Relativistic Classical Mechanics and Electrodynamics (Synthesis Lectures on Engineering, Science, and Technology, Band 1) 168173706X, 9781681737065

This book presents classical relativistic mechanics and electrodynamics in the Feynman-Stueckelberg event-oriented frame

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Relativistic Classical Mechanics and Electrodynamics (Synthesis Lectures on Engineering, Science, and Technology, Band 1)
 168173706X, 9781681737065

Table of contents :
Preface
Symbols
Background
Conceptual Approaches to Spacetime
Point Mechanics in 4D Spacetime
The Two Aspects of Time
The ``Proper Time'' Formalism in QED
The Stueckelberg–Horwitz–Piron (SHP) Framework
Bibliography
Theory
Canonical Relativistic Mechanics
Lagrangian and Hamiltonian Mechanics
The Free Relativistic Particle
The Relativistic Particle in a Scalar Potential
Two-Body Problem with Scalar Potential
Many-Body Problem and Statistical Mechanics
Bibliography
Classical Electrodynamics
Classical Gauge Transformations
Lorentz Force
Field Dynamics
Ensemble of Event Currents
The 5D Wave Equation and its Green's Functions
The Mass-Energy-Momentum Tensor
Worldline Concatenation
PCT in Classical SHP Theory
Bibliography
Applications
Problems in Electrostatics and Electrodynamics
The Coulomb Problem
Contribution to Potential from G_Maxwell
Contribution to Potential from G_Correlation
Liénard–Wiechart Potential and Field Strength
Electrostatics
Plane Waves
Radiation from a Line Antenna
Classical Pair Production
Particle Mass Stabilization
Self-Interaction
Statistical Mechanics
Speeds of Light and the Maxwell Limit
Bibliography
Advanced Topics
Electrodynamics from Commutation Relations
Classical Non-Abelian Gauge Theory
Evolution of the Local Metric in Curved Spacetime
Zeeman and Stark Effects
Classical Mechanics and Quantum Field Theory
Bibliography
Authors' Biographies
Blank Page

Citation preview

Series ISSN 1939-5221 LAND • HORWITZ

Relativistic Classical Mechanics and Electrodynamics Martin Land, Hadassah College, Jerusalem Lawrence P. Horwitz, Tel Aviv University, Bar Ilan University, and Ariel University

RELATIVISTIC CLASSICAL MECHANICS AND ELECTRODYNAMICS

This book presents classical relativistic mechanics and electrodynamics in the Feynman-Stueckelberg event-oriented framework formalized by Horwitz and Piron. The full apparatus of classical analytical mechanics is generalized to relativistic form by replacing Galilean covariance with manifest Lorentz covariance and introducing a coordinate-independent parameter τ to play the role of Newton’s universal and monotonically advancing time. Fundamental physics is described by the τ-evolution of a system point through an unconstrained 8D phase space, with mass a dynamical quantity conserved under particular interactions. Classical gauge invariance leads to an electrodynamics derived from five τ-dependent potentials described by 5D pre-Maxwell field equations. Events trace out worldlines as τ advances monotonically, inducing pre-Maxwell fields by their motions, and moving under the influence of these fields. The dynamics are governed canonically by a scalar Hamiltonian that generates evolution of a 4D block universe defined at τ to an infinitesimally close 4D block universe defined at τ+dτ. This electrodynamics, and its extension to curved space and non-Abelian gauge symmetry, is well-posed and integrable, providing a clear resolution to grandfather paradoxes. Examples include classical Coulomb scattering, electrostatics, plane waves, radiation from a simple antenna, classical pair production, classical CPT, and dynamical solutions in weak field gravitation. This classical framework will be of interest to workers in quantum theory and general relativity, as well as those interested in the classical foundations of gauge theory.

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Relativistic Classical Mechanics and Electrodynamics Martin Land Lawrence P. Horwitz

Relativistic Classical Mechanics and Electrodynamics

Synthesis Lectures on Engineering, Science, and Technology Relativistic Classical Mechanics and Electrodynamics Martin Land and Lawrence P. Horwitz 2019

Copyright © 2020 by Morgan & Claypool

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means—electronic, mechanical, photocopy, recording, or any other except for brief quotations in printed reviews, without the prior permission of the publisher. Relativistic Classical Mechanics and Electrodynamics Martin Land and Lawrence P. Horwitz www.morganclaypool.com

ISBN: 9781681737065 ISBN: 9781681737072 ISBN: 9781681737089

paperback ebook hardcover

DOI 10.2200/S00970ED1V01Y201912EST001

A Publication in the Morgan & Claypool Publishers series SYNTHESIS LECTURES ON ENGINEERING, SCIENCE, AND TECHNOLOGY Lecture #1 Series ISSN ISSN pending.

Relativistic Classical Mechanics and Electrodynamics

Martin Land Hadassah College, Jerusalem

Lawrence P. Horwitz Tel Aviv University, Bar Ilan University, and Ariel University

SYNTHESIS LECTURES ON ENGINEERING, SCIENCE, AND TECHNOLOGY #1

M &C

Morgan

& cLaypool publishers

ABSTRACT This book presents classical relativistic mechanics and electrodynamics in the FeynmanStueckelberg event-oriented framework formalized by Horwitz and Piron. The full apparatus of classical analytical mechanics is generalized to relativistic form by replacing Galilean covariance with manifest Lorentz covariance and introducing a coordinate-independent parameter  to play the role of Newton’s universal and monotonically advancing time. Fundamental physics is described by the  -evolution of a system point through an unconstrained 8D phase space, with mass a dynamical quantity conserved under particular interactions. Classical gauge invariance leads to an electrodynamics derived from five  -dependent potentials described by 5D pre-Maxwell field equations. Events trace out worldlines as  advances monotonically, inducing pre-Maxwell fields by their motions, and moving under the influence of these fields. The dynamics are governed canonically by a scalar Hamiltonian that generates evolution of a 4D block universe defined at  to an infinitesimally close 4D block universe defined at  C d  . This electrodynamics, and its extension to curved space and non-Abelian gauge symmetry, is well-posed and integrable, providing a clear resolution to grandfather paradoxes. Examples include classical Coulomb scattering, electrostatics, plane waves, radiation from a simple antenna, classical pair production, classical CPT, and dynamical solutions in weak field gravitation. This classical framework will be of interest to workers in quantum theory and general relativity, as well as those interested in the classical foundations of gauge theory.

KEYWORDS spacetime, relativistic mechanics, classical electrodynamics, electrostatics, quantum field theory

vii

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

PART I 1

Conceptual Approaches to Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1 1.2 1.3 1.4 1.5

Point Mechanics in 4D Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The Two Aspects of Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 The “Proper Time” Formalism in QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 The Stueckelberg–Horwitz–Piron (SHP) Framework . . . . . . . . . . . . . . . . . . . . 9 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

PART II 2

Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Canonical Relativistic Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.1 2.2 2.3 2.4 2.5 2.6

3

Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Lagrangian and Hamiltonian Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Free Relativistic Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Relativistic Particle in a Scalar Potential . . . . . . . . . . . . . . . . . . . . . . . . . . Two-Body Problem with Scalar Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Many-Body Problem and Statistical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15 17 18 20 21 23

Classical Electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.1 3.2 3.3 3.4

Classical Gauge Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lorentz Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Field Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ensemble of Event Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25 27 29 31

viii

3.5 3.6 3.7 3.8 3.9

The 5D Wave Equation and its Green’s Functions . . . . . . . . . . . . . . . . . . . . . . The Mass-Energy-Momentum Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Worldline Concatenation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PCT in Classical SHP Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

PART III 4

Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Problems in Electrostatics and Electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.1

4.2 4.3 4.4 4.5 4.6 4.7

4.8 4.9

5

33 34 36 38 43

The Coulomb Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Contribution to Potential from GMaxwell . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Contribution to Potential from GCorrelation . . . . . . . . . . . . . . . . . . . . . . Liénard–Wiechart Potential and Field Strength . . . . . . . . . . . . . . . . . . . . . . . . Electrostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plane Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Radiation from a Line Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Classical Pair Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Particle Mass Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.1 Self-Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.2 Statistical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Speeds of Light and the Maxwell Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47 48 50 53 57 62 65 73 82 84 89 93 95

Advanced Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.1 5.2 5.3 5.4 5.5 5.6

Electrodynamics from Commutation Relations . . . . . . . . . . . . . . . . . . . . . . . . 97 Classical Non-Abelian Gauge Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 Evolution of the Local Metric in Curved Spacetime . . . . . . . . . . . . . . . . . . . 110 Zeeman and Stark Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Classical Mechanics and Quantum Field Theory . . . . . . . . . . . . . . . . . . . . . . 116 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

Authors’ Biographies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

ix

Preface This book presents classical relativistic mechanics and describes the classical electrodynamics of relativistic particles following the approach of Stueckelberg, Horwitz, and Piron (SHP). This framework, pioneered by E. C. G. Stueckelberg in 1941 and employed by Schwinger and Feynman in the development of QED, generalizes classical analytical mechanics to relativistic form by replacing Galilean covariance with Lorentz covariance, and introducing a new coordinateindependent evolution parameter  to play the role of Newton’s postulated universal and monotonically advancing time. Fundamental physics is described by the  -evolution of a system point through an unconstrained phase space, in which each event is represented by its covariant spacetime coordinates and velocities or momenta. The full apparatus of analytical mechanics is thus made available in a manifestly covariant form, from Lagrangian and symplectic Hamiltonian methods to Noether’s theorem. This approach to relativistic classical mechanics makes SHP a convenient framework for analyzing the “paradoxes” of special relativity, and in particular provides a clear resolution to the grandfather paradox. Making the free particle Lagrangian invariant under classical gauge transformations of the first and second kind leads to an electrodynamics derived from five  -dependent potentials, described by 5D pre-Maxwell field equations. Individual events trace out worldline trajectories as  advances monotonically, inducing pre-Maxwell fields by their motions, and moving under the influence of these fields. The resulting theory is thus integrable and well posed, governed canonically by a scalar Hamiltonian that generates evolution of a 4D block universe defined at  to an infinitesimally close 4D block universe defined at  C d  . This electrodynamics, and its extension to curved space and non-Abelian gauge symmetry, is the most general interaction possible in an unconstrained 8D phase space. We present examples that include classical Coulomb scattering, electrostatics, plane wave solutions, and radiation from a simple antenna. Standard Maxwell theory emerges from SHP as an equilibrium limit, reached by slowing the  -evolution to zero, or equivalently, by summing the contributions over  at each spacetime point. A feature of SHP not present in standard Maxwell theory is that under certain conditions, particles and fields may exchange mass dynamically, under conservation of total mass, energy, and momentum. As a result, pair processes such as electron-positron creation and annihilation are permitted in classical electrodynamics, implementing Stueckelberg’s original goal. Two processes that tend to restore a particle’s mass to its standard value are described, one a self-interaction along the event trajectory and the other a general result in statistical mechanics. Mass restoration of this type has been found in mathematical simulations of event trajectories. Beyond its usefulness as an approach to electrodynamics, the theory presented in this book provides the basis for a systematic, step-by-step progression from relativistic classical mechanics

x

PREFACE

to relativistic quantum mechanics, many-body theory, and quantum field theory. As an example, we discuss the correspondence of the fifth classical gauge potential to the Lorentz scalar potential used in quantum mechanical two-body problems to obtain manifestly covariant solutions for the bound state, scattering experiments, and relativistic entanglement in time. Similarly, we discuss the implications of the classical relativistic mechanics for quantum field theory. This classical framework will thus be of interest to workers in quantum theory, as well as those interested in its foundations. Martin Land and Lawrence P. Horwitz December 2019

xi

Symbols µ, ν, λ, ρ = 0, 1, 2, 3

4D spacetime indices

α, β, γ, δ = 0, 1, 2, 3, 5

5D formal indices (skipping 4)

ηµν = diag(–1, 1, 1, 1)

4D flat Minkowski metric

ηαβ = diag(–1, 1, 1, 1, η55)

Formal 5D flat Minkowski metric

c

Speed of light associated with x0 = ct

c5

Speed associated with x5 = c5t

{F, G} =

∂F ∂G ∂F ∂G – µ ∂x ∂pµ ∂pµ ∂xµ

[F, G] = F G – GF Dẋµ dẋµ = + Γ𝜈µρẋ𝜈 ẋρ Dτ dτ ∇αXβ =

∂Xβ + XγΓβγα ∂x∂

Γσµλ = ½gµv (∂σg𝜈λ + ∂λg𝜈σ – ∂𝜈gλσ) Φ(τ) = δ(τ) –

(ξλ)2 δ

(τ)

Poisson bracket Commutator bracket Absolute derivative Covariant derivative Christoffel symbol Interaction kernel for electromagnetic field

λ

Parameter with dimensions of time

ξ = ½ 1 + cc5 2 φ(τ) = λΦ–1 (τ)

Numerical factor Inverse function for kernel

PART I

Background

3

CHAPTER

1

Conceptual Approaches to Spacetime 1.1

POINT MECHANICS IN 4D SPACETIME

By one measure of success, Newtonian analytical mechanics continues to outshine the modern physics that has replaced it: the impact of its underlying physical picture on conventional notions of “reality” in the wider culture. Beyond science per se, this picture was absorbed into the foundations of Enlightenment philosophy, expanding into the modern humanities and social sciences, lending it an appearance of self-evident ordinariness. Thus, in his influential textbook Classical Mechanics, Herbert Goldstein introduces the physical framework—space, time, simultaneity, and mass—by writing [1, p. 1] that “these concepts will not be analyzed critically here; rather, they will be assumed as undefined terms whose meanings are familiar to the reader.” This familiarity is understood to flow from everyday experience with Newtonian objects fqn j n D 1;    ; N g defined as positions in an abstract Cartesian space fqni .t / j i D 1;    ; 3; n D 1;    ; N g of infinite extent, whose configuration develops through their functional dependence on the universal time t flowing forward uniformly. Indeed, the Newtonian picture is so central to conventional understandings of the “everyday” that more than one hundred years after Einstein’s annus mirabilis, it is the relativistic character of the Global Positioning System (GPS) found in billions of smartphones that feels distinctly unfamiliar, and “weirdness” still seems an apt term for quantum phenomena. Moreover, it is easy to forget that much of the Newtonian worldview seemed similarly “weird” to many in Newton’s day, especially the uniform linearity of time, a notion seemingly at odds with certain varieties of human experience outside the laboratory, more readily described in the language of nonuniform and cyclical flows of time. As early as 1908, Minkowski [2, p. 34] declared that: “space and time as such must fade away into shadow, and only a kind of union of the two will maintain its reality.” Although initially resistant to Minkowski’s tensor formulation, Einstein’s 1912 exposition of special relativity [2, p. 128] elaborates the advantages of taking the event in 4D spacetime as the fundamental object. Then, in formulating general relativity, the deconstruction of the Newtonian view of space was a crucial step, as emphasized by Einstein in his 1921 lecture at Princeton [3, pp. 2–3]. Arguing that direct experience must be the basis for physical concepts, he declared that, “the earth’s crust plays such a dominant role in our daily life in judging the relative position of bodies that it has led to an abstract conception of space which certainly cannot be defended.” That contemporary textbooks on relativity must still repeat Einstein’s identification of the spacetime

4

1. CONCEPTUAL APPROACHES TO SPACETIME

event as actual experience—superseding antique notions of infinite Euclidean space he deemed illusory—indicates not only the conceptual complexity of relativity, but also a continuing cultural disparity between modern physics and other realms of human knowledge. In 1937, Fock [4] generalized the Newtonian picture to relativistic form by writing events in 4D Minkowski spacetime as fxn ./ j  D 0;    ; 3; n D 1;    ; N g;

where xn0 ./ D ctn ./ represents the time registered for the event on the laboratory clock. These events describe a configuration that evolves as the scalar parameter  , identified by Fock with the proper time, advances monotonically. Writing xP  ./ D

dx  d

he showed that by minimizing the action   Z 1 e S D d mxP 2 C xP  A 2 c

(1.1)

for a point event in an electromagnetic potential A .x/, one obtains the classical relativistic equations of motion. Here and in the rest of the book we take the flat metric in 4D spacetime to be  D diag . 1; 1; 1; 1/ : Fock observed that the elimination of  in favor of t in these equations is generally difficult, but is easily accomplished for the free event satisfying xR  D 0 as   d x=d  u dx D D 0 x. P / D xP 0 ./; xP ./ D u0 ; u H) dt dt=d  u =c  for constant u D u0 ; u . Still, Fock’s generalization was not yet complete. In the Newtonian picture, a point particle whose position is described by the 3-vector trajectory x.t/ may follow any continuous curve. In 1941, Stueckelberg [5, 6] observed that the relativistic generalization described by Fock cannot represent all possible spacetime curves because the evolution parameter is identified with the proper time of the motion. In particular, any worldline whose time evolution reverses direction must cross the spacelike region that separates future-oriented trajectories from past-oriented trajectories. Therefore, in curves of this type the sign of xP 2 . / will change twice and the computed proper time interval 1q 1p 2 ds./ D  dx  dx  D xP . / d  c c

fails as a parameterization. Recognizing a physical meaning in curves of this type, Stueckelberg argued for their inclusion in relativistic mechanics, requiring the introduction of an independent evolution parameter  , analogous to the time t in the Newtonian picture, and related to

1.1. POINT MECHANICS IN 4D SPACETIME 2

2

2

2

the proper time s through the dynamical relation c ds . / D xP ./d  . In this, he followed Einstein’s approach, by deprecating an historical abstraction he saw as an obstruction to clear physical understanding of observed phenomena. Stueckelberg’s interest in general 4D curves can be understood from Figure 1.1 on page 5 (adapted from [5]). In his model, pair annihilation is observed in curve B when the worldline reverses its time direction, because laboratory apparatus registers two events (two points on the worldline) appearing at coordinate time t D t1 but none at t D t2 . The event first propagates forward in t (with xP 0 > 0) and then propagates backward in t (with xP 0 < 0), continuing to earlier times while advancing in space. Stueckelberg’s identification of the xP 0 < 0 piece of the trajectory with an antiparticle observed in the laboratory will be discussed in detail in Chapter 2.

x0 = ct τ = −∞

τ= ∞

τ= ∞

t = t2

t=0

x

t = t1

A

τ = −∞

B

τ = −∞

C

τ= ∞

Figure 1.1: Three types of worldline identified by Stueckelberg.

In a similar way, curve C represents pair creation as two events are observed at t D t2 but none at t D t1 . These curves may thus be seen as the smooth classical equivalent of a Feynman spacetime diagram, and the physical picture they present is known as the Feynman–Stueckelberg interpretation of antiparticles [7, 8].

5

6

1. CONCEPTUAL APPROACHES TO SPACETIME

Stueckelberg recognized that the standard Maxwell field F  .x/ alone would not permit c 2 ds 2 ./ D xP 2 d  2 to change sign and proposed a modified Lorentz force D xP  d xP     D C € xP xP D F  .x/g xP  C G  .x/ D d

(1.2)

 with local metric g and compatible connection € . He also included a new vector field G  .x/ 2 that is required to overcome conservation of xP , as seen through

D 2 D xP  xP D 2xP  D 2xP  G  .x/ D D

! 0:

G ! 0

In the absence of G  , spacetime curves are single-valued in x 0 and may, in principle, be reparameterized by the proper time of the motion. As a simple example we consider a particle in flat space in a constant electric field E D E zO and take G  D 0. Writing the velocity xP D .c tP; 0; 0; zP /, the equations of motion reduce to c tR D F 0i xP i D E zP

zR D F 30 xP 0 D cE tP

with solution 1 sinh E E which can be reparameterized by t as

z. / D

t./ D

z.t/ D z.0/ C

c .cosh E E

c p 1 C .Et/2 E

1/ C z.0/

 1 :

The velocities are tP. / D cosh E

zP . / D c sinh E



confirming that the mass is conserved with xP xP  D c 2 and so  2 "  2 # dt dz c 2 D c 2 tP2 zP 2 D c 2 1 ! d dt

 Pt D 1

v2 c2



1 2

;

where v D dz=dt . Now, by contrast, we consider the particle in a constant field G  D G zO and take E D 0 so that the equations of motion are c tR D 0

zR D G

with solution t./ D 

zD

1 2 1 G D Gt 2 : 2 2

In this case the mass decreases with  as xP  xP  D c 2

G2 2

and the motion may become spacelike (superluminal).

1.2. THE TWO ASPECTS OF TIME

1.2

THE TWO ASPECTS OF TIME

As seen in the previous section, curves B and C in Figure 1.1 cannot be parameterized by the coordinate time because they are double-valued in x 0 , and cannot be parameterized by the proper time of the motion s because s 2 becomes negative in the region of the time reversal point. Realization of the classical Feynman–Stueckelberg picture thus requires the introduction of a parameter  entirely independent of the spacetime coordinates—an irreducible chronological (historical) time, similar in its role to the external time t in nonrelativistic Newtonian mechanics. The simplicity of this picture in accounting for the observed phenomena of pair processes strongly supports the conclusion [9] that time must be understood as two distinct physical phenomena, chronology  and coordinate x 0 . A laboratory clock registers the coordinate time of an event occurrence much as a 3D array of detectors (meter sticks) registers the event’s coordinate position. The chronological time determines the order of occurrence of multiple events, with natural implications for relations of causality. Thus, when laboratory equipment reparameterizes the observed events along curve B in Figure 1.1 by x 0 , two events approaching one another will be observed at t D t1 and again at t D 0. But the underlying physics will be determined by field interactions at the locations of four distinct, ordered events, governed by a microscopic dynamics such as (1.2) and registered sequentially at t D t1 , t D 0, again at t D 0, and later at t D t1 . In this sense, there are no closed timelike curves in this picture. The so-called grandfather paradoxes, by which one may return to an earlier time to interfere with the circumstances that brought about ones own physical presence and agency, are thus resolved. We notice that the return trip to a past coordinate time x 0 must take place while the chronological time  continues to increase. Since the occurrence of event x  .1 / at 1 is understood to be an irreversible process that cannot be changed by a subsequent event occurring at the same spacetime location, x  .2 / D x  .1 / when 2 > 1 , the return trip cannot erase the earlier trajectory. This restriction is analogous to the conceptually simpler observation in nonrelativistic physics that a process may produce new events at any given moment, but cannot delete from the historical record events that occurred at an earlier moment. A more complex problem is the twin scenario, in which a traveler initially at rest in an inertial frame makes a trip of total distance d at speed v , so that the round trip time measured by a clock in this frame is t D d=v . The coordinates assigned to the traveler in the rest frame evolve as 8 < .c ; x0 C u / ; 0    =2 ; x D .ct; x/ D : .c ; x C u u / ; =2     0 where uD

dx d x dt D D v d dt d 

t D

dt  D  d

7

8

1. CONCEPTUAL APPROACHES TO SPACETIME

so that d D u D vt . The coordinates assigned to the traveler in a co-moving frame evolve as  x 0 D ct 0 ; x0 D .c; 0/ and so the elapsed time registered on the traveler’s clock is  D t= . This result is consistent with the usual presentation of the twin scenario.

