Lieb and Thirring’s Argument in “Bound for the Kinetic Energy of Fermions Which Proves the Stability of Matter” (1975) Basic insight: The energy of the Thomas-Fermi model atoms in bulk, with suitably modified constants, can be shown to be a lower bound for the actual N-electron Hamiltonian, HN. Since Thomas-Fermi atoms do not bind to each other, their (Thomas-Fermi) N-electron Hamiltonian is proportional to N. Hence, we have a lower bound proportional to N, the stability of matter. Teller’s argument for No-Binding is, roughly: adding some charge to a ThomasFermi nucleus will raise the potential, and so building up a molecule is harder than building up two well-separated atoms. Now, along the way, we need to bind both the kinetic and the potential energies of H N:

18

1

Introduction

(A) Kinetic Energy Bound: The Thomas-Fermi kinetic energy, K × ρ5/3, where ρ is the Thomas-Fermi averaged charge density of the electrons in bulk matter and K is a constant, turns out to be a lower bound, for some K′, for the actual kinetic energy of N fermions (the electrons) in bulk matter. “Surprisingly,” a statement about the energy levels of a carefully chosen single-particle H1 (for one electron in bulk matter) can tell us about the kinetic energy of N fermions for the correct HN. It is carefully chosen in that the Thomas-Fermi form appears, and surprisingly it works as well for the sum of eigenvalues. 1. The ground state energy of any N-fermion Hamiltonian is less than or equal to its kinetic energy, T, plus potential energy, V, for any antisymmetric wavefunction, ψ: E0 ≤ Tψ + Vψ Eventually, V will be chosen to be the sum of independent- or singleparticle potentials (those of H1) of the form ρ2/3. 2. If the ground state energy is to be as low as possible, the fermions should fill up the energy levels two by one. The ground state energy is bounded by the sum of the negative eigenvalues of H1, 2 × sneg1 ≤ E0. 3. sneg1 ≥ -L × (–V)5/2 by a general argument connecting the eigenvalues to the shape of the drum. If (–V)5/2 is made proportional to ρ5/3 (as promised above), then we have: sneg1 ≥ -L′ × ρ5/3. Moreover, Vψ is ρV, and so, by our choice of V, Vψ, too, is proportional to ρ5/3. 4. Substituting (1) and (3) into (2), we get: K′ × ρ5/3 ≤ T. (Note that T can be computed here for the correct wavefunction for HN.) This bound is just the Thomas-Fermi kinetic energy with modified constant K′. In effect, we have an uncertainty principle connecting momenta to a functional of positions. Any such principle would have a ρ5/3 integral (or at least something with those dimensions), by dimensional analysis, given that T is an integral of the gradient squared of the wavefunction. (B) Bounding the N × N term by an N term: The contribution of electron–electron repulsion to the potential energy, an N 2 term, needs to be bounded by a term proportional to N. The Thomas-Fermi approximation for the repulsion can be converted to a lower bound for HN’s repulsion. So, the total repulsion term is now proportional to N. (The “trick” here is that at a crucial point one interchanges the electron and nucleus coordinates.) 1. The N-particle energy, EN is a sum of the kinetic energy (T ), the attractive energy of electrons and nuclei (A), and the repulsive energy of the electrons with each other (Re) and the nuclei with each other (Rn): EN = T – A + Re + Rn.

Appendix

19

2. By (A) above: K′ × ρ5/3 ≤ T. 3. We make use of No-Binding for Thomas-Fermi atoms, and the interchange of electrons with nucleus coordinates trick, to get Re proportional to N. (a) No-Binding for Thomas-Fermi atoms means: ENTF ≥ N × E1TF. (b) Substituting the Thomas-Fermi version of eq. 1 into eq. 3a, but now for “heavy” electrons of mass m/γ, and for an equal number of nuclei Z = 1, we find: γ TTF – ATF + ReTF + Rn(TF) ≥ N × E1TF/ γZ7/3. TF (c) A and Rn(TF) are functions of the nuclear coordinates, which are arbitrary. So, we might make them the heavy-electron coordinates (there is the same number of nuclei and heavy-electrons). Note that such an “Rn/e(TF)” (Rn(TF) with electron coordinates) is a curious term, essentially the electron–electron interaction but assuming the charge distribution is point like (which it is not). We need to bring in the actual distribution, and so we average over the square of the wavefunction. (d) Applying 3c and averaging eq. 3b over the system, we get: γ TTF –2 ReTF + ReTF + Re ≥ N × E1TF/ γZ7/3 (Note how ATF and Rn/e(TF) are transformed!) Now we have an inequality for Re in TF terms. 4. Substituting (2) and (3d) into (1) EN ≥ [(K′ – γ) TTF – ATF + ReTF + Rn(TF)] + N × E1TF/ γZ7/3. 5. The bracketed expression in 4 is just the Thomas-Fermi energy functional for a system with electrons of mass m/(K′ – γ), and by No-Binding we can replace the bracketed expression with N × E1TF/(K′ – γ), giving EN ≥ N × E1TF × {1/(K′– γ) + 1/ γZ7/3}. We can minimize the braced expression, and we have our result (which turns out to be inversely proportional to K′).

A.5 Subheads and Subtopics of C.N. Yang, “The Spontaneous Magnetization [M] of a Two-Dimensional Ising Model” (1952) I:. “Spontaneous Magnetization” (M ), (Eqns. 1–16) M = h max jΛ2 j maxi, j max i ≈ ðj¯i þ j þ iÞÞ=√2 M as an off-diagonal matrix element of a rotation (30) in spinor space (Eq. 15): h–|Λ2|+i. Note that {U, Λ2} = 0 (anticommutation) for the spin inverting operator U: U|+i = |+i, U|–i = - |–i, for even and odd eigenvectors, respectively.

20

1 Introduction

II:. “Reduction to Eigenvalue Problem” A: (Eqns. 17–25) M as a limit of a trace of a rotation matrix, M = lim trace Λ’ = lim trace Λ2 Λ1’, where lim Λ’ = Λ = Λ2Λ1, where Λ1 is made up of creation or annihilation operators for the even and odd states. In effect, Λ1’ restores the rotational symmetry that is broken by the even-odd matrix element. Eventually, taking the limit will effectively project out half the system (34). [Employing Yang’s notation, Λ2 = V 1/2sV –1/2, Λ1’ = S(T+-1MT-).] B–E: Eigenvalues for an improper rotation. B: (26–33) M as a product of the eigenvalues of Λ' (which is to be blockdiagonalized by ζ). C: (34–44) λα (Eigenvalues for the improper rotation Λ') in terms of those (l) of another matrix, G, which is independent of the limit. D: (45–59) ξαβ (Eigenvectors of the diagonalizing matrix ζ, then taking the limit to get ξ in terms of the surrogate “eigenvector” y). E: Summary. III:. “Limit for Infinite Crystal” A: (59a–69) Going to an infinite crystal, express how G (actually now D+D-) operates on vectors as a contour integral around the unit circle. It is an integral representation of a sum of terms, each of which is expressed in terms of the roots of unity. Set up the eigenvalue problem for l (IIIA): a singular integral equation in terms of Onsager’s phase, exp iδ’. B: (70–75) y (surrogate eigenvector) is to be found by the same eigenvalue problem as IIIA but now for zero eigenvalue (IIIB). Elements of ξ in terms of these eigenvectors. C: (76–80) Second integral equation (IIIB) solved by inspection. M4 (the fourth power “to eliminate the undetermined phase factor” introduced by the definition of the odd and even states) as a product of limiting eigenvalues of rotation (IIIA). IV:. “Elliptic Transformation” [or Elliptic Substitutions] (81–88) Θ(z, =exp iω) = exp iδ’(ω) = cn u + i sn u = [(k –exp iω)/(k –exp –iω)]1/2. Eigenvalues for IIIA; IIIA now expressed as an integral equation (Eq. 84) using the substitution. V:. “Solution of the Integral Eq. (84) (89–95) Analyticity and pole-and-period considerations for elliptic functions give solutions and eigenvalues. The product is now expressed as a q-series, which is an elliptic function. VI:. “Final Results.” M as fourth root (96) NOTE: λα and ξαβ are defined in terms of a limit of a going to infinity on the imaginary axis. Numbers in parentheses refer to equations in the original paper.

Chapter 2

Why Mathematical Physics?

The 2022 Prizes from the International Mathematical Union featured two focusing on mathematical physics: a Fields Medal to Hugo Duminil-Copin, and the Carl Friedrich Gauss Prize for Applications of Mathematics to Elliott H. Lieb. As mathematicians and probabilists, Duminil-Copin (and his teacher, Stanislav Smirnov, 2010 Fields Medal) developed rigorous ways of thinking of a lattice—as in percolation, where we are searching for a path through a lattice—at its critical local connection probability that leads to a path through the lattice (Fig. 2.1). They extended the ways we usually think of smooth environments and their properties— analytic functions—to lattices, employing in effect a discrete complex analysis. Then, they develop ingenious ways of describing that lattice at the critical point, ways amenable to rigorous mathematical proof. Langlands and collaborators (1994–2000) empirically located the critical points for some percolation lattices in an effort to discover the right mathematical variables (“crossing probabilities”) to characterize the lattice. Duminil-Copin’s and Smirnov’s work makes rigorous what was suggested by the Langlands work.1 The mathematicians’ strategy is to go directly to the conformal boundary-value problem, set up definitions of needed objects (for example, “random currents”), and prove theorems. The (mathematical) physicists almost always start out with the model’s Hamiltonian, then the partition function, and they calculate by whatever method they need, proving the scaling symmetry in the limit (of the critical point), and then the rotational invariance. However, we should note that the mathematicians likely would not know what they needed to prove until the physicists had earlier discovered or revealed the basic symmetries of the system.

1

J. Cardy and A. Zamolodchikov had derived the relevant feature—conformal symmetry, perhaps not so mathematically rigorously. It turns out that that mathematical rigor was not only for show, to justify what the physicists had uncovered but also led to deeper physical understanding. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 M. H. Krieger, Primes and Particles, https://doi.org/10.1007/978-3-031-49776-6_2

21

22

2

Why Mathematical Physics?

Fig. 2.1 A square lattice where adjacent lattice points may be randomly connected. If the probability of connection is greater than a critical value (say 0.5) there will be a path of flow from the top to the bottom

In a lifetime of work, Lieb has employed mathematics as the usual physicists’ starting point to reveal the physics.2 For example, Lieb and collaborators T.D. Schultz and D. C. Mattis (1964) showed how that Ising lattice might be thought of as a quantum field theory (with annihilation and creation operators). The lattice as a field is seen to be composed of independent or noninteracting fermions (having antisymmetric commutation rules), suggesting the physics of what made Onsager’s original 1944 solution work. Those fermions were patterned rows of spins, “quasiparticles”—not the original spatially localized individual spins (and so the quasiparticles’ interaction with an external magnetic field is not straightforward). There was a lovely analogy to the Cooper pairs of electrons in the BCS theory of superconductivity. In another remarkable achievement, Lieb and W. Thirring (1975) suggested the actual mechanism and physics behind F. Dyson and A. Lenard’s (1967–1968) amazing mathematical proof of the stability of matter in quantum mechanics and 2

Subsequent to the Gauss Prize (“for deep mathematical contributions of exceptional breadth which have shaped the fields of quantum mechanics, statistical mechanics, computational chemistry, and quantum information theory″): Lieb received the American Physical Society Medal for Exceptional Achievement in Research (for ″major contributions to theoretical physics through obtaining exact solutions to important physical problems, which have impacted condensed matter physics, quantum information, statistical mechanics, and atomic physics″); the Kyoto Prize in Basic Sciences (“Pioneering Mathematical Research in Physics, Chemistry, and Quantum Information Science Based on Many-Body Physics”); and, with Joel Lebowitz and David Ruelle, the Dirac Medal from the International Center for Theoretical Physics (“for groundbreaking and mathematically rigorous contributions to the understanding of the statistical mechanics of classical and quantum physical systems”).

2.1

The Big Ideas

23

electromagnetism, that the energy of bulk matter is bounded from below—Energy ≥ C ✕ Number-of-Particles: Ordinary matter does not implode, say its electrons and nuclei attract each other unstoppably so that the potential energy would go to minus “infinity.” Namely, matter “is” composed of Thomas-Fermi-model atoms (Lieb and B. Simon, 1977). These model atoms’ electrons would appear to live around the nucleus in a cloud of their own making. Since those Thomas-Fermi model atoms do not bind to each other (as shown by E. Teller, 1962), the energy lower bound of matter is proportional to the number of such atoms (rather than the square of that number, which would make matter unstable). Along the way, Lieb and Thirring reduced the Dyson and Lenard proportionality constant, from C ~ 1014 Rydbergs (1 Ry = the binding energy of H, 13.6 eV) to the order of 30. There are subsequent proofs, not so much calculation and modeling, by Lieb and collaborators, where Thomas-Fermi theory yields more general mathematical inequalities. (Still, I believe the Thomas-Fermi analysis is illuminating physically.) Note that my use of “is” above is an actual physical claim in the following sense. When individual objects are living in an environment (here, atoms in a piece of bulk matter), they are altered by that environment. If the mathematical account is effective and so illuminating, it will say how those objects are so altered or may be considered. Hence, we speak of “quasiparticles,” fermions in such an environment, and of “elementary excitations,” bosons in such an environment, and more generally of “bare” and “dressed” particles. One of the payoffs of a rigorous mathematical account is to suggest the nature and mechanism of that dressing. Quantum field theory shows how to compute the dressing by means of what is called renormalization.

2.1

The Big Ideas

There are four ideas already hinted at in these examples: Technique is physical. Namely, the technical devices you use to work out a model or push through a proof are not merely mathematical machinery. Rather, they build in the actual physics of the system. In addition, that machinery is likely deeper mathematically than merely convenient machinery to solve a problem. Rigor is revealing. If you find that in order to make a derivation go through or to prove a theorem you need to be rather more precise than you expected, you will find that the needed rigor has forced you to discover physical features of the system that are otherwise not noticed. Surely, a physicist’s derivation may well be correct. With sufficient rigor and empirical studies, we can be more sure just how it is correct. Tricks are physical. If you need to use a trick to make a derivation go through, it is likely that the trick itself is revealing the physics of the system and may well point to interesting mathematics. As we shall see, R. J. Baxter’s ingenious inventions

24

2

Why Mathematical Physics?

turn out not to be just ad hoc tricks, but they may be seen to be justified and given meaning by Duminil-Copin’s mathematics. And, mathematical physics often leads to deep mathematics otherwise not on your agenda. None of these ideas are new, and Newton and Maxwell provide powerful examples of them. However, the recent prizes remind us of their recurrent force. Calculation is powerful; theory and proof have a subsequent role that is not merely clean-up, for they are illuminating. One other point, in general, the mathematicians already know what they must prove since the physicists have uncovered and thought through what is distinctive in a system. Some physicists’ questions, say what is happening to the degrees of freedom of the system at the critical point, are perhaps not directly addressed by the rigorous mathematics. On the other hand, mathematicians are welcome to show more generally how the partition function being zero, the mass or energy of the particles at the critical point is zero or ε, the spontaneous magnetization starts to appear, and the specific heat is infinite, are mutually consistent and necessary when that is the case.

2.2

Ising in Two Dimensions: An Identity in a Manifold Presentation of Profiles

The two-dimensional Ising model of ferromagnetism, a lattice of magnetic spins considered in a temperature bath, T, allows for a wide variety of methods for deriving the statistical mechanics partition function. Earlier analyses approached the critical point starting with a finite lattice and going to an infinite volume limit, lim N,V!infinity, N/V constant, or starting out with an infinite lattice then getting closer to that critical temperature, or both, and then seeing what happens. That is, the critical point is a limit point. (H. A. Kramers emphasized that thermodynamics is a “limit science.”) Duminil-Copin and Smirnov were able to get directly at the behavior of the system at its critical point, that it exhibited conformal symmetry, by taking the lattice and giving it an expression in terms of complex analysis and analytic functions and holomorphicity, that is, employing a discrete complex analytic formulation, and then going to zero grid separation. In effect, the physicists showed the way, and Smirnov and Duminil-Copin then faced the critical state directly. Smirnov proved the conformal invariance in two dimensions; Duminil-Copin proved the conformal invariance in dimension three. DuminilCopin made rigorous use of the various methods employed by the physicists (for example, the kinds of lattice deformations employed by Baxter, using the startriangle relation). And, what would appear as Baxter’s idiosyncratic and ingenious inventions, now have a deeper meaning and motivation. Baxter’s “Z-invariance,” in effect moving a line of bonds, appears to become the “isoradial” nature of the lattice, that each face is inscribable in a circle of radius 1.

2.3

Ising Susceptibility

25

While the Ising model in dimension two has proved comparatively tractable to mathematical physical analysis, it still does not have an exact solution in dimension three. Duminil-Copin and collaborators have created ways of conceiving of Ising in three dimensions to further discover its properties and how in four or more dimensions it becomes generic or Gaussian. Moreover, they showed how to describe the continuity of the phase transition and the sharpness of that transition without having a solution in dimension three. Now that lattice, at the critical point, has specific symmetries (conformality: scaling, rotational invariance, angle preserving)—thereby becoming a very different physical object, in effect surpassing the various earlier formulations which approached the critical point as a limit but were not there. Duminil-Copin and Smirnov’s rigorous mathematics and modeling allowed them to see the lattice directly at the point.3 I should mention that in computer science, there are modes of “smoothly” analyzing algorithms (Spielman and Teng, 2001–2004), namely, varying the algorithms and their test inputs, so that one might obtain a more reliable account of average, typical, and worst-case execution times. Much as Duminil-Copin and Smirnov are able to go from a discrete lattice situation to a continuous one, here smooth analysis goes from discretely different algorithmic contexts to a smoothed average of these contexts.

2.3

Ising Susceptibility

To derive a magnetic property of the Ising lattice, actually the spontaneous magnetization (that is, with no applied magnetic field), hence a “permanent” magnet, at one point C.-N. Yang (1952) uses a trick, an artificial limiting process, to convert an expression into a “commuting plane rotation,” that is, very roughly exp a becomes lim iaε !a exp iaε, and the latter term is a rotation. In another derivation (Yamada, 1983–1986), that artificial rotation is essentially a consequence of L’Hôpital’s rule applied to a fraction whose numerator and denominator are both zero in the limit. That is, what appears as an ingenious and artificial trick reflects a natural move in another derivation. Other derivations of the spontaneous magnetization allow for other perspectives that are less tricky (keeping in mind that Yang’s was the first published one). In time, tricks are likely to become natural, rigorous, and meaningful in themselves. A bit further on in his derivation, Yang solves an eigenvalue problem by inspection. He is in effect doing what is called a “Wiener-Hopf factorization,” when a Fourier transform would not work since here one is concerned only with positive frequencies. Wiener-Hopf later appears explicitly in T. T. Wu’s work. Now,

3

Again, the earlier physical accounts by J. Cardy and by A. Zamolodchikov did directly explore the conformality by approaching and thus suggesting (albeit correctly) what it looked like.

26

2

Why Mathematical Physics?

T. T. Wu, B. M. McCoy, C. A. Tracy, and E. Barouch (1976) performed a lengthy and complicated calculation of the Ising spin–spin correlation function, in what is called the scaling regime, when the size of the lattice approaches infinity and we are very close to the critical point. (See McCoy and Wu, 2014 for earlier and subsequent developments.) They use Wiener-Hopf methods and Toeplitz determinants (determinants of matrices, where aij is the iminusjth coefficient in a Fourier transform of a nice function, already present in Wu, McCoy, and Tracy’s earlier work.) Remarkably, at the end of this calculation they end up with a Painlevé transcendent, a distant cousin of the trigonometric functions. Over subsequent years, Painlevé, Toeplitz, and Wiener-Hopf have become recurrent related themes, as pointed out in recent surveys of the work of Harold Widom (2022) on Toeplitz operators and in the work on random matrices and the Tracy-Widom probability distributions (of dependent random variables) by P. Deift, A. Its, K. Johansson, and others (1988–2013). Namely, the surprising appearance of what might be called peculiar mathematical objects in a derivation in mathematical physics may turn out to be a manifestation of systematic and beautiful mathematics more generally, with the individual objects being intimately connected not only generally but also particularly and physically in that situation. Moreover, in McCoy and Wu’s 2014 book on Ising, they present and review further work that makes the methods and results of their earlier “lengthy and complicated” 1976 calculation part of a panoply of results (the so-called exponential and form-factor expansions), with Painlevé, Toeplitz, and Wiener-Hopf playing an ongoing role, with their own logic and mathematical issues, leading, in the hands of J.-M. Malliard and collaborators, to further developments concerning differential equations and singularities. That is, the peculiarities of the physicists’ computational work point to interesting mathematical problems.4 It is interesting that the mathematicians’ rigor (that is, Duminil-Copin’s) may in retrospect be seen to motivate the physicists’ solutions. The mathematicians’ focus on the interfaces separating plus and minus clusters of the lattice points back to the polymeric/combinatorial solutions; their “fermionic observables” point to the fermions in the algebraic solutions; and the isoradial lattice points to Baxter’s methods.

2.4

Where’s the Physics?