1.3

THE “PROPER TIME” FORMALISM IN QED

Although this book focuses on relativistic classical mechanics, we make a brief digression into the application of spacetime parameterization methods by Schwinger and Feynman in developing quantum electrodynamics. In his 1951 calculation of vacuum polarization in an external electromagnetic field, Schwinger [10] represented the Green’s function for the Dirac field as a parametric integral and formally transformed the Dirac problem into a dynamical theory in which the integration variable acts as an independent time. Applying his method to the Klein– Gordon equation, we express the Green’s function as GD

1 eA=c/2 C m2

.p

i

so that writing 0

0

G.x; x / D hxjGjx i D i

Z

1

dse

i.m2 i /s

0

hxje

i.p eA=c/2 s

jx 0 i

(1.3)

the function G.x; x 0 I s/ D hx.s/jx 0 .0/ihxje

satisfies i

 @ hx.s/jx 0 .0/i D p @s

i.p eA=c/2 s

jx 0 i

e 2 A hx.s/jx 0 .0/i c

(1.4)

with the boundary condition lim hx.s/jx 0 .0/i D ı 4 .x

s!0

Schwinger regarded x  .s/ and   .s/ D p  .s/ that satisfy canonical relations

e  A .s/ c

x 0 /:

as operators in a Heisenberg picture

Œx  ;    D i

Œ  ;    D

dx  ds

i Œ  ; K D

i Œx  ; K D

ie  F c d  ; ds

(1.5) (1.6)

1.4. THE STUECKELBERG–HORWITZ–PIRON (SHP) FRAMEWORK

where K D .p

2

eA=c/ . Using (1.5) and (1.6) we find     e 2    xP .s/ D iŒx ; K D i x ; p A D 2 p c

e  A c

(1.7)

and so may perform the Legendre transformation   Z Z Z  1 2 e  ds L D ds xP p K D ds xP C xP  A 4 c whose classical limit takes the form of the Fock action (1.1). Although Schwinger found this representation useful because the scalar parameter s is necessarily independent of x  and xP  , and so respects Lorentz and gauge invariance, it is known [8] as the Fock-Schwinger “proper time method.” DeWitt [11] regarded (1.4) as defining the Green’s function for a Schrodinger equation  e 2 @ A (1.8) i s .x/ D K s .x/ D p s .x/ @s c which he used for quantum mechanical calculations in curved spacetime. Similarly, Feynman [12] used (1.8) in his derivation of the path integral for the Klein–Gordon equation. He 2 regarded the integration (1.3) of the Green’s function with the weight e im s as the requirement that asymptotic solutions of the Schrödinger equation be stationary eigenstates of the mass operator [email protected] . To pick the mass eigenvalue one extends the lower limit of integration in (1.3) from 0 to 1, and adds the requirement that G.x; x 0 I s/ D 0 for s < 0. Feynman noted that this requirement, equivalent to imposing retarded causality in chronological time s , leads to the Feynman propagator F .x x 0 / whose causality properties in t are rather more complex. Related issues of causality arise in classical relativistic field theory.

1.4

THE STUECKELBERG–HORWITZ–PIRON (SHP) FRAMEWORK

In 1973, Horwitz and Piron set out to systematically construct a manifestly covariant relativistic mechanics with interactions. They observed that the principal difficulties in previous efforts arose when attempting to define observables that respect a priori constraints associated with the presumed dynamics. For example, although it may seem natural to choose the proper time of the motion as the worldline parameterization, Stueckelberg showed that this choice prohibits a classical account of observed pair phenomena. Worse still, in the Fock–Schwinger formalism identification of spwith the proper time clashes with the formulation of quantum observables, p since dx 2 D xP 2 ds does not commute with x  , rendering the relations (1.5) and (1.6) difficult to interpret rigorously. A closely related question is reparameterization invariance. Although one might regard the parameter  as arbitrary, the Fock action (1.1) is clearly not invariant under  !  0 D f ./

9

10

1. CONCEPTUAL APPROACHES TO SPACETIME

because the Lagrangian is not homogeneous of first degree in the velocities. Invariance is often restored by replacing the quadratic term in the action with a first-order form such as Z  p  e S D d  mc xP 2 C xP  A c which leads to fixed particle masses p D

@L xP  e D mc p C A  @xP c xP 2

 p

!

e 2 A D c

m2 c 2

and restricts the system dynamics to the timelike region by imposing xP 2 < 0. Although the Fock action permits mass exchange, the mass of individual particles is fixed for interactions governed by Stueckelberg’s force law (1.2) when G  D 0. Similarly, in the Fock– Schwinger formalism (1.7) shows that xP 2 D 4K and thus corresponds to a classical constant of the motion. Thus, fixed mass is demoted from the status of a priori constraint to that of a posteriori conservation law for appropriate interactions. Rejecting such a priori restrictions, Horwitz and Piron postulate that classical particles and quantum states can be described in an unconstrained 8D phase space   E ;p x D .ct; x/ pD c with canonical equations xP  D

dx  @K D d @p

pP D

dp D d

@K ; @x 

where K is a scalar function that determines the system dynamics and its conservation laws. This framework is seen to include Newtonian mechanics by imposing the restrictions K D H.x; p/

t D

E

which leads to dx i @H D dt @pi

dp i D dt

@H @xi

@H dE D ; dt @t

where i D 1; 2; 3. To describe a free relativistic particle one may write KD

p2 2M

!

xP  D

p M

and

pP  D 0

so that dt =d  D E=M c 2 and d x=dt D pc 2 =E . In particular, for a timelike particle, xP 2 D

p2 D M2

m2 c 2 D constant; M2

1.5. BIBLIOGRAPHY

11

2

where the dynamical quantity m ./ is conserved because @[email protected] D 0. Similarly, a relativistic particle in a four-potential A .x/ is characterized by K D .p ce A/2 =2M with results comparable to the classical limit of the Fock–Schwinger system. Moreover, Horwitz and Piron considered a two-body problem with a scalar interaction characterized by the Hamiltonian KD

p2 p12 C 2 C V .jx1 2M1 2M2

where V .jx1

x2 j/ D V ./ D

p

.x1

x2 j/ ;

x2 /2

.t1

t2 /2

generalizes action at a distance to action at a spacelike interval. As in nonrelativistic mechanics, the center of mass and relative motion may be separated as KD

where 

p D

p  p P  P C C V ./; 2M 2m

P  D p1 C p2 M2 p1

 M1 p2 =M

M D M1 C M2 m D M1 M2 =M:

The center of mass motion is thus free, satisfying PP  D 0. For the relative motion, one has pP  D

@K D @x

@V @x

(1.9)

in which case we may identify @V [email protected] with G  in (1.2) so that individual particle masses are no longer necessarily fixed. In this framework, Horwitz and Arshansky found relativistic generalizations for the standard central force problems, including scattering [13, 14] and bound states [15, 16]. This formulation of the relativistic two-body problem can be extended to many bodies in the context of classical gauge theory, providing the basis for the SHP approach to classical relativistic mechanics.

1.5

BIBLIOGRAPHY

[1] Goldstein, H. 1965. Classical Mechanics, Addison-Wesley, Reading, MA. 3 [2] Einstein, A. 1996. Specielle Relativitätstheorie, George Braziller, New York, English and German on facing pages. 3 [3] Einstein, A. 1956. The Meaning of Relativity, Princeton University Press, Princeton, NJ. DOI: 10.4324/9780203449530. 3 [4] Fock, V. 1937. Physikalische Zeitschrift der Sowjetunion, 12:404–425. http://www.neo-cl assical-physics.info/uploads/3/4/3/6/34363841/fock_-_wkb_and_dirac.pdf 4

12

1. CONCEPTUAL APPROACHES TO SPACETIME

[5] Stueckelberg, E. 1941. Helvetica Physica Acta, 14:321–322 (in French). 4, 5 [6] Stueckelberg, E. 1941. Helvetica Physica Acta, 14:588–594 (in French). 4 [7] Halzen, F. and Martin, A. D. 1984. Quarks and Leptons: An Introductory Course in Modern Particle Physics, John Wiley & Sons, New York. DOI: 10.1119/1.14146. 5 [8] Itzykson, C. and Zuber, J. B. 1980. Quantum Field Theory, McGraw-Hill, New York. DOI: 10.1063/1.2916419. 5, 9 [9] Horwitz, L., Arshansky, R., and Elitzur, A. 1988. Foundations of Physics, 18:1159. 7 [10] Schwinger, J. 1951. Physical Review, 82(5):664–679. https://link.aps.org/doi/10. 1103/PhysRev.82.664 8 [11] DeWitt, B. 1965. Dynamical Theory of Groups and Fields, Gordon and Breach, New York. DOI: 10.1119/1.1953053. 9 [12] Feynman, R. 1950. Physical Review, 80:440–457. 9 [13] Arshansky, R. and Horwitz, L. 1989. Journal of Mathematical Physics, 30:213. 11 [14] Arshansky, R. and Horwitz, L. 1988. Physics Letter A, 131:222–226. 11 [15] Arshansky, R. and Horwitz, L. 1989. Journal of Mathematical Physics, 30:66. 11 [16] Arshansky, R. and Horwitz, L. 1989. Journal of Mathematical Physics, 30:380. 11

PART II

Theory

15

CHAPTER

2

Canonical Relativistic Mechanics 2.1

LAGRANGIAN AND HAMILTONIAN MECHANICS

In many ways, the picture underlying classical relativistic mechanics is a generalization of its Newtonian predecessor, with the replacements 8 9 4D spacetime 3D space ˆ > ˆ > < = ! chronological time  chronological time t ˆ > ˆ > : ; Lorentz covariance Galilean covariance made in an analogous canonical structure. A spacetime event x  refers to the 4-tuple .ct; x/ of coordinate observables that can, in principle, be measured by a clock and an array of spatially arranged detectors in a laboratory.1 Each event occurs at a chronological time  such that for 2 > 1 the event x  .2 / is said to occur after the event x  .1 /. Event occurrence is an irreversible process—a given event cannot be influenced by a subsequent event, although laboratory equipment may present the history of events in the order of their recorded values of x 0 D ct . Following Fock and Stueckelberg, we consider a relativistic particle to be a continuous sequence of events traced out by the evolution of a function x  ./ as  proceeds monotonically from 1 to 1. The chronological time  is taken to be an external universal parameter, playing a role similar to that of t in Newtonian physics. In Stueckelberg–Horwitz–Piron (SHP) theory, event dynamics are defined on an unconstrained 8D phase space .x  ; p  / by the canonical equations xP  D

dx  @K D d @p

pP  D

dp  D d

@K ; @x

(2.1)

where K.x; p; / is a Lorentz invariant Hamiltonian. This framework thus inherits the canonical structure of Newtonian analytical mechanics, with the additional complexity of Lorentz covariance. Defining Poisson brackets as fF; Gg D 1 Although

@F @G @x  @p

@F @G @p @x 

this description oversimplifies the measurement process, it will be sufficient here.

16

2. CANONICAL RELATIVISTIC MECHANICS

we have for any function on phase space, dF @F dx  @F dp @F @F @K D  C C D  d @x d  @p d  @ @x @p

@F @K @F @F D fF; Kg C C @p @x @ @

generalizing the result in nonrelativistic mechanics. Since fK; Kg  0, the Hamiltonian is a constant of the motion unless K depends explicitly on  . Because of its unconstrained canonical structure, the conditions for the Liouville–Arnold theorem apply: the 4D system is integrable— solvable by quadratures—if it possesses 8 independent conserved quantities Fi ; i D 1;    ; 8 satisfying fK; Fi g D 0 and fFi ; Fj g D 0. Performing the Legendre transformation from the Hamiltonian to the Lagrangian L D xP  p

variation of the action ıS D ı

Z

K;

d  L .x; x; P / D 0

leads to the Euler–Lagrange equations d @L d  @xP 

@L D0 @x 

in familiar form. Under transformations x ! x 0 D f .x/ that leave the action invariant, the Noether theorem follows in the usual manner, so that for infinitesimal variation ıx we find   d @L  ıx D0 d  @xP  leading to the conserved quantity [email protected][email protected]  / ıx  . In particular, since L is a scalar invariant under Lorentz transformations ƒ with antisymmetric generators M x 0 D ƒx

the quantity

!

ıx D x 0

x ' ı!  M x

   l D @[email protected]  M x D x p

x p

is conserved, and the Poisson bracket relations ˚ l ; l D g l C g l C g l C g l express the Lie algebra of the Lorentz group. The components of l can be split into Li D ij k x j p k

Ai D x0 pi

so that l 2 D 2 L2

A2



xi p0

(2.2) (2.3)

2.2. THE FREE RELATIVISTIC PARTICLE

17

generalizes the conserved nonrelativistic total angular momentum in central force problems. We write the velocity of a general event as   x. P / D c tP; xP D u0 . /; u. / with no restrictions on its orientation—xP may be timelike, lightlike, or spacelike. In the timelike case, an observer can boost to a co-moving frame in which  x./ P D c tP; 0

2 and so by Lorentz invariance, xP 2 D c tP in any instantaneous frame. Still, while tP D 1 may be a dynamical result in the rest frame, it is not an a priori constraint.

2.2

THE FREE RELATIVISTIC PARTICLE

As in the earlier work of Fock and Schwinger, the free particle Hamiltonian is taken to be KD

p2 2M

generalizing the nonrelativistic form. Applying the canonical equations (2.1), the equations of motion are @K p @K xP  D D pP  D D0 @p M @x with solution

p  M as seen previously in Section 1.4. From p  D M xP  , a Legendre transformation leads to the free particle Lagrangian 1 L D xP  p K D M xP 2 2 and so naturally, 1 p2 L D M xP 2 D K D D constant: 2 2M Given the absence of constraints, the sign of p 2 depends on its spacetime orientation. Introducing the mass m2 D p 2 =c 2 for a timelike event, we have xP 2 D m2 c 2 =M 2 and we generally take m D M so that tP D m=M D 1 in the rest frame. For this case, "  2 #  dx 1 2 2 0 2 2 P2 c D xP xP D c t 1 ! tP D ˙ p D ˙ ; dt 1 ˇ2 x  D x0 C u  D x0 C

where ˇ D v=c , v D d x=dt , and is the usual relativistic dilation factor.

18

2. CANONICAL RELATIVISTIC MECHANICS

For a timelike free event evolving forward in coordinate time (tP  1), we choose tP D C and recover the standard representation of relativistic velocity:    E p 0 xP D u ; u D .c; v/ D ; ; Mc M where E > 0. Choosing tP D produces a solution of particular interest to Stueckelberg, the timelike free event evolving backward in coordinate time (tP  1),   jEj p D .c; v/ D ; Mc M describing a negative energy event tracing out a trajectory that when reordered by the laboratory clock describes an antiparticle. The general solution xP  D p  =M for a free particle can also accommodate tachyon (p 2 > 0) and lightlike (p 2 D 0) worldlines with no loss of generality.

2.3

THE RELATIVISTIC PARTICLE IN A SCALAR POTENTIAL

Adding a scalar potential V .x/ to the Hamiltonian KD

p2 C V .x/ 2M

leads to the equations of motion xP  D

@K p D @p M

@K D @x 

@V : @x 

M xR  D

@V : @x

pP D

Equivalently, the Lagrangian formulation is LD d @L d  @xP 

@L D0 @x 

1 M xP 2 2

V .x/

!

As seen in (1.9), this problem may describe the reduced interaction of a two-body problem in relative coordinates. As a simplified but suggestive model, we consider the scalar potential V .x/ D Ma  x;

2.3. THE RELATIVISTIC PARTICLE IN A SCALAR POTENTIAL

19

where a is a constant timelike vector. We choose a frame in which a D .cg; 0; 0; 0/

V .x/ D

!

M cgx 0

providing an analogy in the time direction to the approximate nonrelativistic gravitational field close to earth. The equations of motion are M xR  D

@V D @x

Ma

becoming in this frame M xR 0 D

M xR D 0

Mcg

with solution

1 2 g x ./ D x0 C u0 ; 2 where g , t0 and tP0 are taken as positive constants. We recognize this parabolic trajectory as describing the pair annihilation process shown in curve B of Figure 1.1. For simplicity, we now take t0 D 0 and x0 D 0. Thus, the event velocity is t . / D t0 C tP0 

tP . / D tP0

xP ./ D u0 D constant

g

and the trajectory reverses t -direction at t D tP02 =2g when  D tP0 =g . From p D

@L D M xP  @xP 

!

p0 D

E D Mc tP c

!

E D Mc 2 tP

we see that the event propagates forward in t with E > 0 for  <  and backward in t with E < 0 for  >  . The  >  portion of the trajectory corresponds to Stueckelberg’s interpretation of an antiparticle. The velocity remains timelike except near 0 in the interval c 2 tP0

g

2

u20 < 0

!



ju0 j ju0 j <  <  C ; cg cg

where it becomes spacelike (tachyonic). The event trajectory recorded in the laboratory may be reordered according to t . Thus, at coordinate time t D 0, two events will be recorded, a positive energy event that occurred at  D 0 and a subsequent a negative energy event at  D 2 . From this perspective, the two pieces of the worldline appear as a pair of events approaching one another and mutually annihilating at t D t , with no events recorded with t > t . In a similar way, taking tP0 and g to be negative constants, this solution describes a pair creation process. Although this account of pair processes is not physically realistic, we will present a more accurate description in Section 4.6 using the full apparatus of classical SHP electrodynamics.

20

2. CANONICAL RELATIVISTIC MECHANICS

2.4

TWO-BODY PROBLEM WITH SCALAR POTENTIAL

As we showed in Section 1.4, the two-body problem with scalar interaction can be written as an equivalent one-body problem KD

p12 p2 C 2 C V .x1 2M1 2M2

x2 / D

P  P p  p C C V .x/; 2M 2m

(2.4)

where the center of mass motion satisfies PP  D 0. Arshansky [1] studied classical problems of this type (for the extension to quantum mechanics, see [2]), generalizing the standard nonrelativistic central force problems by taking p  p  V .x/ D V x2 ! V .x/ D V x2 c 2 t 2 for spacelike separations, x 2 > 0. Restriction to the spacelike region can be accomplished through a representation in hyperspherical coordinates of the type 2 3   sin  cos  sinh ˇ rO D 4 sin  sin  5 xD rO 2 D 1: cosh ˇ rO cos  But it was found that reasonable solutions lie in a subspace of the full spacelike region, found by choosing a spacelike unit vector n and solving the equations of motion in the O(2,1)-invariant restricted space ˚ x 2 x j Œx .x  n/n2  0 for which the component of x orthogonal to n is itself spacelike. Arshansky has described this as a classical case of spontaneous symmetry breaking leading to a lowering of the energy spectrum. Taking n D .0; 0; 0; 1/ this region has the representation 2 3   sinh ˇ sin  qO xD qO D 4 cosh ˇ cos  5 qO 2 D 1: (2.5) cos  cosh ˇ sin  In addition to the O(3,1) invariant l 2 defined in (2.3), the O(2,1) invariant N 2 D L23

A21

A22

with components defined in (2.2) is also conserved and plays a role in characterizing the solutions. In these coordinates, the first integrals KD

p2 1 l2 C V .x/ D M P2 C C V ./ D  2M 2 2M2

2.5. MANY-BODY PROBLEM AND STATISTICAL MECHANICS

21

which is cyclic in ˇ and  , and l 2 D M 2 4 P 2 C

N2 sin2 

provide a separation of variables. As in nonrelativistic mechanics, but with an additional degree of freedom, solutions can be found from the four first-order equations ˇP P

D 0

D 0 s

P D

P

D

2 M

 

1 M2 . /

s

V ./

l2

l2 2M2

N2 sin2 



!

D Z

!

Z

r

d D M2 . /

d 2 M

Z

 

V ./

q l2

l2 2M2



d N2 sin2 

providing an example of Liouville integrability. In the quantum case, Horwitz and Arshansky [2–4] solved the bound state problem, leading to a mass spectrum coinciding with the non-relativistic Schrodinger energy spectrum. For small excitations, the corresponding energy spectrum is that of the non-relativistic Schrodinger theory with relativistic corrections.

2.5

MANY-BODY PROBLEM AND STATISTICAL MECHANICS

The many body problem and classical and quantum statistical mechanics, along with applications to bound states, scattering, and relativistic statistical mechanics, are covered extensively in [5]. Here we provide a brief introduction to the subject as preparation for discussion of mass stabilization in Section 4.7.2. The generalization of (2.4) to N -bodies is KD

N X pi  pi  i D1

2Mi

C V .x1 ; x2 ; : : : ; xN /

for which case one may define center of mass coordinates P X Mi xi  M D Mi X D i M i

P D

and relative coordinates pOi D pi

.Mi =M / P 

xO i D xi

X

X i

pi

22

2. CANONICAL RELATIVISTIC MECHANICS

satisfying

X i

pOi D 0

X i

Mi xO i D 0

for the phase space. The Poisson brackets are ˚   xO i ; pOj D  ıij

fX  ; P  g D 

Mj =M



and although the relative coordinates do not satisfy canonical Poisson bracket relations, these relations become canonical in the thermodynamic limit N ! 1 for which Mj =M ! 0. The invariant Hamiltonian takes the form KD

 P  P X pOi pOi C V .x1 ; x2 ; : : : ; xN / C 2M 2Mi i

so that for relative forces, V .x1 ; x2 ; : : : ; xN / D V .xO 1 ; xO 2 ; : : : ; xO N / and the center of mass motion decouples from the interacting system. The equations of motion P XP  D M P xOP i D M

PP  D 0 @K pOPi D D @xO i

@V @xO i

are canonical in form. In statistical mechanics, one regards the N events as elements in a relativistic Gibbs ensemble. As a generalization of the nonrelativistic formalism, we set a mass shell condition K D  , however this is not a sufficient restriction because integration over the hyperbolic 4D phase space may run to infinity for finite p  p . We must therefore also set an energy shell condition P 0 i Ei D E , where Ei D pi (we take c D 1 in this section). Fixing the energy shell is equivalent to choosing a Lorentz frame for the system relative to the measurement apparatus, without which we could not give meaning to the idea of temperature. The microcanonical ensemble of events at fixed energy is then defined as Z €.; E/ D d ı.K /ı.†Ei E/; where d D

Y i

d 4 pi d 4 xi D d 4N p d 4N x

is the infinitesimal volume element in the phase space of the many-body system. The entropy and temperature are given by S.; E/ D ln €.; E/

T

1

D

@S.; E/ ; @E

2.6. BIBLIOGRAPHY

23

where we take the Boltzmann constant kB D 1. We may construct a canonical ensemble by extracting a small subensemble €s from its environment €b (the bath), and summing over all possible partitions of energy and mass parameter between the subensemble and the bath, Z   €.; E/ D d  0 dE 0 €b   0 ; E E 0 €s  0 ; E 0 ; where both mass and energy may be exchanged. Similarly, a grand canonical ensemble may be constructed by summing over all possible exchanges of event number and volume between the subensemble and the bath. We return to the relativistic statistical mechanics in Section 4.7.2 to show that a particle represented as an ensemble of events possesses a mass that tends toward a stable equilibrium, even under perturbations.

2.6

BIBLIOGRAPHY

[1] Arshansky, R. 1986. The classical relativistic two-body problem and symptotic mass conservation. Tel Aviv University preprint TAUP 1479-86. 20 [2] Horwitz, L. P. 2015. Relativistic Quantum Mechanics, Springer, Dordrecht, Netherlands. DOI: 10.1007/978-94-017-7261-7. 20, 21 [3] Arshansky, R. and Horwitz, L. 1989. Journal of Mathematical Physics, 30:66. [4] Arshansky, R. and Horwitz, L. 1989. Journal of Mathematical Physics, 30:380. 21 [5] Horwitz, L. P. and Arshansky, R. I. 2018. Relativistic Many-Body Theory and Statistical Mechanics, 2053–2571, Morgan & Claypool Publishers. http://dx.doi.org/10.1088/9781-6817-4948-8 DOI: 10.1088/978-1-6817-4948-8. 21

25

CHAPTER

3

Classical Electrodynamics 3.1

CLASSICAL GAUGE TRANSFORMATIONS

Historically, classical electrodynamics proceeded from experiment to theory. The Maxwell equations (1860s) were initially posed as a summary of discoveries in the laboratory, including the Cavendish experiments in electrostatics (1770s), Coulomb’s studies of electric and magnetic forces (1780s), and Faraday’s work on time-varying fields (1830s). But the importance of Maxwell’s mathematical theory was not fully recognized [1, p. xxv] until its prediction of electromagnetic waves traveling at the speed of light was verified by Hertz in 1888. It was the successful incorporation of optics into electrodynamics that provoked Einstein to study the spacetime symmetries underlying Maxwell theory in 1906 and led Fock to associate potential theory with gauge symmetry in 1929 [2]. Building on the success of such considerations, the Standard Model of fundamental interactions was developed by requiring invariance under more complex symmetry groups, as were the many candidates for a successor theory. As discussed in Chapter 1, Stueckelberg recognized that the perception of a worldline as a sequence of events following dynamical laws could lead to pair annihilation processes in classical mechanics. Such worldlines moving in the positive or negative direction of the Einstein time t should be parameterized by an invariant  , progressing monotonically in the positive direction. Horwitz and Piron generalized this notion to make the parameter  universal, and in this way were able to study the relativistic classical dynamics of many body systems. In this chapter, we approach classical electrodynamics in a similar manner. Instead of restricting the formalism to the known features of Maxwell theory, we begin with the Lorentz invariant Lagrangian description of a free event and introduce the maximal U(1) gauge invariance applicable to the action, leading to a generalization of the Stueckelberg force law (1.2). We construct an action for the field strengths, again applying general principles of Lorentz and gauge invariance, and obtain  -dependent Maxwell-like equations. The resulting framework can be understood as a microscopic theory of interacting events that reduces to Maxwell electrodynamics in a certain equilibrium limit. Thus, as we explore SHP theory, our points of comparison will be with the Maxwell theory we hope to generalize. The action for a free event

SD

Z

d L D

Z

d

1 M g xP  xP  2

26

3. CLASSICAL ELECTRODYNAMICS

with g .x/ a local metric, is invariant under the addition of a total  -derivative L

! LC

d @ @ ƒ .x;  / D L C xP   ƒ .x;  / C ƒ .x;  / d @x @

(3.1)

on condition that ƒ .x;  / vanishes at the endpoints of the action integral. In analogy to x 0 D ct , it is convenient to introduce the notation x 5 D c5 

xP 5 D c5

@5 D

1 @ c5 @

and adopt the convention ˛; ˇ; ; ı;  D 0; 1; 2; 3; 5

; ; ; ;  D 0; 1; 2; 3;

where we skip ˛ D 4 to avoid confusion with older notations for ct . In this notation (3.1) can be written in the compact form L

! L C xP ˛ @˛ ƒ .x;  /

suggesting a five dimensional symmetry acting as x 0˛ D L˛ˇ x ˇ . But we insist that in the presence of matter, x  and  belong to vector and scalar representations of O(3,1). Still, free fields may enjoy a 5D symmetry, such as L 2 O(4,1) for metric

˛ˇ D diag. 1; 1; 1; 1; C1/

L 2 O(3,2) for metric

˛ˇ D diag. 1; 1; 1; 1; 1/

which contain O(3,1) as a subgroup. Nevertheless, the higher symmetry does play a role in wave equations, much as nonrelativistic pressure waves satisfy   1 @2 2 p .x; t / D 0 v D speed of sound r v 2 @t 2 suggesting a 4D symmetry not physically present in the theory of acoustics. In light of (3.1), we introduce the five potentials a˛ .x;  / into the Lagrangian and note that the action   Z   Z  1 e ˛ 1 e    S D d M xP xP  C xP a˛ D d  M xP xP  C xP a C c5 a5 2 c 2 c is invariant under the 5D local gauge transformation a˛ .x;  /

! a˛0 .x;  / D a˛ .x;  / C @˛ ƒ .x;  / :

As a brief quantum aside, we may write the canonical momentum @L e 1  p D  D M xP  C a ! xP  D p @xP c M

e  a c

(3.2)

3.2. LORENTZ FORCE

27

to find the Hamiltonian

1   e   e  ec5 p a p a a5 2M c c c showing that under (3.2) the Stueckelberg–Schrodinger equation  ec5  e 2 1  .x;  / D i„@ .x;  / D K .x;  / ! i„@ C a5 p a c 2M c enjoys the symmetry [3]   ie .x; / ! exp ƒ.x;  / .x; / „c K D p xP 

LD

.x; /

expressing the local U(1) gauge transformation in the familiar form introduced by Fock.

3.2

LORENTZ FORCE

To study the interaction of an event with the gauge potentials a˛ .x; /, we write the Lagrangian as e ec5 1 a5 (3.3) L D M g xP  xP  C g xP  a C 2 c c with local metric g .x/. Applying the Euler–Lagrange derivative to the kinetic term we obtain     @ 1 1  @g d @     @g  Mg xP xP D M xR C M g g xP  xP  : g d  @xP  @x  2 @x  2 @x  Using the symmetry of the first term in parentheses under  $    @g   1  @g  @g  g  x P x P D g C g xP  xP  @x  2 @x  @x 

we find g

where





d @ d  @xP 

@ @x 



1 D xP   Mg xP  xP  D M xR  C M € xP  xP  D M ; 2 D

  1  @g @g @g D g C 2 @x  @x  @x  is the standard Christoffel symbol and D xP  =D is the absolute derivative of xP  along a geodesic. For the interaction term       @xP   @ d @ d @   g xP a C c5 a5 D g  a g xP  a C c5 a5 d  @xP  @x  d @xP @x  da D xP  @ a c5 @ a5 d  D xP  @ a @ a C @ a c5 @ a5  D xP  @ a @ a C xP 5 [email protected] a @ a5 /  €

28

3. CLASSICAL ELECTRODYNAMICS

so that the Lorentz force is  M xR  C M € xP  xP  D

D

e   g xP @ a c

 @ a C xP 5 [email protected] a

@ a5 /



 e  g f xP  C f5 xP 5 ; c

(3.4)

where we have introduced f˛ˇ .x;  / D @˛ aˇ .x;  /

@ˇ a˛ .x;  /

(3.5)

as the gauge invariant field strength tensor. We note that (3.4) reduces to the Stueckelberg force (1.2) if we put f5 .x;  / ! G .x/

f .x;  / ! F .x/

and so may be said to generalize the Stueckelberg ansatz, for which it provides a foundational justification in gauge theory. In analogy to Maxwell theory, we may take a D 0 in (3.3) and approximate .ec5 =c/a5 .x;  / '

.e=c/.x/ D V .x/

to identify the fifth potential with the scalar potential V .x/ used in Section 2.3. We put the Lorentz force into a more compact form as D xP  e  D M xR  C M € xP  xP  D g  f˛ xP ˛ D c

(3.6)

and notice that the index  runs to 3, while the index ˛ runs to 5. The fifth equation is found by evaluating D D



1 M xP 2 2



D

xP  M

D xP  D D

xP 

 c5 e  g f xP  C f5 xP 5 D ef5 xP  ; c c

(3.7)

where we used f55  0. This expression shows that the f5 field, expressing the action of a5 .x;  / and the  -dependence of a .x;  /, permits the non-conservation of xP 2 and must play a role is classical pair processes. We will see in Section 3.6 that this non-conservation represents an exchange of mass between particles and fields, where total mass-energy-momentum of particles and fields is conserved. Notice that the mass exchange is scaled by the factor c5 =c . As we shall see in Section 4.8, this factor is a continuous measure of the deviation of SHP electrodynamics from Maxwell theory, which is recovered in the limit c5 =c ! 0. We will generally take this factor to be small but finite.

3.3. FIELD DYNAMICS

3.3

29

FIELD DYNAMICS

To construct a dynamical action for the fields we first rewrite the interaction term as Z 1 xP ˛ a˛ .x;  / ! XP ˛ a˛ .x;  / ! d 4 x j ˛ .x; /a˛ .x; /; c where the event current j ˛ .x;  / D c XP ˛ ./ı 4 x

X./



(3.8)

is defined at each  with support restricted to the spacetime location of the event at x D X./. The standard Maxwell current, representing the full worldline traced out by evolution of the event X. /, is found from Z Z  J  .x/ D d  j  .x; / D c d  XP  . /ı 4 x X./ (3.9) as seen for example in [4, p. 612]. This integration is called concatenation [5] and can be understood as the sum at x of all events occurring at this spacetime location over  . The choice of kinetic term for a field theory is guided by three principles: it should be a Lorentz scalar, gauge invariant, and simple (bilinear in the fields with the lowest reasonable order of derivatives). From experience with the Maxwell theory, we first consider the electromagnetic action containing a term of the form f ˛ˇ .x;  /f˛ˇ .x;  / originally proposed by Saad et al. [3]. However, low-energy Coulomb scattering trajectories calculated in this theory [6] cannot be reconciled with Maxwell theory or experiment (we return to this point in Section 4.1). A satisfactory theory is found by generalizing the kinetic term so that the action takes the form [7]  Z Z i e ˛ ds 1 h ˛ˇ 4 Sem D d xd  2 j .x; /a˛ .x; / f .x; /ˆ. s/f˛ˇ .x; s/ ; c  4c where  is a parameter with dimensions of time. This may be written more compactly as   Z 1 ˛ˇ e ˛ 4 f .x;  / f˛ˇ .x; / ; (3.10) Sem D d xd  2 j .x; /a˛ .x;  / c 4c ˆ where fˆ˛ˇ .x; /

D

Z

ds ˆ. 

s/f ˛ˇ .x; s/

is a superposition of fields, non-local in  . The field interaction kernel is chosen to be Z i d h 2 00 ˆ./ D ı . / ./ ı . / D 1 C ./2 e i ; 2 where the factor

  c 2  1 5 1C D 2 c

(3.11)

(3.12)

30

3. CLASSICAL ELECTRODYNAMICS

insures that the low-energy Lorentz force agrees with Coulomb’s law. Integrating by parts the term in (3.10) produced by the factor ı 00 . s/ in (3.11), Z Z   ˛ˇ 00 d  ds @ f ˛ˇ .x; / ı 0 . s/f˛ˇ .x; s/ d ds f .x;  /ı . s/f˛ˇ .x; s/ D Z   D d  @ f ˛ˇ .x; / @ f˛ˇ .x;  / so that Sem D

Z



e d xd  2 j ˛ a˛ c 4

   1 ˛ˇ ./2  ˛ˇ f f˛ˇ C @ f @ f˛ˇ 4c 4c

(3.13)

and the higher derivative in  is seen to break the 5D symmetry of f ˛ˇ f˛ˇ to O(3,1), leaving the gauge invariance of f ˛ˇ unaffected. It remains necessary to give meaning to raising and lowering the 5-index through f 5˛ D 55 f5˛ . Expanding f ˛ˇ f˛ˇ D f  f C 255 f5  f5

we see that we may interpret 55 D ˙1 as the sign of the f52 term in the action, sidestepping any necessary interpretation as an element in a 5D metric. Variation of the electromagnetic action (3.10) with respect to the potentials a˛ .x;  / leads to the field equations e (3.14) @ˇ fˆ˛ˇ .x;  / D j ˛ .x;  / c describing a non-local superposition of fields fˆ˛ˇ .x; / sourced by the local event current j ˛ .x; /. In order to remove ˆ. / from the LHS, we use the inverse function Z d e i  1 jj= e (3.15) './ D ˆ 1 ./ D  D 2 1 C ./2 2 which satisfies Z

ds ' . 

s/ ˆ .s/ D ı. /

Integrating (3.14) with (3.15), we obtain Z e @ˇ f ˛ˇ .x;  / D ds ' . c

Z

d ' . / D 1: 

(3.16)

e ˛ j .x;  / c '

(3.17)

s/ j ˛ .x; s/ D

which describes a local field sourced by a non-local superposition of event currents. While the event current (3.8) has sharp support at one spacetime point, the current Z  ds j sj= P ˛ ˛ X .s/ı 4 x X.s/ (3.18) j' .x;  / D c e 2

3.4. ENSEMBLE OF EVENT CURRENTS

31

can be interpreted as the current induced by a smooth ensemble of events distributed in a neighborhood  of a spacetime point. This interpretation is discussed further in Section 3.4. Because the field strengths are derived from potentials, the Bianchi identity (3.19)

@˛ fˇ C @ f˛ˇ C @ˇ f ˛ D 0

holds. We see that (3.17) and (3.19) are formally similar to Maxwell’s equations in 5D, and are known as pre-Maxwell equations. Expanding the field equations in 4D tensor, vector and scalar components, they take the form @ f 

e 1 @ 5 f D j' c5 @ c

@ f C @ f C @ f D 0

@ f 5 D @ f5

e 5 j c '

@ f5

1 @ C f D 0 c5 @

(3.20)

which when compared with the 3-vector form of Maxwell’s equations r B

e 1 @ ED J c @t c

r BD0

e 0 J c 1 @ r EC BD0 c @t r ED

suggest that f 5 plays the role of the electric field, whose divergence provides the Gauss law, and f  plays the role of the magnetic field. It follows from (3.17) that @˛ j ˛ D @ j  C

1 @ ˛ j D0 c5 @

(3.21)

so that j 5 .x;  / D c5  .x;  / plays the role of an event density, and Z Z d d 4 x  .x;  / D d 4 x @ j  .x;  / D 0 d shows the conservation of total event number over spacetime, in the absence of injection/removal of events at the boundary by an external process.

3.4

ENSEMBLE OF EVENT CURRENTS

The function '. / smooths the current defined sharply at the event, over a range determined by . For  very large, '. / ' 1 for all  , producing a current ensemble associated with a large section of the worldline, approximating the standard Maxwell current. For  ! 0, we approach the limit '. /= ! ı. / which restricts the source current to the instantaneous current produced by a single event.

32

3. CLASSICAL ELECTRODYNAMICS

Rewriting the current (3.18) as Z Z 1 ˛ ˛ j' .x;  / D ds' . s/ j .x; s/ D ds e 2

jsj=

j ˛ .x; 

s/

we recognize j'˛ .x;  / as a weighted superposition of currents. Each of these currents originates at an event X  . s/ along the worldline, occurring before or after the event X  . /, depending on the displacement s . The superposition may thus be seen [8] as the current produced by an ensemble of events in the neighborhood of X  ./, a probabilistic view encouraged by the functional form of the weight '.s/. Consider a Poisson distribution describing the occurrence of independent random events produced at a constant average rate of 1= events per second. The average time between events is  and the probability at  that the next event will occur following a time interval s > 0 is just '.s/= D e s= =, which may be extended to positive and negative values of the displacement. The current j'˛ .x;  / is constructed by assembling a set of event currents j ˛ .x;  s/ along the worldline, each weighted by '.s/, the probability that the event occurrence is delayed from  by an interval of at least jsj. We will see that the causality relations embedded in the pre-Maxwell equations select the one event from this ensemble for which an interaction occurs at lightlike separation, preserving relativistic causality. We may also regard j'˛ .x;  / as a random variable describing the probability of finding a current density at x at a given  . The correlation function for the event density is Z ˝ ˛ 1  ./  .s/ D d 4 x  .x;  /  .x; s/ ; N where N is a normalization. In the case of an event X  . / D u  with constant velocity u , the unsmoothed event current (3.8) leads to Z ˝ ˛ c 2 ı 3 .0/ c2 d 4 x ı 4 .x u / ı 4 .x us/ D ı. s/  . /  .s/ D N ju0 jN showing that the currents at differing times  ¤ s are uncorrelated. For the ensemble current defined in (3.18) the correlation becomes Z ˝ ˛ c2   ' ./ ' .s/ D d  0 ds 0 d 4 x '.  0 /'.s s 0 /ı 4 x u 0 ı 4 x us 0 N Z c 2 ı 3 .0/ D d  0 '.  0 /'. 0 s/ ju0 jN Z c 2 ı 3 .0/ 0 0 D 2 0 d  0 e j  j= j sj= : 4 ju jN Taking  > s and evaluating the integral over three intervals punctuated by s ,  0 , and  leads to   ˝ ˛  s c 2 ı 3 .0/ 1C e . s/= ' . / ' .s/ D 4ju0 jN 

3.5. THE 5D WAVE EQUATION AND ITS GREEN’S FUNCTIONS

33

with a time-dependence characteristic of an Ornstein–Uhlenbeck process with correlation length . This correlation suggests that the current ensemble may be seen as the set of instantaneous currents induced by an event undergoing a Brownian motion that produces random displacement in  under viscous drag along the worldline.

3.5

THE 5D WAVE EQUATION AND ITS GREEN’S FUNCTIONS

Using (3.5) to expand (3.17) leads to the wave equation     55 2 ˛ ˛ˇ ˛ ˇ ˇ ˛ ˇ ˛  @ˇ f D @ˇ @ a @ a D @ˇ @ a D @ @ C 2 @ a D c5

e ˛ j ; c '

(3.22)

where we work in the 5D Lorenz gauge @ˇ aˇ D 0. As discussed above, this form partially preserves 5D symmetries broken by the O(3,1) symmetry of the event dynamics. A Green’s function solution to   55 2  @ @ C 2 @ G.x;  / D ı 4 .x/ ı . / c5 can be used to obtain potentials in the form Z e ˛ a .x;  / D d 4x0d  0 G x c

x0; 

   0 j'˛ x 0 ;  0 :

(3.23)

The Green’s function can be expressed as the Fourier transform Z Z ik˛ x ˛ 1 1 1 5 e G.x; / D d k D d 4 k d  e i.kxCc5 55  / 2 5 ˛ 5 .2/ C k k˛ .2/ C k C 55  2 over an appropriate contour C . To break the 5D symmetry present in the wave equation, we leave the  integration for last, writing Z  1 G.x; / D d  e ic5 55   x; 55  2 ; 2 where .x; m2 / is Schwinger’s principal part Green’s function [9] associated with the Klein– Gordon equation for a particle of mass m. Carefully repeating the steps of Schwinger’s derivation, while allowing 55 to be positive or negative, we are led to  Z  ˇ ˇ1=2    @ 1 ic5 55  2 2 : J0  ˇx 2 ˇ G.x;  / D d e ı x C  55 x @x 2 .2/2 Now performing the  integration, the pre-Maxwell Green’s function becomes G.x;  / D

1 ı.x 2 /ı./ 2

c5 @ 1 . 55 g˛ˇ x ˛ x ˇ / q 2 2 @x 2 55 g˛ˇ x ˛ x ˇ

(3.24)

34

3. CLASSICAL ELECTRODYNAMICS

so that both terms have units of distance 2  time 1 . The first term contains the O(3,1) scalars x 2 and  separately, and is called GMaxwell . It has support at instantaneous  and, as in Maxwell theory, along lightlike separations. The second term, called GCorrelation , has support determined by 8  x 2 C c52  2 D c 2 t 2 x2 c52  2 > 0 ; 55 D 1 < 55 ˛ˇ x ˛ x ˇ D  : 2 x c52  2 D x2 c 2 t 2 c52  2 > 0 ; 55 D 1

on timelike separations for 55 D 1 and spacelike separations for 55 D 1. Contributions from GCorrelation are generally smaller than those of GMaxwell and drop off faster with distance from the source. To avoid singularities, particular care must be taken in handling the distribution functions. The derivative in GCorrelation produces two singular terms  ! ı x 2 c52  2 c5 1 . x 2 c52  2 / GCorrelation .x;  / D 1=2 2 2 2 x 2 c 2  2 3=2 x2 c2 2 5

5

but these singularities cancel when first combined under integrals of the type (3.23) prior to applying the limits of integration. This order of operations expresses an aspect of the boundary conditions posed by Schwinger in deriving the Klein–Gordon Green’s function.