In the 1976 Wu et al. paper, the calculations and algebraic manipulations would seem to be very far from anything physical. Sato et al. (1977) developed a rather different way of getting to the scaling regime, in effect a two-dimensional version of the Dirac equation. Again, they get to the continuum by going to the scaling regime,

4

The Ising model has a rather more extensive life than I have indicated here. Duminil-Copin has written, “100 Years of the (Critical) Ising Model on the Hypercubic Lattice” as his Fields Medal presentation, where connections with probability theory, percolation, and the “conformal bootstrap” are developed.

2.5

Dedekind-Weber and Reciprocity

27

rather than being at the critical point to start with. In addition, again, the calculations and algebraic manipulations would seem to be very far from anything physical. On the other hand, the rigorous Duminil-Copin work strikes me as rather closer to the actual physical situation, with the calculations not obscuring the physics. Put differently, rigorous mathematics may actually be revealing of the physics. Again, you often have to do the not-so-rigorous physicists’ derivation, to know what to aim for in doing a mathematically rigorous derivation. The physicists’ ways of thinking of a system, say as having degrees of freedom, each of which, if excited, has an energy of kBT/2, may not be so manifest in the mathematicians’ formulation.5

2.5

Dedekind-Weber and Reciprocity

Again, it is notable just how many very different methods may be used to solve the Ising model. They might be thought of as arithmetic and combinatorial, as in Kac and Ward’s work, or as in polymeric solutions, as in McCoy and Wu; algebraic and function theoretic, as in Onsager, employing commuting transfer matrices; and analytic, geometric, and topological, as in Baxter’s functional equations and the renormalization group. In effect, we encounter a trifold parallel that Dedekind and Weber (1882) find in their use of what is known about algebraic numbers (arithmetic) to study algebraic functions, thus giving a presumably more rigorous or algebraic account of Riemann’s geometrical account of algebraic functions. In effect, the physicists have provided a concrete instance of Dedekind’s trifold parallel, the mathematicians thus accounting for why there are various Ising solutions. As for Duminil-Copin and Smirnov, who are not solving the Ising model in general, their methods remind me of the polymeric and Baxter. In this prize-winning work, all four themes come to the fore: technical matters are physical, rigor is revealing, tricks are physical, and mathematical physics leads to deep mathematics and that deep mathematics may well be revealing more physics. Perhaps none of this is so surprising, at least no more so than that our language allows us to write poetry. Moreover, each of the different methods, physical or mathematical, illuminates other of the methods, giving meaning, and context to what is otherwise peculiar or idiosyncratic.

5

Which are the microscopic degrees of freedom excited in the conformally symmetric critical system? How are they related to the particles at the critical point having zero energy, in Schultz, Mattis, and Lieb’s account?

Chapter 3

Learning from Newton

In trying to account for the greatness of limewood sculptures of Renaissance Germany, Michael Baxandall suggests the following: . . . art history often takes its main bearings on outstanding individual performances, artists or works, . . . [As for standards of quality we might attend to] the interest of a work’s ambition, its success in fulfilling it, its novelty or its classic representativeness, its observed influence, the complexity and coordination we experience in it, its transparency through to a human condition . . . [We shall] centre very much indeed on outstanding individuals or workshops. [O]nly very good works of art, the performances of exceptionally organized men, are complex and co-ordinated enough to register in their forms the kinds of cultural circumstances sought here. The range of quality among [the carvers] is very wide, the distance between the very good and the run-of-the-mill astonishingly great. [That] has a relationship with such things as the circumstances of production, the dominance of certain genres, and attitudes to prowess in the general culture. (p. 10)

To reduce the mystery around creativity and the claims of genius, one might concentrate on the work itself (often comparatively little being known about the craftsperson), as Sennett has pointed out. Now, even then, there is much that cannot be explained or is mysterious. (Referring to magnetism, Newton calls such a mystery of the creativity of Nature, magnesia.) Therefore, for Newton, there is an “active” Nature. Matter is creative: here think of fermentation or what is called the “vegetative,” about which we might know comparatively little. Moreover, here we shall be concerned with what might be taken as evidence for God’s presence in the world, God’s Providence: God’s creativity—Newton’s major preoccupation. Newton will “discover” gravitation, exhibiting God as “the most perfect mechanic of all.” We might say that one of Newton’s great achievements was developing the apparatus needed to show mathematically that gravitation is universal and exhibits a one-over-R-squared force proportional to the quantity of matter in objects. Through that mathematical apparatus and mathematical proof, he shows that with the assumption of gravity’s properties he can account for what is seen in experience. As Newton puts it in the Author’s Preface to the Reader of the Principia:

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 M. H. Krieger, Primes and Particles, https://doi.org/10.1007/978-3-031-49776-6_3

29

30

3

Learning from Newton

For the basic problem of philosophy seems to be to discover the forces of nature from the phenomena of motions and then to demonstrate the other phenomena from these forces. (p. 382 Cohen edition)

3.1

Lessons from Newton

Mathematics (more generally, but surely the tradition of mathematical physics represented by Newton) is a powerful means of investigating the nature of the world, the world’s creativity. In so far as mathematics allows for a certain kind of precision, it also allows us to say something rather precise about how the world is made up—The Creation. Therefore, for example, modern proofs of “the stability of matter” tell us that electrons must be fermions, particles characterized by the fact you cannot have two of the exactly same kind (space, time, etc.)—the Pauli Exclusion Principle. Of course, “the stability of matter”—the notion itself—will need a more precise mathematical statement, allowing for a certain kind of precision. The archetypal model here is Newton’s claims about gravity. Therefore, mathematics is not merely for show, nor is it for more precise calculation of one more decimal place— although that is sometimes the case. Mathematics is philosophical in that it shows why things are the way they are. Whether it be for the Ancients (the Babylonians, the Greeks) or for Newton, mathematical physics, as we understand it, described the way things are and must be. For Newton, this is evidence of God’s Providence. My second point reflects a reading of the scholarly literature and the literature about that literature (the scholars not only fighting but giving an account of the fight in the past and in the present). When scholars give an account of the sources of Newton’s mathematical/physical work, they are tempted to allow for some influences and exclude others. So, for example, we might say that Newton’s mathematical physics comes out of a tradition of earlier mathematical work, and there is no documentary evidence that his other preoccupations (religion, theology, alchemy, politics) had much to do with it. On the other hand, we might argue that Newton’s alchemical work influenced his notion of an active space, and so allowed for an “action at a distance” in his law of gravitation. As far as I can tell, and I am surely an outsider without the qualification of knowing the original manuscripts, each explanation of the sources of Newton’s mathematical work has some merit, but the temptation to single out one major source of his creative work—maybe his mathematical physics is the source of his alchemy—is an ideological temptation, reflecting time-bound notions of what is respectable work and the particular inclinations of each group of scholars. In Newton’s case, the story is surely multicausal, historical, and very interesting. Newton is not about to be used for your purposes. My third point is that apocalypticism is a recurrent temptation of very strong scientists. Not all, and surely there are some much weaker scientists who are so tempted. However, what interests me is Newton’s or Freeman Dyson’s attempts to find a rigorous account of the end of the world, where “the end of the world” needs to

3.2

Creativity

31

be scientifically specified. For Newton, the science is hermeneutic, sorting and collating data, much as he does for the motion of the celestial bodies. In each case, he is seeking a univocal reading of the Book of Nature or of Scriptures. For Dyson, the science is physics with a healthy dose of an extraordinary imagination. I should note that some of our everyday terms—scientist, mathematical physics, theology, natural philosophy—are not for all time, and their meanings have changed dramatically over the centuries. My usage here will reflect our current use, and I will try to be careful when I apply them to Newton’s times. My last point is perhaps too cute by half, but perhaps worth stating: Hard work matters. Craftsmanship matters. As far as I can tell, really strong scientists work very very hard, as do strong basketball players. (Bill Bradley said that when he did not practice he knew that his opposition was practicing.) My favorite analogy is dairy farming, where the cows have to be milked every day—whatever your mood or the situation. Newton would have been a farmer of the year, and his strongest works would be seen as a product of his unending practice with all of his skills.

3.2

Creativity

Surely, I might be in awe of Newton and of many modern mathematical physicists. I find painters like Frances Bacon or Jasper Johns amazing, in awe of their work and their changing styles and problems. Now, I know there is a large literature on creativity in psychology and management and art. However, it does not help me with Newton, or with Bacon or Johns. (Now, I should note that notions such as creativity and genius are comparatively modern, defined and understood as we understand them well after Newton’s time—but we are surely welcome to apply those notions to earlier times and study how Newton’s reputation as creative or as a genius changed over the decades and centuries.) I think I understand what people mean by God’s Creativity, as in Genesis. And it may be that very talented people may or may not exhibit signs of the highest creativity and of what might be called genius, dependent on the time they are born in, the community within which they work (and so the cognitive and intellectual resources of their community, and their interactions with others), and the problems that are available for being worked upon. But none of these notions of creativity or genius are usually what people want to mean by creativity or genius, with an emphasis on individual distinctiveness. So, I shall be asking a rather more practical question: Scholars, using manuscript sources, and a knowledge of the skills and crafts of the time, can show us how Newton did it, whatever it is: filling in the steps only hinted at in the public materials. Perhaps the scholars find notes, notebooks, and earlier sources. In effect, they provide as many intermediate steps as they can, given the sources and reasonable inference, showing how ordinary folks might have done the same thing—if they had only thought to do so. That is, the amazing work is now shown to be a matter of step-

32

3 Learning from Newton

by-step incremental moves. So, you now understand historically and logically what must or might have taken place. To some extent, the work has now been demystified. One’s admiration for Newton is unlikely to diminish even if you have shown how the work must have been done step-by-step or you do it yourself. One’s admiration for Dyson and Lenard’s proof of the stability of matter does not diminish, even after much shorter and perhaps more perspicuous proof is provided in the subsequent decades. In part, first movers or first doers do not know that it can be done. Their persistence, courage, and inventiveness are admirable. In part, a step-by-step account makes it clear that one would have to be extraordinary to have followed that path and not given up. And in part, creativity as most people conceive of it is not meant to be explained by the scholar’s strategy.

3.3

Mathematical Physics

Now that I have punted on the questions of creativity and genius, let us return to Newton. Given the vast scholarship of late on his alchemy and his religion and theology, it is perhaps old-fashioned to focus on his work in mathematical physics. However, once we understand mathematics as philosophical, perhaps we are not so old-fashioned after all. A major problem here is our access to work in mathematical physics. I am not sure one can understand work in a field—its motivations, its inventiveness, just how it was done—without being able to actually do that work, albeit not at the highest levels. Can you appreciate how a painter did her work without having painted yourself? (This is very different than the work of most art historians or critics, who are concerned with the place of the work in history and culture and its meaning.) My claim, to repeat, is that mathematical physics is not merely for show or for greater precision. It is substantive, metaphysical, and philosophical about the nature of the world—naturally, perhaps even theologically. A precise mathematical description of the world, a description we are willing to believe is a good abstraction and careful proof, allows us to learn about the nature of the world. Roughly contemporary with Dyson and Lenard and Lieb and Thirring, Kenneth G. Wilson was trying to understand forces between nuclei, which are called strong interactions. The conventional techniques of perturbation theory (developed to do celestial mechanics, and truly Newtonian) cannot be used in this context since the perturbations are so pervasive. His deep insight was to divide up the interactions into various scales, sum them up at each scale, and then use that sum for the next larger scale. It is a physics not of perturbations but of scales. In talking to his Cornell colleagues, he was told about similar problems but now in understanding how a magnetic material such as an iron bar becomes permanently magnetized once its temperature is below about 1000 degrees K, and how a metal might be influenced by a magnetic impurity (the Kondo effect). How do you sum up all the interactions among the atomic magnets to determine the bulk magnetization of a piece of iron or

3.3

Mathematical Physics

33

the influence of an impurity on a metal? His technique was to employ an insight of Leo Kadanoff, namely, group the atomic magnets (here in two dimensions) into blocks of say 9 atoms (blocks of 3 × 3), and group the blocks of 9 atoms into larger blocks of 9 blocks themselves, and so on. In effect one has a cascade of scales.1 Now, the problem is to figure out how to convert such an insight into physics you can calculate. Surely you needed some symbolic abstract mathematics, but in the end what worked for Wilson was a numerical calculation using computers for much of the work, doing that sum of sum of sums. Wilson had experience using digital computers to work out numerical solutions to difficult problems, and he was not afraid of complex situations and many special cases. What was crucial was the algorithm to be employed to do that iterative summation at all scales, and actually it was not so much an algorithm but a judicious keeping of higher order terms or corrections—many of them, just what might horrify an algorithmer. And what we discover from Wilson’s (and earlier) work is that we might well think of the world as composed of similar processes at all scales, from the atomic to the bulk. A precursor to Wilson’s work, first formulated in the mid-1920s in Ising’s dissertation, was what we have called the Ising model. One had a grid of spins or atomic magnets in one or two or three dimensions, and one asked, if the temperature were low enough (but above absolute zero), was there an abrupt transition to order. Ising could show there was no such transition in a row of spins, that is, in one dimension. The spins bounced around so much due to thermal motion that they never could get themselves lined up. In two dimensions in a grid, the spins have four rather than two nearest neighbors. Perhaps the greater adjacent-spin interactions could then conquer thermal vibration? In 1936 Rudolph Peierls showed that this should happen. (More mathematically, see van der Waerden, 1941.) In 1944, Lars Onsager showed in a rigorous proof that in two dimensions one could get such a sharp transition. Initially, what he did was to start out with two infinite rows of spins, then three, then four, solving the problem in each “toy” case. He noticed some regularities in the solutions, and then using this intuition he solved the problem for an infinite number of rows using an algebraic analysis that is quite impressive. Along the way, in solving the problem, Onsager employs elliptic functions (essentially, generalized versions of the trigonometric functions). For those not acquainted with such, there appears to be some legerdemain involved in the paper, but in fact Onsager knew Whittaker and Watson’s Modern Analysis (1902, 1927) rather well and it is all there. More to the point, elliptic functions build in the scaling symmetries later found by other methods. That is, those elliptic functions do not just solve the problem; they build in deep physical facts about the system. In later work, Ising matter was shown in effect to be composed of pairs of particles, not unlike the Cooper pairs of electrons that make for superconductivity.

1

Kolmogorov, earlier, had indicated that this might be the way turbulence works, with eddies composed of smaller eddies composed of smaller eddies—a cascade of scales.

34

3 Learning from Newton

Or, Ising matter exhibits the kind of scaling symmetry we mentioned earlier. Yet, when we look back, perhaps it is all there in Onsager already. Keep in mind that this work was not experimental. It was, as for Newton, Dyson and Lenard, Lieb and Thirring, Wilson, or Onsager, it was mathematical, solving a problem. In solving a problem, the mathematical apparatus pointed to crucial features of the Ising system that surely had physical meaning. It turns out that there is no exact solution to the Ising model in three dimensions, and one of the reasons for Wilson’s work was to find a way of calculating the properties of Ising matter in three dimensions. The story gets even better. If two-dimensional Ising matter is finite in extent, there will be no sharp transition to permanent magnetization. You need the infinity in at least two directions to pin down the spins to each other. In 1952, Lee and Yang showed that it is crucial that the series of approximations to an infinite-size lattice does not converge uniformly to the infinite lattice. (Think of uniform convergence as a property of a series in which the limit point is much like the intermediate points.) The lack of uniformity allows an infinite series of approximations, each of which does not have a sharp transition, to converge to a solution in which there is a sharp transition. Put differently, we understand the sharp transition to order in terms of a fairly technical mathematical feature of an infinite series of approximations. And we might understand that feature in physical terms. Mathematical devices and technologies build in deep ideas, whether we appreciate those ideas ahead of time, if ever. They are not merely machinery for calculation; they are machinery that is substantive and philosophical and metaphysical, in effect, sneaking in important features of the world, for free, allowing us to solve the problem. It pays to discern those features, for they tell us deep facts about the world we live in. If electrons were not fermions, matter as we know it would not be stable. More generally, we know that the language we use to describe the world will affect what we can say and how we say it. If mathematics is the language in which the Book of Nature is written, it is perhaps not surprising that that language is not so passive a device. It takes enormous effort to read some of the papers I have been describing, and often they are known by reputation rather than for having been read. Wilson read Lieb’s version of the solution to the two-dimensional Ising model rather than Onsager’s original paper, written as Lieb’s was in modern quantum-field-theoretic language. Nevertheless, the payoff of reading the original paper is discovering the richness of the thinking of these mathematical physicists and their sheer persistence. However, to appreciate this work you do need to be rather well trained. Again, it seems to matter that you be able to do this sort of work if you are to appreciate how inventive and persevering these scientists are. I am told by colleagues that the problem with many critical studies of film or theater is that the writers do not much appreciate the everyday issues on a movie set, the way decisions are made, the ad hoc nature of much of what goes on. That does not devalue observations by the critic, but those observations may have had little to do with the intentions of the creators.

3.4

3.4

Influence

35

Influence

Enormous scholarly energy has been devoted to understanding the sources of and influences on Newton’s thought. At least as much energy has been devoted internally to understanding the interplay of Newton’s lifelong mathematical-physical interests, theological interests, and his alchemical interests. Everyone seems to agree that, for Newton, these three interests were coequal and apparently inseparable, part of an effort to understand God’s Providence and God’s nature—the nature of the universe—to use modern terms. However, there is a very lively debate on the hierarchy of influence, whether and how the mathematical-physical, the theological, and the alchemical influenced each other. One might tell a complex story of mutual influence, presumably changing over time and subject matter. Documents need to be interpreted and found, inferences from documents or the lack thereof must be made, and historical context must be elaborated. In addition, matters of economy, politics, class, and technology might well be important. As we come to better understand seventeenth-century alchemy, theology, and natural philosophy (“science” as we know it is not of much use here, it appears), the story will surely be modified. Such a story rarely if ever answers the questions that concern many scholars: those who want to show any single influence was primary, or any single aspect was immune to others. Kenneth Wilson tells his own origin story: we learn about his father (a Harvard chemistry professor), his advisor (Murray Gell-Mann), and his early work on strong interactions. He mentions his facility with computers, a recurrent theme in this work. He tells us how he came upon the problem of understanding phase transitions in bulk matter (from his Cornell colleagues), his having been trained as an elementary particle physicist. He points out that computers and computational tricks of this generation become the mathematics of the next generation. He likes to find numbers by doing good approximate calculations, often demanding lots of computer usage. (Hans Bethe, his Cornell colleague, also wanted to find the numbers to compare with experiment.) No religion, no theology, no chemistry, although class and gender played a role. Charles Townes will write of his Christian faith, but we learn nothing of its influence on his work on masers, lasers, or many other wonderful experiments.2 Nowadays, most men and women who are scientists do not credit their religious and faith commitments to their achievements. Newton would have seen it all in a piece. So times have surely changed. The typical argument concerning Newton is how and whether his mathematical physics, as we now understand that subject, was influenced by his theological commitments or his alchemical studies. Action-at-a-distance, embodied in the inverse square law of gravitation, was strange, and so it was at times proposed that there was an active material world between well-separated objects gravitating toward each other. 2

Townes, much like Newton, saw science and religion as on the same quest.

36

3

Learning from Newton

Could alchemical notions of active substances or an active nature have played a role in his thinking? If we focus on the Principia, as did I. B. Cohen, we might argue that there are no such documents attesting to such an influence, while if we focus on Newton’s alchemical work, as did B. J. T Dobbs, such an influence is quite credible. My impression is that Cohen very much wants to keep the mathematical physics separate from Newton’s other activities, while Dobbs wants to claim it is all of a piece. Now it could be that science might influence theology, and there is a very long tradition of that sort, perhaps now more lively than ever. If Nature is God’s Providence and is evidence of Her Being, or Nature is God’s Providence and so God must be the God who produced Nature, the more we learn about Nature, the more we learn about God. God’s infinity is subject to our imagination about the nature of the infinite. When Georg Cantor proposed his notions of the infinite in the later nineteenth century, he consulted with Jesuits to be sure he was not being unorthodox. Cantor defined a first infinity as the number of natural numbers, 1, 2, 3, . . . He then defined a sequence of strictly larger infinities, the aleph-N’s, where the first infinity is0‫ א‬.. Another infinity, strictly larger than aleph-nought, is the number of real numbers, defined by an infinite decimal expansion including repetitions and zeros. We still do not know if this infinity is equal to aleph-one, although Woodin has made some interesting speculations in the last decade, namely, the continuum is equal to 2‫א‬. Concrete mathematical speculation on this issue has proved to be extraordinarily fruitful. A century after Cantor, mathematicians have defined a baroque variety of infinities, and they have asked and shown whether and how they form a hierarchy. The principles they use, such as the principle of reflection (properties of the universeof-all-sets are “reflected” down, or evidenced, in a smaller set), strike me as Scholastic and wonderful. I would very much like to hear Newton’s thoughts on God’s infinity, given our articulated notions of the infinite (whether numerically, as I have described, or geometrically for space). Cantor was not at all idiosyncratic in his concern for authoritative approval of his scientific work. Newton was a religious man of his time, but his theology was unorthodox. He was sure that the early Church had gone awry, missing the true meaning of the Scriptures. He was a unitarian (an Arianist), denying the Trinity, although respecting Jesus. God was above All, the Lord of Dominion, with full and arbitrary powers. Such heterodoxy was to be kept to himself if he were to have a fellowship at Cambridge. So, when Newton writes a justification, at the end of the Principia, a General Scholium, it is both orthodox and hyperbolic. Yes, we have a good account of celestial mechanics in the Principia, and so we know some of God’s ways. However, the initial conditions, the orderliness of the solar system “proceed[s] from the counsel and dominion of an intelligent and powerful Being.” God is duration and space, substance and meaning. (Jesus and the Holy Spirit are never mentioned.) As for gravitation, we may know how it makes its presence felt and how it operates, but we have no idea of its nature or source. To a modern physicist or mathematical physicist, the challenge remains powerful.