3.6

THE MASS-ENERGY-MOMENTUM TENSOR

Under transformations x ˛ ! x 0˛ D x ˛ C ıx ˛ that leave the action invariant, a field undergoes   .x/ !  0 x 0 D  .x/ C ı0  .x/ C ıx  .x/ D  .x/ C ı0  .x/ C ıx ˛ @˛  .x/ ; where ı0  .x/ D  0 .x/

 .x/

is a variation in the form of the field at a fixed point x and ıx  .x/ D ıx ˛ @˛  .x/

is a variation induced in the fixed form of the field by the variation of x . The action undergoes Z Z 4 0 0 0 ıSem D d x d L d 4 x d  L; ƒ0

0

ƒ

where ƒ ! ƒ is the change of volume induced by the variation in x . Expanding the first term, this becomes   Z Z @L 4 ˛ 4 ˛ @ˇ  .x/ ıx ˇ ıSem D d x d  [email protected]˛ L/ ıx C d x d  @˛ Lg ˇ @ [email protected] / ƒ ƒ   Z @L 4 ˇ d x d  @˛ ıx ıˇ  ; @ [email protected] / ƒ

3.6. THE MASS-ENERGY-MOMENTUM TENSOR

35

where we used the Euler–Lagrange equations   @L @L @a D 0: @ @ [email protected] / Since ıS D 0 and the variations are arbitrary, we obtain Noether’s theorem   @L ˛ˇ ˇ @˛ Lg @  .x/ D @˛ Q˛ˇ D 0 @ [email protected] / for the conserved current Q˛ˇ . The electromagnetic Lagrangian can be written Lem D

1 ˛ˇ f .x; /f˛ˇ .x;  / ; 4c ˆ

e ˛ j .x; /a˛ .x;  / c2

where fˆ˛ˇ

.x;  / D

Z

ds ˆ. 

s/ f ˛ˇ .x; s/

is the non-local convolved field. Under translations, ıx ˇ D "ˇ

! ıa˛ D 0

and so the conserved current is Qˆ˛ˇ D

@L  @ ˇ a @ @ ˛ a

Lg ˛ˇ D

1 ˛ˇ g c



1 ı" f fı" 4 ˆ

e j a c



1 ˛ ˇ f @ a : c ˆ

This current may be made symmetric in the indices by adding the total divergence ˆ˛ˇ D

e 1 1  ˛ ˇ  @ fˆ a D 2 j ˛ aˇ C fˆ˛ @ aˇ ; c c c

where the second form follows from the inhomogeneous pre-Maxwell equation. Now, the symmetric current is i e h ˛ˇ ˆ˛ˇ D Qˆ˛ˇ C ˆ˛ˇ D ˆ0 C 2 j ˛ aˇ j  a g ˛ˇ ; c where ˛ˇ ˆ0

 1 f ˛ f D c ˆ

ˇ

1 C fˆı" fı" g ˛ˇ 4



is the source-free current. By explicit calculation, using the homogeneous pre-Maxwell equation, we find e ˇ˛ @˛ Tˆ˛ˇ D f j˛ ; c2

36

3. CLASSICAL ELECTRODYNAMICS

where Tˆ˛ˇ D

˛ˇ ˆ0 D

 1 f ˛ f ˇ c ˆ

1 ˛ˇ ı" g fˆ fı" 4



is the conserved mass-energy-momentum tensor. Writing the ˇ D 5 component of the conservation law @˛ Tˆ˛5 D

e 5˛ f j˛ c2

(3.25)

and using j ˛ .x; / D c XP ˛ ./ı 4 x

X. /



for the single particle current leads to @˛ Tˆ˛5 D

e 5˛ f .x;  / XP ˛ . /ı 4 x c

 X./ :

Integrating the LHS over spacetime leaves the  -derivative Z Z Z Z 1 d 1 d 5 4 55 4 ˛5 4 d x Tˆ D d 4 x Tˆ55 d x @ ˛ Tˆ D d x @  Tˆ C c5 d  c5 d  and integrating the RHS gives Z e d 4 x f 5 .x;  / XP  ./ı 4 x c

 X. / D

e 5 f .X. /;  / XP  ./: c

Recognizing this expression from the fifth Lorentz force equation   1 ec5 5 d 2 M xP D 55 f xP  d 2 c the RHS and LHS combine as Z   d 1 4 55 2 d x Tˆ C 55 M xP D0 d 2 demonstrating that the total mass of fields and events is conserved. Since M xP 2 has units of energy (xP 2 D c 2 ), we see that Tˆ55 has units of energy density (energy per 4D spacetime volume).

3.7

WORLDLINE CONCATENATION

We saw in (3.21) that the source current satisfies @˛ j'˛ .x;  / D 0, and so the vector part j' .x;  / cannot be a divergenceless Maxwell current. However, Stueckelberg noticed that under the boundary condition j'5 .x;  / ! 0 !˙1

3.7. WORLDLINE CONCATENATION

we have @

Z

d  j' .x;  / C

1 c5

Z

37

d  @ j'5 .x;  / D @ J  .x/ D 0;

where using (3.16) we confirm Z Z Z ds J  .x/ D d  j' .x;  / D d  ' . 

s/ j ˛ .x; s/ D

Z

ds j  .x; s/

in agreement with (3.9). Again, this integration, called concatenation [5], represents the sum at the spacetime point x of all events occurring over time  . Saad and Horwitz [3] extended Stueckelberg’s argument, showing that under the additional boundary condition f 5 .x;  /

! 0

!˙1

 -integration of the pre-Maxwell equations leads to Maxwell’s equations in the form @ˇ f ˛ˇ .x;  / D

e ˛ j .x;  / c '

@Œ˛ fˇ  D 0

9 > > > > > =

Z

> > > > > ;

@˛ j ˛ D 0

where F

˛

.x/ D

Z

d 

!

8 e ˆ @ F  .x/ D J  .x/ ˆ ˆ c ˆ ˆ < @Œ F D 0 ˆ ˆ ˆ ˆ ˆ : @ J  .x/ D 0;

d  ˛ f .x;  /: 

Under concatenation, F  become Maxwell fields while F 5 decouples from the Maxwell system. In addition, integrating the Green’s function (3.24) for the pre-Maxwell wave equation Z Z 1 d  GMaxwell D D.x/ D ı.x 2 / d  GCorrelation D 0 (3.26) 2 recovers the 4D Maxwell Green’s function. When concatenating GCorrelation , the two singular terms arising from the derivative must once again be subtracted prior to applying the limits of integration. As we have seen, SHP electrodynamics can be understood as a microscopic theory of events interacting at time  . We saw in Section 3.6 that during these interactions, particles and pre-Maxwell fields may exchange mass under conservation of total mass-energy-momentum. As mentioned in Section 1.3, Feynman recognized that mass exchange of this type is also permitted, in principle, in QED. He interpreted integration over the evolution parameter, the final step in Equation (1.3) that describes the quantum Green’s function for scalar particles, as extraction of asymptotic mass eigenstates from these complex interactions. In much the same way, we will see that concatenation—integration of the pre-Maxwell field equations over the evolution

38

3. CLASSICAL ELECTRODYNAMICS

parameter  —extracts from the microscopic event interactions the massless modes in Maxwell electrodynamics, expressing a certain equilibrium limit when mass exchange settles to zero. Thus, we will frequently compare the concatenated form of results in pre-Maxwell electrodynamics with the corresponding formulation in Maxwell theory, as a means of maintaining contact with established phenomenology.

3.8

PCT IN CLASSICAL SHP THEORY

We recall from Section 1.1 that Stueckelberg’s initial motivation for consideration of  -evolution was his desire to formulate a classical electrodynamics that includes antiparticles and describes pair processes through the dynamic evolution of a single type of evolution x  ./. And as we saw in Section 2.3, particles and antiparticles differ only in the direction of their t -evolution, specifically the sign of xP 0 ./, or equivalently the sign of the energy E D p 0 c . In quantum field theory the relationship of particles and antiparticles, characterized by a charge conjugation operation C , is significantly different. This operation is understood as a third discrete symmetry of the field equations, along with the improper Lorentz symmetries—time reversal T and space reversal P —that we expect to hold for a Lorentz covariant system. An antiparticle is obtained by acting with C and T to produce a particle with both the sign of the charge and the temporal ordering of its evolution reversed. Operator implementation of C generally requires a quantum formalism with complex wavefunctions, and the combined C T operation is anti-unitary. We require electrodynamics to be symmetric under the improper Lorentz transformations T and P , and first find the transformations of the fields under these operations. We then consider a C operation, which in Wigner’s original sense of “reversal of the direction of motion” [10], acts on the  -evolution. The Lorentz equations in explicit three-vector form are   d 2x0 e dx 0 M D e .t; x;  /  55  .t; x;  / d2 c d   d 2x e dx 0 dx M 2 D C  b .t; x;  / 55  .t; x;  / e .t; x;  / d c d d and under space inversion P    x D x 0 ; x ! xP D xP0 ; xP D x 0 ; x P

become   d 2 xP0 e d xP 0 M D eP .tP ; xP ;  /  55 P .tP ; xP ;  / d2 c d  dxP0 e d xP d 2 xP .t / D e ; x ;  C  bP .tP ; xP ;  / M P P P d2 c d d

55 P .tP ; xP ;  /



3.8. PCT IN CLASSICAL SHP THEORY

39

so that   d 2x0 e dx 0 M eP .tP ; xP ;  /  55 P .tP ; xP ;  / D d2 c d dx 0 d 2x e dx M 2 D eP .tP ; xP ;  / C  bP .tP ; xP ;  / d c d d



55 .  .tP ; xP ;  // :

Invariance under P , understood as form invariance of the interaction, requires that bP .tP ; xP ;  / D b .t; x;  / P .tP ; xP ;  / D  .t; x;  /

eP .tP ; xP ;  / D e .t; x;  / P0 .tP ; xP ;  / D  0 .t; x;  /

and as we would generally expect, the vectors e and  change sign, while the axial vector b and 0-component  0 are unchanged. Under time inversion T ,    x D x 0 ; x ! xT D xT0 ; xT D x 0 ; x I T

we similarly write   d 2 xT0 e d xT 0 M D eT .tT ; xT ;  /  55 T .tT ; xT ;  / d2 c d  dxT0 e d xT d 2 xT .t / D C  bT .tT ; xT ;  / M e ; x ;  T T T d2 c d d

55 T .tT ; xT ;  /



so that    d 2x0 e dx 0 M D 55 T .tT ; xT ;  / eT .tT ; xT ;  /  d2 c d  e dx d 2x dx 0 M 2 D eT .tT ; xT ;  / C  bT .tT ; xT ;  / 55 T .tT ; xT ;  / : d c d d

Now form invariance requires eT .tT ; xT ;  / D e .t; x;  / T0 .tT ; xT ;  / D  0 .t; x;  /

bT .tT ; xT ;  / D b .t; x;  / T .tT ; xT ;  / D  .t; x;  /

and here we notice that  0 and  transform as expected for components of a 4-vector, but the transformations of e and b are opposite to the behavior generally attributed to the electric and magnetic 3-vectors under time inversion. This can be attributed to our having respected the independence of x 0 . / as a function of  , not constrained by the mass-shell condition dx 0 D Cq d

1 1

.d x=dt/

: 2

40

3. CLASSICAL ELECTRODYNAMICS

In general, all of the field components transform tensorially as components of the f  and   . From the transformation properties for the field strengths, we may deduce the transformation properties of the 5-vector potential components. First, we have   ePi D ei H) @0 aPi @i aP0 D @0 ai @i a0 and so we conclude that aP0 D a0

aPi D

ai

which is consistent with bPi D bi D "ij k @j ak :

(3.27)

Similarly, eiT D

ei

@0 aTi

H)

@i aT0 D

@0 a i

@i a0



so we see that aT0 D

a0

aTi D ai

again consistent with (3.27). For the second vector field,  Pi D i H) @5 aPi @i aP5 D along with aPi D ai leads to

@5 ai

@i a 5



aP5 D a5

which is consistent with P0 D  0 D @5 a0

@0 a5 :

Similarly, Ti D i

along with aTi D ai leads to

H)

@5 aTi

@i aT5 D @5 ai

@i a 5

aT5 D a5 :

Thus, the 4-vector and scalar components of the potential transform tensorially under space and time inversion. The pre-Maxwell equations in 3-vector form, as given in (4.17) and (4.18), r e

1 @ 0 e  D j'0 D e'0 c5 @ c

r b

1 @ e c @t

r C

1 @ e  D j' c5 @ c

1 @ 0 e ec5  D j'5 D ' c @t c c r 0 C

r bD0 r eC r 

1 @ 1 @  C 55 eD0 c @t c5 @

1 @ bD0 c @t

55

1 @ bD0 c5 @

3.8. PCT IN CLASSICAL SHP THEORY

41

are seen to be invariant under P and T using the transformations of the fields, under the choices jP0 .tP ; xP ;  / D j 0 .t; x;  / jP .tP ; xP ;  / D

jT0 .tT ; xT ;  / D

j .t; x;  /

j 0 .t; x;  /

jT .tT ; xT ;  / D j .t; x;  /

jP5 .tP ; xP ;  / D j 5 .t; x;  /

jT5 .tT ; xT ;  / D j 5 .t; x;  / ;

where again the 4-vector and scalar components of the current transform tensorially under space and time inversion. In order to discuss charge conjugation, we must make another short digression into quantum mechanics. As in Section 3.1, we may write the Stueckelberg–Schrodinger equation as  ec5  1   e   e  .x;  / D .x;  / i @ C a5 p a p a c 2M  c   c 1 ie  ie .x;  / D @ a @ a 2M c c and, taking the complex conjugate, observe that this system will be form invariant under a charge conjugation C that operates as e .x;  /  a .x;  / a5 .x;  /

! eC D

C

!

C

e

C .x;  /

! C D

C



D

.x; /



 ! aC .x;  / D a .x; /

C

5 ! aC .x;  / D

a5 .x; /

C

if these transformations can be made consistent with the pre-Maxwell equations and Lorentz force. As we now show, this consistency can indeed be established. Leaving aside the quantum wavefunction and returning to classical mechanics, transformations of the potentials lead to field strength transformations e k D @0 ak

@k a0

b k D "kij @i aj

 k D 55 @ ak

 0 D 55 @ a0

@k a5 @0 a5

!

C

!

C

!

C

!

C

ek bk k 0

so that this operation reverses the sign of tensor quantities carrying a scalar index. Under these transformations, the pre-Maxwell equations remain form invariant as long as    j 0 ; j; j 5 ! j 0 ; j; j 5 C D j 0 ; j; j 5 C

42

3. CLASSICAL ELECTRODYNAMICS

which is again a reversal of the scalar component. Similarly, the Lorentz force   d 2 xC0 e d xC 0 M D e    C 55 C c d C d C2   dxC0 d xC d 2 xC e C  bC 55 C M D eC c d C d C d C2 undergoes   d 2x0 e M D e d2 c  d 2x e M 2 D e d c

   dx 0 55  d   dx 0 dx C b d d

55 . /



becoming d 2x0 M D d2 d 2x M 2 D d

  e dx 0 e 55  c  d e dx 0 dx C b e c d d

 55  ;

thus implementing classical charge conjugation. We see that current conservation  @ j  C @ j 5 D 0 ! @ j  C . @ / j 5 D 0 C

5

is preserved, but since j is interpreted as the number of events in a localized spacetime volume at a given  , the meaning of jC5 D j 5 must be examined carefully. In standard relativistic mechanics, the continuity equation leads to a conserved charge through integration over volume in space as Z Z  d dQ @ J  D 0 ! D d 3 x eJ 0 D c d 3 xr  .eJ/ D 0 d d and since 0

J .x/ D c

Z

d  XP 0 . / ı 4 .x

X.//

cannot change sign in this approach, only the conjugation e ! e can account for charge reversal. But in SHP, charge conservation follows from Z Z  dQ d ˛ 4 5 @˛ j D 0 ! D d x ej D c5 d 4 x e @ j  D 0; d d where it is the event density j 5 .x/ D c XP 5 ./ ı 4 .x

X. // D cc5 ı 4 .x

X. //

3.9. BIBLIOGRAPHY

43

that cannot change sign. But the effective charge of an event interacting through the Lorentz force is associated with ej 0 .x/ D ec XP 0 . / ı 4 .x X. // which can change sign through XP 0 ./ according to Stueckelberg’s prescription. Thus, the operation e ! e is not a required symmetry. Following Stueckelberg, we disentangle the symmetries of the coordinate time t from those of the chronological parameter  by making the following interpretations of the discrete reflections. 1. Space inversion covariance P implies certain symmetric relations between a given experiment and one performed in a spatially reversed configuration. 2. Time inversion covariance T implies certain symmetric relations between a given experiment and one performed in a t -reversed configuration, which is to say one in which advancement in t is replaced by retreat, and so a trajectory with xP 0 > 0 is replaced by a trajectory with xP 0 < 0. Thus, we expect symmetric behavior between pair annihilation processes and pair creation processes. 3. Charge conjugation covariance C implies certain symmetric relations between a given experiment and one in which the events are traced out in the reverse chronological order and carry opposite charge. The operations P and T are improper Lorentz transformations and therefore must be symmetries of any (spinless Abelian) relativistic electrodynamics. But we do not regard the operation C defined here as connecting symmetrical dynamical evolutions. Rather, we associate the reversal of temporal order performed by C with the re-ordering of events performed by the observer in the laboratory, who interprets events as always evolving from earlier to later values of t . Thus, charge conjugation exchanges the viewpoint of the events under interaction with the viewpoint of the laboratory observer. The charge inversion (associated with the gauge symmetry) under this exchange reinforces the view of antiparticles in the laboratory, but does not influence the event dynamics.

3.9

BIBLIOGRAPHY

[1] Born, M. and Wolf, E. 1999. Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, Cambridge University Press, Cambridge. 25 [2] Jackson, J. D. and Okun, L. B. 2001. Review of Modern Physics, 73:663. 25 [3] Saad, D., Horwitz, L., and Arshansky, R. 1989. Foundations of Physics, 19:1125–1149. 27, 29, 37

44

3. CLASSICAL ELECTRODYNAMICS

[4] Jackson, J. 1975. Classical Electrodynamics, Wiley, New York. DOI: 10.1063/1.3057859. 29 [5] Arshansky, R., Horwitz, L., and Lavie, Y. 1983. Foundations of Physics, 13:1167. 29, 37 [6] Land, M. 1996. Foundations of Physics, 27:19. 29 [7] Land, M. 2003. Foundations of Physics, 33:1157. 29 [8] Land, M. 2017. Entropy, 19:234. http://dx.doi.org/10.3390/e19050234 32 [9] Schwinger, J. 1949. Physical Review, 75(4):651–679. https://link.aps.org/doi/10. 1103/PhysRev.75.651 33 [10] Wigner, E. P. 1959. Group theory and its application to the quantum mechanics of atomic spectra, Pure Applied Physics, Academic Press, New York (translation from the German). https://cds.cern.ch/record/102713 38

PART III

Applications

47

CHAPTER

4

Problems in Electrostatics and Electrodynamics 4.1

THE COULOMB PROBLEM

Introductory treatments of electromagnetism quite naturally begin with the static Coulomb force between two point charges at rest. However, in the framework of Stueckelberg, Horwitz, and Piron, this seemingly simple configuration requires some clarification. A timelike event in its rest frame can be given with velocity XP 2 D

c2

!

XP D .c; 0/

so that this “static” event evolves uniformly in  with coordinates X./ D .ct; X/ D .c.t0 C /; X0 /

and the displacement .ct0 ; X0 / at  D 0 plays a role in interactions with other events. Taking X0 D 0, so that the event simply evolves along the t -axis in its rest frame, the associated event current is 8 0 2 3 ˆ ˆ j .x;  / D c ı .ct c.t0 C  // ı .x/ ˆ < j ˛ .x;  / D c xP ˛ ı 4 .x X. // ! j .x;  / D 0 ˆ ˆ ˆ : 5 j .x;  / D cc5 ı .ct c.t0 C  // ı 3 .x/ with support restricted to the spatial origin—as in Maxwell theory—and to the time t D t0 C  . The source for the pre-Maxwell field is the smoothed ensemble current 8 0 j' D c' .t .t0 C // ı 3 .x/ ˆ ˆ ˆ Z < j' D 0 j'˛ .x;  / D ds ' . s/ j ˛ .x; s/ ! (4.1) ˆ ˆ ˆ : 5 j' D c5 ' .t .t0 C // ı 3 .x/ which varies continuously in t , and as  advances has its maximum at t D t0 C  . The potential induced by this current may be found, as in (3.23), by integration with the Green’s function (3.24), containing two terms, G D GMaxwell C GCorrelation . We first treat the Maxwell term, which

48

4. PROBLEMS IN ELECTROSTATICS AND ELECTRODYNAMICS

produces a potential with the expected 1=R dependence, multiplied by a time-dependent form factor found from ' . We then find the contribution from the correlation term which is scaled by the small factor c5 =c and drops off as 1=R2 .

4.1.1 CONTRIBUTION TO POTENTIAL FROM GMaxwell The leading term in the potential is Z  2  ret   e a .x;  / D d 4x0d  0 ı x x0  ı.  0 /' t 0 .t0 C  0 / ı 3 x0 2 Z   2  e cdt 0 ı c 2 t t 0 x2  ret ' t 0 .t0 C / D 4    e R D '  t t0 4R c a .x;  / D 0 0

a5 .x;  / D

c5 0 a .x;  / ; c

where we insert  ret D  .x x 0 / to select retarded spacetime causality, and write R D jxj. As observed from a spacetime point x D .ct; x/, this field will grow as  ! t t0 R=c from below and then decrease. Since the time coordinate of the source is tevent D t0 C  , the maximum occurs if the observer is located at time t D tevent C R=c , representing a delay equal to the signal transmission time at the speed of light. Put in a more familiar way, the time coordinate of the event detected at time t is tevent D tretarded D t R=c . To study the “static” Coulomb problem, we consider a test event evolving uniformly at x D .c C ct0test ; x/, where x is constant. Inserting these coordinates and using (3.15), the potential experienced by the test event becomes   e R 1 e 0 a .x;  / D ' t0 C e jt0 R=cj= (4.2) D 4R c 2 4R c5 0 a5 .x;  / D a .x;  / ; c where t0 D t0test t0 defines the mutual t -synchronization between the events. From (3.12) we may take  ' 1=2 for c5 =c  1, and so recover the Coulomb potential a0 .x;  / D

e 4R

in the particular case that t0 D R=c . By contrast, if t0 D 0, then a0 takes the form of a Yukawa potential e a0 .x;  / D e 2jRj=c (4.3) 4R

4.1. THE COULOMB PROBLEM

49

suggesting a semi-classical interpretation in which the photons carrying the pre-Maxwell interaction have mass m c 2  2„=. Taking m to be smaller than the experimental error on the mass of the photon (10 18 eV =c 2 ) [1], we may estimate  > 104 seconds. In this approximation c will be larger than any practical distance in the problems we consider. The field strength components found from the Yukawa-type potential with t0 D 0 are f k0 .x;  / D @k

e 1 e 4R 2

R=c

f k5 .x; / D

c5 k0 f .x;  / c

f ij .x; / D f 50 .x;  / D 0

so that the test event will undergo Coulomb scattering e M xR D f k xP  c k

ec5 5k 55 f D c

  c52 e k0 0 f xP C 55 c c

according to the Lorentz force (3.4). Since the test event velocity is x./ P D .c; 0/ this becomes ! !   2  c 2  1 C 55 cc5 e R=c e2 e R=c 5 2 1 C 55 r D e ; M xR D 2 r 2 c 4R 4R 1 C cc5 where we used (3.12) for  . Now suppose the source event were an antiparticle event evolving backward in time with XP 0 D c . This would change the signs of a0 .x; / and f k0 .x; / but not the signs of a5 .x; / or f k5 .x;  /. We can thus write the Coulomb force for both cases as ! 2 1 ˙ 55 cc5 e R=c .C= / 2 F D e ; 2 r 4R 1 C c5 c

where the upper sign is for a particle event and the lower sign is for an antiparticle event. Since 55 D ˙1, this expression provides an experimental bound on c5 =c , given by " 2 #2  1  55 cc5  e C eC ! e C eC D 1 ˙ experimental error ' ; 2  .e C e ! e C e / 1 C c5 c

where  is the total classical scattering cross-section at very low energy. The action (3.13) recovers the usual first-order kinetic term f ˛ˇ f˛ˇ in the limit  ! 0, in which case 1 jj= lim e D ı. / !0  and the source of the pre-Maxwell field reduces to j'˛ .x;  / ! j ˛ .x;  /. If we take  very small but finite, then from (4.2) we have   e R 0 a .x;  / ' ı t0 4R c

50

4. PROBLEMS IN ELECTROSTATICS AND ELECTRODYNAMICS

for the potential experienced by the test event. Now the support of the potential is restricted to a lightline between the events, and for any synchronization t0 ¤ R=c there will be no interaction. As we remarked in Section 3.3, a solution for Coulomb scattering can be found in this case [2], but the delta function potential leads to a discontinuous trajectory that is difficult to reconcile with classical phenomenology. This discontinuity is a primary motivation for introducing the interaction kernel. We mention in passing that this difficulty is not present in SHP quantum field theory because the definition of asymptotic states with sharp mass implies the loss of all information about the initial t -synchronization of the scattering particles. The significance of the small  limit appears in a number of places. As discussed in Section 3.4,  characterizes the section of a worldline over which the event current is smoothed. In this sense,  can be seen as the correlation length of a statistical process that assembles the current from an ensemble of events occurring along the trajectory. When  is small, the interaction between an event trajectory and a test event is determined by a small number of points along the worldline, including only one point when  D 0. Moreover, the mass spectrum m c 2  2„= of the electromagnetic field associated with the Yukawa-like potential (4.3) becomes large. By contrast, if  is large, then the source j'˛ .x;  / of the pre-Maxwell field is assembled from a large ensemble of events along the worldline, locally approximating the concatenation of the worldline performed in constructing the Maxwell current J  .x/. In this case, the mass spectrum m c 2  2„= of the electromagnetic field is small, approaching zero in the limit  ! 1. From (3.16) and (4.1) the concatenated current is 0

J .x/ D c

Z

d ' .t 

.t0 C  // ı 3 .x/ D cı 3 .x/

J .x/ D 0

describing a static Maxwell charge at the origin, and the concatenated potential is e A .x/ D 4R 0

Z

 d '  

 t

t0

R c



D

e 4R

A .x/ D 0

describing the static Coulomb potential. As required, J  .x/ and A .x/ are independent of t0 and invariant under a shift of the event x  . / along the time axis. The microscopic interaction between the events is thus seen to be sensitive to the t -synchronization t0 of the interacting events, a parameter not accessible by the standard Coulomb law.

4.1.2 CONTRIBUTION TO POTENTIAL FROM GCorrelation Up to this point, we have treated only the potential found from the leading term GMaxwell in the Green’s function. To consider the potential found from GCorrelation again we take as source the event X D .ct0 C c; 0/, but simplify the calculation by taking t0 D 0 and approximate '. 0

4.1. THE COULOMB PROBLEM

s/ D ı.

0

51

s/ so that Z

 d 4 x 0 d  0 GCorrelation x x 0 ;   0 c 2 ı ct 0 Z  D ec d  0 GCorrelation .ct c 0 ; x/;   0 :

0

a .x;  / D

e c

We introduce the function g.s/ to express terms of the type    .x X.s//2 C c52 . s/2 D ..ct; x/ .cs; 0//2 C c52 . where s/2

g .s/ D .t

c52 . c2

R2 c2

 c52 C D 1 2 2 c so that the potential can be written as

BD

ec5 .c; 0; c5 / a .x;  / D 2 2 c 3

Z

ds

2 t 

 s/2 D c 2 g .s/ ;

s/2 D C s 2 C Bs C A

and

2 D

  c 0 ı 3 x0

2 



A D t2

R2 c2

 1  .g .s// ı .g .s//  .t 2 g 3=2 .s/ g 1=2 .s/

2  2

s/ :

The zeros of g .s/ are found to be

s˙ D

p B ˙ B2 2C

4AC

t D



2  ˙

r

R2 .1 2 / C 2 .t c2 .1 2 /

/2

(4.4)

and since we assume 2 < 1 there will be roots for any values of t and R. In addition, the condition  ret D .t s/ requires t > s . Attempting to set t < s leads to  q R2 .1 2 / C 2 .t /2 t 2  R2 c2 2 2 .t / t< )   > .1 2 / c2 and so t  s is a condition of integration for the  term. Similarly, attempting to set t > sC leads to  q 2 t 2  C Rc 2 .1 2 / C 2 .t  /2 R2 ) 2 .t / > 2 t> 2 .1  / c leading to the condition .g.s// .t

s/ ¤ 0

)

s  s  t  sC

52

4. PROBLEMS IN ELECTROSTATICS AND ELECTRODYNAMICS

from which a .x;  / D

 Z c5  1 ec5  1; 0; 2 2 c 3 c 2

s

ds 1

Z

1 g 3=2 .s/

1

ı .g .s//  .t g 1=2 .s/

ds

1

 s/ :

Using the well-known form [3] Z .C x 2

2 .2C s C B/ dx D ; 3=2 C Bx C A/ q.C x 2 C Bx C A/1=2

where B2

q D 4AC

we notice from (4.4) that p B B2 s D 2C and so 1 2

Z

4AC

p

B

D

q

p

)

2C

q D 2C s C B

ˇ 2C s C B 2C s C B ˇˇ D ds 3=2 g .s/ qg 1=2 .s / qg 1=2p.s/ ˇ 1 1 p q 2 C C D 1=2 qg .s / .2C s C B/2 p 1 1 2 1 C Dp 2 2 R2 .1 2 / C 2 .t qg 1=2 .s /

s

1

c

The second term is

Z

Z

1

Since

ds

ds

1

and using the identity

we can evaluate Z 1

1

ı .g .s//  .t g 1=2 .s/

s/ D

jg 0

(4.5)

:

s/

ˇ f .s / ˇˇ ds f .s/ ı .g .s// D 0 jg .s /j ˇs

ı .g .s//  .t g 1=2 .s/

 /2

Dg

1 .0/

 .t s / 1 D 0 : 1=2 .s /j g .s / jg .s /j g 1=2 .s /

0 g 0 .s / D C s 2 C Bs C A D 2C s C B D

p

q

we see that this term cancels the singularity in the first term, leaving p Z Z 1 1 2 1 s 1 ı .g .s// 1 ds 3=2 ds 1=2  .t s/ D 2 2 1 g .s/ 2 R2 .1 2 / C 2 .t g .s/ 1 c

/2

4.2. LIÉNARD–WIECHART POTENTIAL AND FIELD STRENGTH

and

53

r

e c5  .c; 0; c5 / a .x;  / D 2 4 c R2 1

c5 1 c c5  c5 2 C c .t c c

 /2

:

We notice that the potential has units of c=distance2 D 1/distance, as does the potential associated with GMaxwell . This contribution to the potential is smaller by a factor of c5 =c than the Yukawa potential found in (4.3), and drops off faster with distance, as 1=R2 compared to 1=R. This term may be neglected when the contribution from GMaxwell is significant, but as we will see in Section 4.7.1, it may lead to qualitatively important phenomena when the leading term vanishes.

4.2

LIÉNARD–WIECHART POTENTIAL AND FIELD STRENGTH

We now consider an arbitrary event X ˛ . / for which the smoothed current is Z ˛ j' .x;  / D c ds ' . s/ XP ˛ .s/ ı 4 Œx X .s/ and the Liénard–Wiechert potential found from GMaxwell is Z  2  ret   e a˛ .x;  / D d 4x0d  0ı x x0  ı   0 j'˛ x 0 ;  0 2c Z   e ds ' . s/ XP ˛ .s/ ı .x X .s//2  ret ; D 2

(4.6)

where  ret imposes retarded x 0 causality. Writing the line of observation as z D x

X  .s/

!

z 2 D Œx

X .s/2

and using the identity Z

ds f .s/ ı Œg .s/ D

f .R / jg 0 .R /j

we obtain a˛ .x;  / D

e ' . 4

R / ˇ ˇ.x 

;

R D g

1

.0/

XP ˛ .R /

ˇ; X  .R // XP  .R /ˇ

where the retarded time R satisfies z 2 D Œx

X.R /2 D 0

 .x

X .R // D 1:

(4.7)

(4.8)

54

4. PROBLEMS IN ELECTROSTATICS AND ELECTRODYNAMICS

Introducing the notation for velocity u D XP  . /

ˇ D

XP  c

u5 D XP 5 D c5

and the scalar length RD

1 d Œx 2c d R

X .R /2 D

z  u D jz  ˇj c

(4.9)

the potential becomes a .x;  / D

e ' . 4R

R / ˇ 

a5 .x;  / D

e ' . 4R

R /

c5 ; c

(4.10)

where R is nonnegative because u is timelike and z  is lightlike. Thus, a .x;  / takes the form of the usual Liénard–Wiechert potential from Maxwell theory multiplied by the factor ' . R / which separates out the  -dependence of the fields. To calculate the field strengths, we need derivatives of the Liénard–Wiechert potential. Since 1 d " . R / d ' . R / D e j R j= D ' . R / ; d R 2 d   where " . / D signum. /, we obtain the  -derivative 1 1 e @ a .x;  / D 'P . c5 c5 4ju  zj

R / u D

e ' . R / " . R / u 4c5 ju  zj 

directly from (4.10). The spacetime derivative is most conveniently found by applying the identity (4.7) to expression (4.6) Z   e ds ' . s/ XP ˛ .s/  ret @ ı .x X .s//2 @ aˇ .x;  / D 2 Z h i e ds '. s/XP ˇ .s/  ret ı 0 .x X .s//2 Œ2 .x  X  .s// D 2 Z i XP ˇ .s/ Œx  X  .s/ ret d h e ds '. s/ ı .x X .s//2 D  2 ds XP .s/  .x X .s// and integrating by parts to obtain " d '. ds ds " e 1 d D '. 4 ju  zj ds

e @ a .x;  / D 2  ˇ

Z

XP ˇ .s/ Œx  s/ XP .s/  .x 

s/

ˇ

z .s/u .s/ uz

X  .s/ X .s// #

:

sDR

#

h  ret ı .x

X .s//2

i

4.2. LIÉNARD–WIECHART POTENTIAL AND FIELD STRENGTH

55

Using zP  D

u

RP D

d zu D cˇ 2 d c

z  ˇP

we find the field strengths as f



 e ' . R / .z  ˇ  z  ˇ  / ˇ 2 " . R / z  ˇ  z  ˇ  .x; / D 4 R R2 c R    9 z  ˇP  z  ˇP  R C .z  ˇ  z  ˇ  / ˇP  z = cR2

f 5 .x; / D c5

e ' . R / 4 cR



;

" . R / z  C ˇ  Rc 2 =c52 c R  9 z  ˇP  z = C : cR2 ;

(4.11)

zˇ2 C ˇR R2

(4.12)

It is convenient to express the fields as elements of a Clifford algebra [4] with basis vectors e˛  eˇ D ˛ˇ

e˛ ^ eˇ D e˛ ˝ eˇ

eˇ ˝ e˛

(4.13)

and Clifford product e˛ eˇ D e˛  eˇ C e˛ ^ eˇ :

Separating spacetime and scalar quantities as X. / D X  ./e

X 5 D c5  1 @5 D @ c5

d D @ e 

and writing   D f 5 , the field strength tensors f D

1  f e ^ e 2

f 5 D f 5 e5 ^ e D e5 ^ 

are Clifford bivectors, (3.5) takes the form f Dd ^a

D

@5 a

da5 :

In this notation, the pre-Maxwell equations (3.20) are d f

@5  D

d ^f D0

e j' c

d  D

e 5 j c '

d ^  C @5 f D 0;

(4.14)

56

4. PROBLEMS IN ELECTROSTATICS AND ELECTRODYNAMICS

where we may evaluate d  f and similar terms using the Clifford identity a  .b ^ c/ D .a  b/ c

.a  c/ b:

R , Defining the dimensionless quantities associated with acceleration ˇP D X=c   P .ˇ  z/ ˇ ˇP  z ˇ P P ˇz ˇR ˇcQ QD W D D c c c

the field strengths become   z ^ ˇ ˇ2 R3

f .x; / D

e ' . 4

.x;  / D

c5 e ' . R / c 4 R

R /



" . R / R c

zˇ 2 C ˇR R2

in which the factors ' . R / and " . Z d ' ./ D 1 



" . R / c

W





z c2 Cˇ 2 R c5



zQ R2



R / contain the  -dependence. Since Z Z d " ./ d 0 ' ./ D ' ./ D 0   

the concatenated fields are found by replacing ' . R / ! 1 and " . R / ! 0, in agreement with the standard Maxwell result. We mention again that these field strengths were obtained using only the leading term GMaxwell in the Green’s function, and neglect the smaller contributions from GCorrelation . Although the neglected terms vanish under concatenation, they may dominate the dynamics when the leading contribution is zero. In particular, while GMaxwell has support on the lightcone, GCorrelation has timelike or spacelike support (depending on the choice of 55 ) and so becomes significant in self-interactions. Taking c  R and neglecting mass transfer, so that ˇ 2 D u2 =c 2 D 1, we may approximate e z f .x; / D ' . R / 3 ^ .ˇ C W / (4.15) 4 R .x;  / D

c5 e ' . c 4

R /

z .1

Q/ R3

ˇR

and split the field strengths into the short-range retarded fields f ret D

e ' . 4

R /

z^ˇ R3

 ret D

c5 e ' . c 4

R /

z

ˇR R3

that drop off as 1=R2 , and the radiation fields f rad D

e ' . 4

R /

z^W R3

 rad D

c5 e ' . c 4

R /

zQ R3

4.3. ELECTROSTATICS

57

associated with acceleration that drop off as 1=R. As elements of a Clifford algebra, the field strengths admit geometrical interpretation. The factor z ^ ˇ in f ret represents the plane spanned by the velocity ˇ and the line of observation z . Similarly, we recognize z

ˇR D

ˇ 2 z C ˇ .z  ˇ/ D

representing the projection of ˇ onto the z f ret D

e ' . 4

R /

.z ^ ˇ/  ˇ

ˇ plane, and so we have z^ˇ R3

 ret D

c5 ret f ˇ c

for the retarded fields. Similarly, using a  .b ^ c ^ d / D .a  b/ c ^ d

and z 2 D 0, we see that

.a  c/ b ^ d C .a  d / b ^ c

  z ^ W D z ^ ˇ ^ ˇP  z

in f rad represent the projection of z onto the volume spanned by z , ˇ , and ˇP . Similarly,  ret is proportional to zQ D .ˇP  z/z=c , the projection of z onto the acceleration ˇP .

4.3

ELECTROSTATICS

The covariant equivalent of a spatially static charge is a uniformly evolving event  X ./ D u D u0 ; u with constant timelike velocity XP D u D ˇc , which in its rest frame simply advances along the time axis as t D ˇ 0  . As a result, and given the geometrical interpretation of the Clifford forms, the field strengths are essentially kinematical in structure. Writing the timelike velocity ˇ in terms of the unit vector ˇO ˇ2 < 0

ˇ D jˇj ˇO

ˇO 2 D

1

ˇ2 D

jˇj2

the observation line z can be separated into components     zk D ˇO ˇO  z z? D z C ˇO ˇO  z which satisfy

 2  2 zk2 D ˇO 2 ˇO  z D ˇO  z  2  2  2 2 z? D z 2 C 2 ˇO  z ˇO  z D ˇO  z D  2 .ˇ  z/2 D jˇj2 ˇO  z D jˇj2 zk2 :

zk2

(4.16)

58

4. PROBLEMS IN ELECTROSTATICS AND ELECTRODYNAMICS

The condition of retarded causality z 2 D c 2 R2 ˇ 2

2cR ˇ  x C x 2 D 0

relates the field to the location of the event along the backward lightcone of the observation point. This implicit choice of R and its gradient  0 D d.z 2 / D 2 c 2 R d R ˇ 2 cR ˇ cd R ˇ  x C x D 2 ŒcRd R C z lead to the following expressions: d R D

z cR

.ˇ  d / R D

d .ˇ  z/ D d ˇ  x d

1 D . 1/n Rn

.z  d / R D

z2 D0 cR

ˇ ˇ z?  .ˇ  z/ ˇ ˇ 2 z ˇ 2 R D D ˇˇ 2 ˇ ˇz cR ˇ ˇ   n .ˇ  z/ ˇ ˇ 2 z n ˇˇ 2 ˇ z? D RnC2 .ˇ  z/nC2

d  z D d  .x d ^ z D d ^ .x d ^ zO D d ^

ˇz D1 ˇz

z D jzj

cˇR / D d  x

cˇ  d R D 3

cˇR / D cd R ^ ˇ D 1 ˇ^z jzj R

zO ^

z jzj

2

D

ˇ^z R ˇ ^ zO : R

Using these expressions, the pre-Maxwell equations (4.14) can be easily verified for the case of a uniform velocity event [5]. For example, recalling ' 0 D '"=, the exterior derivative of f is   e z^ˇ 2 z^ˇ 0 . / d ^ ' . R / ˇ C '  d ^f D R 4 R3 cR2 which produces terms of the type: d' .n/ ^ .z ^ ˇ/ D

' .nC1/

z ^ .z ^ u/ D 0 cR

ˇ^z d ^ .z ^ ˇ/ D .d ^ z/ ^ ˇ D ^ˇ D0 R " ˇ ˇ #   n ˇˇ 2 ˇ z? 1 d n ^ .z ^ u/ D ^ .z? ^ u/ D 0 R RnC2

and thus we recover d ^f D0

4.3. ELECTROSTATICS

59

from kinematics. It is convenient to write the field strengths in 3-vector and scalar form .e/i D f 0i

.b/i D "ij k f j k

./i D f 5i

 0 D f 50

for which the field equations split into four generalizations of the 3-vector Maxwell equations r e

1 @ 0 e  D j'0 D e'0 c5 @ c

r b

1 @ e c @t

1 @ e  D j' c5 @ c

r bD0 1 @ r eC bD0 c @t

(4.17)

and three new equations for the fields  and  0 r C

1 @ 0 e ec5  D j'5 D ' c @t c c

r 

55

1 @ bD0 c5 @

1 @ 1 @  C 55 e D 0: r C c @t c5 @

(4.18)