3.5

The Apocalypse

37

No Theory of Everything accounts for initial conditions, and if and when quantum gravitation is accounted for, there will be similar problems, all the way down. To an outsider, it would appear that the scholarly arguments about science, religion, theology, and alchemy are being used to play out contemporary issues. Newton’s Nachlass was huge, and for a very long time little studied or appreciated. God only knows what we might find that would be grist for someone’s mill. One more example is Newton’s “sex life,” as we might call it. People have testified that Newton was chaste, and that might well have been the case, given the rules for his professorship at Cambridge. In any case, we have no documents, no Excel spreadsheets listing his encounters, no love letters, and no Leporello who would provide us with a list of his 1003 conquests (as in Don Giovanni). Not quite, actually—we do have some letters. We do have evidence of his close personal and emotional relationship with a much younger, fairly talented, gentleman, Nicolas Fatio de Duillier. He wanted to share a room with Fatio and have their beds in the same room. Perhaps we know much less about the practices of the time, especially in terms of intimacy. What we discover in reading the commentary and history based on these documents is perhaps more about the writers than about Newton, for again there is no smoking gun, so to speak. As historians have told us, our notions of emotional intimacy, homosexuality, heterosexuality, gay, straight, etc., are of fairly recent origin. If you want to keep Newton “pure”—untouched by sexuality, or by theology, or by alchemy, you do need to separate him into disjoint parts with little congress among them. And if you want to make Newton a unified whole, you will have to modify your notions of sexuality, theology, alchemy, and mathematical science, to suit Newton rather than our own times. Presumably, the Templeton Foundation would have given Newton one of its Prizes. However, they would be honoring a man whose theology was highly unorthodox and who surely was not a reliable advocate for the Spiritual.

3.5

The Apocalypse

Much of mathematical physics, and even some of pure mathematics (as in the nature of the infinite and implicitly God’s infinitude), pursues by different means what were once natural philosophical (and often theological) issues. The largest payoff of our post-WWII investment in particle accelerators, telescopes, and the training of physicists is the account we now might give of the early universe—what we call The Big Bang and perhaps The Inflationary Scenario. In those first few minutes, beginning in those first 10-30 s, the variety of particles we now find froze out of the quark-gluon plasma, much as a confection might separate into layers as it cools down. In the subsequent billions of years, the more complex particles, the elements, formed in stars and were then dispersed by stellar explosions. It is a remarkable origin story, where time and structure and substance parallel each other in a homology that is profound. At the least, the first day of Genesis is well accounted for.

38

3

Learning from Newton

The Apocalypse, the revelation at (presumably) the end of time, turns out to be a recurring preoccupation as well. Now, Judgment Day and The End of the Universe are by no means the same, although they are often treated as cotemporal. If we set off an atomic bomb, will the atmosphere catch fire and burn up the earth (teaching us a lesson)? (Konopinski calculated and suggested NO.) Herman Kahn asked, Will the survivors envy the dead? after a nuclear war. (He thought not.) In one of the particle accelerators, if we create a quark-gluon plasma, or perhaps a black hole, will we be absorbed into the hole or be eaten up by the Higgs particle? We can actually calculate such contingencies with some confidence, given what we know. We know that very high-energy cosmic rays have yet to create a sufficiently long-lasting black hole. It is perhaps more difficult to know much about The End of Time or Time Without End. Dyson has done some rough calculations of the long-term future (especially for consciousness), albeit they are fairly straightforward once we allow our imaginations (were they so supple as is Dyson’s) to think practically so far ahead. It is rather more difficult to say what it would be like to be around “after” the end of time. What scientists have done is give a substantive theory of space and time, general relativity, with time measured by the frequency of a Cesium clock and space measured by the speed of light. They are much less sure what to say about being around after the end of time, outside of space— Now, Newton too was concerned about the Apocalypse, albeit his main source was The Book of Revelation. His problem was to find a consistent univocal reading of the text that told the truth. His hermeneutic was as modern as it gets, seeking the correct interpretation of the text and its events and symbols. Perhaps characteristic of modern scientists, he cannot brook with multiple truths. There is only one story to be told, although there may be various provably equivalent versions. We are presented with apocalyptic visions and predictions, whether of nuclear war or nuclear winter, environmental catastrophe and doom, population bombs, or global warming. Yet, it is difficult to predict doom (or assign it a probability), given that a prediction might make us change our behavior to avoid that doom. So, the prediction must make us even blinder to what we might do, or we might become aware of the danger after it is too late to make a difference. Applying probability theory, it would even appear that doom’s probability is zero-or-one. In any case, it would still seem that the most monitory and cogent apocalypticists are in the religious and literary realm. Newton’s advantage is that he could honestly employ theological methods that he saw as paralleling his mathematical-physical arguments in their search for a single truth. Newton’s various inquiries—mathematical, theological, and alchemical—were part of a quest to understand the nature of the world, God’s Providence. Moreover, I am convinced that current work, no matter how stripped it is of metaphysical language, provides the kind of proof of the nature of the world that Newton would have found cogent and important. I also believe that mathematics and mathematical physics are strong enough to survive Newton’s theological and alchemical tendencies. They have often implicitly absorbed them. Unfortunately, for natural scientists, their ability to sketch powerful apocalyptic visions is rather weak, given their own standards.

3.5

The Apocalypse

39

We might learn something rather more pedestrian from Newton. It helps to be indefatigable, to live a long time, to be persistent and ambitious and sufficiently strong to conquer local difficulties in your project. While mathematics may be one of the languages of nature, it would seem to be a powerful way of teaching us about nature. But not everything is encompassed by the unreasonable effectiveness of mathematics in the natural realm. Newton’s work tells us that.

Chapter 4

Primes and Particles

Mathematics and physics have been borrowing notions from each other for a very long time. Some mathematical object or construction provides a superb model of the physical world. Some physical account instantiates a more general mathematical object or construction, avant la lettre, that mathematical notion only then discovered. These analogies have been recurrent, and there are periods when the fields do not talk much to each other, and then there are periods when they are virtually the same. Abstract mathematical constructions are found to be useful by physicists, sometimes decades after the mathematicians have lost interest in them. Physical theories develop as they will, while across town, there is mathematics that will enlighten the physicist—but by chance such enlightenment may take place, if it does, decades later than would be ideal. This is also true within mathematics, the archetypal cases of Artin and Hasse both being at Hamburg, or Riemann and Dedekind at Göttingen, where the work of one of their colleagues would have illuminated that of the other, and although they knew each other well there is no sign of such influence. There is another sort of analogy that is rather more general. The physicists’ account of the most elementary of particles has a structure much like the mathematicians’ account of the integers and the numbers (such as √–1 or √5) needed to solve simple equations (x2–5 = 0). Here, it is not a matter of physicists or mathematicians borrowing from each other. Rather they are borrowing from a warehouse of thinking and structures available to all. The analogy is quite generic, of course, instantiated in each area in a distinctive way. Moreover, the way physicists think of elementary particles in their families and in their interactions are much like the anthropologists’ account of family, kinship, and marriage.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 M. H. Krieger, Primes and Particles, https://doi.org/10.1007/978-3-031-49776-6_4

41

42

4.1

4

Primes and Particles

The Thermodynamics and Music of the Numbers

A function packaging information may have properties, emergent properties, not at all apparent from what it packages. Hence, the zeta function, ζ(s), as a packaging function of the primes, seems to have a connection with a modular function θ(z,τ), which as well packages, in a different way, numbers that can be made by a square of an integer (1, 4, 9, . . .). The statistical mechanics partition function for the canonical ensemble (fixed temperature, T, and volume), packaging the weighted (by exp –E/ kBT, E being the energy of a way) number of ways of making a macroscopic system from its many microscopic components, proves to be exp –F/kBT, where F is the thermodynamic free energy and kB is Boltzmann’s constant. Or, if we add up the suitably weighted resonant frequencies of a drum, we can then get a measure of its area and perimeter: that is, Spectrum ≈ Geometry.1 Or, quantum mechanical amplitudes for a process are defined as the sum of amplitudes for each of the various ways or paths that process might be composed (“a sum of histories”)—and again those total amplitudes have surprising properties, and they may be deduced by methods making no reference to the ways or paths. So, we might speak of: The Thermodynamics of Numbers; The Sounds of Primes and Music(al Notes); The Shape of the Harmonies of Nature; or, as we shall see, An Explosion’s Ejecta. In each case, we have a packaging or generating function defined in terms of what it apparently packages. Yet, that packaging function has properties that are not apparent from what it packages. Often, that packaging function may be defined and computed in an entirely different manner, making no reference to the original packaged combinatorial objects. Moreover, properties of the packaging function we might readily prove in terms of how and what its packages are very different than other properties of that packaging function we might readily prove from the way we understand it as a whole, as an emergent object, or even as an object composed of very different parts than in the counting-up we did. In Ising, we may count up individual spins in interaction with each other. Or, the lattice may be seen to be composed of different-sized islands of up-spins or down-spins, and we count the interactions along the islands’ boundaries between concentrations of up-spins and of down-spins. Moreover, typically, we have an overarching goal. In number theory, we want to be able to solve equations, and so factor polynomials into monomials (x–a) since each of those factors points to one of the solutions to the equation, here a, which hopefully is a prime or is otherwise a product of primes. In particle physics, we want to be able to discern the fundamental particles, their interactions, and their families.

1

Or, L functions, generalizations of the zeta function, which are defined by combinatorial numbers (say the number of solutions to a polynomial equation, modulo a prime number), or by the traces of matrix representations of the Galois group of the equation—turn out to have as well definitions in terms of automorphic objects, that is, objects with scaling symmetry, apparently having nothing to do with the combinatorial numbers.

4.1

The Thermodynamics and Music of the Numbers

43

We have empirically discovered that what would seem to be elementary is often composite, so that in a number system that includes not only the rational numbers but also solutions to equations such as x2–2 = 0, numbers that are prime among the integers are no longer prime but are in fact composite: here 2 is composite: 2 = √2 × √2. Similarly, if we raise the energy of interaction of elementary particles we discover that many of them would seem to be composite, harboring other particles or composed of more elementary particles. They might have internal structure, so that they may be excited into new states, much as an atom may be excited to states with higher main quantum numbers, n. On the other hand, we now know that the muon is not best thought of as a heavy electron, for the muon heads its own family of particles much as do the electron and the tau. Or, we might say that the new particles are present “virtually,” to show themselves only when there is enough energy for them to emerge. Or, in collisions of particles, new particles emerge if the energy of collision is large enough. To be elementary is context dependent. At another level, there are objects that we might consider rather more elementary. In the physics of the Standard Model, the most elementary of the particles are quarks and leptons. In mathematics, the most elementary of the numbers are usually taken to be the integers or the primes. Ideally, we would like some hints of these problematic nonprimes or nonelementary particles when we worked just with the integers or the rational numbers or at lower energies. We might say that number systems and nature have hidden symmetries revealed only when new numbers are introduced or the energy becomes great enough. One of the goals of algebraic number theory is to understand primes in enhanced number systems by studying just the rational numbers or the integers. One of the goals of particle physics is to find hints of emergent symmetries before they appear full-blown, since these symmetries and particles, even if they are virtual and energetically too demanding, may well affect what we see at lower energies, albeit marginally. (Hence, there is talk of “Beyond the Standard Model” and “New Physics.”) One strategy that is often revealing is to create a dense “explosive” situation and see what emerges. Mathematicians study solutions to equations modulo a prime for all primes at the same time. Physicists create a hot plasma of particles (say, crashing together two protons or two gold atoms at very high energy), and in studying the emergence of jets of particles they might find hints of what is beyond the capabilities of their more subtle methods. (Put differently, that hot plasma is in the realm of thermodynamics, the jets are expressions of the boiling off of material, and the jets’ composition is in the realm of particle physics and statistical mechanics.) In all these endeavors, we are in effect producing or deciphering another Rosetta Stone, the same text in different languages. The actual texts are different versions of what we take as the same message, sometimes detailed and particular, sometimes gorgeous and rhetorical, and sometimes a theme and variations. Yet, those different versions suggest differing responses and pathways. The Standard Model requires the input of as many as 28 independent parameters. [The masses of the 6 leptons, the 6 quarks, the Higgs, the three coupling constants such as the electron’s charge e, and mixing angles, . . .] These parameters are not explained by the

44

4

Primes and Particles

Standard Model; their presence implies the need for an understanding of Nature at an even deeper level. Nonetheless, processes described by the Standard Model possess a remarkable insulation from signals of such New Physics. (Donoghue, p. 1) Why Galois representations should be the source of Euler products [ζ(s)=Π(1–ai–s)–1)] with good functional equations [ζ(s) ≈ ζ(1–s)] is a complete mystery. (Taylor, p. 13) Riemann’s insight was that the frequencies of the basic waveforms that approximate the [Dedekind] psi function [a counting function of the primes, ψ(n) = n Π p2P: p|n (1 + 1/p), where P is the set of all primes, and the product is over all primes dividing n] are determined by the places where the zeta function is equal to zero. To me, that the distribution of prime numbers can be so accurately represented in an harmonic analysis is absolutely amazing and incredibly beautiful. It tells of an arcane music and secret harmony composed by the prime numbers. (Bombieri)

There is a speculative analogy between how mathematicians understand prime numbers and how physicists understand elementary particles. Such an analogy must reflect a deeper strategy of dealing with hierarchical systems in which there is coherence among apparently independent objects, such as each prime number, and coherence among apparently distinct levels in the hierarchy, as in algebraic number theory or in the families of elementary particles and their fundamental forces (strong, electromagnetic, weak, gravitation); where objects are composed into more natural units or are decomposed into simpler ones, which may then be decomposed themselves; where we may learn to add up those objects, actually their properties, and where enumeration leads to remarkable symmetries; and where discerning the “right” objects allows for powerful synthesis in theorizing. (The physicist Steven Weinberg says that if you choose the wrong objects, “you’ll be sorry.”) What we discover is that the objects’ and levels’ independence is only apparent, and the connections among them are multiple and amazing, and that linearizing such a system is a key to understanding it. The pervasive theme is symmetry and orderliness, and when that symmetry is broken or, conversely, when that symmetry is encased in a higher symmetry. (The basic objects are modeled by matrices (linear “irreducible representations”), with particular properties to embody those symmetries.) Surprisingly, enumeration of the objects may lead to smooth functions that have lovely symmetries of their own. Moreover, anthropologists’ account of kinship and marriage, with prescribed and proscribed exchanges of women and material (that is, marriage), parallel particle physicists’ accounts of particles in interaction, the particles in families, and some exchanges of particles and so interactions being forbidden and others are allowed. Along the way, some remarkable mysteries present themselves, both in mathematics and in physics. These Why questions are likely to be answered by a theory that shows How. Why does enumeration of the prime numbers eventually lead to a nice function, employed by Fourier to understand heat flow and earlier in the work of Bernoulli and Euler? Why does the particle system in the Standard Model of elementary particles allow for a kind of scaling symmetry (the renormalization group), much as in statistical physics the partition function may scale with a

4.2

A Potted History

45

temperature-like variable (usually called k). What accounts for our ability to handle just one level of the energy hierarchy at a time (effective field theory and perturbation theory)? Why do analytic objects allow one to better understand arithmetic objects (the theta function and the system of primes)? Why do particles and primes, in their interactions (with other primes, with other particles), still yield to other objects that behave much as do primes and particles? And why are particles and number systems to be seen as irreducible linear representations (those matrices), or as groups that operate on something like vector spaces—in each case linearizing the system? And why do very different mathematical objects appear to have a relationship to each other? Together, they lead to an analogy of analogies—what is called a syzygy (a relation of relations), of the mathematics analogy with the mathematical physics analogy. What the mathematicians desire in great detail and specificity, the physicists appear to have achieved explicitly in a model system. What the physicists might like in an explanation of the diversity of the solutions to the Ising model is hinted at by the rather more general mathematical program, that Rosetta Stone. We might epitomize these considerations by rubrics: SYZYGY, the analogy of analogies (Weil’s Rosetta Stone); AUTOMORPHY OF ENUMERATION, where the primes are related to each other as in reciprocity, and the shape of the functions is recurrent; HIERARCHY AND EMERGENCE of incommensurable objects, and classifications that are perspicuous when you focus on the RIGHT OBJECTS; and MACHINERY, going from group representations to facts about the world.

4.2

A Potted History

In the nineteenth century, mathematicians figured out how to think about prime numbers when one enlarged the rational number system by including Gaussian integers, such as x + yi, where i is √–1 and x and y are integers, or by including irrationals such as √2, as in a + b√2. The discovery that prompted their concern was that a prime such as 3 actually factors if we add in √2 and √–1 to our number system: namely, 3 = (1 + i√–2) × (1–i√–2). Moreover, if we add in √5, for composite numbers such as 6 (= 2 × 3), there is no unique factorization, for (1 + √–5) × (1–√– 5) = 6. The question they answered was, How does a prime number in the original system then decompose, if it does, into “prime” factors (that is, the prime numbers in the enlarged number system), and then how to think about the restoration of the uniqueness of factorization by primes. Correspondingly, a physicist wants to understand how new particles appear in collisions of particles we already know as we raise the energy of those colliding particles, and how the new particles are both elementary and might be said to be composed of other particles. They want to restore the distinctiveness of elementarity and the distinctive properties that identify a particle. So developed notions of “ideal numbers” in that enlarged number system (to take care of uniqueness, one spoke of sets of numbers, “prime ideals”). As for the

46

4 Primes and Particles

equations that are the sources of the extensions, such as x2–2 = 0, one attends to the Galois groups attached to them that express the intrinsic symmetries of those equations (in their roots, so that for x2 + bx + c, whose roots are r1 and r2, b = r1 + r2 and c = r1 × r2, and the interchange of the roots leaves the equation unchanged) and so the solvability or not of those equations by an algebraic formula. In the nineteenth and twentieth centuries, physicists determined how to think about elementary particles if new forces were introduced that affected those particles. If we think of an array of springs, each spring (as a particle) might be characterized by its frequency of motion. However, if those springs interact with each other (because they are linked as in a mattress), the frequencies of motion of that mattress have shifted from the frequencies of the individuals’ motions to what are called the “normal modes” of vibration or the collective modes. How do we express those individual frequencies in terms of the new mattress frequencies? Moreover, they discovered that some of the time, most remarkably, a particle in interaction with external forces might become “dressed up” by the new forces (so it is now to be called a “quasiparticle”), having an altered mass but otherwise having the same spin and other such properties—as in the actual physical electron when compared with the “bare” electron, or an electron in a solid vs. an electron in a vacuum. The symmetries of the new forces are expressed in the new frequencies and dressings. (The mathematicians would say, “what can we say about the field K (here, by analogy, the interacting system) given what we know about the field k (the system of individuals), where K includes k and is an “extension” of k.) So, for example, and a preview and analogy to our concern about uniqueness of prime factorization: The K-zero particle, K0, and its antiparticle, the K-zero-bar, K0bar, have the same mass, but the right (or analogously “prime”) particles of a decaying K-zero system (one in which different forces are in effect than the forces that produced the K0s) are in fact closer to K01 and K02 defined by: K 0 þ K 0 bar = √2 K 0 1 and K 0 K 0 bar = √2 K 0 2 , each a quantum mechanical coherent sum. The K01 and K02, are known as CP eigenstates, CP meaning the charge conjugation and parity transformations. The K01 and K02 have very different lifetimes (due to their different symmetry under CP, +1 and -1) and so whether they can decay into two (+1) or three (-1) pions. In fact, these are not quite the right particles here, for one of the actual decaying particles, now known as the K0long, is an admixture of K02 and a small amount of K01 (that is, it is a mixture of the CP eigenstates K01 and K02). The other is K0short, an admixture of K01 and a small amount of K02. That is, their decays violate CP symmetry. So K0long can sometimes, ~10-3, decay into two pions. The masses of the K0long and K0short differ by a very small amount (unlike the equal mass of the antiparticles K0 and K0bar); they are not the antiparticles of each other. [Unfortunately, mathematicians use K to indicate an extension field, and physicists use K to designate a particle about 1000 times as heavy as an electron, a “strange” particle, the kaon, and K is also used to indicate the force between spins in the Ising model.]