0

Writing d D e0 @0 C r and f D e0 ^ e C 21 f j k ej ^ ek we find that 8 ˆ < r bD0 d ^f D0 ! 1 @ ˆ : r eC bD0 c @t expressing the absence of electromagnetic monopoles. In the rest frame of a charged event, we may set tP D 1 ! ˇ D e0 , so for an observation point x D .ct; x/ 8 jxj ˆ ˆ ˆ R D t ˆ c < z2 D 0 ! R D e0  .x cR e0 / D jxj ˆ ˆ ˆ ˆ : z D .c .t R / ; x/ D R .e0 C xO / and the field strengths reduce to e f .x;  / D ' . 4 .x; / D

c5 e ' . c 4

  e0 ^ xO " . R / R / 1C R D e0 ^ e.x;  / R2 c R /



    xO " . R / c2 C e 1 C C x O : 0 R2 cR c52

60

4. PROBLEMS IN ELECTROSTATICS AND ELECTRODYNAMICS

We thus find that the magnetic field b is zero, while   e ' . R / ' 0 . R / eD xO 4 R2 R and 0

 .x;  / D

e ' 0 . R / 4 R



D

c5 c C c c5



c5 e c

(4.19)

(4.20)

:

Because we obtained f .x;  / using only the leading term GMaxwell in the Green’s function, we expect errors on the order of the neglected term GCorrelation . In particular, we notice that   55 1 55 @ @ C 2 @2 GMaxwell D ı 4 .x/ ı ./ ı.x 2 / ı 00 ./; 2 c52 c5 where the second term on the right is canceled when GCorrelation is included in the wave equation. As a result, calculating   c5 e ' 00 . R / r  D ' . R / ı 3 .x/ ; c 4 cR where we use r  .Ox=R2 / D 4ı 3 .x/, and

e ' 00 . R / 1 @ 0  D c @t 4 cR





c5 c C c c5

leads to the Gauss law as c5 e 1 @ 0  Cr  D ' . c @t c 4

R / ı 3 .x/ C

c e ' 00 . R / c5 4 cR

exposing an error at the order of ı 00 . R /. We now consider a long straight charged line oriented along the z -axis, with charge per unit length e . In cylindrical coordinates p x D .; z/  D .x; y/ D O  D x2 C y2 p the fields  and e are found by replacing R D 2 C z 2 in (4.19) and (4.20) and integrating along the z -axis to find   1 0  1=2 1=2 .2 Cz 2 / .2 Cz 2 / 0 Z '  t C '  tC c c B C e C .; eD dz B @ A O z/ 2 2 3=2 4 c . C z / .2 C z 2 /

0 D

e c5 4 c

Z

 '0  dz

1=2

tC

.2 Cz 2 /



c

c .2 C z 2 /1=2

:

4.3. ELECTROSTATICS

61

To get a sense of these expressions, we may use (3.15) to approximate '.x/ D ı.x/ which permits us to easily carry out the z -integration to obtain 1 0 eD

e B  .t =c /  @  3=2 2 c .t  /2 2 =c 2

which vanishes  > R D t Z

ı .t =c  / C q A O 2 2 2 .t  /  =c

=c as required for retarded causality. Since Z d d 0 ' ./ D 1 ' . / D 0  

the concatenated electric field is found as Z Z e d e 1 O z/ D O 0/ .; .; E.x/ D e.x; / D dz 3=2 2 2  4 2 . C z /

in agreement with the standard expression. To obtain the field of a charged sheet in the x y plane with charge per unit area  , it is convenient to start from the potential from a charged event, and integrating over x and y with p 2 R D x C y 2 C z 2 . Thus,   q Z '  t C 1c .x x 0 /2 C .y y 0 /2 C z 2 c dx 0 dy 0 a0 .x; / D q 4 c .x x 0 /2 C .y y 0 /2 C z 2 and a5 .x; / D .c5 =c/a0 .x;  /. Changing to radial coordinates .x; y/ ! .;  / we obtain   p 1 Z 2 C z2 '  t C  c c a0 .x; / D dd p 2 4 c  C z2 p which by change of variable  D 1c 2 C z 2 becomes Z c 1 0 a .x; / D ' . t C / d : 2 jzj=c We calculate the fields from e.x; / D

ra0 D

  '  2

where .x; / D .c5 =c/e.x; / and 55  D @ a 0 C c5 c  D 55 c c5 0

tC

   jzj r jzj D ".z/'  c 2

  c 1 1 c5 5 @t a D 55 @ a0 c  c c5  c c5 jzj '  tC : c c

tC

jzj c



zO ;

62

4. PROBLEMS IN ELECTROSTATICS AND ELECTRODYNAMICS

By concatenation, we recover  Z Z  E.x/ D d  e.x;  / D d  ".z/'  2

jzj tC c



zO D

 ".z/ zO 2

in agreement with the Maxwell field from a charged sheet. We notice that, as expected, the space part of the electric fields change sign at the plane of the sheet, pointing out at each side. Consequently, an event passing through a charged sheet of equal sign will decelerate in space on its approach and then accelerate as it retreats. However, unlike the field of a point event, the temporal part  0 is an even function of spatial distance and so the event may accelerate along the time axis on both its approach to the charged sheet and its retreat. In such a case, the spatial motion will asymptotically return to its initial condition, while the event acquires a net temporal acceleration, corresponding to a shift in energy and mass.

4.4

PLANE WAVES

From the wave equation (3.22) for j ˛ .x;  / D 0 we may write the field in terms of the Fourier transform [6] Z Z 1 1 0 ˛ˇ 5 ikx ˛ˇ f .x;  / D d ke f .k/ D d 4 k d  e i.kxCk0 x C55 c5  / f ˛ˇ .k; /; 5 5 .2/ .2/ where  D k 5 D 55 k5

is understood to represent the mass carried by the plane wave, much as k 0 and k represent energy and 3-momentum. This interpretation is supported by the wave equation which imposes the 5D constraint k ˛ k˛ D k2 .k 0 /2 C 55  2 D 0 H) 55  2 D .k 0 /2 k2 (4.21) expressing  in terms of the difference between energy and momentum. Under concatenation, the field becomes Z Z Z d  ˛ˇ d 4 k ik x  1 ˛ˇ d 4 k i k x  ˛ˇ ˛ˇ f .x; / D e e F .k/ F .x/ D f .k; 0/ D  .2/4 c5 .2/4 and recovers the 4D mass-shell constraint k  k D 0 for the Maxwell field. In the transform domain, the sourceless pre-Maxwell equations take the form ke

55  0 D 0

ke

k0b D 0

k

b D 0

kbD0

k

k00 D 0

k  b C k0e

e C k 0 

55  D 0 k 0 D 0

4.4. PLANE WAVES

63

which can be solved by taking k and e? as independent 3-vector polarizations, and writing ek D 55

 k k0

? D

 e? k0

0 D

1 k  k k0

bD

1 k  e? k0

for the remaining fields. Unlike Maxwell plane waves, for which E, B, and k are mutually orthogonal, the pre-Maxwell electric fields e and  have both transverse and longitudinal components. When  ! 0, we find that e, b, and k become mutually orthogonal and  becomes a decoupled longitudinal polarization parallel to k. We use (3.11) to write the convolved field as Z Z ds 1 0 ˛ˇ ˛ˇ fˆ .x;  / D ˆ. s/f .x; s/ D d 4 k d  e i.kx k0 x C55 c5  / fˆ˛ˇ .k; /; 5  .2/ where fˆ˛ˇ .k; / D

1 C .c5 /2 ˛ˇ f .k; / 

introduces a multiplicative factor that will appear once in each field bilinear of Tˆ˛ˇ . In terms of the 3-vector fields, the mass-energy-momentum tensor components are  1  0 e  eˆ C b  bˆ C 55   ˆ C  0 ˆ 2c i 1 e  bˆ C 55  0 ˆ c 1 e  ˆ c i 1   bˆ C  0 eˆ c  1  0   ˆ  0 ˆ C 55 .e  eˆ b  bˆ / : 2c

Tˆ00 D Tˆ0i D Tˆ50 D Tˆ5i D Tˆ55 D

For the plane wave, the energy density is Tˆ00 D

which, since e2? D

1 2

 1 C ./2 1 2 e? C 55 k2 c 

 e2? C b2 , is equivalent in form to the energy density in Maxwell theory  00 D

 1 E2 C B2 2c

with the addition of the independent polarization k . The mass density is found to be Tˆ55 D

2   2 00 2  2 2 1 C ./ C   D e T 55 k  ck02 ? k02 ˆ

64

4. PROBLEMS IN ELECTROSTATICS AND ELECTRODYNAMICS

expressing energy density scaled by the squared mass-to-energy ratio for the field. The energy flux—the standard Poynting 3-vector—is Tˆ0i ! T0ˆ D

k 00 T k0 ˆ

expressing the energy density Tˆ00 flowing uniformly in the direction of the momentum normalized to energy. Comparing the proportionality factor to that for a free particle k k0

!

p 1 M d x=d  v D D E=c c M dt =d  c

which will not generally be a unit vector unless  D 0, as it must be for Maxwell plane waves. The mass flux vector—a second Poynting 3-vector—can be written Tˆ5i ! T5ˆ D

k 55 T  ˆ

expressing the mass density Tˆ55 flowing uniformly in the direction of the momentum normalized to mass. Finally, k 0 55  Tˆ50 D Tˆ D 0 Tˆ00  k so that Tˆ5 can be written as Tˆ5 D

k  k  55 Tˆ D 2 Tˆ00  k0

expressing the mass density Tˆ55 flowing in the direction of the 4-momentum. In this sense, Tˆ50 represents the flow of mass into the time direction. We notice that when  ! 0, as is the case for Maxwell plane waves, k=k 0 becomes a unit vector and Tˆ5˛ D 0, so that mass density and flow vanish. The interpretation of plane waves carrying energy and momentum (energy flux) uniformly to infinity is thus seen to generalize to mass flow, where mass is best understood through (4.21) as the non-identity of energy and momentum. Suppose that a plane wave of this type impinges on a test particle in its rest frame, described by x ˛ ./ D .c; 0; c5  /. Since xP D 0, the wave will interact with the event through the Lorentz force (3.6) and (3.7) as M xR  . / D

 e  f 0 .x; /xP 0 ./ C c5 f 5 .x;  / c

which for k ¤ 0 ) k  k ¤ 0 becomes tR D

55 e

c5 1 k  k Mc 2 k 0

xR D

d . d

1 M xP 2 / 2

D

ec5 55  0 xP 0 c

c  i e h  c5   5 C  C  e? 1 C 55 k M c k0 c k0

4.5. RADIATION FROM A LINE ANTENNA

65

d 1 . 12 M xP 2 / D 55 ec5 0 k  k d k showing that the incident plane wave will initially accelerate the test event in such a way as to transfer mass. If the plane wave is a far field approximation to the radiation field of an accelerating charge, then the resulting picture describes the transfer of mass by the radiation field between charged events.

4.5

RADIATION FROM A LINE ANTENNA

The radiation from a dipole antenna is treated generally in Maxwell theory [7] by approximating the oscillating current as the separable current density J .x; t / D J .x/ e i!t

r  J .x/ C i! D 0;

where the second equation expresses represents current conservation, and of course we take the real parts of all physical quantities. This approximation may be justified by posing a collection of oscillating charges with position 4-vectors  Xn ./ D ctn . / ; an e i ! which for nonrelativistic motion includes tn ./ D t D  for each particle. The Maxwell current for this collection is XZ P n ./ ı 4 .x Xn .// d  cX J .x; t / D n XZ  D d  i ! can e i! ı .ct c / ı 3 x an e i! "n # X  i !t 3 i !t D i! an ı x an e e n

so that replacing the term in square brackets with its time average over one cycle of oscillation T D 2=! we obtain " Z # X  1 T J .x; t / ' dt i ! an ı 3 x an e i!t e i!t D J .x/ e i !t : T 0 n Thus, J .x/ approximates the time-dependent current density by a time averaged static configuration in space, rendering the antenna problem tractable. To treat the dipole antenna in SHP electrodynamics [8] we cannot make use of this approximation because the microscopic current X X  P n . / ı 4 .x Xn . // D j .x; t;  / D c X i! an e i ! ı .t / ı 3 x an e i ! n

n

66

4. PROBLEMS IN ELECTROSTATICS AND ELECTRODYNAMICS

is not integrated over  , and so time averaging cannot be performed in any meaningful way. Instead, in analogy to this approximation, we pose a current of the form   j 0 .x;  / D c 0 .x/ C  .x/ e i!t  . j .x;  / D J .x/ e i !t  .

t/

t/

  c5 0 j .x;  / D c5 0 .x/ C  .x/ e i!t  . c

j 5 .x;  / D

t/ ;

where 0 .x/ is a background event density. The function  . t / expresses a correlation between t and  , inserted by hand in place of a time averaging procedure. In this sense, the replacement j .x; t;  /

!

J .x/ e i !t  .

t/

may be less precise than the comparable approximation in Maxwell theory, and we must be attentive to artifacts introduced by the model. In analogy to (3.15), we choose  .

1 t/ D e 2

j tj=

which imposes a correlation 

 .

t/ !

D

Z

d! ˆ .!/ e i !. 2

t/

ˆ .!/ D

1 1 C .!/2

t '  through

8 < strong correlation:  ! 0 )  . : weak correlation:

 ! large ) t

t/ ! ı .

t/ ) t D 

 evenly distributed:

Notice that in the strong correlation limit, the potential found from the Green’s function Z  2 2   e 0 a .x;  / D d 3 x 0 d.ct 0 / ı x x0 c2 t t 0 J x0 e i !t ı  2   Z  e 1 jx x0 j i! 3 0 D e d x ı  tC J x0 0 4c c jx x j

t0



describes a Coulomb-like potential oscillating in  simultaneously across spacetime, rather than a wave propagating with phase .kr !t /. This suppression of the expected wavelike behavior can be characterized by the dimensionless parameter 1 T antenna period D D ! 2 correlation time

4.5. RADIATION FROM A LINE ANTENNA

67

which we take to be small but greater than zero. The total number of events in this system at time  is found from the spacetime integral Z 1 N ./ D d 4 x j 5 .x;  / c5 Z Z Z Z 3 3 D d x 0 .x/ dt  . t/ C d x  .x/ dt e i !t  . t/ D N0 C

N e i!

1 C .!/2

where N0 D

Z

;

3

d x 0 .x/

N D

Z

d 3 x  .x/

given as a background event number with an oscillating perturbation. We must have N0 >

N 1 C .!/2

to unsure that the event number remains positive. Similarly, the total charge is given by Q ./ D Q0 C

Q e i! 1 C .!/2

;

where Q0 D eN0 and Q D eN , so that the total charge does not change sign, which would suggest pair creation and annihilation processes. Since the background density 0 .x/ is independent of t and  , conservation of the 5D current becomes 0D

 1 @ 5 @  i!t  1 @ 0 j Cr jC j D  .x/ e  . t/ C r  J .x/ e i!t  . c @t c5 @ @t @ C  .x/ e i !t  . t/ @ D Œi! .x/ C r  J .x/ e i !t  . t/

so that i ! C r  J D 0

! e

and we identify pD

Z

Z

d 3 x J .x/ D

d 3 x x e  .x/

e

Z

d 3x x r  J D e

Z

t/

d 3 x x .i!/

i !p D Id dO

as the dipole moment p of the charge distribution  .x/, so that i!p can be written as a constant current I along a dipole of length d in the direction dO . The total current density is Z e i! pe i! J . / D d 4 x J .x/ e i!t  . t/ D c 1 C .!/2

68

4. PROBLEMS IN ELECTROSTATICS AND ELECTRODYNAMICS

representing an oscillating dipole. The induced potential found from the Green’s function GMaxwell is     Z e 1 jx x0 j ˛ 3 0 ˛ 0 a .x;  / D d x ;x ; j c t c 4 jx x0 j c so that writing x D r rO , we make the far field approximation 1=2 ˇ ˇ  2 R D ˇx x0 ˇ D r 2 C x0 2r rO  x0 'r

rO  x0

and the dipole approximation ˇ ˇ ˇkOr  x0 ˇ < kd D 2d  1 ) e ikOrx0 ' 1 

r

 1

rO  x0 r ' c c

d rO  xO r 0



'

to obtain a0 .x;  / '

Q Q0 C e 4 r 4 r

a .x;  / ' p a5 .x;  / D

ik e 4 r

i .kr !t /

i .kr !t /

  

  

tC

tC

r c

r c

c5 0 a .x;  / : c

We define the spherical wave factor  .x;  / D

e

i .kr !t /

4 r

  

tC

r c

and split the field strengths into spacetime and polarization factors, as b D r a Db b eD

1 @ a c @t

 D 55

ra0 D

1 @ a c5 @

 0 D 55

b bD

 b e D ik"1 QOr

Q0 rO Cb e 4 r 2

c5 0 c5 Q0 ra D rO Cb  c c 4 r 2

1 @ 0 a c5 @

ikId "1 rO  dO

1 @ 5 a Db  0 c @t

b  D ik

hc

5

 Id dO

i "1 QOr C i"2 Id dO

c hc i 5 b  0 D ik "1 C i"2 Q; c

where we used 1=kr  1 and define "1 D 1 C

".

t C R=c/ i !

"2 D

55

c ". c5

t C R=c/ : !

r c

4.5. RADIATION FROM A LINE ANTENNA

69

We drop the static Coulomb terms produced by Q0 , as these do not contribute to radiation. Since !  small tends to suppress wavelike behavior, but c5 =c  1 [9], we approximate "1 ' 1 but leave "2 unchanged. Taking the orientation of the antenna to be dO D zO the polarizations then simplify to b b e ' ik .QOr Id zO / b ' ikId rO  zO i hc hc i 5 5 b  ' ik b  0 ' ik C i "2 Q QOr C i"2 Id zO c c and we notice that terms containing 1=! appear only in the components of b   . Such terms are artifacts of modeling the time correlation by . t /, and can be understood as the contribution to the fields required to impose this correlation across spacetime. As was seen for plane waves, these fields will accelerate a test event initially at rest through the Lorentz force in such a way as to transfer mass to the event. The mass-energy-momentum tensor will contain bilinear field combinations of the type   h h  i  i 1 1 ı" ˛ T ˛ˇ D fˆ fı" g ˛ˇ fˆ˛ f ˇ ! Re A C i B˛   Re Cˇ C i Dˇ  c 4 and it is convenient to separate the resulting products as T ˛ˇ all terms containing 1=! . We designate  !2   t C cr 2 S .x;  / D k sin2 .kr 4 r  !2 r   t C c cos2 .kr C .x;  / D k 2 4 r  !2   t C cr 2 X .x;  / D k 2 sin .kr 4 r

D T0˛ˇ C T˛ˇ , where T˛ˇ includes

!t /

!t /

!t / cos .kr

!t /

and note that these functions drop off as 1=r 2 and so will produce nonzero surface integrals at large r , as is characteristic of radiation fields. Using these functions, the components of T0˛ˇ are      c 2 1 5 2 2 2 00 2 .Q C Id cos / C 2 .Id / 1 .cos  / C 255 T0 D Q S .x;  / c 2   c 2   5 T00 D Id .Q Id cos  / zO C Id .Id Q cos / C 55 Q rO S .x;  / c c5 T050 D Q .Q Id cos / S .x;  / c c 5 T50 D Q .Q Id cos / S .x;  / rO D T050 rO c 1 55  .Q Id cos /2 S .x;  / T055 D 2

70

4. PROBLEMS IN ELECTROSTATICS AND ELECTRODYNAMICS

all of which have spacetime dependence S .x;  /. The components of T˛ˇ are T00 D T0 D T50 D T5 D T55 D

i i 1 h 2h c5 "2 .Id /2 C Q2 C .x;  / 55 "2 Q ŒQ C Id cos  X .x;  / 2 c     c5 c 55 "2 Q Id X .x;  / C 55 C .x;  / zO C Q X .x;  / rO c c5 "2 Id .Id

Q cos / X .x;  /

h "2 Id ŒQ

 Id cos  zO C .Id /2

1 h 2 " .Id /2 2 2

 i Q2 rO X .x;  /

 c5 Q2 C .x;  / C "2 Q .Q c

Id cos  / X .x;  /

i

whose spacetime dependence is determined by C .x;  / and X .x;  / and is thus out of phase with the T0˛ˇ . As expected from the transfer of mass made possible by the fields   , we find a nonzero mass density T 55 and mass flux T0 and T5 into time and space. Moreover, integrating over a sphere of radius r , the net mass flux into space will be of the form Z P D d  r 2 rO  T50 Z h c i 5 D d  r 2 rO  Q .Q Id .cos // S .x;  / rO c  !2 Z c5 2 2   t C cr 2 DQ r k sin .kr !t / d  ŒQ Id cos   c 4 r k 2 c5 2   r 2 2 Q   tC sin .kr !t / D 4c c and thus nonzero wherever  ' t r=c . Just as the energy radiated by a Maxwell dipole antenna must be provided by the amplifier that drives the oscillating current density, the mass radiated by an SHP antenna is continuously provided by an amplifier that creates events and drives them into the antenna. For a center-fed antenna of length d oriented along the z -axis, the charge density may be described by ( d  z  d2 ı .x/ ı .y/ z .z/ ; 2  .x/ D 0 ; otherwise; where z .z/ D

1 1 Œz .z/ C z . z/ C Œz .z/ 2 2

z . z/ D C .z/ C  .z/

4.5. RADIATION FROM A LINE ANTENNA

71

divides the charge density into even and odd parts. The total oscillating charge is

QD

Z

3

d x e .x/ D 2e

Z 0

d 2

dz C .z/

and the dipole moment is

Id dO D i!e

d

Z

d 3 x x .x/ D 2i e! zO

Z2

dz z  .z/

0

showing that C describes a net charge Q driven symmetrically into the left and right segments of the antenna, while  describes a dipole moment produced by shifting charge from one antenna segment into the other segment. Since Q D eN , we see that the amplifier driving net charge into the antenna must be driving new events into the antenna as well, accounting for the radiated mass. Taking Q D 0 so that the amplifier shifts charged events between antenna segments without injecting new events, the fields reduce to b eD

b bD

ikId zO

b 0 D0

ikId rO  zO

b  D k"2 Id zO

so that the effect of the waves on a test event at rest reduces to d . d

1 M xP 2 / 2

D

e55 c5  0 D 0

and there is no transfer of mass. Similarly, the components of become T0˛ˇ  T000 D .Id /2 1 T00 D .Id /2

 1 cos2  S .x;  / 2  cos  zO C rO S .x;  /

T050 D 0 T50 D 0 1 55  .Id /2 cos2  S .x;  / T055 D 2

describing no transfer of mass into space or time.

72

4. PROBLEMS IN ELECTROSTATICS AND ELECTRODYNAMICS

The components of T˛ˇ also simplify to T00

D

1 2



c 0 !c5 

2

.Id /2 C .x;  /

T0 D 0

T50 D

0 c 55 .Id /2 X .x;  / !c5 

T5 D T 50 .Oz rO /   c 0 2 1 .Id /2 C .x;  / ; T55 D 2 !c5 

where we replaced "R = with  0 = . These expressions involve no transfer of energy but do describe nonzero transfer of mass into space and time directions. Once again, we understand this transfer as an artifact of the time correlation model that enters through the derivative of  . t/, rather than an inherent feature of radiation from an oscillating charge. In particular, all of the nonzero terms in the expression for mass conservation contain  0 , so that these terms are separately conserved among themselves. To see this we expand (3.25) as @˛ T ˛5 D

e 5˛ f j˛ c

!

1 @ 50 1 @ 55 T C r  T5 C T D c @t c5 @

e   j c

which becomes  1 @ 50 1 @ T C r  T5 C T055 C T55 D c @t c5 @

e   j c 

(4.22)

because T050 D T50 D 0. We also write the field as  because it contains the factor "2 . Finally, we note that because T055 depends on  only through the factor of  2 in S .x;  /, the derivative @ T055 must similarly contain  0 . Thus, each term in (4.22) enters through the derivative of the time correlation model, and these terms are conserved among themselves with no corresponding energy transfer. Integrating the energy Poynting vector T0 over the surface of a sphere of radius r we must evaluate  rO  T0 D .Id /2 rO  cos  zO C rO S .x;  /  D .Id /2 cos2  C 1 S .x;  / D .Id /2 sin2  S .x;  /

4.6. CLASSICAL PAIR PRODUCTION

73

to find the instantaneous radiated power Z P D d  r 2 .Id /2 S .x;  / sin2   !2 Z 2 Z  t C cr 2 2 2   D .Id / r k sin2 .kr !t / d d sin3  4 r 0 0  !2 r   tCc 8 sin2 .kr !t / D .Id /2 k 2 4 3   2 2   2 r k .Id /   tC sin .kr !t / : D 6 c  Since we have assumed that 1=! is small, we may take   t C cr as effectively constant over one cycle of the wave, so that the average radiated power over one cycle is Z k 2 .Id /2   r 2 1 T r 2 k 2 .Id /2   PN '   tC dt .sin .kr !t //2 D   tC 6 c T 0 12 c which agrees with the standard result up to the factor of  2 . The neutral antenna radiates energy in agreement with the Maxwell result and radiates no mass (leaving aside the derivatives of the arbitrarily chosen function  ).

4.6

CLASSICAL PAIR PRODUCTION

A standard technique for pair creation in the laboratory is the two-step process by which Anderson [10] first observed positrons in 1932: high energy electrons are first scattered by heavy nuclei to produce bremsstrahlung radiation, and electron/positron pairs are then created from the radiation field. The Bethe-Heitler mechanism [11] describes this technique as the quantum process, e C Z ! e CZ C Z C ! Z C e C eC

involving a quantized radiation field and the external Coulomb field of the nuclei. We now calculate the classical trajectories that produce this two-step process, as shown in Figure 4.1. Because the electromagnetic interaction is instantaneous in  , we may take both stages of the Bethe-Heitler process as occurring at 2 : (1) the scattering of particle-2 by a nucleus at t1 (Ein > 0 ! Eout > 0) and (2) the absorption of the resulting bremsstrahlung radiation by particle-1 at t2 (Ein < 0 ! Eout > 0). In the Stueckelberg picture, the E < 0 (antiparticle) trajectory of particle-1 must have been produced at the earlier chronological time 1 < 2 . To examine the conditions that might produce this initial negative energy trajectory, we describe particle-1 scattering in the Coulomb field of another nucleus at t D t3 and emerging with negative energy moving backward in t .

74

4. PROBLEMS IN ELECTROSTATICS AND ELECTRODYNAMICS

t Z τ1 E>0 1

E0



i) nt (a

pa rt ic le



1

t3

t2 t1

E>0

τ 2 > τ1 E>0

Z

Figure 4.1: Bethe–Heitler mechanism in classical electrodynamics.

In the laboratory, where events are recorded in the order determined by clock t , the process appears as particle-2 scattering at t D t1 and emitting bremsstrahlung, followed by the appearance at t D t2 of a particle/antiparticle pair. Then at t D t3 , the antiparticle encounters another particle causing their mutual annihilation. Our analysis is carried out in three parts. We first consider the Coulomb scattering of a slow incoming particle by an oppositely charged nucleus. To produce the pair annihilation observed at 1 , the outgoing particle must have E < 0, while at 2 the interaction must lead to E > 0. We identify the condition that allows the energy of the outgoing particle to change sign. In the second part, we compute the radiation field produced by the acceleration of the scattered particle at 2 using the Liénard–Wiechert potential for an arbitrary trajectory. In the third part we again use the Lorentz force to treat the acceleration of the E < 0 particle absorbing the radiation at 2 , and find the condition for its return to an E > 0 trajectory. With the function '. 1 / in the field strengths, the Lorentz force is a set of coupled nonlinear differential equations. By taking the correlation time  to be small we may again approximate '. 1 /  ı. 1 /, so that interactions are limited to a range R  c and outgoing scattering trajectories are easily obtained by integration of the Lorentz force. This solution provides a reasonable qualitative description of the classical Bethe–Heitler process, which may be refined by numerical solution of the exact Lorentz force equations.

4.6. CLASSICAL PAIR PRODUCTION

75

Initially (at time  ! 1), the target nucleus Z and incoming particle are widely separated. We set the nucleus at rest at the origin of the laboratory frame, XZ . / D .ctZ ; xZ / D .c; 0/ 

and from some point x the line of observation zDx

XZ . / D .ct; x/

.c; 0/ 

satisfies  /; x/2 D 0

z 2 D .c.t

!

c.t

/ D R D jxj

  O ; z D R 1; R

!

where R is the scalar length defined in (4.9) as uz D c

XP Z  z D c

  O .c; 0/  R 1; R c

D R:

For the observation point we use the location of the incoming particle-1, approaching the nucleus on the trajectory  x D Xin . / D .ct; x/ D u C s D tPin .c; v; 0; 0/  C s t ; 0; sy ; 0 ; where uD

dx dx D tPin D v tPin d dt

d .ct; x; y; z/ d

tPin D

dt 1 Dp d 1 ˇ2

and ˇ D v=c . The scattering takes place in the plane z D 0 and since the nucleus is at the origin, we can write the spatial distance between the incoming particle and the target as q p 2 2 2 R . / D jxj D x C y D v tPin  C sy2 : Putting   R.1 /, the support of the fields is narrowly centered around the retarded time 1 , so that 1 is determined from the causality conditions for the initial trajectories, ŒXin .1 /

XZ .1 / 2 D 0

Xin0 .1 /

XZ0 .1 / > 0:

These equations have the solution

1 D

v tPin

 q 1  v s t C s t2 1 2v

sy2 1

2v

 

!

vc

q s t2 v

sy2

;

76

4. PROBLEMS IN ELECTROSTATICS AND ELECTRODYNAMICS

where we introduce the smooth parameter    1 1 0; v D 0 v D 1 ! P 1; v D c: v tin Notice that the 0-component s t of the impact parameter must be positive in order for the interaction to take place. The location of the incoming particle at the time of interaction is now O x .1 / D R R

where RD  O D R

v s t C

1 1

2v

t .1 / D tPin 1 C s t =c;

 q v s t2

 2v C s t

sy2 1



q s t2

   sy2 1 2v ; 1 2v sy ; 0 q  s t C v s t2 sy2 1 2v

! st

vc

0s

[email protected] 1

vc

1 sy2 sy ; ; 0A : s t2 s t

Applying the Coulomb potential calculated in Section 4.1.1, the potential induced by the target nucleus in this approximation is Ze ı . 4R so that the nonzero field strengths a0 .x;  / D 

e i D f 0i D @0 ai

@i a0

are eD

ra

0

D

a5 .x;  / D

 i D f 5i D @5 ai c5 ra D e c 1 / D



dt d

@i a 5

55  D c5

5

where we used @ ı. @t

ai D 0

1 /

0



1 tDtin

@ ı. @

c5 0 a .x;  / c

 0 D @5 a 0

@0 a5

  c52 1 1C 2 @ a0 ; c tPin 1 / :

The nucleus and the incoming particle have opposite charge, so the Lorentz force   e e M xR 0 D f 0i xP i C f 05 xP 5 D e  xP 55 c5  0 c c   e  k0 e M xR D f xP 0 C f k5 xP 5 D ec tP 55 c5  c c on the incoming particle becomes     c52 1 ı . 1 / Ze 2 x P  r 1 C @ tR D 2 2 Mc c tPin 4R   c2 ı . 1 / Ze 2 tP 55 52 r : xR D M c 4R

4.6. CLASSICAL PAIR PRODUCTION

77

The delta function enables immediate integration of the force equations as tPf

Ze 2 tPin D Mc 2

1 C=2 Z

d

 xP  r

1 =2

   c2 1 ı . 1 / @ 1 C 52 c tPin 4R

2

Ze 1 xP .1 /  r Mc 2 4R  Ze 2 O D xP .1 /  R M c 2 4R2 D

xPf

Ze 2 xP in D M

1 C=2 Z

d

 tP

1 =2

D

 Ze 2 M 4R2

 tP .1 /

 c52 ı . 1 / r 2 c 4R  c2 O 55 52 R; c

(4.23)

55

(4.24)

where the velocities are evaluated at the interaction point as  i   1h tP; xP f C tP; xP in : tP; xP .1 / D 2 We introduce the dimensionless parameter for Coulomb scattering ge D

 Ze 2 c Ze 2 1 correlation length interaction energy D  D  2 2 Mc 4R R 4R Mc impact parameter mass energy

which appears in (4.23) and (4.24) as the factor controlling the strength of the interaction. Writing 1 1 ˛x D ge RO x ˛y D ge RO y 2 2 we can expand the Lorentz force as components in the form 2 3 2 32 3 2 32 3 1 ˛x ˛y c tPf 1 ˛x 0 c tPin 0 2 c 4 ˛x 1 0 5 4 xPf 5 D 4 ˛x 1 0 5 4 v tPin 5 C 255 52 4 ˛x 5 c ˛y 0 1 ˛y yPf ˛y 0 0 0 and solve for the final velocity, 82 3 2 3 ˆ c tPf c tPin ˆ < 1 7 6 6 P 7 v tin 5 4 xPf 5 D 1 2 ˆ4 1 4 ge ˆ : yP 0 f

where we neglect c52 =c 2  tPin .

39 2 3 c > v RO x =  7> 6  1 6 7 2 2 2 7 ; O O ge tPin 4 c RO x 5 C ge tPin 6 4 v Rx Ry 5> 4 > ; O c Ry 2v RO x RO y 2

(4.25)

78

4. PROBLEMS IN ELECTROSTATICS AND ELECTRODYNAMICS

Before considering pair annihilation, we examine the low velocity and low interaction energy limit of this result. Taking tPin ! 1

jPxj D v  c

v ! 0

ge  1

the initial velocity reduces to XP in ./ ! .c; v; 0; 0/ ;

the final velocity becomes tPf  tPin

xPf  xP

0s

O [email protected] 1 R

O ge c R

1 sy2 sy ; ; 0A s t2 s t

R D st

and the scattering angle can be found as O  xP xPf  xP xP 2 ge c R v ge c RO x ˇ ˇ ˇ ˇ : cos  D ˇ ˇ D D ˇxPf ˇ jPxj ˇxPf ˇ jPxj ˇxPf ˇ

If we also wish to impose the nonrelativistic condition for conservation of energy, we obtain a new constraint in the form h i2 O xP 2 D v 2 D xPf2 D xP ge c R ) 2v RO x D ge c in which case

1 h cos  D ˇ ˇ v ˇxPf ˇ

i ge c RO x D 1

2RO x2 :

Now, using the definition of ge we find r 2s t 4M v 2 sy  1 C cos  RO y sy 2v D D  cot D D 2 1 cos  s t ge c v Ze 2 RO x which recovers the Rutherford scattering formula if 2s t D 1: v

(4.26)

But for low energy we have s t D R .1 / which we assumed to be comparable to c . Since we cannot have v  c in this low velocity case, (4.26) cannot be maintained. This result is unsurprising because the short-range potential cannot provide an adequate model of nonrelativistic Rutherford scattering. Removing these restrictions and returning to the relativistic case, the condition for pair annihilation at 1 is that particle-1 scatters to negative energy, that is tPf < 0 for some value of ge which we call g1 . From (4.25), tPf D tPin

1

g1 .v=c/RO x C 14 g12 1

1 2 g 4 1

4.6. CLASSICAL PAIR PRODUCTION

79

and we see that for small values of g1 , tPf

! tPin  1:

Since v < c and Rx < 1, the numerator has discriminant .vRx =c/2

1 2 M c2 4R meaning that the interaction energy is greater than the mass energy of the annihilated particles. As g1 approaches 2 from below tPf becomes very large. After g1 passes this critical value, tPf decreases from large negative values, taking the limiting value  tPf ! tPin C 2 H) Ef D .Ein C 2Mc 2 / g1 !1

so that the outgoing trajectory is timelike for all values of g1 > 2. Having found the condition for pair annihilation at time 1 we now consider the scattering at time 2 , which we also treat as an incoming particle approaching a nucleus of opposite charge. Therefore we may apply the general expression (4.25). Particle-2 approaches a second nucleus along some trajectory xin . / and emerges from the interaction along trajectory xf ./ with positive energy. The scattering and acceleration of particle-2 produces a radiation field which can be evaluated at some point of observation y  using the Liénard-Wiechert potential for an arbitrary trajectory. The support of '. 2 / is narrowly centered on 2 , and so the line of observation z  must be a lightlike vector, which we write as z D y

x  .2 / D O

O ; O 2 D 1: O D .1; /

We express the initial and final 4-velocities of the scattered particle as ˇin D xP in =c

ˇf D xPf =c

and define ˇ D ˇf ˇin ˇ . / D ˇin C ˇ  . 2 / ˇP . / D ˇ ı . 2 /  1 ˇ .2 / D ˇN D ˇf C ˇin : 2

80

4. PROBLEMS IN ELECTROSTATICS AND ELECTRODYNAMICS

From (4.11) and (4.12) express the radiation fields produced by an arbitrary trajectory as      5 frad D e'. 2 /F  z; ˇ; ˇP frad D e'. 2 /F 5 z; ˇ; ˇP ; where

F 

 2 z ^ ˇP  D4

  3 .z ^ ˇ/ ˇP  z 5 4c3

F 5

 3 2 Pz z ˇ c5 4 5 D c 4c3

and  D z  ˇ is the scalar distance from the scattered particle to the point of observation. As pictured in Figure 4.1, the radiation emitted by the scattering of particle-2 is absorbed by the negative energy particle-1 arriving at y  . Using the Lorentz force equations we calculate the change in velocity of particle-1 caused by the incoming radiation. Since each term in the P field strengths contains ˇ./ D ˇ ı . 2 / and '.0/ D 1=2, the change in velocity yP  .2 / of particle-1 is Z 1 h i e  5 yP  D d  frad yP C frad yP5 Mc 1 Z h    i e2 1 D d  '. 2 / F  z; ˇ; ˇP yP 55 c5 F 5 z; ˇ; ˇP Mc 1     e2  5 N N D (4.27) F z; ˇ; ˇ yP C 55 c5 F z; ˇ; ˇ 2Mc expressed in terms of the velocity change ˇ and average velocity ˇN . These are found from (4.25) to be 3 2 3 2 1 ˇ RO x 1 2 7 6 7 6 g ge 6 RO x 7 C 2 e tPin 6 ˇ RO 2 7 P ˇ D t in x 5 5 1 1 g2 4 1 41 ge2 4 4 e ˇ RO x RO y RO y

ˇN D

3

1 4 ˇ 5 P t in 1 2 g 4 e 0

1 1

2

2

3 2 0 ˇ RO x 1 1 2 6 7 6 g g e 2 4 e O2 P 6 tPin 6 RO x 7 5 C 1 1 g 2 ˇ tin 4 Ry 1 14 ge2 4 4 e RO x RO y RO y

3

7 7; 5

O is the unit vector from the second nucleus to incoming particle-2 at the moment where now R of scattering. Since particle-2 scatters at 2 to an E > 0 outgoing trajectory, we may take v  c and so we set ge D g2 < 1 and g22  0 for this interaction. From (4.27) the Lorentz force acting on particle-1 at 2 can be written " #  .z ^ ˇ/  z ^ ˇN .ˇ  z/ e2  yPf C yPf 2Mc 4c3 " #  z ^ ˇN .ˇ  z/ e 2 .z ^ ˇ/   D yPin yPin 2Mc 4c3

4.6. CLASSICAL PAIR PRODUCTION

O  O D 0, we find neglecting the term .c52 =c 2 /F 5 . Making the simplifying choice R    1 O N ˇ  z D tPin 1 v Ox C g2 Rx ˇ  z D g2 tPin vRO x ; 2

where again we take g22  0. Defining a second dimensionless factor for radiation gR D

1 e2 1 1 interaction energy D 2 2 4 Mc 2 mass energy

using 

.ˇ  z/ z ^ ˇN

.z ^ ˇ/ 



 yP D z .ˇ  y/ P  .z P ˇ   y/   .ˇ  z/ ˇN  yP z .z  y/ P ˇN

and now taking ˇ  0, the Lorentz force splits into the 0-component yPf0

O  yPf D yPin0 C g2 gR R O  yP in g2 gR R

and the space component h   O  yPf O C yP 0 yPf g2 gR R f

 i O D O  yPf R h  O  yP in O C yPin0 yP in C g2 gR R

 i O : O  yP in R

We write the velocity of incoming negative energy particle-1 as yPin0
1 which requires that g D g2 gR > 1

!

2Mc2 e2 > ; 4 g2

where g2 < 1 and so the energy absorbed from the bremsstrahlung emitted from the scattering at 2 must be at least the total mass of the particle creation event observed in the laboratory. This provides a classical equivalent of the Bethe–Heitler mechanism in Stueckelberg–Horwitz–Piron electrodynamics.

4.7

PARTICLE MASS STABILIZATION

As we have seen, under the right circumstances a particle and an interacting pre-Maxwell field may exchange mass. In practical examples, such as pair creation and annihilation, the mass shift will be symmetric under evolution, so that the initial and final masses will be equal. As another model of mass shift, consider an event propagating uniformly on-shell as  x ./ D u D u0 ; u u2 D c 2

4.7. PARTICLE MASS STABILIZATION

83

until it passes through a dense region of charged particles inducing x . / D u C X . / ;

where X ./ is a small stochastic perturbation. If the typical distance scale between force centers is d then the perturbation will be roughly periodic with a characteristic period d a very short distance D D a very short time, a moderate velocity juj

a fundamental frequency

!0 D 2

juj D very high frequency, d

and an amplitude on the order of jX  . /j  ˛d

for some macroscopic factor ˛ < 1. The perturbation can be represented in a Fourier series X X sn e in!0  an e in!0  D ˛d Re X . / D Re n

n

with four-vector coefficients

  an D ˛dsn D ˛d sn0 ; sn D ˛d csnt ; sn ;

where the sn represent a normalized Fourier series (s0  1). The perturbed motion is of scale d , but the perturbed velocity X 2 n sn ie in!0  xP  . / D u C XP  . / D u C ˛ juj Re n

is of macroscopic scale ˛ juj. The unperturbed mass is m D M xP 2 . / =c 2 D M and the perturbed mass is !2 X M xP 2 . / M in!0  mD D u C ˛ juj Re 2 n sn ie c2 c2 n ! X t in!0  ' M 1 C 4 ˛ juj Re n sn ie ; n

2

where we neglect terms in ˛ . This kind of interaction may produce a macroscopic mass shift   X m m m !m 1C D 4 ˛ juj Re n snt ie in!0  m m n that remains significant after the interaction. Two approaches have been suggested to explain why such mass shifts are not observed: one involving a self-interaction of the particle and its radiation field under mass shift, and the second a more general argument in statistical mechanics.

84

4. PROBLEMS IN ELECTROSTATICS AND ELECTRODYNAMICS

4.7.1 SELF-INTERACTION We consider an arbitrarily moving event X  . / at the origin of a co-moving frame so that  XP . / D c tP ./ ; 0

X . / D .ct . / ; 0/

and a change in mass m D M XP 2 =c 2 D M tP2 can only result from a change in energy through acceleration of t . We say that the event is on-shell if tP D 1. The Green’s function permits us to compute the field at some point x induced by the evolving event. If the motion at time  produces an observable field at time   >  at some point x D X.  / along the trajectory of the event itself, then the event will experience a self-force. Because GMaxwell D 0 on the event’s timelike trajectory, only a contribution from GCorrelation can produce such a self-interaction, and, as seen from (3.24), only if 55 D C1. We approximate '. 0 s/ D ı. 0 s/ as in Section 4.1.2, introduce the function g.s/ to express terms of the type      2 c52  2 2  2 2  2  2 t  t .s/ s/ D c c g .s/ D X. / X.s/ C c5 . . s/ c2 and write a

˛

X 





;





ec5 D 2 2 c 3

Z

1  .g.s// ds X .s/ 2 .g.s//3=2 P˛

ı .g.s// .g.s//

1=2

!

 ret

for the self-field experienced by the event. We designate the two terms as   a˛ X   ;   D a˛ C aı˛ : For an event evolving uniformly on-shell we have    c52 t  D  g.s/ D 1 .  c2

s/2

and using identity (4.7) are led to Z    ec5  .c; / a X  ;  D 0; c ds   s 5 2 20 c3    c52 2  1 c 2 . s/ B1  B B   3=2 @2 c52 2  1 c 2 . s/ ec5 .c; 0; c5 / D  3=2 c52 2 3 2 c 1 c 2

Z



0

 ı 1  1

1 1 ds @  2 . s/ 3 1

c52 c2 c52 c2





.