4.3

Symmetry and Orderliness

47

In each case, symmetries we might discover in the original number system or original physical system determine how the original primes or particles now decompose into the natural objects (the primes, the comparatively stable albeit decaying particles) of the enlarged system. In the case of particles, we often only discover those symmetries when they appear full-blown. As far as I can tell, the strategies developed by the mathematicians and the strategies developed by the physicists did not influence each other. It is surely the case that the mathematicians knew a great deal about the mattress frequency problem: how to decompose a system into its fundamental frequencies and how to think about the shifts in those frequencies from their original values. In the 1850s, Riemann connected the prime numbers to a function that exhibited a scaling or automorphic symmetry (that is, looking the same at different scales or was of the same sort in transformations of the independent variable). Namely, Riemann created an arithmetic function that combined (“packaged” is the term of art) the integers and so the primes: the zeta function, ζ(s) = Σn 1/ns = Πover the primes (1– p–s)–1. He then showed that one could understand that package in terms of another function: the theta function, θ(z,τ). Theta was modular or automorphic (being of the same form): namely θ(0,–1/τ) is proportional to θ(0,τ), the proportionality factor, √– iτ, is called the “modulus.” Now, theta was employed by Fourier to account for the behavior of the temperature in a metal bar at very early and much later times after it is put in contact with a heat source. Namely, θ(z,τ) is like θ(z,1/τ): if t = iτ, θ(z,t/i) is like θ(z,i/t), the temperature in the metal bar is the same form in early and much later times [much as in high-school trigonometry, sin 2x = 2 sin x sin (π–x)]. Notably, we have gone from a function of the prime numbers to a function that describes the flow of heat. We know that that flow is due to random motions of molecules and so theta must tell us about a random walk. And theta’s Fourier expansion is in terms of the squares of the integers, actually q, q4, q9, . . . More generally, if f[(az + b)/(cz + d )] = (cz + d)k f(z), where ad–bc = 1 and a, b, c, and d are integers, f is called a modular function. While it was not so easy to compute zeta, it was straightforward to compute theta to any accuracy one wishes. Namely, there was some sort of regularity in the prime numbers that allowed one to think of the primes in terms of the symmetry or automorphy of a cousin of their packaging function and its Fourier decomposition. We have gone from the primes to automorphy to heat flow to a random walk to the squares of the integers! As for particles, as far as I am aware, there is no such path yet.

4.3

Symmetry and Orderliness

Symmetry and orderliness are opposites of each other. If an object is symmetric, say independent of the angle you examine it (it exhibits “isotropic” symmetry), it has no angular order. If an object is orderly, say exhibiting a preferred direction, it is not

48

4 Primes and Particles

symmetric angularly. Of course, that direction may describe another symmetry, say around that axis of the preferred direction. The world of particles or of primes is hierarchical: one in which different particles are distinguished at different levels of interaction energy, starting out with what we see every day in the protons and neutrons and electrons of ordinary matter, and then muons and pions and resonances; where number systems, that is, different fields of “primes” have more and more adjuncted numbers such as √–1 or √5, while starting out with the integers or the rational numbers. At each level of the hierarchy (in that level’s lowest or ground state or “vacuum,” that is), there are no new particles present yet, much as in algebraic number theory there are no nonrational numbers (such as √5 or √–1) present in the ground field, k, the rationals or the integers. The elementary objects (particles, integers, or rational numbers) that then appear have integrity and exhibit certain symmetries (or, conversely, orderlinesses). At the next level, that symmetry is broken and new orderliness is discerned. (So, water will freeze, giving up disorder and isotropic symmetry, becoming orderly and crystalline with the more limited crystalline symmetry). Or primes at one level prove composite at the next level, and one has to discern the right primes at that next level. So, the once elementary K0 and K0bar become K0long and K0short in the decaying system. What is desired is a theory that enables one to describe the symmetries present in one level that are now broken in the next level (or the other way around). The mathematicians say that they want to be able to describe how the primes of the field k behave in K, an extension of k, an extension say by √5 or √–1, using what we know about the properties of k alone (and what we know are congruences: a = bmod m, if b‐a = N × m, that is, “a is congruent to b modulo m,” where a, b, N, and m are integers). For the physicists, symmetries they do not pay attention to at one level are then revealed as they are broken at the next level—and ideally one would describe those symmetries at the lower level so that you can understand their being broken at that next level. Given a system that is rotationally symmetric, that rotational symmetry might be broken by an applied magnetic field and so new particles appear (here, energy levels and spectral lines, giving fine structure to the original line). What is crucial is that the different levels are in effect incommensurable: a transition takes place to a new symmetry, or new prime numbers are introduced, a symmetry that has no place in the previous level. We go from K0 and K0bar, to K01 and K02, to K0long and K0short. Although they may be defined in terms of each other, the physics they evidence is quite different, from CPT (for antiparticles), to CP for the charge-parity conservation, to CP violation. More generally, particles or states with the just right symmetries are particles that are elementary at that level. (“Right” needs to be explained.) For the mathematician, the ideals that allow for primes and unique factorization need to be described.

4.4

4.4

Coherence

49

Coherence

When we look at a list of prime numbers, there is no apparent orderliness, no rules that tell you how to find the next prime number,2 no principle that links, say, 11 with 17. Yet, 11 and 17, for example, are linked by so-called reciprocity laws that, at their most basic level, connect the congruence properties of the primes: namely, the properties of a prime, p, modulo a prime q, are directly related to the properties of q modulo p, suggesting that the primes are intimately connected to each other. They are called reciprocity laws reflecting the notation (pq), meaning the properties of a prime, p, modulo a prime q, since one is linking (pq) to (qp). The Riemann Hypothesis says that the zeros of the zeta function are at s = ½ + tni, suggesting all sorts of regularities in the primes. Empirically, by calculation, the Riemann Hypothesis seems to be very well supported by those calculated zeros. And finally, and remarkably, the spacing of the separations of the tn seems to be well accounted for, empirically, by the same function that accounts for the spacing of nuclear energy levels, the random matrix theory of Wigner. Again, the enumeration of the primes in Riemann’s zeta function can be connected to another function, that theta function, and that latter function is very orderly in the coefficients of its Fourier expansion.3 Namely, there is a coherence to the system of prime numbers that is not at all apparent but would seem to be quite deep. As for the elementary particles, the Standard Model of particle physics systematically orders the elementary particles by their properties (mass, spin, strangeness, . . .), each one belonging to a family, the families belonging to larger families—so there are families headed by electrons, by muons, and by tau particles. Historically, before the Standard Model was envisaged, various rules were discovered that linked particles into families defined by symmetries and suggested missing family members that were often then discovered. A century ago, the spectral lines of the elements were found to have regularities that would be encapsulated in what are called series (such as the Balmer series) rather than families, and rules were developed that linked the members of a series. Bohr’s quantum model of the atom suggested the source of those rules. It was Heisenberg’s matrix mechanics (which worked directly in terms of the frequencies of the spectral lines) and, a bit later the Schrodinger equation for quantum mechanics showed that the spectral lines of an element were a coherent 2

Yet, one added to the product of all the primes smaller than and including a prime p is prime. Personally, that I knew this fact got me into an advanced first year math class at Columbia. 3 Formally, ζ(s) = Σn,integers 1/ns which, by unique prime factorization, = Πover the primes (1 – p–s)– 1, which latter form is called an Euler product (here it may be useful to think of sin πx = π × Πn,integers (1–n2/x2), although not just the primes alone), is nicely connected to θ(z,q), which is Σ1 to 1 qn × n exp.(2πinz), where q = exp.(iπτ), τ being the imaginary ratio of the quasiperiods of θ. For solutions to the heat equation, the time, t, is set to iτ, and so the exponential falloff of the temperature in time. Namely, the primes (or the integers) packaged as in zeta, ζ, are connected to a function theta, θ, which has regular Fourier coefficients (qn × n), namely q, q4, q9, . . ., in effect packaging the squares of the integers, n × n.

50

4

Primes and Particles

system and how they were connected, not merely discrete independent objects. Quantum mechanics connected the specific features of an atom to those lines and that coherent system. In the background, a different sort of coherence is hinted at in the Planck radiation law of a black body (a heated element). To be elementary is context-dependent. At another level, there are objects that we might consider rather more elementary. In physics, that is, the Standard Model, the most elementary of the particles are quarks and leptons. In mathematics, the most elementary numbers are the integers.4

4.5

Decomposition

In number theory, the recurrent issue is how to restore unique prime factorization when, in an extended field, K, that uniqueness has failed. Which are the primes in K, and when and how do former primes from k (of which K is an extension) now decompose, if they do. In physics, the recurrent issue is to provide unique names for the particles or spectral lines when they initially appear. Under other forces their names or properties do not explain their behavior: likely their names are not unique and are shared with other particles (and if their masses or energies are identical, they are “degenerate states” under the original forces). For number theory, the algebraic number theory initiated by Dedekind (and Kummer, and later, the class field theory of Hilbert, Takagi, and Artin) provides a way of finding out how the primes in a number field k behave in the extended field K by examining congruences in k. That is, we can know about all about K from studies within k. The particle physicist is not so fortunate. Some of the time, we might have an isotropic system and turn on an electrical or magnetic field (so we have an “extended” system), and so we might be able to say what will happen in that extended system, how the spectral lines or particles will split, no longer so elementary. But for the most part, when we are within a system, say the Standard Model of particle physics, we have many speculations as to the new states and symmetries in the extended system, but we do not yet know which extension will turn out to be the

4 It has been suggested that there is the following analogy between the physics and the mathematics, each reflecting the overall coherence of the objects being studied, particles and primes:

1. Represent the elementary particles or a number system in terms of irreducible linear representations (matrices): representations of the Lorentz group (relativistic invariance), or representations of the absolute Galois group. 2. Then put those group representations through some Machinery: Quantum Field Theory and the Standard Model, or Reciprocity Laws. 3. And out comes some properties of the system being studied. See A. Ash and R. Gross, Fearless Symmetry: Exposing the Hidden Patterns of Numbers. Princeton University Press (Princeton NJ, 2008).

4.5

Decomposition

51

case. We can probe the original system and hope to see deviations from predictions of the Standard Model, deviations detected by behavior that is “new physics,” parametrized by symmetries beyond those of the Standard Model. Historically, some of the time, this strategy has worked wonderfully. When there were indications that something was peculiar in the kaon system (the “tau-theta puzzle”) as it (actually, the K+) decayed through the weak interaction, the force responsible for beta-radioactivity, careful theoretical speculation suggested that right-left parity might be violated in weak-interaction particle decays. And one could say what would be seen in such a case. That was seen both in the decay of radioactive nuclei and in the decay of the muon (as well as that of the kaon). Some of the time, again in the kaon system, we are totally surprised by results—seeing two pions when we expected three pions in the decay (K0long is not just K02), and then we have to backtrack in our understanding of the original system to see how an extension of the original theory would produce these phenomena.5 One other strategy available to the particle physicist is inverse scattering. One probes a particle and sees what comes out. From those outgoing particles, and given the nature of the incoming one, the physicist tries to understand the structure of the probed particle or of its force field. Perhaps this is not so different than the question asked in algebra, where one is given a quotient group, G/N, and N, and one wants to infer the nature of G. The problem here is to specify a G in terms of G/N and N.6

5

•

•

• 6

None of these methods have the nice specificity of congruences in the number theory case. Effective Field Theory: The intermediate states and loop diagrams are for all energies, including particles that are not directly probed, especially those of high energy and are heavy. Now, effects from high energy appear local when viewed from low energy, due to the Uncertainty Principle: ΔE Δt > h/2π, h is Planck’s constant, so that if the energy is high then Δt is small, so c × Δt is small. The effects are incorporated into the coupling constants we measure, or the effects are depressed by a power of the heavy-mass/energy (Donoghue, p. 119). So, at any level or energy, the other levels are present but are either phenomenological and so measured in the coupling constants such as the electric charge, or they are depressed. The renormalization group gives an account of integrating out the higher-energy extended effects. We might probe for high-energy effects by adding suitable Standard Model symmetry violating terms to the lagrangian, usually depressed by 1/Λ, Λ being the higher energy scale. Sometimes, you have a potential degeneracy, two particles with the same properties but different mass, and that may be a hint of what to look for. On the discovery of the muon (which was expected to be just a heavy electron), I. I. Rabi said: “Who ordered that?,” asking why should we have this object, what forces are we not recognizing? Actually, what we now call the muon was discovered earlier, and was thought to be the pion mediating Yukawa’s force. Only when the actual pion was seen and recognized as such, years later, was that earlier muon recognized for what it was. The muon does not interact strongly, while the pion does interact strongly (with nuclei for example). Or, you make a model, such as the rotating/vibrating nucleus, which then suggests particles (energy levels) that will be found, much as you have n, l, j for atoms. Such is the mathematicians’ cohomology.

52

4.6

4

Primes and Particles

Hierarchy

In both the mathematics and the physics, what is striking is that there are wellseparated levels in a hierarchy, much as clans and families are well-separated in many societies. In mathematics, there is, typically, a tower of fields, the higher members incorporating the lower ones, the base is the ground field, usually the rational numbers. In physics, the forces of nature described in the Standard Model are of very different strengths, and so their effects are of very different magnitude. The “strong” force, such as between quarks and gluons, is much stronger than electromagnetic forces among charged particles and photons, and these are somewhat stronger than are “weak” forces among leptons such as muons and neutrinos and W and Z particles, and these are very much stronger than are gravitational forces. Typically, forces lower down in the hierarchy (that is, much weaker) are treated as ignorable or as perturbations of those higher up. At each level in the hierarchy, the world is treated as if there were nothing there, the ground state or vacuum (which is made by the stronger forces), and then one sees the appearance of the relevant new objects or particles as energy becomes available. There are “effective field theories” that focus on one level, the effects of higher energy levels incorporated into their measured coupling constants, such as the electric charge, the effects of the lesser or equal energies being the focus. So, you can study the behavior of electrons among atoms, what is in effect all of chemistry, without worrying about the much weaker force of gravity, or the weak radioactive decay of particles, or on the other hand the strong or nuclear forces. We have some idea of the sources of the hierarchies of the number systems, namely the adjunction of more and more roots of prime numbers. However, we do not know why the forces of nature are so hierarchized and so different from each other, both in strength and in quantum numbers that characterize each force. We might get a hint from the behavior of ordinary condensed matter, which may vaporize into a gas, or freeze into a solid, or melt into a liquid. Many of these phase transition points are quite sharp in terms of temperature or pressure, so we might speak of the freezing point of water. We have come to understand such phase transitions as the collective effect of the interactions of all the atoms or molecules that compose that bulk matter, namely, the forces of random motion are defeated or not by those interactions, as the temperature changes (the random motions become stronger as the temperature is elevated), and it is the possibility of enough attractive forces among the molecules to defeat randomness. When there is just enough to defeat the random forces (now decreased since the temperature has been lowered), matter freezes. The freezing point is sharp largely because a very large number of molecules are coherently acting together. So, we might say that at a high enough energy, the various kinds of forces are equally strong, but as the energy becomes closer to the ones we have in our laboratories, the forces have in effect made a phase transition into distinct types and of very different strengths. What was roughly equal and indistinguishable at the Big Bang has in its ejecta become the hierarchy of forces we now see.

4.7

Adding-Up and Linearity

53

For another example, consider a solid at a very low temperature. While there is surely the quantum mechanical zero-point energy, the particles that then appear as the temperature rises, such as sound particles or phonons, can appear because they are vibrational modes of the orderliness of the crystalline lattice of the solid, and as the temperature rises there is energy available to produce them copiously. (That is, there are now vibrations of the lattice, linear springlike (or harmonic) and not-solinear as when the springs are stretched too far (or anharmonic), that in effect are the source of these particles.)

4.7

Adding-Up and Linearity

In both mathematics and physics, one is always looking for the “right” objects. Which are the primes in this number system? Which are the individuated objects in this physical system? The primes need to act like primes, indecomposable and constituting all the other objects. The individuated objects in their interactions with each other or with external forces account for the phenomena we see and compose all the other objects that are supposedly composed of them. So, most everyday phenomena are explained by some heavier particles (perhaps having a charge or a magnetic moment), protons, neutrons, and nuclei; lighter ones, electrons; and massless neutral particles, photons. With the right objects and the correct interactions among them, theorizing makes sense, and one can give an account of the world in terms of the kind and number of such objects and their interactions with each other. One is trying to take a complicated system and by discerning the right objects one is able to decompose the system or the objects linearly as a vector of properties in a larger vector space. One’s goal is to restore the right objects to center stage and to describe the world in terms of a “sum” of the various sorts of objects in various dimensions. Moreover, one wants such objects to have some sort of additivity: we can add integers, and, at low energies or low temperatures and so modest interactions, we can count the number of elementary particles, and in chemistry, we can surely count the number of electrons in a reaction. Physicists usually have some sort of conservation law, say it be of a charge or of an energy. In the case of charge, the conservation is expressed through a “gauge field,” that connects the local to the global, so what happens here is consistent with what is out there. Conservation of electric charge, ρ, is expressed through the electric field, E: for example, Gauss’s law, that the outflow of electric field (divergence of E) is proportional to the charge density, div E = 4π ρ. To top it off, it turns out that zeroes of these partition functions or zeta- or L-functions or quantum mechanical amplitudes may be significant. The zeros of the statistical mechanics partition function indicate a phase transition. The zeros of the zeta function and its relatives may tell us about the geometry of a space, and given the Riemann Hypothesis about the location of those zeros, we learn even more.

54

4 Primes and Particles

The zeros of the quantum mechanical amplitude indicate interference and the absence of a signal. Parenthetically, another interesting case of enumeration being so powerful is the “Weyl asymptotics,” which tell you that “you can hear the shape of a drumhead”— by counting the number of resonant tones of that drumhead, up to a certain frequency, and dividing by that maximum frequency, and so obtaining an estimate of the drum’s area (in effect, you are just counting cubes in phase space (of position and momentum, of x and p), one resonant tone or eigenvalue per cube). Keep in mind that in the case of the primes or of the interacting particles, we are in effect counting up all the various interactions. In such counting or enumeration, there was no obvious automorphy or functional equation. Yet, automorphy is ubiquitous in how we understand the primes and their roles in solving equations and in understanding how new forces or interactions affect what were once the elementary particles (leading to new more elementary ones).

4.8

Divisibility

By divisibility, I mean that we have found the right constituents that allow us to divide a complex system into nice components—those right objects that divide numbers but cannot be divided themselves (namely, the primes); those quarks that would appear to be indivisible (if you try to pull them apart they generate identical objects, quark-antiquark pairs, the energy from the color force of gluons becoming greater as you try to separate the original quarks to about 10-15 cm so allowing for the production of those pairs)—but which compose less elementary objects. We want families composed of well-defined individuals, those individuals having an enduring integrity. Again, the kaon system is archetypal, where the K0 and K0bar are antiparticles, but the right particles for studying the decay of kaons, the right divisors so to speak, are subtle unequal mixtures of the K0 and K0bar. We seek unique prime factorization or the right particles because we want to solve equations and provide physical theories that are rather simpler and more perspicuous, and those numbers and particles behave as we have come to expect of primes that are integers and particles that are indecomposable. The integers and the primes, the elementary particles, are each linked deeply and systemically to the other primes or to the other elementary particles, respectively. In our theorizing, we try to discern those deep and systemic linkages, and we want to be able to see the world beyond what we have now by examining that world we have now, examining the primes, examining the everyday elementary particles (the ones

4.8

Divisibility

55

we have access to in matter, as well as at an accelerator, and in astrophysical phenomena and cosmic rays).7 Earlier in this chapter I wrote: Why does enumeration of the prime numbers eventually lead to a nice function, employed by Fourier to understand heat flow, and earlier in Bernoulli and Euler? [There is a structure to the system of primes.] . . . .8 I suspect that these questions are improperly posed, at least for now. Rather than asking Why?, we need to continue to inquire about Just How? In detail, in a particular context, are these the ways of the world and of numbers? We are like Newton, contemplating the mystery of the creativity of an active Nature, magnesia, when what we know at best are the mechanisms and manifestations of those wonders. Perhaps we need to go well beyond the Standard Model of particles, or the known mathematics in number theory, for an overarching account. For example, George Mackey (1977, 1980, . . .) used harmonic analysis and representation theory as such an overarching account, albeit more historical than technical. Robert Langlands provided another overarching account, rather more technical, at roughly the same time. I am not claiming that Mackey or Langlands provide the right account, so much as the scale of ambition and overview likely needed to answer the above questions, if they have satisfactory answers.

7

Langlands’ program of connecting counting functions (“motivic” objects) with automorphic objects, tells us that enumeration and self-similarity are deeply connected. More speculatively, we are using analytic means for arithmetic purposes, much as we might solve a cubic equation using trigonometric functions. (If we let cos θ be equal to a function of the equation’s coefficients, the trigonometric identity cos(3θ) = 4cos3(θ) - 3cos(θ) will lead to the solutions of the cubic.) Or, we employ analytic means to do arithmetic work, using distinguished values of the exponential exp (2πi/n) or of elliptic functions, to extend the rationals to imaginary quadratic number fields (extensions otherwise achieved by adding in √–p, p being a prime)—what is called “Kronecker’s Jugendtraum.” 8 Or, for example one provided in a survey on Galois representations: algebraic varieties over the rationals, algebraic automorphic forms, and geometric representations of the Galois group over the rationals, where the connections are for the most part still conjectures.

Chapter 5

So Far and in Prospect

It may be helpful to collect in one place some of the themes and the physics we have recurrently referred to and will return to later.

5.1

Kinship and Particles

The anthropologists’ account of kinship is a story of allowed and forbidden relationships (marriage) and material exchanges, with the proviso that what is not forbidden will occur (what has been called plenitude). The physicists’ account of particle interactions is again a story of allowed and forbidden interactions as well as exchanges of intermediate particles, and again, what is not forbidden will occur as often as it might. A similar account is found in the chemists’ account of reactions, with the proviso that the system is usually at a particular temperature and that will affect the rate of reaction so that plenitude applies in this more restricted sense.

5.2

Primes and Particles

The mathematicians’ account of prime numbers indicates that if you add a number such as the square root of minus one (√–1 = i) to the integers, numbers that were primes might no longer be so: 5 = (2 + i) × (2–i) = 4 + 1. Algebraic number theory accounts for such effects, saying that if we examine features of the integers, we can predict how the primes will be affected if we add such nonintegers. The physicists’ story of elementary particles gives an account of whether a particle remains elementary or proves to be composite if the energy increases, “composite” so there might be ionization of an atom (into electrons and a nucleus or ion) if the temperature is raised, and whether there might appear new apparently elementary particles if the energy is raised (often called “discoveries”). More © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 M. H. Krieger, Primes and Particles, https://doi.org/10.1007/978-3-031-49776-6_5

57

58

5

So Far and in Prospect

generally for high energy particle physics, particles that might actually appear were the energy sufficiently large, may still make their presence felt at a lower energy in some process if they might act “virtually.” (In effect, the virtual particles live for a very short time, a time given by the Uncertainty Principle, time ~h/mass, where h is Planck’s constant. In that time, they make their presence felt.1) That is, the virtual particles are intermediates in that process and affect its rates. Rates calculated without assuming virtual particles lead to predictions denied by the experiment. Those intermediate processes, for example, contribute to the magnetic moment of the electron, and unless we include them, we will get an incorrect result in our calculation. Of course, we might not know what those virtual particles might be, but if we suggest they exist we can calculate how they would affect lower energy processes.

5.3

Effective Field Theory

When we study chemistry, we do not have to worry about elementary particle theory, since the characteristic energies of particle physics are much greater than the energies involved in chemical bonds—millions of electron-volts in most particle physics vs. an electron-volt in chemical reactions. More generally, when we are trying to understand some phenomenon, there is usually a characteristic energy. Phenomena with much greater characteristic energy are in effect hidden in the setup of the phenomenon, and those phenomena with characteristic energies that are much smaller do not make their appearance (we are insensitive to them in general) or their effects are small. We are able to effectively pay attention to the phenomena at hand, the others either hidden in the setup or much too small to be seen by us. On the other hand, sometimes a wide range of energy scales are relevant, and then we need to learn to pay attention to that range through a field theory that allows for scaling.

5.4

Packaging Functions Connecting Spectra to Surprising Symmetries

Perhaps the best-known packaging function, also called a generating function, is the series for sin x = x – x3/3! + . . . = Σodd integers (-1)n–1xn/n! in effect packaging the odd factorials, whose behavior as sin x is not at all apparent from the factorials it packages. Another such function is the Fourier transform which packages the frequency components of a signal into a message or perhaps some music.

If h is 6.6 × 10-34 kg m2/s, and an electron’s mass is about 9 × 10-31 kg (½ Mev/c2), and c is 3 × 108 m/s, then the time associated with an electron, h/(me c2) is about 10-20 s.

1

5.4

Packaging Functions Connecting Spectra to Surprising Symmetries

59

More generally, mathematicians and physicists package features (or a spectrum) of a system: a) the prime numbers as Π(1–pi–s)-1; b) the energies of different configurations of the molecules in a gas as Σ exp –βEi; c) the frequency spectrum of a drum sound; d) the Feynman sum of histories of particle interactions—into a packaging function: a) the zeta (ζ)- or L-function; b) the partition function (whose logarithm is proportional to the thermodynamic Free Energy); c) the Weyl asymptotics; d) the Schwinger Green’s function.

Those packaging functions have lovely properties (symmetries, for example) not at all apparent from what the package: a) zeta, ζ (s), can be shown to be the Fourier-like Mellin transform, M[θ] = 0,1 xs–1 θ(x) dx of the theta function, θ, θ describing the flow of heat, where symmetries of θ, manifest when we think of heat flow (not at all apparent when we think of the primes or ζ, lead to a relationship of ζ(1–s) to ζ(s); b) the interaction of molecules in a gas leads to the equation of state of an ideal gas; c) the area and circumference of the drum; d) and connections between different but now related interactions of particles), or connections between the interactions of particles with other particles and related interactions of their anti-particles.

5.4.1

Multiple Ways of Computing Packaging Functions, Revealing Other Symmetries in the Spectrum

There may be several distinct ways of computing a packaging function. The thermodynamic partition function of a two-dimensional lattice of magnetic spins (the Ising model) may be computed by adding up the right particles: the individual spins, by adding up the different strings of like-pointing spins, or by inferring the symmetries of that partition function and not doing any adding-up at all. We might label these ways as algebraic, arithmetic and combinatorial, and analytic, respectively. Each of these various distinct ways tells us different and distinct facts about the thermodynamic features of that lattice, not readily seen by the other ways. There is here an identity (that lattice) in the manifold of profiles or perspectives.

5.4.2

Algebraic, Arithmetic, Analytic: An Analogy of Analogies: Syzygies

The elliptic curves and elliptic functions, cousins of the circle/parabola/hyperbola (the conic sections) and the trigonometric functions, have algebraic (function theory), arithmetic, and analytic (geometry) aspects. It turns out that those different aspects are analogous to different ways of thinking of algebraic functions, the roots of polynomials with polynomial coefficients, ways understood by Dedekind and Weber in 1882: algebraically, arithmetically, and analytically. They showed how Riemann’s geometric and analytic way of thinking

60

5

So Far and in Prospect

of algebraic functions might be understood algebraically by modeling their account on the arithmetic account of algebraic numbers (roots of polynomials with integer coefficients). We have an analogy of analogies, what is called a syzygy. Andre Weil (1940) called this analogy a Rosetta Stone, with three columns: Algebra (Italian), Arithmetic (German), and Geometry (Riemann), the parenthesized terms referring to the language and nationality of the relevant mathematical school.2

5.5

The Right Particles or Parts

The “right” particles or parts make it comparatively easy to give an account of phenomena. If you choose the wrong parts, you are unlikely to be able to give any such account. In a system with many objects, the right particles need to be discovered—likely they are not those objects. In one way of computing the partition function for the lattice of magnetic spins, the right particles to add up are not the individual spins. Rather, the right particles (called quasiparticles) are pairs of patterned rows of spins. Similarly, superconductivity is best explained if we think of the electrons in a metal as (Cooper) pairs, with their spins canceling out.

5.5.1

Fermions

In chemistry, we learn that each of the electrons surrounding an atomic nucleus has a unique name (n, l, j, the quantum numbers of their orbits), describing its “path” around the nucleus. That is why the Periodic Table looks like it does. The electrons are arranged in shells, and within a shell, each electron has a unique name in addition to its shell number, n. We learn that chemical reactions often depend on electrons in the outer or valence shell. Moreover, in a solid such as a metal, again electrons have unique names and they cannot all bunch up (for then their names could not be unique). Rather, they are “pushed out” to higher energies so to speak, much as racers are distinguished by their race-times. On the other hand, the photons or light particles between the mirrors of a laser may well have the same names, so that the light is “coherent” and the intensity of the emitted light goes as the square of their number. Imagine worshippers in a congregation, singing together, all of whom are equally important. They can sing together and their prayer will be much louder than were we just to add up the sound of each of

2

Those distinct ways have parallels in the Langlands program in number theory connecting packaging functions derived from arithmetic to packaging functions derived from symmetries of functions (“automorphy”).

5.5

The Right Particles or Parts

61

their prayers when they are alone. (They stimulate each other to sing more strongly.) These are “bosons.” Or, think of elephants marching on a bridge, in time. Particles such as electrons and protons and neutrinos and quarks are fermions. Particles such as photons, W’s, Higgs, and gluons are bosons, and they transmit the forces between the fermions.

Chapter 6

Creation: When Something Appears Out of Nothing

Many physical phenomena would seem to appear at a point, much as The Creation happens in the Hebrew Bible. There is Nothing, what we call a vacuum. And then there is Something: orderliness, when there was symmetry; particular particles when there was none of that sort before. At a particular temperature, energy, magnetic field, when interesting things (begin to) happen: new energy levels or particles appear, ice defrosts, a smoothly flowing fluid shows signs of incipient turbulence, an electron beam at an accelerator starts to create new particles or there is an abrupt increase of cross-section (because new possibilities are available in the beam-target system). A vacuum then becomes occupied. The recurrent problem is to find a notion of such a point and such a pregnant vacuum that is just precise enough. I shall refer to that “at” as a point, but as we shall see, it is often not what is conventionally thought to be a point in mathematics, and the vacuum will often be populated with fluctuations that are part of Nothing. Moreover, at such a point there may be an in-between state, different from states characterized by points smaller or larger, and a new sort of vacuum. So, at the critical point of the Ising model, in that in-between, the lattice exhibits a symmetry that is isotropic, translationally invariant, and homothetic (a sort-of scaling symmetry), a “conformal” symmetry, and a new sort of vacuum. Or, put differently, the critical point is a coherent sum, (|+ > + |–>)/2, of states above and below, very different from either of them. On the other hand, at the freezing point of water, there is both ice and water, in varying proportions, and the in-between is perhaps not so exotic. What kinds of points and vacua and what formal definitions might the physicists need to provide? 1. Lim N→ or lim N → 1 N/Volume is constant or lim N → 1 and t (t = (T–Tc)/Tc) → 0 • Limit: “the infinite volume limit,” when the volume goes to infinity, but the density remains constant, or perhaps a limit as N goes to infinity and as well the temperature gets very close to the critical temperature (called the “scaling

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 M. H. Krieger, Primes and Particles, https://doi.org/10.1007/978-3-031-49776-6_6

63

64

6

Creation: When Something Appears Out of Nothing

regime”). Again, such a vacuum has symmetries very different than when one is not so close to the limit. That limit point may converge pointwise or uniformly, and in the latter case, one will not find a phase transition. The vacuum remains unchanged since on the way, the system is finite and does not have such an emergent discontinuity or phase transition state at the limit. 2. lim k → 1 Σakn, for a sum of positive numbers, ak, each say to the nth power • And we expect there to be zeros of that sum. That only becomes possible if n is a complex number, n + iδ, and we are in effect adding up vectors, ~exp iδ, and that sum may well “pinch” the real axis. So, to see a phase transition as a zero of the partition function, knowing that one is adding up positive numbers—that is, Boltzmann factors exp—βΕi—one adds a small imaginary part to the inverse temperature, so β becomes β + iδ, and eventually δ is to be set to zero. 3. (|+ > + |–>)/2 • A point might be at resonance, a coherent sum of states between just above the point’s value and just below. So Montroll described the critical point of the Ising model. Again, the vacuum is very different than either the |+ > or the |–>) states: it is their coherent sum. 4. Fluctuations vs. Damping, Pressure in a bubble vs. Pressure in the water. • A point might be just when fluctuations start to appear, their onset. Almost always, there are fluctuations below (or above) that point, so we have to say it is a point when fluctuations actually grow rather than be damped and shrink back to nothing. There is a conflict between the sources of the fluctuation and the environment’s capacity to tame the fluctuation. • In turbulence, what seems to occur is that at such a point (say, at a sufficiently high Reynold’s number) the fluctuations do not die out; rather, they grow and thrive. In fact, that is not so much a point as a range of values, with probabilities assigned to each value that there will be an onset of turbulence. Water on the way to boiling will have small bubbles until at the boiling point there is a robust boil, so to speak. In effect, most of the small bubbles below the boiling point are defeated by the water pressure around them. • The momentum p of the target particles has a width Δp (perhaps as in the Uncertainty Principle, h/2πΔx, or perhaps due to thermal motion). Or, a vacuum may be populated with fluctuations, zero-point quantum-mechanical fluctuations, or in a metal with electrons near the Fermi level, or quark momenta inside a proton, so that interactions may take place even if there would seem to be insufficient energy, the extra energy provided by the target particles’ latent momentum. We may want to distinguish fluctuations that are internal, as are virtual particles in quantum field theory, from fluctuations that are actual, as in the statistical mechanics of a gas.

6

Creation: When Something Appears Out of Nothing

65

• Unstable elementary particles will have a mass width rather than an exact mass, a Breit-Wigner distribution, as in resonances and particles that will decay. They can have a range of energies, the width, and just when they decay and the mass we measure is a random process. 5. A point when new phenomena appear. • And it is those phenomena that we attend to. The vacuum has new symmetries. The onset of order in an Ising lattice would seem to be sharp at the critical temperature, but that onset would appear less sharp were we to attend to a range of cluster sizes. If we have a finite lattice, the critical temperature will depend on the lattice size. That order may be indicated by an interface, clustering, or both, but the manifest indicator of the point is when the specific heat becomes “infinite” or when the spontaneous magnetization begins to rise. In percolation, the crossing probability goes from zero to one, abruptly it would seem, but only for an infinite lattice. For a finite lattice, the onset is not quite so sharp, and the critical crossing probability is not sharp and is more like a logistic. There will be crossings some of the time, even when there is a lower than the critical probability, although such crossings are more infrequent as the lattice becomes much larger. 6. Scaling appears (say, a critical exponent α) • The system would seem to appear the same at a very wide range of scales. Again, the vacuum’s symmetry seems to change. 7. A surface is planar • But as the temperature rises the surface roughens; molecules randomly escape their “proper” place and land elsewhere. Or, conversely, say a lattice has interactions that are not only local but also link to next-to-nearest neighbors. It turns out that at the critical point, the crossover matters little and the lattice is apparently planar. Here a vacuum emerges. 8. An infinite specific heat • If we have a system of particles, an “infinity” of them, then perhaps at the critical point their energies are small or zero, ε or 0. The specific heat can be infinite, since “all” the degrees of freedom are then excited, that is exp—βε = 1– βε ~ 1. By equipartition, each mode or degree of freedom has energy kBT/2. 9. A singularity which is a point seen in retrospect • When a star burns up all its nuclear sources of energy, self-gravitation is still present. The star may collapse onto itself (since now there is nothing to stop gravity’s force), and depending on what halts that collapse, the star becomes a white dwarf, a neutron star, or a black hole. Were we to know the star’s mass and composition exactly, we might be able to point to a moment when it will collapse. More likely, fluctuations trigger the collapse. A black hole is

66

6

Creation: When Something Appears Out of Nothing

known by its mass, angular momentum, and charge, and it is in effect a blackbody at 10-9 degrees Kelvin, much colder than the cosmic blackbody radiation of about 3 degrees. Having become a blackbody, it emits radiation and slowly, very slowly, disappears. For a supernova, the point when it would collapse would seem to be quite sharp but unlikely to be predictable. 10. A point when new phenomena appear in a specific form • At the critical point in an Ising lattice, its spontaneous magnetization is small but grows as |t|1/8, where t = (T–Tc)/Tc. For a finite-sized lattice, the specific heat maximum grows as approximately ½ ln N + 0.21, while the critical point is shifted by approximately Tc /N, where N is the size of the lattice. One wants to show that the singularity of the specific heat and the onset of spontaneous magnetization occur at the same temperature.

6.1

Points

Each of the above points is distinctive, conceptually and often practically. In actual experiments, physicists know that a point is never that exact, dependent as it is on the size of the sample, and just how close we might get to just that point in our setup. (Actually, we know we are there when new phenomena begin to appear.) They endeavor to set up their experiment so that the sample is large enough, say, and the temperature is as close as they can make it to the point’s temperature or other parameter (such as beam energy). The sample being less than infinite in size but quite large and the temperature not being perfectly controlled should not make too much of a difference in what they see if they are well within a macroscopic regime. They may need to scan the parameter to find the maximum. However, the point is presumably a point, at least in their models and calculations for large samples, that is, the “infinite volume limit.” More generally, they need mathematical notions of a point or a surface, say, to do their calculations, at least at first. The actual sort-of point is a provisional mathematical point. As is the case for the vacuum. Now, it may be that a phenomenon, such as permanent magnetization, begins to appear at a temperature point and grows and continues to appear at lower temperatures. Does it appear immediately, or does it slowly appear as one moves away from that point? If our theory uses as its model an immediate appearance, we might imagine magnifying our theoretical lens and seeing that in actuality the phenomena appear incrementally, a curve with a finite slope to start with (or second derivative, or perhaps as ~|t|1/8).1

1

Maybe that sharp moment of appearance is actually a matter of our cutting through a space, so that a smooth curve in space is cut and one sees a sharp point. So, a “catastrophe” depends on our reducing the dimension of our space. Conversely, a sharp intersection point might be “blown up” to

6.2

Vacua

67

That is, we almost always believe there is a microscopic source of these macroscopic phenomena, but we also believe that our setup has washed over that microscopic detail.

6.2

Vacua

We have a notion of Nothing, from which Something appears. Again, that appearance might be gradual or abrupt. That Nothing is called a physical vacuum. Whether it is due to quantum effects or thermal effects, there are unavoidable fluctuations. In general, those fluctuations do not grow, but some of the time they do. The closer we are to the point, the more likely they are to grow. Quantum mechanically, we speak of virtual particles (fluctuations, particles that appear for times comparable to say 10-22 s for strongly interacting particles, perhaps 10-10 s for weakly interacting particles). In fact, the fluctuations will alter what we measure, so we find the remarkable power of quantum electrodynamics in providing perhaps ten digits of corrections to some points. At the energy threshold for an interaction, when the interaction should not occur if the incoming particle has less than a minimal energy, we find that just below that minimal energy sometimes interactions will occur, and above that threshold, interactions become plentiful. In scattering electrons or muons off protons, if the electrons or muons have just enough energy, they will “allow” for the appearance of the next heavier quark (say from the light quarks to the next heavier quark, the charm quark), and there is then a suitable jump in the cross-section. In effect, the vacuum has transformed. Particles or states that are otherwise forbidden start to appear at a threshold, at first in small quantities and then copiously. Again, the specific heat of bulk matter will become larger as more degrees of freedom are excited: as the temperature is raised to a critical point and the states’ energies become small at that point, their probability of excitation is proportional to exp‐Ei =kB T:

have a more continuous structure, much as we might loosen a knot. (So, the mathematician can do “resolution of singularities.”)

68

6.3

6

Creation: When Something Appears Out of Nothing

Mathematical Sleight of Hand: So to Speak

Mathematically, the problem is how, formally, to describe a Nothing which in fact can have Somethings in it, at first “virtually” and rarely but influentially, and eventually numerous and highly structured. In effect, how is one to mathematically define points and fluctuations in such a way that they behave physically. Physicists do such defining as a matter of course in their work. The challenge is even more demanding. In effect, one notes that fluctuations also cause dissipation, so that there is a measurable consequence of those fluctuations, whether it be particle interactions below threshold, or residual viscosity in an apparently ivicid fluid. Specifically, such as “anomalous dissipation.” If we think of a viscous fluid whose viscosity is then slowly driven to zero (say, by changing its composition), we find that even at zero viscosity there is dissipation. Namely, the avenues to dissipation fully in force when the viscosity was larger still allow for dissipation when the viscosity has been reduced to zero. Put differently, particle interactions take place even when it would seem that there is insufficient energy available—except when there are energy fluctuations. Fluid dynamicists say that there is “spontaneous stochasticity,” in effect fluctuations, that leads to anomalous dissipation. In turbulence, there are nonunique solutions (or flows), so we cannot go backward in time to the originating trigger. (And, particles are separated ~ (time)3, independent of where they started out. That time asymmetry is what we mean by dissipation.) The empty vacuum, that Nothing, would seem to have structure left over from when it was occupied by excitations or particles; it has a latent structure that is then revealed by particles or modes of dissipation that appear in that vacuum.

6.4

Points, Again

We need models of such points and vacua (Nothings), models that allow for fluctuations and some sort of apparent “memory” or hysteresis. In the course of their work, physicists create such models. Again, it is of physical interest whether a point is punctiform or just an arbitrary designation in a process. Each of the points I have sketched above may be seen in either way. More generally, one is giving an account of how Something might appear in Nothing, where fluctuations are ever-present.2 Curiously, in the social sciences and the humanities, these points when Something and Nothing are both there, are modeled all the time. An object is “transitional,” both real and imaginary, both-and, liminal. A conversion experience is apparently instantaneous, as is related by Augustine or Paul, but the historian tells I should note that mathematicians allow for a notion of 0 + ε as a genuine number, but as far as I can see this is not to the point here (but it is relevant in describing the calculus ala Leibniz).

2

6.4

Points, Again

69

a rather more back-and-forth account, only retrospectively made punctual. When we are uncertain—when we do not know what we do not know, a decision cannot be a matter of probabilities and costs vs. benefits, with probabilities attached, or even be strategic in that it has a higher goal that demands sacrifice. Rather, we must act first and think later, that initial action not being arbitrary but informed by our experience. And we may be tragically mistaken. And much of what we call virginal nature is actually designed. National Parks are invented experiences, rather guided and pruned, or second or third growths, for the initial growth has been destroyed by natural disasters and human habitation.

Chapter 7

Packaging “Spectra” (as in Partition Functions and L/ζ-Functions) to Reveal Symmetries in Nature and in Numbers

ΝΟΤΕ: Symmetry means that some feature does not matter, so circular symmetry means that the compass direction does not matter. If that symmetry is broken or violated, then the system is now ordered; in this case, it points in a particular direction. Note that if it points in a particular direction, it may be said to be symmetric if the system is rotated around that compass direction (but note that the rotation cuts through the plane of the compass).

7.1

Geometry and Harmony

The world as we see and experience it might be said to be geometrical (recall Galileo). In about 1860, earlier actually in Faraday and contemporaneously in Thompson and Tait, Maxwell understood this to say that to understand geometry is to understand the forces of nature. Here, we might think of a magnet and its lines of force revealed by iron filings. There is a corresponding harmony of nature so that shape determines sound, here think of a musical instrument, the shape and size of a drum determining its sound spectrum. Conversely, those harmonies, when considered as a whole, point to their source (say, its shape and structure), and any orderliness we find in that harmony tells about the deep features of that source. Atomic spectra, revealed when we heat up matter to a gas, tell us about the structure of the atoms that compose that matter. We might think of that harmony as a package of information about the source, and often features of that package (say symmetries) tell us otherwise less apparent features of the source. We may discern other spectra as a whole, such as the prime numbers pointing to their source and its structure, and elementary particles (and their properties) pointing to the fundamental forces of nature and their structure. Or, again, considering the

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 M. H. Krieger, Primes and Particles, https://doi.org/10.1007/978-3-031-49776-6_7

71

7 Packaging “Spectra” (as in Partition Functions and L/ζ-Functions). . .

72

sound of a drum, we might package those sounds into a function that will tell us about its shape. Those structures have symmetries, shapes so to speak, not at all manifest from the individual objects, the primes, the particles, or the tones—those symmetries are in effect emergent from considering the harmony as a whole.

7.2

Parts and the Right Parts

A bit of matter is composed of atoms and molecules, much as a number is a product of the prime numbers (12 = 2 × 2 × 3). It is essential that we have the right parts, so the composition makes sense and is useful (to think with). So, we might decompose something into its more elementary parts (and those parts may then be decomposed). We shall discern the rules of composition, whether they are additive, one-by-one, or multiplicative, or the more complex rules of musical composition. Or, they may be parts that might be seen as components of the whole, where those parts may be parts of an automobile or organs of an animal. Namely, there are parts that compose a more complex whole, and those parts are themselves composed of even more elementary parts. Each part on its own scale has the integrity of an individual, a carburetor, or an ingredient in a recipe. In general, larger more encompassing parts are more specific than their components. They are in effect more ordered, but they are more symmetrical than the things they compose.

7.3

Plenitude

Given a set of parts (numbers, atoms, people) there are rules for how they are to be put together, and regulations forbidding some combinations. Given those rules and regulations, any combination of parts is allowed and is instantiated. Hence, the variety of recipes in a cookbook, the chemicals we see in nature, the organization of a society in terms of its marriage and kinship rules or in terms of its economy (what is exchangeable for what). By the way, we might not explicitly appreciate those rules of composition. We discover more about those rules when new compositions prove successful or are drastically wrong. We may then discern what the rules allow and what they forbid. For example, many notional schemes of power production violate the conservation of energy, and they are shown to be impractical.

7.5

7.4

Layers

73

Manifold Perspectives or Profiles

There is value in having diverse ways of thinking about nature, even if in the end they come to the same result. One way may build in orderliness or symmetry in an explicit way. Another way is to attend in detail to the components of what you are studying. So that same result may be shown to have orderliness built-in, or it may be shown how the different parts work together to get that same result. Again, we can package the prime number, pi, into a function ζ (s) = Π (1– pi–s)-1. We can show that that function (of s), packaging the primes, is related (through a Laplace-like Mellin transform) to another function θ, which plays a role in understanding the flow of heat, where the primes would seem to play no role. (And the transform would seem not to introduce the primes behind our backs.) Now, θ has all sorts of lovely symmetries that make sense when we think of heat flow. (And in thinking of heat flow, we realize that it takes place by molecules colliding with each other, what is called a random walk, as a model of diffusion of heat, and that introduces new symmetries into θ.)

7.5

Layers

We might have a tower of objects, lower objects less complex than higher ones. It would be nice to be able to say something about a higher object by looking at just the lower objects. Given what we know of the elementary particles, might we be able to say something about particles that might appear if the energy were much higher? Given what we know about integers and primes, might be able to say what would happen if we added in the square root of minus one (=i)? (Again, some primes would no longer be prime (2 = (1 + i) × (1–i) = 1 × 1 – i × i = 1 + 1), and perhaps there would be new primes.) To repeat: Another layer-like phenomenon: When we study chemistry, we never have to worry about elementary particle theory, since the characteristic energies in particle physics are much greater than the energies involved in chemical bonds— million of electron-volts vs. one electron-volt. More generally, when we are trying to understand some phenomenon, there is usually a characteristic energy of that phenomenon. Phenomena with much greater characteristic energy are in effect hidden in the setup of the phenomenon, and those phenomena with characteristic energies that are much much smaller do not make their appearance (we are insensitive to them in general). We are able to effectively pay attention to the phenomena at hand, the others either hidden in the setup or too small to be seen by us. On the other hand, sometimes a wide range of energy scales are relevant, and then we need to learn to pay attention to that range through a theory that allows for scaling.

74

7.6

7

Packaging “Spectra” (as in Partition Functions and L/ζ-Functions). . .

Fermions

To repeat section E1 in the previous chapter: In chemistry, we learn that each of the electrons surrounding an atomic nucleus has a unique name (n, l, j, the quantum numbers of its orbit), describing its “path” around the nucleus. That is why the Periodic Table looks like it does. The electrons are arranged in shells, and within a shell each electron has a unique name besides its shell number. We learn that chemical reactions often depend on electrons in the outer or valence shell. Moreover, in a solid such as a metal, again electrons have unique names and they cannot all bunch up (for then their names could not be unique). Rather, they are “pushed out” to higher energies so to speak, much as racers are distinguished by their race times. On the other hand, the photons or light particles between the mirrors of a laser may well have the same names, so that the light is “coherent” and the intensity of the emitted light goes as the square of their number. They may march in step. Imagine worshippers in a congregation, singing together, all of whom are equally important. If they attend to harmony, their prayer will be much louder than were we just to add up each of their prayers when they are alone. (Moreover, they stimulate each other to sing more strongly.) These are “bosons.” Particles such as electrons and protons and neutrinos and quarks are fermions and have a spin 1/2. Particles such as photons, W’s, Higgs, and gluons are bosons, have spin 0 or 1, and they transmit the force between the fermions.

7.7

A Concrete Realization of the Dedekind-Weber Program

Again, in the last 80 years, physicists have been computing the properties of a two-dimensional lattice of magnetically interacting spins (The Ising model), as a classical system, employing a variety of methods. Their goal is to obtain the partition function, PF = Σ exp. –βEi where β is an inverse temperature and Ei is the energy of the lattice for a particular configuration of spins, and the sum is over all such configurations. The thermodynamic free energy, from which the various physical properties of interest might be readily computed, is (-1/β) × ln PF. There are at least three methods of calculating that same object, the partition function, and of course they must come to the same conclusion, with moments of recognition of an aspect of one calculation mirrored in another. The physicists’ computations instantiate an analogy of analogies, what is called a syzygy. One method of computing the partition function of the lattice focuses on the thermal symmetries, such as PF(k) ~ PF(1/k), an automorphy or retention of form, where k, here, is a temperature-like variable. A second method is essentially algebraic, diagonalizing a Hamiltonian, in the end finding the right particles or excitations of the lattice (patterned rows of spins), while along the way noting that the k-symmetry points to algebraic commutation rules. And a third method might be

7.8

Another Multiplicity

75

called arithmetic, finding systematic ways of adding up the contribution of the individual spins for each configuration, in effect a topology of paths. Keep in mind that the analogy in the mathematician’s case is a very different sort of analogy than in the physicist’s, for one is an analogy of theories, and the other is an analogy of methods of calculation (although the theories behind each method might be said to be what is analogized). Fortunately(!), all methods get the same result. But each method provided insights that were more difficult to infer through the other methods: automorphy, the right particles and a group representation mirroring automorphy, and an analogy with the sum-of-histories formulation of quantum mechanics.1 Mathematicians working in this tradition might be encouraged to recognize the Ising model solutions as a model for their work—as an analogy to their analogy. As for the physicists, the mathematicians’ analogy gives coherence to their diverse methods. It is not by chance that there are the three methods, rather they would seem to be more deeply connected (not only in the Ising model, but more generally).

7.8

Another Multiplicity

Decisions are sometimes transformative, and their value is difficult to anticipate, we say they are “big.” There is, as well, uncertainty, where we do not know what we do not know, yet we must act. Acting has consequences that are informative about what to do next, we are never at rest, and sacrifice is the occasion for invention and for revelation of our situation’s otherwise hidden aspects. If we make careful descriptions of our situation, we will reveal what has been unsaid and hidden. The trick is not to depend on a single description but rather to have a multiplicity of such descriptions from various perspectives or methods so that we understand our situation as a whole. We will run into blockages, which in their totality reveal further structural features of our situation. We might think of our situation as a balance of forces—what holds it together, what dissipates it. We might call the most concentrating force gravity, only to be stopped by what is intrinsically incompatible (“fermions” cannot be squeezed together too much). There are facticities that cannot be denied. The world having been so compressed—think of a massive hot star—there will be an explosion, a supernova, the ejecta providing surprises. (In this case, heavier elements.)

1

More generally, elliptic curves and elliptic functions exhibit the same analogy—having analytic/ geometric, algebraic, and arithmetic aspects, intimately connected since they are presumably about the same object.

76

7

Packaging “Spectra” (as in Partition Functions and L/ζ-Functions). . .

There is surprise, multiplicity, and identity, so there is coherence, but there is resistance and blockage as well. If you choose the right individuals, you will find lovely packages, often indicating levels of compositeness of the components of that level. What is remarkable is that we may discern an order among the objects not at all apparent in their individuality, in effect a legacy of the package.

Chapter 8

Legerdemain in Mathematical Physics: Structure, “Tricks,” and Lacunae in Derivations of the Partition Function of the Two-Dimensional Ising Model and in Proofs of The Stability of Matter

Many mathematical or physical paper would seem to magically go from one line to the next, the reader unable to figure out the logic of the transition. Such legerdemain, whether it be magical or in doing physics, is no less impressive if you know how it is done, for you yourself would have to train extensively to actually perform these sleights of hand. What you might have ignored turns out to have needed informed careful attention. Yet, to be struck by legerdemain you must have actually read the paper, so that the device or method would stop you cold. Where did that come from? How do you get from line A to line A + 1? (I should note that much of the discussion below will benefit from having those papers in front of you.1) To recall, mathematical physics employs rigorous mathematics to develop physical theories and to solve physical problems. Those solutions may not be exact and only provide approximations (but with rigorously derived errors). In any case, in the process of doing the mathematical work, one often discovers conditions needed to make the mathematics do the needed work and those conditions are often of physical significance. On the other hand, one may provide an exact solution, but not having done a careful job with your deltas and epsilons, or with your claim of analytic continuation, the solution is not mathematically rigorous. (Some of Baxter’s exactly solved models are not derived rigorously. In Kaufman and Onsager’s, 1949, solution for spontaneous magnetization, they did not publish the results since Onsager did not know how “to fill out the holes in the mathematics—the epsilons and deltas” concerning limits of Toeplitz matrices.2 But Onsager did reveal his answer.) 1

For diagrams of such, see Doing Mathematics (2015), Figs. 4.4–4.6, pp. 137–138; Figs. 4.7–4.9, 150–152; Fig. 4.13, 175; and, Fig 4.18, 197. See the Appendix to Chap. 1. 2 Onsager never published his derivation of this result, although in 1949 he and Kaufman produced a paper on the short-range order or, more correctly, on the set of pair-correlation functions of the square lattice Ising model. It was left to C.-N. Yang to derive the result independently. Only 20 years later, at the Battelle Symposium in Gstaad, did Onsager reveal fully that in computing the long-range order he had been led to a general consideration of Toeplitz matrices and determinants © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 M. H. Krieger, Primes and Particles, https://doi.org/10.1007/978-3-031-49776-6_8

77

78

8

8.1

Legerdemain in Mathematical Physics: Structure, “Tricks,” and Lacunae. . .

Examples

Here I shall again be concerned with two examples of mathematical physics. Solutions for the partition function of the two-dimensional Ising model and proofs of the stability of matter: The Ising Model 1. 2. 3. 4. 5. 6. 7.

The meaning of the numbers (Ising partition function, numerically) An amazing invention (Yang’s commuting plane rotation) Employing a device of the past (Jordan-Wigner transformation) Where did that come from (WMTB ending up with PIII) “A Useful Identity, Easily Seen” Signposting the along the way (Baxter) “Further Details of Simplifications Like This Will Not Be Reported”

The Stability of Matter 1. “Hacking Through a Forest Of Inequalities” (Dyson-Lenard). 2. “Thomas-Fermi Atoms Do Not Bind” (Lieb-Thirring). 3. “An elementary identity, Fourier analysts are quite familiar with it. Gruesome details, nasty and ghastly calculations, modulo oversimplifications, applying general nonsense” (Fefferman). Again, in reading a paper, it is not unusual to find that the transition from line A to line A + 1 is not clear or obvious, at least from what the author says. Eventually, you figure it out or trust the author so you can get on with the rest of the argument or proof. Or, a curious and surprising object is defined, with perhaps little motivation provided at that point. As for the object, its effectiveness may be revealed further down in the paper, or perhaps you have seen it employed elsewhere by that author or in other work. Moreover, the structure of the argument or proof may seem so unmotivated, except that it does the work, and you wonder where it came from. One suspects that that structure is discovered in the proving, and then the proof is presented in a much more lovely fashion than the author originally followed, yet the meaning of that structure, other than it serves the exposition, is elusive.3 Elsewhere (for which see the Appendix to Chap. 1), I have been concerned with the overall structure of the papers, rather than the line-by-line transitions and the but did not know how “to fill out the holes in the mathematics—the epsilons and the deltas.” By the time he had done this, he found that “the mathematicians” had got there first—although, in fact, the generality and depth of Onsager’s results were not matched for many years. 3 As for the structure of the argument, I have elsewhere (see my Doing Mathematics, edn. 2 (DM) and The Constitutions of Matter (CM)) described the structure of some of these papers. See also the Appendix to Chap. 1.

8.2

The Two-Dimensional Ising Model

79

legerdemain at various points that allow the paper to actually work. Usually, the authors tell you about the structure, early on in the paper, so you can follow their argument, but what they tell may not be explicit enough, nor accompanied with a diagram, say, so that you actually see that structure. What might seem like an argument driven by line-by-line inference is actually part of a strategy and structure that not only makes strategic sense but also makes physical sense. That may be a matter of realizing the essential physics of the problem, or perhaps the proving leads to the author’s realizing the essential physics of the problem and how to present the proof in a strategic fashion.

8.2

The Two-Dimensional Ising Model

Computing the partition function or sum over states is a problem in classical statistical mechanics, the technical problem being to enumerate the possible states and their energies in an automatic a way as possible: algebraically, combinatorially, symmetrically, or graph theoretically.

8.2.1

The Meaning of the Numbers

It is sometimes useful to get into the inside baseball, even when we have fine formula solutions to problems. By “inside baseball” I mean the quantitative, numerical, details—and noting how they might be illuminating. Lee and Yang, 1952, showed that at the critical point, in the infinite volume limit, the grand partition function of statistical mechanics has a zero (Grand Partition Function = Σ exp – β(μNi – Ei) = exp –βG, Ei is energy, N is number of particles, S is entropy, and μ is fugacity, G being the grand potential). But to see the approach to that zero, one had to compute the relevant part of the grand partition function for a finite number of particles, as the number of particles goes to infinity, or to compute that same part of the GPF at the infinite particle level for values of the interaction near the critical point. Here, the lacunae hides the legerdemain, legerdemain which is achieved so-tospeak mechanically. If we examine exact solutions to the Ising model in two dimensions, for ln PF, and the correct formal and numerical calculations for a finite-size lattice, and then for an infinite lattice as approximated by integrals, we will uncover more of what is going on in these calculations, their actual physics rather than their merely being formal or mathematical. 1. The two parts of the differing formulas for the ln PF split into a constant and a sum, the sum representing a high-degree polynomial. 2. Yet all must come to the same number, and so we see how one addend makes up for the other (except in Baxter, where they are not obviously separated).

80

8

Legerdemain in Mathematical Physics: Structure, “Tricks,” and Lacunae. . .

3. The different formulas have different ways of adding up the relevant physical objects that make up the Ising lattice, the quasiparticles’ energies. Onsager and McCoy/Wu have the same objects it would seem (albeit of different energy scales). 4. The formulas actually produce the correct high and low-temperature values for lnPF, as we might hope. 5. The numerical approximations are in fact physical, taking into account an increasing number of spins in the Ising lattice. The sums are sums of quasiparticle-free energies. 6. Lee and Yang teach us that the partition function should have a zero at the critical point, but as they point out, that is only seen in the polynomial factor and only in the infinite-N limit. For it is only then that a monic polynomial with all positive coefficients could have a real zero. The formulas below from Onsager are for the ln PF, the logarithm of the partition function (per spin) for the two-dimensional Ising model with the same coupling horizontally and vertically, K. At the critical point, K = Kc = 0.440687. . . (sinh22Kc = 1). The first formula is exact to the relevant sums, which Onsager necessarily provides, for those sums are what Onsager actually calculates. The integrals are for an infinite number of particles. (Note that exp 0 to π/2 ln is a geometric mean of the integrand.) Onsager 1 n lnð2sinh 2K Þ 2 π ð2r–1Þ 1 acosh cosh 2K cosh 2K * – cos ] þ 2n 2 r, 1 to n

for n2 spins

Onsager 2 ½ lnð2sinh 2K Þ þ

1 2π

π

γ ðωÞdω, per spin, where sinh 2K *

0

¼ 1= sinh 2K andγ ðωÞ ¼ acoshðcosh 2K cosh 2K * – cos ωÞ, Onsager calls the integrand γ(ω), referring to an angle ω in a hyperbolic triangle, whose sides are 2 K and 2 K*, γ(ω) being the third side and opposite the angle ω. He then explains the integrand’s behavior in terms of the triangle. Schultz, Mattis, and Lieb indicate that these γ(ω) are energies of quasiparticles, in effect orderly rows. The integrand for the Onsager formula may be seen to be the energies (in kBT units) of the spins that make up the lattice.4 4

Hurst points out that, in particle physics, at the singular point, a meeting of two singularities, there is the threshold for the appearance of particles. At the critical point the quasi-particles’ energies begin at zero, allowing for an infinite specific heat.

8.2

The Two-Dimensional Ising Model

81

K Fig. 8.1 ln PF/per spin vs. K

If we plot ln PF/per spin vs. K, we expect ln 2 (=0.69) for small K and 2 K for large K at high and low temperatures, respectively. That is, at high temperatures up and down spin directions are equally likely, while at low temperatures there should be alignment (all up or all down) (Figs. 8.1 and 8.2). As for Baxter and McCoy/Wu, Baxter 1 1 2

1 2p

1 ln 2 cosh 2K þ k 2

j,1 to 2p

π j– 12 1 þ k –2k cos 2p

1 2

2

,

per spin, where k ¼ 1= sinh 2 2K and the lattice has m rows of 2p sites Baxter 2 ð1=2πÞ

0 to π

FðωÞ dω = ð1=2πÞ

ln 0 to π

2 cosh 2 2K þ ð1=k Þ 1 þ k 2 –2k cos 2ω

1=2

Þ

where k = 1= sinh 2 2K: McCoy/Wu ln √2 cosh 2 K þ ð1=πÞ = 2 sinh2K= cosh 2 2 K

0 to π=2

ln 1 þ √ 1–κ2 sin 2 ω

dω, where κ

dω,

8 Legerdemain in Mathematical Physics: Structure, “Tricks,” and Lacunae. . .

82

Baxter γ(ω) 2.5 2 1.5 1 0.5

1 16 31 46 61 76 91 106 121 136 151 166 181 196 211 226 241 256 271 286 301 316

0

↑ Baxter Formula, ω: 0→π

1 14 27 40 53 66 79 92 105 118 131 144 157 170 183 196 209 222 235 248 261 274 287 300 313

2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

↑ Onsager Formula for γ(ω), ω: 0→π 0.8

0.7 0.6 0.5 0.4 0.3 0.2 0.1 1 8 15 22 29 36 43 50 57 64 71 78 85 92 99 106 113 120 127 134 141 148 155

0

↑ The McCoy/Wu Formula for “γ(ω),” ω:0→π/2 Fig. 8.2 Baxter, Onsager, and McCoy/Wu integrands. " Baxter Formula, ω: 0!π, " Onsager Formula for γ(ω), ω: 0!π, " The McCoy/Wu Formula for “γ(ω),” ω:0!π/2

8.2

The Two-Dimensional Ising Model

83

By eye, the (1/2π) × Baxter area is about 0.9; (1/2π) × Onsager is about 0.5; and (1/π) × McCoy/Wu is about 0.2, where the lengths of the x-axis are π, π, and π/2, respectively. When the constant term is added in, all should come to the same value for lnPF. We expect that ln PF at the critical point (Kc = 0.440687) to be about 0.93 (0.929695288 = ln 2.533737 = ln(√2 × exp.(2G/π)), where G is Catalan’s constant. Constant Onsager 0.35 = 0.5 × ln2 McCoy/Wu 0.69 = ln2 Baxter

Divide integral by 2π→ π→ 2π→

→ to get 0.5 0.2 0.9

and the Σ is 0.85 0.89 0.9

Note, again, the difference in the Onsager and McCoy/Wu sums is nicely compensated for by the difference in the constants. It would be interesting to show how the integrands are related, and in some papers the authors show how their solution corresponds to Onsager’s original solution. McCoy/Wu actually obtained Onsager’s solution for the double integral and then derived the single integral. I do not believe Baxter tries to show equivalence. In the case of the Pfaffian and combinatorial solutions, as in McCoy/Wu, they show how the matrices of Kac and Ward are related to theirs, and Kac and Ward show how their formula is similar to Onsager’s and Kaufman’s. Again, Lee and Yang (1952) point out that at the critical point, the partition function should exhibit a zero in the infinite volume limit. Namely, one might think of the PF as being a ConstantΝ × Polynomial (here in tanh K ), the polynomial’s degree being the number of particles or spins. Basically, in the partition function we have products of terms (cosh K+ σσ′sinh K ) = cosh K (1+ σσ′tanh K ), and so we have a polynomial from the product of the (1+ σσ′tanh K ) terms. (In other situations, one wants to avoid computing exponentials, and (eK+ σσ′e–K) becomes eK(1+ σσ′e– 2K ), and defining K* by sinh2K* = 1/sinh2K, we again get a (1+ σσ′tanhK*) term.) Isolating the integral (now N is infinite) in Onsager, we get a good handle on the polynomial, the additional term ln(cosh2Η) or 1/2 ln(sinh2Η) coming from the creation of the (1+ σσ′tanh K*) terms. Below are plots of the sums for Onsager as a function of K. They have minima at the critical value of K, Kc = ~0.44, noting again that we are dealing with ln PF (Fig. 8.3). Another approach might be: If we examine the above formulas for the lnPF, to get the sum it makes sense to compute lnPF–ln√sinh2K (or –ln coshK ). Below we have shown (ln(PF/coshK ))^10 (for ten spins) for k = 0.8, 0.9, 1.0, 1.1, 1.2, 1.3. At the critical value, k = 1.0, there is a minimum (Fig. 8.4).

5

Yang did not draw from Onsager’s unpublished work on the spontaneous magnetization.

84

8

Legerdemain in Mathematical Physics: Structure, “Tricks,” and Lacunae. . .

Fig. 8.3 Sums for Onsager, minimum at K = ~0.44

Onsager sum vs. K 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0

0.2

0.4

0.6

0.8

1

1.2

Fig. 8.4 Onsager lnPF vs k for ten spins. Minimum at k = ~1

8.2.2

An Amazing Invention

For reference, in the Appendix to Chap. 1 is an outline of Yang’s paper. Yang’s work draws in part from Kaufman’s and Onsager’s earlier work, using a spinor analysis rather than Onsager’s (1944) quaternions.5 Onsager’s paper was generally known to use methods unfamiliar to most physicists, and his student Kaufman’s reformulation in the comparatively more familiar terms of spinors and commuting plane rotations enabled Yang to do his work, drawing as well on intuitions about spatial rotations more generally. Yang’s invention appears in the middle of an ingenious and swerving derivation of the spontaneous magnetization, M, of the Ising model in two dimensions. Within a not unfamiliar structure provided by the headings of the sections: M as an off-diagonal matrix element of a rotation in spinor space; Reduction to an Eigenvalue Problem; Limit for Infinite Crystal; Elliptic Transformation [or Elliptic Substitutions]; Solution of the Integral Equation;

8.2

The Two-Dimensional Ising Model

85

Final Results. M as fourth root . . .

Aside from that ingenious device of temporarily restoring a rotational symmetry by means of an artificial limiting process, a symmetry otherwise broken by the even and odd states, the rest of the paper would appear to be a matter of tricky and tedious manipulation and invention: “we notice that,” “rearranging rows and columns,” etc., solving an eigenvalue problem, finding the right diagonalization. We notice that . . .

At one point in his derivation of the spontaneous magnetization of the Ising model, Yang needs to rotate a matrix so as to diagonalize it. What he does is quite inventive, a way forward that is not at all obvious. Later derivations perhaps make his move a bit more understandable, for which see the note.6 He says that, 1 + Cn does not induce a rotation. But we notice that 1 + Cn = lim a! i1 cos-1a (cos a – i Cn sin a) = lima! i1 cos-1a exp.(–iaCn), and exp.(–iaCn) does induce a rotation. (p. 810)

Yang says that this was the longest calculation in his career. His noticing-that, before it was realized, was likely one of the slowdowns in his calculation. (He was not so worried about rigor, it would appear.) He employed “we notice that” again later in the paper. As for swerving, Yang says: To summarize the results of this section [II]: the spontaneous magnetization [“I,” which I denote by M] is given by (31), in which the λs [of section IIC] are related through Eq. (36) to the eigenvalues l of Eq. (44), and in which ξ11 and ξ12 are the first and the (n + 1)th element of the column matrix ξ1 [of IID] calculated through (48) from the column matrix y which in turn is determined by (54) and (56). ξ1 is to be normalized according to (59) [IIE].

Schematically: M = ð31Þ : ð32Þ → λ → ð36Þ → lð44Þ and ð33Þ → ξ11 , ξ12 ← ξ1 ð48Þ ← yð54, 56Þ: Corresponding to Sections: IIB IIB IIC IIC IIB IID Sections III and IV of the paper set out to fulfill this agenda.

IID

See Yang, “The Spontaneous Magnetization,” p. 811, Equation 34. Yamada suggests another perspective on the “artificial limiting process.” He introduces another such process which then expresses the correlations as a ratio of determinants, but which determinants are zero in the limit (columns are equal). So he applies L’Hôpital’s rule, and takes the derivatives of both numerator and denominator. However, such a derivative, d/dω, is in effect an infinitesimal rotation. Baxter claims to be following Yang in his more recent algebraic derivation. The artificial limiting process is given by Baxter’s exp(-γJ ), γ → 1 (this is not Onsager’s γ), where J is in effect the one-dimensional Hamiltonian. The transfer matrix is represented by a Hamiltonian, exp - αH, that H being much like Onsager’s A + k-1B. As for Kaufman and Onsager’s work, Baxter has unearthed the original manuscripts. He describes the two ways they did the correlation function (Szegő limit theorem, integral equation), and provides a third way of his own invention. 6

86

8

8.2.3

Legerdemain in Mathematical Physics: Structure, “Tricks,” and Lacunae. . .

Employing a Device of the Past

Lieb, Mattis, and Schultz (1961), studying one-dimensional models in mathematical physics, employed a transformation that turned mixed fermion/boson operators into fermion operators, a transformation attributed to Jordan and Wigner (1928). Schultz et al.’s (1964) redoing of Onsager’s algebraic derivation, use notions of second quantization and diagonalizing a quadratic form—rather more familiar to contemporary physicists. They are then confronted with operators, the Pauli spin operators, σx,yi, that anticommute at a site, but for different sites, the spin operators commute. It is nevertheless well known how to change such operators to ones obeying a complete set of anticommutation rules [fermions]. We introduce annihilation and creation operators by the formation Cm = [exp(π i Σj,1 to m–1σj+σj-] σ-m [obviously not at all local as an operator].7

8.2.4

Where Did That Come From?

Even more strenuous and hairy and lengthy were Wu et al.’s, 1976, account of the correlations of Ising spins when very close to the critical point and for very large lattices (the “scaling regime”). The length and complexity of the paper are striking, ending with a distant cousin of the trigonometric functions (for example, already discussed earlier in Ince’s textbook Ordinary Differential Equations, 1926, pp. 345ff). I have no idea of the source of their persistence and stamina, although I am told that patience is crucial in this sort of work. Following Wu and collaborators’ earlier detailed calculations, years later Wu et al., 1976, after a long and complicated calculation, found that the correlation in the scaling regime of the Ising model may be expressed using one of the Painlevé transcendents, PIII, distant cousins of the sines and cosines. As far as I can tell that was not at all expected until WMTB arrived there. The authors are a sequence of descendants of an expert on antennas and electromagnetic theory, Ronold King, and presumably Wu learned his skill from his teacher, one of whose other students, John Myers, in working on antenna theory, studying the scattering of electromagnetic waves from a strip, had concluded his dissertation work with that same distant cousin of the trigonometric functions. Myers had encountered an integral similar to the one found by WMTB. The integral is a to b K0(|x–x’|)φ(t) dt, K0 being a Bessel function, solvable by Wiener-Hopf techniques, techniques Wu employed elsewhere in his work on Ising. The genealogy is:

7

Lieb, Schultz, and Mattis, p. 861.

8.2

The Two-Dimensional Ising Model

87

King ðPhD 1932, HarvardÞ ! Wu ð1956Þ ! McCoy ð1967Þ ! Tracy ð1973, Stony BrookÞ ! Myers ð1963Þ

8.2.5

“A Useful Identity, Easily Seen”

The impetus for writing about legerdemain was a blog post that explained the transition from Eq. 106–Eq. 108 in Onsager, 1944, the exact solution of the Ising model, supplying as well a proof of Eq. 107, a formula needed for the transition. In Onsager (1944), there are moments when the transition between lines may need a bit more elaboration. For example, his formula (106) for the partition function, PF (his λ), in the limit of an infinite number of particles, might be shown to be symmetrical between the horizontal and vertical couplings, Onsager’s H and H′, of the spins on the lattice. (106) ln PF =1/2 (ln 2 sinh 2H ) + (1/2π) 0toπ γ(ω)dω where cosh γ(ω) = cosh 2H′ cosh 2H* – sinh 2H′ sinh 2H* cos ω, and where H* is defined by sinh 2H sinh 2H* = 1

He says, There are several ways to show that (106) actually describes a symmetrical function of H and H′. For example, with the aid of the useful identity (107) 0 to 2π ln (2 cosh x – 2 cos ω) dω = 2 π x, we can convert (106) into the double integral (108) [the symmetrical form]. It seems rather likely that this result could be derived from direct algebraic and topological considerations without recourse to the operator method used in the present work. (pp. 248–249) ð108Þ ln ðPF=2Þ ¼ 1= 2π 2 ×

π

ln cosh 2H cosh 2H ′ –sinh 2H cos ω–sinh 2H ′ cos ω ′ dω dω ′

0

In all subsequent discussions and derivations of the work, it seems that nowhere is (107) discussed, or is the derivation of (108) from (106) using (107) shown. Now, other derivations (except for Baxter’s) get the double integral form directly, so they need not address the issue. (They often then go to the single integral formula.) The details are provided in a blog post, both the proof of (107) and the derivation of (108). As for proving (107) that will involve a statement about the geometric mean of (1+ r exp. iω). And then for (108), from (106) using (107), where x is now γ(ω), and so cosh x (that is, cosh γ(ω)) can now be under its own integral in (106), and so eventually we have (108). (Note that if H = H′, cos ω + cos ω′ may be transformed

88

8

Legerdemain in Mathematical Physics: Structure, “Tricks,” and Lacunae. . .

into cos ω × cos ω′, making for a nicer integral. That is, cos (ω + ω′) = cos ω cos ω′ + sin ω sin ω′, and then doing the same for ω–ω′, and so ω becomes (ω + ω′).)8

8.2.6

Signposting Along the Way

As for intermediate results, Baxter is direct and honest: I knew from experience how many sheets of paper go into the waste-paper basket after even a modest calculation: there was no way they could all appear in print. I hope I have reached a reasonable compromise by signposting the route to be followed without necessarily giving each step. . . . I discuss the functions k(α) and g(α) in some detail since they provide a particularly clear example of how elliptic functions come into the working. I merely quote the result for the spontaneous staggered polarization and refer the interested reader to the original paper: its calculation is long and technical, and will probably one day be superseded . . .9

Fefferman, below, signposts using colloquial expressions and oversimplified abstracted versions of a calculation.

8.2.7

“Further Details of Simplifications Like This Will Not Be Reported Here”

Kenneth Wilson, 1975, came to understand some of the mysteries of quantum field theory by actually calculating results numerically on a computer, employing simpler models still possessing the mystery. Actual numerical calculations (again computationally) allowed him to understand systems with infinite degrees of freedom, not by 8 Then, Onsager, using Kramer and Wannier’s, 1942, variable κ, symmetric around the critical temperature, “expand[s] the logarithm [in 108] in powers of κ and κ′ and integrate term by term, but now to get a series in κ 2 n, when K and L are the same so κ and κ′ are the same—with binomial-like coefficients (109c): lnPF–ln(2cosh2K ) = ln(1– (1/16)κ2 – (1/256)4κ4 . . .). At the critical point ln (2cosh2K) = 1.04, and ln(series) = -0.089.” κ =2√k/(k + 1), k = 1/(sinh2 2 K ), when K = L. K + W’s and Onsager’s kappa is ¼ of the kappa I use here. In his Appendix, using an expression Φ(u) whose integral is proportional to 0 to π γ(ω)dω, Onsager says, “It is easily seen that Φ(u) satisfies the standard conditions for development in partial fractions. . .” and then one might “readily perform[ed]” the integral of Φ(u) and so in effect 0 to π γ(ω)dω. (p. 258). “To obtain an expansion suitable for computation in [that] region” of the critical point, one finds a version of Φ(u) that “is a periodic function of u.” “Its Fourier series is easily derived [my italics] with the aid of the identity . . . from those of the Jacobian elliptic functions,” for which he gives a reference to Whittaker and Watson’s Modern Analysis. Earlier in the paper, Onsager uses the theory of group representations to diagonalize a pseudoHamiltonian. Subsequently, Kaufman, 1949, redid Onsager, for a finite lattice, using the apparatus of spinors and spin-representations of the orthogonal group, which made Onsager’s method more recognizable to many physicists. 9 Baxter, Exactly Solved, p. v.

8.3

The Stability of Matter

89

radically truncating those degrees of freedom but by finding the crucial ones. The meaning of the formulas and models was found numerically, namely by which approximations actually worked. (Onsager had guessed how to go about an exact solution by starting with lattices with 2, 3, and 4 infinite rows of spins and solving them by hand.) Wilson is reporting on his application of the renormalization group to solving the two-dimensional Ising model, using an algorithm and a computer and approximations along the way. He says, The sum over all configurations of the old spins was carried out sequentially, . . . only the coupling of s15 to nearby spins was included in the calculation. The “nearby spins” are those shown in Fig. 9 . . . . In practice it was possible to reduce the nearby spins to the subset C of 13 spins shown in Fig. 9. . . . Further details of simplifications like this will not be reported here.

In the actual practice of programming and of discovering which interactions might be ignored (for they had little actual effect on the numbers) the “details” become apparent, perhaps in retrospect made motivated and scientific. Just which terms in an expansion are retained, which left behind, is a craft skill. Wilson, it seems, had always been interested in approximations that allow you to solve problems and get out actual numbers. Computation and programming has been one of his methods, the other being perturbation theory in quantum field theory using Feynman diagrams. I suspect that some of the time, the author finds ways of moving forward in their argument, paying more attention to the mathematics than the physics, and only later, if ever, is the physics behind the devices and tricks revealed. Yet, that physics is almost always of great interest, displaying features of the system that were otherwise not appreciated.

8.3

The Stability of Matter

The stability of matter is something we take for granted, but a rigorous proof eluded all since the beginning of quantum mechanics. Namely, the ground state energy of a clump of matter, namely due to the Coulomb electrical forces among the electrons and nuclei, should have a lower bound proportional to the number of atoms or molecules in that clump (rather than say N2). Onsager, 1939, had provided an argument that was rough, not rigorous but physically correct. Later, Fisher and Ruelle revived the problem.

90

8

8.3.1

Legerdemain in Mathematical Physics: Structure, “Tricks,” and Lacunae. . .

“Hacking Through A Forest Of Inequalities”

Lenard interested Dyson in his problem, deriving from Lenard’s work in plasma physics. Dyson’s training as a classical analyst served him well, and their strategy put the calculation on the road. The Dyson/Lenard proof demanded a sequence of inequalities, each with a small proportionality constant, but the product of those constants was quite large, about 1014 Rydberg. (1 Rydberg = 13.6 eV) Dyson referred to their proof in terms of “hacking,” but in fact the proof is physically well motivated, the headings of the sections being informative, albeit it does not have the enormous advantage of the physics provided in Lieb and Thirring.

8.3.2

“Thomas-Fermi Atoms Do Not Bind”

Lieb and Thirring, 1975, were able to redo the proof, presumably suspecting that it could be done using the Thomas-Fermi model of an atom, especially familiar to Lieb from his earlier work with Simon, 1973. Lieb and Thirring’s, 1975, has a constant that is perhaps 10-13th of Dyson-Lenard, largely because they get the physics right. Namely, Teller had shown that in the Thomas-Fermi model of an atom, there is no binding. So if TF atoms (with a different proportionality constant) could be shown to be a lower bound of the energy of matter, then the total energy would be proportional to N atoms, and so stability would be proven.

8.3.3

“An Elementary Identity, Fourier Analysts Are Quite Familiar with It. Gruesome Details, Nasty and Ghastly Calculations, Modulo Oversimplifications, Applying General Nonsense”

To give an account of atoms and of stability, Fefferman and collaborators (1986–1997) employed rigorous classical mathematical analysis to achieve a deeper understanding and better bounds on the binding energy/particle. Fefferman is a master of the devices of classical analysis. He had experience with “nasty” calculations and persistence—shown in his earlier work. Fefferman says at one point: we can start with the elementary identity 5 3 |x–x’|-1 = (1/π) yεR^3, R > 0, χx, x’ ε B(y, R) dy dR/R for x, x’ ε R (25) . . .[except for 1/π] identity (25) is forced by the fact that both sides transform in the same way under translations, rotations, and dilations.10

10

“On the Dirac . . .,” Advances in Mathematics, 1994, p. 6.

8.4

Genealogy Reconsidered

91

where R3 is three-dimensional space, y is the center of a Ball of radius R, χ is the indicator function for when x and x’ are within the Ball.11 Elsewhere he says of a related formula that “it is easily verified by using the Fourier transform. Fourier analysts are quite familiar with it. . .” He refers to the “gruesome details” of the calculation. Fefferman constantly provides previews and motivations, often designating them in colloquial terms. In Fefferman’s Bergman kernel paper, he speaks of “nasty” and “ghastly” calculations; “the idea is to transfer the Bergman kernel”; “We apply this general nonsense. . .”, both meant to motivate and set up a complex calculation; “as a sum of terms which (hopefully) transform to something recognizable when expressed in ζ-coordinates”; “Modulo oversimplifications, this is the plan of our proof of Theorem 1.”, at the conclusion of a detailed motivating introduction.

8.4

Genealogy Reconsidered

One explanation of such legerdemain is to understand the genealogy of the authors and their previous work and teachers. It would appear that Elliott Lieb and Ronold King were significant sources and influences. The notion of the transfer matrix and Kac and Ward’s combinatorial solution were formative. (I have not found the source, if there is one, of Baxter’s techniques.) So (Fig. 8.5): In sum, legerdemain is a matter of what you happen to know, dogged persistence, ingenious invention apparently out of nowhere but likely from past work you have done, learning that something can be done so it makes sense to try to do it better using your repertoire of skills and knowledge, and the capacity for approximations to retain the crucial features of a system.

Let V(x,y) be the integral over all z in R3 and r > 0 of the indicator function of the set where x and y both lie within distance r of the point z, divided by r5 . Then, one sees from the definition of V(x,y) that V(x,y) = V(x + w,y + w) for any vector w, therefore, V(x,y) = V(x–y,0). Moreover, V(Tx,0) = V (x,0) for any rotation T of R3, as one also sees from the definition of V(x,y) . Therefore, V(x,y) is a function of the distance from x to y. Again from the definition of V(x,y), one checks easily that V(tx, ty) = V(x,y)/t for any positive real number t. Therefore, V(x,y) has the form constant/(distance from x to y). Evaluating V(x,y) for one particular pair of points, one finds that the constant is π. (as provided by CF to MK 3Ap18). 11

92

8

Legerdemain in Mathematical Physics: Structure, “Tricks,” and Lacunae. . .

Lieb →Jordan-Wignerwork with Mattis on one-dimensional models→SML

Dyson/Lenard ↓

Lieb→Thomas-Fermi modelwork with Simon→Lieb/Thirring Lieb→Baxter (at MIT 1968-1970)

Lieb/Thirring↓ Lieb/Simon→Fefferman←E.Stein

Kramers&Wannier + Montroll →transfer matrix, critical pointLieAlgebras,WhittakerWatson→Onsager→spinorsKaufman → Yang

Kaufman→Kac/Ward→Montroll/Potts/Ward,Hurst/Green,Kastelyn→PfaffianMethod Peierlsgraph combinatorics→van der Waerden ↑

*RonoldKing→TaiWuToeplitz+Cheng→B.M.McCoy→C.A.Tracy→WMTB →Myers→PainlevéIII↑

Matthewscomputation↓ QuantumFieldTheory →GellMan/LowRenormalizationGroup→Wilsoncomputational field theory, strong coupling,slices in p-space Kadanoff,Widom↑

*King was Wu’s advisor, Wu Mccoy’s, McCoy Tracy’s. RONOLD KING→PHDHARVARD,1956WU→PHDHARVARD,1967MCCOY→PHDSTONYBROOK,1973TRACY RONOLD KING→JOHN MYERS PHDHARVARD,1963

Fig. 8.5 Some of the genealogy of the Ising model’s physicists. SML = Schultz, Mattis and Lieb.

Chapter 9

Mathematical Physics

I have been describing some of the practice of mathematics applied to physics, more particularly mathematical physics.1 The demands for explicitness and rigor in this work suggest that the features of this practice are in effect unavoidable—although derivations of the same fact may go by different routes. Rigor matters substantively. Mathematical physicists apply mathematics and its technology to prove or derive features of physical systems such as phase transitions or the stability of matter. They may well develop systematic theories of quantum fields or of classical mechanics, rigorous and precise, although that is not my concern here. Theoretical physicists may well use mathematics in their work, but they are rather more concerned with explaining empirical observations or developing theoretical structures, although mathematical physicists may then take on those theoretical developments and make them mathematically rigorous and so may provide a deeper understanding of the physics. I do not believe this rigorization and formalization is just for show or just to satisfy the mathematical physicist’s standards. I should note that not all mathematical physics would be considered rigorous by mathematicians and that the precise definitions employed are driven by an attempt to give a mathematical account of a physical phenomenon. Moreover, often what is provided is an exact solution, rather than a rigorous one, or an approximation is provided with presumably rigorous error bounds. As for the notion of mathematical rigor, let us say that it is what mathematicians do that they find convincing among themselves. What we see again and again is that physics yields to mathematical technologies and methods that have already been developed or suggest that such technologies need to be worked out. Wigner’s fabled unreasonable effectiveness of mathematics in the natural realm is evidenced here, but of course not all physical problems are so 1 I do not believe I have anything to say about the philosophy of mathematics or physics that would be generic, divorced from my main examples, that would in any sense be profound. Or, about more general issues of explanation, knowledge, and ontology. I would hope that my exposition would be of use to those who wish to make philosophical arguments, but that may be a misplaced hope.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 M. H. Krieger, Primes and Particles, https://doi.org/10.1007/978-3-031-49776-6_9

93

94

9

Mathematical Physics

amenable, so that “unreasonableness” is perhaps too strong a claim. One must keep in mind how much work the mathematical physicist must do to find a way of successfully modeling the physical situation and how they call in the mathematicians for what they need at times. There are no miraculous events here. Rigor is usually essential, although an exact solution that might be made rigorous may be a remarkable achievement. (Here I am thinking of Rodney Baxter’s work.) The fabled story is that of Kaufman and Onsager’s late 1940s solution of the spontaneous magnetization of the Ising model. They had the solution, but the derivation was not rigorous; Onsager did not know what to do about the deltas and the epsilons. Onsager was at Yale and asked Kakutani, who then asked the expert, Szegő. Onsager said that the mathematicians got there first. Still, it would seem in fact that Onsager and Kaufman had actually done the requisite work. The mathematical physicist makes precise what one might mean by a phase transition or by the stability of matter, by the fact that their notions of such then are provable and are empirically accurate, and they capture our intuitions of what they should mean. Moreover, in the process of doing a derivation or a proof, various mathematical features prove essential. So for our usual example, to model a phase transition, your mathematical model must not converge uniformly. It must converge only pointwise, for then the limit of smooth functions need not be smooth.2 At that point, usually a symmetry change takes place abruptly. And “infinity,” for the mathematical demonstration, is perhaps a matter of the upper limit of an integration being infinity or of a limiting process. Moreover, an infinite or very large sum of nonzero complex numbers can “pinch” the real temperature axis (that is, be almost zero), so indicating a phase transition. Or, the fact that electrons are fermions, governed by antisymmetric wavefunctions, is crucial in the proof of the stability of matter. Were electrons bosons, and governed by symmetric wavefunctions, there would not be such stability. When you have many derivations, each from a different perspective, you might think that such is nice, but So what? This also happens in mathematical work. However, it turns out that each such derivation reveals more of what is going on, physically and mathematically, in the system and in the mathematics. In effect, one has “an identity in a manifold presentation of profiles”—the way phenomenologists describe it (Sokolowski). Moreover, if you examine each of the proofs or derivations, you can see how each builds in features of the mathematics or the physics that prove crucial, features only revealed in your examining that range of proofs or derivations. This point is generic, but only in examining actual rigorous work do you appreciate it. Now, it is not unusual for a scientist to have an intuition or an idea about how to go about doing things. However, until one actually does that going about, that doing, one does not know if it will actually work. That electrons are fermions might matter

2

That is, the epsilon-delta of a limit is not independent of the point as you go to a macroscopic system. In practice, macroscopic means the number of particles, N, approaches, say, about Avogadro’s Number (6 × 1023). Often, in practice, 108 is large enough.

9

Mathematical Physics

95

to the stability of matter, it would seem; but you only know just how that idea plays a role, and that it does play a crucial role, when the calculation is actually done, with no loopholes or lacunae. And, that it works when you show that if electrons were not fermions that stability would not be possible. Moreover, if one starts out without any good ideas and does what might be called a brute-force calculation, you are unlikely to achieve your goal if along the way you do not begin to have ideas about what is going on. One of the recurrent features of my two main examples is the attempt to link smaller scales to larger ones. Namely, you know the interactions of the parts, but how they are evidenced in the macroscopic world is what you need to prove or show. To prove the stability of matter one divides space into parts (cubes inside of cubes, spheres inside of spheres) and goes from the parts to the whole of space. With the proper choice of such parts the interstitial energy of interaction, of the spaces between parts and the smaller-cube to smaller-cube interactions’ contributions of the energy, goes to zero. One can do a similar division in the case of the Ising model, dividing the two-dimensional grid into blocks or squares composed of smaller blocks or squares. Other proofs were developed that focused on the fact that many-particle interactions could be parsed into a sum of two-particle interactions. As I have indicated, initial proofs and derivations and calculations are often ugly, brute-force endeavors, albeit with lots of good ideas along the way. Yet, those proofs usually have an inner beauty only appreciated after many complementary proofs are provided. Moreover, in the case of the Ising model, one might understand that proof and subsequent new proofs as participating in an analogy that is analogous to another analogy, a mathematical one out of Dedekind and Weber. Again, these generalities only make sense in terms of a set of specific calculations, in which you can identify the details and the analogy. Along the way, we might ask how mathematics and physics are made useful for each other. Can we say what is special about certain physical (or biological or chemical) systems that make them amenable to mathematics? How is abstraction available both to the intuitions we have and to the precision required by mathematics? What is left out? What makes it possible for rigor to reveal the physics? Just how are ugly initial proofs made beautiful in time? What do the many methods of working out a problem each reveal? And why are spectrum and shape connected, why is harmonic analysis connected to scaling and automorphy, just how are the local and the global connected? How does linearization play a role here? These questions appear again and again in phenomena, in explanations, and in mathematical physics. They have specific answers for particular systems. It is less clear to me there are more general answers. Now, much of mathematics connects local and global phenomena. And we know that the exact nature of a molecule is only roughly reflected in a solid made up of those molecules, and vice versa. In effect, there are “emergent phenomena,” such as a crystal, that forget or hide some of the features of their microscopic constituents and have features that are surprising given what we know about the microscopic parts. Mathematical physics often explains this emergence and sometimes even

96

9 Mathematical Physics

connects the local with the global by pointing to the appropriate variables that transcend the difference continuously. Again and again, physicists find ways of adding up objects, using suitable mathematical technologies—vector spaces, symmetry groups, poles, and branch cuts. A deeper understanding is not about that “unreasonable effectiveness.” It is about how what is otherwise hidden is revealed by mathematical work that is exact, sometimes rigorous, and usually precise. Physicists’ explanations launch mathematical technologies (think of Newton and calculus), those technologies fixing and cleaning up the physicists’ work. And mathematics is sometimes recruited to do physical work, and in doing so mathematics is further developed (by mathematicians and by physicists). None of this is profound, in general. For only in the specific workings of the mathematics and the physics is the connection evidenced and articulated. However, it may be that in a suite of specific cases a more general connection is hinted at, more deeply connecting the mathematics and the physics in a precisely articulated analogy. Different modes of solution to a problem likely point to equivalent mathematics or different aspects of that same Nature. Numbers and quantities matter, and formalism is not merely for show. Whether it be human families, families of particles, or primes in different number fields and their relationships—parallels are ubiquitous, likely interesting and informative, but rarely are they compelling. There are other parallels and analogies, and they too might be as cogent and compelling. Put differently, the world of Nature and mathematics could be otherwise, but it happens to be this way. Transcendent and transcendental accounts to justify the way things happen to reveal more about our questions than about Nature or mathematics. Yet, once in a while we hit paydirt, as in the calculus and in group theory.

Bibliography

Aldous, D., & Diaconis, P. (1995). Hammersley’s interacting particle process and longest increasing subsequences. Probability Theory and Related Fields, 103, 199–213; (1999) Longest increasing subsequences: From patience sorting to the Beik-Deift-Johansson Theorem. Bulletin (New Series) of the American Physical Society, 36, 413–432. Baxandall, M. (1982). The limewood sculptors of renaissance Germany. Yale University Press. Baxter, R. J. (1982). Exactly solved models in statistical mechanics. Academic. Baxter, R. J. (2008). Algebraic reduction of the Ising Model. Journal of Statistical Physics, 132, 959–982. Baxter, R. J. (2011). Onsager and Kaufman’s calculation of the spontaneous magnetization of the Ising model. Journal of Statistical Physics, 145, 518–548; (2012) Onsager and Kaufman’s calculation of the spontaneous magnetization of the Ising model: II. Journal of Statistical Physics, 149, 1164–1167; (2010) Some comments on developments in exact solutions in statistical mechanics since 1944. Journal of Statistical Mechanics: Theory and Experiment, P11037, 26 p. Baxter, R. J., & Enting, I. (1978). 399th solution of the Ising model. Journal of Physics A, 11, 2463–2473. Benettin, G., Gallavotti, G., Jona-Lasinio, G., & Stella, A. L. (1973). On the Onsager-Yang-value of the spontaneous magnetization. Communications in Mathematical Physics, 30, 45–54. Bombieri, E. (September 1992). Prime territory. The Sciences, 32. da Costa, F. A., de Oliveira, R. T. G., & Viswanathan, G. M. Fermionization of the 2-D Ising model: The method of Schultz, Mattis and Lieb. https://gandhiviswanathan.wordpress.com/2015/12/ 03/fermionization-of-the-2-d-ising-model-the-method-of-schultz-mattis-and-lieb/ Dauben, J. W. (1979). Georg Cantor. Harvard University Press. Dedekind, R., & Weber, H. (1930). Theorie der algebraische Funktionen einer Veränderlichen, pp. 238–350 (with a comment by E. Noether, “Erläuterungen zur verstehenden Abhandlung,” p. 350, on the threefold analogy) in Bd. 1, R. Dedekind, Gesammelte mathematische Werke (eds. R. Fricke, E. Noether, and O. Ore) Braunschweig: Vieweg. Originally, Journal für reine und angewandte Mathematik (“Crelle”) 92 (1882): 181–290. ____, Theory of Algebraic Functions of One Variable (trans. J. Stillwell), Providence RI: American Mathematical Society, 2012. André Weil, “Une lettre et un extrait de lettre à Simone Weil,” (1940), vol. 1, pp. 244–255; “De la métaphysique aux mathématiques,” [Science (Paris): 1960, pp. 52–56] vol. 2, pp. 408–412, in Oeuvres Scientifiques, Collected Papers (New York: Springer, 1979). Translated as, “A 1940 Letter of André Weil on Analogy in Mathematics,” Notices of the American Mathematical Society 52 (March 2005): 334–341.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 M. H. Krieger, Primes and Particles, https://doi.org/10.1007/978-3-031-49776-6

97

98

Bibliography

Donoghue, J. F., Golowich, E., & Holstein, B. R. (2014). Dynamics of the standard model (2nd ed., p. 1). Cambridge University Press. Dyson, F. J. (1979). Time without end. Reviews of Modern Physics, 51, 447–460. Dyson, F. J., & Lenard, A. (1967). Stability of Matter. I. Journal of Mathematical Physics, 8, 423–434. Lenard, A., & Dyson, F. (1968). Stability of matter. II. Journal of Mathematical Physics, 9, 698–711. Eyink, G. L., & Drivas, T. D. (2015). Spontaneous stochasticity and anomalous dissipation for Burgers equation. Journal of Statistical Physics, 158, 386–432. Fefferman, C. (1974). The Bergman kernel and biholomorphic mappings of pseudoconvex domains. Inventiones mathematicae, 26, 1–65 , at pp. 4, 45, 9, 19, 37. Fefferman, C. (1986). The N-body problem in quantum mechanics. Communications on Pure and Applied Mathematics, 39, S67–S109, at p. S90, p. S88. Hurst, C. A., & Green, H. S. (1960). New solution of the Ising problem from a rectangular lattice. Journal of Chemical Physics, 33, 1059–1062. Kac, M., & Ward, J. C. (1952). A combinatorial solution to the two-dimensional Ising model. Physical Review, 88, 1332–1337. Kadanoff, L. P. (1966). Spin-Spin correlations in the two-dimensional Ising model. Il Nuovo Cimento, B44 , 276–305, especially pp. 289–293, at p. 292. Kastelyn, P. (1963). Dimer statistics and phase transitions. Journal of Mathematical Physics, 4, 281–293. Kaufman, B. (1949). Crystal statistics II: “Partition function evaluated by spinor analysis”. Physical Review, 76, 1232–1243. Kaufman, B., & Onsager, L. (1949). Crystal statistics, III: short range order in a Binary Ising Lattice. Physical Review, 76, 1244–1252. For a critical history see, Baxter, R. J. (2011). Onsager and Kaufman’s calculation of the spontaneous magnetization of the Ising model. Journal of Statistical Physics, 145, 518–548; (2012). Onsager and Kaufman’s calculation of the spontaneous magnetization of the Ising model: II. Journal of Statistical Physics, 149, 1164–1167. Krieger, M. H. (1987). The elementary structures of particles. Social Studies of Science, 17, 749–752. Krieger, M. H. (1989). Big decisions. In Marginalism and discontinuity. Russell Sage Foundation. Krieger, M. H. (1995). Could the probability of doom be zero-or-one? Journal of Philosophy, 92, 382–387. Krieger, M. H. (1996). Constitutions of matter. University of Chicago Press. Krieger, M. H. (2015). Doing mathematics. World Scientific, chapters 3–5. Krieger, M. H. (2022). Riding uncertainty. Journal of Planning and Education Research, 42(3), 482–486. Lebowitz, J. L., & Lieb, E. H. (1969). Existence of thermodynamics for real matter under coulomb forces. Physical Review Letters, 32, 631–634. Lieb, E. H., & Lebowitz, J. L. (1972). The constitution of matter: Existence of thermodynamics for systems composed of electrons and nuclei. Advances in Mathematics, 9, 316–398. Lee, T. D., & Yang, C.-N. (1952). Statistical theory of equations of state and phase transitions: Part 2, lattice gas and Ising model. Physical Review, 87, 410–419. Lieb, E. H., & Mattis, D. C. (1966). Mathematical physics in one dimension. Academic , a note on p. 469, in an article by Lieb, Schultz, and Mattis, p. 410, provides the history of the JordanWigner transformation. Lieb, E. H., & Seiringer, R. (2009). The stability of matter in quantum mechanics. Cambridge University Press. provides a survey. Lieb, E. H., & Simon, B. (1977). Thomas-fermi theory of atoms, molecules and solids. Advances in Mathematics, 23, 22–116. Lieb, E. H., & Thirring, W. (1975). Bound for kinetic energy which proves the stability of matter. Physical Review Letters, 35, 687–689. Lovejoy, A. O. (1936). The great chain of being. Harvard University Press.

Bibliography

99

McCoy, B. M., & Wu, T. T. (1973). The two-dimensional Ising model, 2nd edition. Harvard University Press; Dover, 2014. McKean, H. P., & Moll, V. (1997). Elliptic curves: Function theory, geometry, arithmetic. Cambridge University Press. Montroll, E. W. (1941). Statistical mechanics of nearest neighbor systems. Journal of Chemical Physics, 9, 706–721. Montroll, E. W., Potts, R. B., & Ward, J. C. (1963). Correlations and spontaneous magnetization of the two-dimensional Ising model. Journal of Mathematical Physics, 4, 308–322. Newell, G. F., & Montroll, E. (1953). On the theory of the Ising model of ferromagnetism. Reviews of Modern Physics, 25, 353–389, at p. 376. See also, Brush, S. G. (1967). History of the LenzIsing model. Reviews of Modern Physics, 39, 883–893. Onsager, L. (1939). Electrostatic interaction of molecules. Journal of Physical Chemistry, 43, 189–196. Onsager, L. (1944). Crystal statistics I. Physical Review, 65, 117–149. Peierls, R. (1936). On Ising’s model of ferromagnetism. Proceedings of the Cambridge Philosophical Society, 32, 477–481. Ruelle, D. (1967). States of classical statistical mechanics. Journal of Mathematical Physics, 8, 1657–1668. Sato, M., Miwa, T., & Jimbo, M. (1977). Studies on holonomic quantum fields. I. Proceedings of the Japan Academy, 53A, 6–10. Schultz, T. D., Mattis, D. C., & Lieb, E. (1964). The two-dimensional Ising model as a soluble model of many fermions. Reviews of Modern Physics, 36, 856–871. Schweber, S. (2008). Einstein and Oppenheimer: The meaning of genius. Harvard. Sennett, R. (2009). The craftsman (p. 290). Yale University Press. Spielman, D. A., & Teng, S. H. (2009). Smoothed analysis: An attempt to explain the behavior of algorithms in practice. Communications of the ACM, 52(10), 76–84. Taylor, R. (2002). Notes on Galois Representations, p. 13, re the enumeration/automorphy question ICM , vol I, pp. 449–474. Teller, E. (1962). On the stability of molecules in the Thomas-Fermi theory. Reviews of Modern Physics, 34, 627–631. van der Waerden, B. (1941/1942). Die lange Reichweite der regelmassigen Atomordnung, Zeitschrift für Physik, 118, 473–488. Whittaker, E. T., & Watson, G. N. (1990). A course of modern analysis. Cambridge University Press. [edn. 4, 1927; 1902]. Widom, B. (1965). Equation of state in the neighborhood of the critical point. Journal of Chemical Physics, 41, 3898–3905. Widom, On Harold (2022). see Bulletin of the American Mathematical Society, 59, 155–190; and (2022). Notices of the American Mathematical Society, 69, 586–598. Wigner, E. P. (1970). The unreasonable effectiveness of mathematics in the natural sciences. In Symmetries and reflections. MIT Press. Wilson, K. G. (1975). The renormalization group: Critical phenomena and the Kondo problem. Reviews of Modern Physics, 47, 773–840. Woodin, W. H. (2001). The continuum hypothesis, Part I and Part II. Notices of the American Mathematical Society, 48, 657–676, 681–690. Wu, T. T., McCoy, B. M., Tracy, C. A., & Barouch, E. (1976). Spin-spin correlation functions for the two-dimensional Ising model: Exact theory in the scaling region. Physical Review, B13, 315–374. Yamada, K. (1986). Pair correlation function in the Ising Square Lattice, Generalized Wronskian Form. Progress of Theoretical Physics, 76, 602–612, at pp. 603, 608. Yang, C.-N. (1952). The spontaneous magnetization of a two-dimensional Ising model. Physical Review, 85, 808–816. Commented on in Yang, C.-N. (1983). Selected papers 1945–1980, with commentary. Freeman.

Index

B Barouch, E., 26 Baxter, R., 3, 4, 7, 8, 23, 24, 26, 27, 77–79, 81– 83, 85, 87, 88, 91, 94

C Cardy, J., 21, 25

D Dedekind-Weber, 3, 8–9, 74–75 Deift, P., 26 Duminil-Copin, H., 2, 21, 24–27 Dyson, F.J., 3, 9–11, 15–17, 22, 23, 30–32, 34, 38, 78, 90

F Fefferman, C., 12, 78, 88, 90, 91

K Kac, M., 8, 27, 83, 91

L Langlands, R., 21, 55, 60

Lieb, E.H., 2, 3, 6, 7, 9–13, 17–19, 21–23, 27, 32, 34, 78, 80, 86, 90–92

M Mattis, D., 7, 22, 27, 80, 86, 92 McCoy, B.M., 26, 27, 80–83

O Onsager, L., 3, 6–9, 13, 20, 22, 27, 33, 34, 77, 78, 80–89, 94

R Riemann, B., 3, 8, 13–14, 27, 41, 44, 47, 49, 53, 59, 60

S Schultz, T., 6, 7, 22, 27, 80, 86, 92 Simon, B., 23, 90 Smirnov, S., 21, 24, 25, 27

T Thirring, W., 11, 12, 17, 22, 23, 32, 34, 90 Thomas-Fermi, 11, 12, 17–19, 23, 78, 90 Tracy, C., 26, 87, 92

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 M. H. Krieger, Primes and Particles, https://doi.org/10.1007/978-3-031-49776-6

101

102 W Ward, J., 8, 27, 83, 91 Widom, H., 26, 92 Wigner. E., ix, 4, 7, 49, 86, 93 Wilson, K.G., 3, 32–35, 88, 89, 92 Witten, E., viii–ix Wu, T.T., 25–27, 86, 87, 92

Index Y Yang, C.-N., 3, 7, 19, 20, 25, 34, 77, 79, 80, 83–85

Z Zamolodchikov, A., 21, 25