. 



s/ s/ 2

2

 1

C C 1=2 C A

ı .  s/  .  ˇ ˇ ˇ  ˇ s/2 ˇ ˇ.

1 s/ A

:

4.7. PARTICLE MASS STABILIZATION

85

Since Z



ds 1

1 . 

s/ 3

D

1 2 . 

ˇ  ˇ 1 ˇ D lim ˇ 2  s! s/ 2 . s/ 2 1

and Z



ds

ı . 

s/  .  s/2

. 

1

s/

D lim s!

 .  . 

s/ s/2

D lim s!

1 2

. 

s/2

we find that for uniform on-shell motion a X 





;





!

1 2

1

ec5 .c; 0; c5 / lim D s! 2 2 c 3

2 . 

s/ 2

s/2

. 

D0

the self-force vanishes. In general, because XP i D 0 and a˛ .X .  / ;   / does not depend on X i , we have ai D 0

@i a0 D @i a5 D 0

f  D f 5i D 0

)

and so the field reduces to f 50 D @5 a0

@0 a 5 D

1 1 @  a0 C @ t a5 ; c5 c

where the partial derivative @  only acts on the explicit variable (not on t .  / or  ret ). Similarly, the velocity XP ˛ .s/ is constant with respect to @  . Inserting the potential we find

@5 a0

@0 a5 D

Z

  .t .  /

t .s//2

c52 .  c2

s/2



 ret  ds  5=2    ; s c2 .t .  / t .s//2 c52 .  s/2   c52 2   2 Z ı .t . / t .s// . s/ c2  ec5 c5 ds  ret    ; s ;   2 3 3=2 2 c c c2 .t .  / t .s//2 c52 .  s/2

3ec5 c5 4 2 c 3 c

where     ; s D tP.s/. 

s/

t 



 t .s/

86

4. PROBLEMS IN ELECTROSTATICS AND ELECTRODYNAMICS

characterizes the energy acceleration in the rest frame, which will be associated with mass shift. Similarly, the derivatives of aı produce   c52 2   2 Z ı .t . / t .s// . s/ c2  ec5 c5 ret     ; s @5 aı0 @0 aı5 D ds   3=2 2 2 c 3 c c2 .t .  / t .s//2 c52 .  s/2   c52 2 0   2 Z 2ı .t . / t .s// . s/ c2  ec5 c5  ret    ; s ds   2 3 1=2 2 c c c2 .t .  / t .s//2 c52 .  s/2 and combining terms we find f 50 D f50 C fı50 C fı50 0 ;

where

f50 D

fı50 D

fı50 0 D

Z

  .t .  /

2

c52 .  c2

2



t .s// s/  ret  ds  5=2    ; s c2 .t .  / t .s//2 c52 .  s/2   c52 2   2 Z ı .t . / t .s// . s/ c2  e c52 ret  ds    ; s   3=2  2 c4 c2 .t .  / t .s//2 c52 .  s/2   c52 2 0   2 Z ı .t . / t .s// . s/ c2  e c52 ds   ret    ; s :  2 4 1=2  c c2 .t .  / t .s//2 c52 .  s/2

3e c52 4 2 c 4

(4.28)

(4.29)

(4.30)

Notice that if the particle remains at constant velocity (in any uniform frame), then  0   u0  u  u0 0 0  x ./ D u  !   ;s D . s/  s D0 c c c and so the self-force f 50 vanishes. For any smooth t . /, we may approximate t 



1 s/ C tR.s/.  s/2 C o .  2  1 s/ C tR.s/.  s/2 C o .  s/3 2

t .s/ D t .s/ C tP.s/.  D tP.s/. 

s/3



t .s/

4.7. PARTICLE MASS STABILIZATION

87

so the function     ; s D tP.s/. 

t 

s/



 t .s/ D

1 tR.s/.  2

s/2 C o . 

s/3



is nonzero only when the time coordinate accelerates in the rest frame, equivalent to a shift in the particle mass. As a first-order example, we consider a small, sudden jump in mass at  D 0 characterized by 8 8 < 1 <  ;  0 and calculate the self-interaction. Since  ret enforces t.  / > t.s/, it follows that  < 0

)

s 0 and s > 0

tP.  / D tP.s/ D 1 C ˇ

)

.  ; s/ D 0:

)

But when   > 0 and s < 0, .  ; s/ D tP.s/. 

s/

t 



 t .s/ D  

s





.1 C ˇ/  



 s D

ˇ 

and f 50 can be found from the contributions (4.28)–(4.30). Writing g .s/ D t  



t .s/

2

and solving for g.s  / D 0, we find

c52  . c2

s/2 D .1 C ˇ/  

0

B s  D @1 C

ˇ 1

s

2

c52  . c2

s/2

1

C   c5 A  >  c

so that g.s/ > 0 in the region of interest s < 0 <   and there will be no contribution from the terms (4.29) or (4.30). Thus, 2 Z 0 1  3e c5 50 50 f D f D . ˇ / 2 4 ds  5=2 4 c 1 c52  2 .t .  / t .s//2 . s/ c2 2 Z 0 1  3e c5 D . ˇ / 2 4 ds  5=2 : 2 4 c 1 c 2 5 ..1 C ˇ/   s/ .  s/2 c2

88

4. PROBLEMS IN ELECTROSTATICS AND ELECTRODYNAMICS

Shifting the integration variable as x D   Z

0

s the integral becomes

1

ds  1 ..1 C ˇ/  

s/2

c52  . c2

s/2

Z

D

5=2



1

dx .C x 2

C Bx C A/5=2

where c52 c2

C D1

B D 2ˇ 

A D ˇ 

2

which can be evaluated using the well-known form [3] Z

dx

D

.C x 2 C Bx C A/5=2

where q D 4AC

2.2C x C B/ p 3q C x 2 C Bx C A



1 8C C 2 C x C Bx C A q



;

B 2 . We finally find the field strength in the form f

50

  c52 1 e Q ˇ; 2 ; D 4 2 c52 .ˇ  /3 c

  c2 where Q ˇ; c52 is the positive, dimensionless factor 2

6 6 6 6 6    2 6 c5 Q ˇ; 2 D 6 62 1 c 6 6 6 6 4

0

B B B B B  3=2 2 B c5 B1 B 2 c B B B B @

0

11

1=2 B CC ˇ B CC B1 C  CC  @ c52 A C C 1 C c2 2 31=2 C C C C 2 6 7 2ˇ ˇ C 61 C 7 C C 2 25 4 A c5 c5 1 1 2 2 c c

 1

c52 c2

ˇ2 C

 1

c52 c2

2 c52 B B1 C c5 c2 @ c2

1=2

2

6 61 C 4

3

1

0

ˇ 1

c52 c2

2ˇ C c52 1 1 c2

7 7 7 7 7 7 33=2 7 7 7 7 ˇ2 7 7 7 5 25 c

C C A

5

c2

;

4.7. PARTICLE MASS STABILIZATION

89

which is seen to be finite for c5 < c , with   c2 Q ˇ; 52 c

c5 !0

!2 1

1Cˇ

Œ1 C 2ˇ C ˇ 2 1=2

!

D 0:

Since f  D 0, the Lorentz force induced by this field strength is then M xR  D ef ˛ xP ˛ D ef 5 xP 5 D

ef 5 xP 5 D

55 ef 5 xP 5 D

ef 5 c5

and since f 5i D 0 M xR i D 0 M xR 0 D c5 ef 50 D

8 ˆ < ˆ :

0

;  < 0

  c52 e 2 1 Q ˇ; 2 ;  > 0 4 2 c5 .ˇ  /3 c

which causes the 0-coordinate to decelerate. When the event returns to on-shell propagation the function .  ; s/ and field strength f 50 again vanish. The mass decay can also be seen in the Lorentz force for the mass     c52 d 1 c e 2 Q ˇ; M xP 2 D ef 5 xP  D ef 50 xP D ecf 50 tP D tP: d 2 4 2 c52 .ˇ  /3 c2 We notice that if ˇ < 0 then f 50 changes sign so that the self-interaction results in damping or anti-damping to push the trajectory toward on-shell behavior. Although this model is approximate, it seems to indicate that the self-interaction of the event with the field generated by its mass shift will restore the event to on-shell propagation.

4.7.2 STATISTICAL MECHANICS In Section 3.4 we saw that a particle, as observed through its electromagnetic current, can be interpreted as a weighted ensemble of events '.s/x  . C s/ selected from a neighborhood of event x  . / (along a single timelike trajectory) determined by '.s/. Here we model a particle as an ensemble xi ./ of N mutually interacting event trajectories given at a single  . Constructing the canonical and grand canonical ensembles without an a priori constraint on the total mass of the system, the total mass of the particle is determined by a chemical potential. Under perturbation, such as collisions for which the final asymptotic mass of an elementary event is not constrained by the basic theory, the particle returns to its equilibrium mass value. Here we provide here a brief summary of the full model given in [12, 13]. As described in Section 2.5, we first construct a canonical ensemble by extracting a small subensemble €s (the particle system) from its environment €b (the bath ensemble). Summing

90

4. PROBLEMS IN ELECTROSTATICS AND ELECTRODYNAMICS

over all possible partitions of energy and mass parameter between the particle and bath Z €.; E/ D d b d s d b d s ı.Kb b /ı.Ks s /ı.Es C Eb E/ı.s C b / Z D d  0 dE 0 €b .  0 ; E E 0 /€s . 0 ; E 0 / in which both mass and energy may be exchanged. We suppose that the integrand has a maximum over both variables  0 ; E 0 , providing an equilibrium point for the system. By analyzing the partial derivatives, it can be shown that no saddle point configuration is possible in the neighborhood of the maximum. The conditions for equilibrium can then be written €b .

1 0; E

E 0/

@€b . @E

0; E

E 0 /jmax D

1 @€s 0 0 1 . ; E /jmax  0 0 €s . ; E / @E T

€b .

1 0; E

E 0/

@€b . @

0; E

E 0 /jmax D

1 1 @€s 0 . ; E/jmax  ; 0 0 €s . ; E / @ T

and

defining temperature in the usual way, and a new effective “mass temperature” T . Writing Sb .; E/ D ln €b .; E/

Ss .; E/ D ln €s .; E/

it follows that at maximum @Ss 1 @Sb D D @E @E T

@Ss 1 @Sb D D : @ @ T

By additivity of entropy, the total entropy of the system is independent of  0 ; E 0 in the neighborhood of the maximum, and for  0 and E 0 small compared to  and E , €b .

0; E

E 0 / D e Sb .

 0 ;E E 0 /

Š e Sb .;E /

in this neighborhood. Then Z €.; E/ D d  0 dE 0 €s . 0 ; E 0 /e Sb .;E / e

0 T

e

0

@Sb @

E0 T

E0

@Sb @E

D e Sb .;E / e

D e Sb .;E /

Z

d s e

Ks T

0 T

e

e

E0 T

Es T

leading to the partition function QN .T ; T / D

Z

d e

K T

e

E T

;

where the overall factor Sb .; E/ cancels out in any computation of average values. The Helmholtz free energy A is defined through Z A.T ;T /=T QN .T ; T / D e d e K=T e .A E /=T D 1

4.7. PARTICLE MASS STABILIZATION

91

from which it follows that A D hEi C T

@A D hEi @T

TS

SD

@A @T

and

T2 @A : T @T Under the canonical distribution, corresponding to an equilibrium of both heat and mass, without exchange of particles with the bath, we therefore obtain a mean value for hKi, the effective center-of-mass mass of the subensemble, which is determined by T and T . Computing the fluctuations in energy, one finds hKi D

˝ .E

˛ @ hEi hEi/2 D T 2 @T

˝ .K

˛ @ hKi hKi/2 D T 2 @T

showing that the mean mass rises with the mass temperature. (Since K is proportional to a negative mass in this metric, T is a positive number, to be identified with a “mass temperature.”) Repeating the above for the grand canonical ensemble, in which the system (particle) ensemble may exchange events and volume with the bath, one decomposes the full microcanonical in terms of its canonical subsets N Z X QN .V; T; T / D d s e Ks =T e Es =T QN Ns .V Vs ; T; T /; Ns D0

where QN Kb D K

Ns .V

Ks and Eb D E

Vs ; T; T / D

Z

d b e

Kb =T

e

Eb =T

Es . Making the usual identifications @A D @V

@A D @N

P

and defining the new mass chemical potential @A D @K



leads to the grand partition function

Q.V; T; T / D e VP =T D where QNs .T; Ks ; Es / D

Z

d  s  Ks e

Es =T

N X

z Ns QNs .T; Ks ; Es /;

Ns D0

z D e =T

De

 O  =T

:

92

4. PROBLEMS IN ELECTROSTATICS AND ELECTRODYNAMICS

It follows that

@ @ ln Q hKi D  ln Q: @z @ Modifying the Helmholtz free energy for the grand canonical ensemble, hN i D z

T ln Q

A D hN i T ln z C hKi T ln 

leads to

QDe

A=T

z



:

It follows that the internal energy is U  hEi D A

  T @ ln Q hKi C T ln Q C T 2 hN i  C  C T @T

and using the thermodynamic relation U D A C TS

one finds

@ .T ln Q/ C SD @T Finally, the Maxwell relations are  ˇ @A ˇˇ SD @T ˇV;hN i;hKi @A D @ hN i

At the critical point in hKi

@A D0 @ hKi



 1 C T T



P D @A D @ hKi !

hKi 

 hN i : T

ˇ @A ˇˇ @V ˇ;;T

  T  C : T T D T



(4.31)

and so  is positive since T is negative. The particle in this model is a statistical ensemble which has both an equilibrium energy and an equilibrium mass, controlled by the temperature and chemical potentials, thus assuring asymptotic states with the correct mass. The thermodynamic properties of this system, involve the maximization of the integrand in the microcanonical ensemble, where both the energy and the mass are parameters of the distribution. A critical point in the free energy is made available by the interplay of the equilibrium requirements of the canonical ensemble (where the total mass of the system is considered variable) as for the energy, and the equilibrium requirements of the grand canonical ensemble (where a chemical potential arises for the particle number). The particle mass is controlled by a chemical potential, so that asymptotic variations in the mass can be restored to a given value by relaxation to satisfy the equilibrium conditions.

4.8. SPEEDS OF LIGHT AND THE MAXWELL LIMIT

4.8

93

SPEEDS OF LIGHT AND THE MAXWELL LIMIT

As discussed in Section 3.7, concatenation—integration of the pre-Maxwell field equations over the evolution parameter  —extracts from the microscopic event interactions the massless modes in Maxwell electrodynamics, expressing a certain equilibrium limit when mass exchange settles to zero. In this picture, the microscopic dynamics approach an equilibrium state because the boundary conditions hold pointwise in x as  ! 1, asymptotically eliminating interactions that cannot be described in Maxwell theory. The Maxwell-type description recovered by concatenating the microscopic dynamics may thus be understood as a self-consistent summary constructed a posteriori from the complete worldlines. We have assumed that 0  c5 < c and we must check that SHP theory remains finite as c5 ! 0. First we notice that c5 appears explicitly three times in the pre-Maxwell equations (3.20) 1 e e c5 @ f 5 D j' @ f 5 D j'5 D e' @ f  c5 c c c @ f C @ f C @ f D 0

@ f5

@ f5 C

1 @ f D 0 c5

twice in the form c15 @ and once multiplying the event density ' . The derivative term poses no problem in the homogeneous pre-Maxwell equation, which is satisfied identically for fields derived from potentials. Specifically, the fields f5 contain terms of the type @5 a D c15 @ a that cancel the explicit  -derivative of f , evaluated before passing to the limit c5 ! 0. However, the homogeneous equation does impose a new condition through  c5 @ f5 @ f5 C @ f D 0 ! @ f D 0 c5 !0



requiring that the field strength f become  -independent in this limit. For the fields derived in Section 4.2 this condition is violated by the multiplicative factor '. R / unless we simultaneously require c5 ! 0 )   1=c5 ! 1, in which case '.x; / ! 1=2 D 1, using (3.12) for  . This requirement effectively spreads the event current j'˛ uniformly along the particle worldline, recovering the  -independent particle current Z Z ! ds 1  j  .x; s/ D J  .x/ j' .x;  / D ds ' . s/ j  .x; s/ j'5 .x;  / D

Z

ds ' .

s/ j 5 .x; s/

@ j' .x;  / C

1 @ j'5 .x;  / c5

!

Z

ds j 5 .x; s/

! @ J  .x/ D 0

associated with Maxwell theory. Generally, because the  -dependence of the potentials and fields is contained in ' , the condition  ! 1 eliminates all the terms in the pre-Maxwell equations containing @ . Similarly, the photon mass m  „=c 2 must vanish.

94

4. PROBLEMS IN ELECTROSTATICS AND ELECTRODYNAMICS

We saw that f 5 is generally proportional to c5 for fields of the Liénard–Wiechert type. Therefore, we can write the inhomogeneous pre-Maxwell equations in the finite form   e  1 5 e  @ f D j' f D ' ; @ c c5 c where we see that f 5 decouples from the field f  that now satisfies Maxwell’s equations. To find the limiting form of the electromagnetic interactions, we consider an arbitrary event X  ./, which induces the current Z ˛ j' .x;  / D c ds ' . s/ XP ˛ .s/ ı 4 Œx X .s/ : From the field strengths found in Section 4.2 the Lorentz force on a test event moving in the field induced by this current can be written  e  M xR  D f  .x;  /xP  C f 5 .x; /xP 5 c D

where

F  .x; / D

e2 e 4c

j R j=

F  .x;  /xP  C c52 F 5 .x;  / ; 1 C .c5 =c/2

   .z ˇ e z  ˇ  / ˇ 2 " . R / z  ˇ  z  ˇ  4R R2 c R  9   z  ˇP  z  ˇP  R C .z  ˇ  z  ˇ  / ˇP  z = R2

F 5 .x; / D

e 4cR



 2



z ˇ Cˇ R R2





" . R / z C ˇ Rc c  9 R  P z ˇz = : C cR2 ;

2

=c52

;

In the limit  ! 1 and c5 ! 0, we see that c52 F 5 .x;  / ! 0, and so the Lorentz force interaction reduces to the  -independent expression M xR  D

e2 F  .x/xP  4c

recovering the Lorentz force in the standard Maxwell form. The parameter c5 =c thus provides a continuous scaling of Maxwell’s equations and the Lorentz force to the standard forms in Maxwell theory. The combined limit  ! 1 and c5 ! 0 restricts the possible dynamics in SHP to those of Maxwell theory, as a system in  -equilibrium [9].

4.9. BIBLIOGRAPHY

4.9

95

BIBLIOGRAPHY

[1] Tanabashi, M., Hagiwara, K., Hikasa, K., Nakamura, K., Sumino, Y., Takahashi, F., Tanaka, J., Agashe, K., Aielli, G., Amsler, C., Antonelli, M., Asner, D. M., Baer, H., Banerjee, S., Barnett, R. M., Basaglia, T., Bauer, C. W., Beatty, J. J., Belousov, V. I., Beringer, J., Bethke, S., Bettini, A., Bichsel, H., Biebel, O., Black, K. M., Blucher, E., Buchmuller, O., Burkert, V., Bychkov, M. A., Cahn, R. N., Carena, M., Ceccucci, A., Cerri, A., Chakraborty, D., Chen, M. C., Chivukula, R. S., Cowan, G., Dahl, O., D’Ambrosio, G., Damour, T., de Florian, D., de Gouvêa, A., DeGrand, T., de Jong, P., Dissertori, G., Dobrescu, B. A., D’Onofrio, M., Doser, M., Drees, M., Dreiner, H. K., Dwyer, D. A., Eerola, P., Eidelman, S., Ellis, J., Erler, J., Ezhela, V. V., Fetscher, W., Fields, B. D., Firestone, R., Foster, B., Freitas, A., Gallagher, H., Garren, L., Gerber, H. J., Gerbier, G., Gershon, T., Gershtein, Y., Gherghetta, T., Godizov, A. A., Goodman, M., Grab, C., Gritsan, A. V., Grojean, C., Groom, D. E., Grünewald, M., Gurtu, A., Gutsche, T., Haber, H. E., Hanhart, C., Hashimoto, S., Hayato, Y., Hayes, K. G., Hebecker, A., Heinemeyer, S., Heltsley, B., Hernández-Rey, J. J., Hisano, J., Höcker, A., Holder, J., Holtkamp, A., Hyodo, T., Irwin, K. D., Johnson, K. F., Kado, M., Karliner, M., Katz, U. F., Klein, S. R., Klempt, E., Kowalewski, R. V., Krauss, F., Kreps, M., Krusche, B., Kuyanov, Y. V., Kwon, Y., Lahav, O., Laiho, J., Lesgourgues, J., Liddle, A., Ligeti, Z., Lin, C. J., Lippmann, C., Liss, T. M., Littenberg, L., Lugovsky, K. S., Lugovsky, S. B., Lusiani, A., Makida, Y., Maltoni, F., Mannel, T., Manohar, A. V., Marciano, W. J., Martin, A. D., Masoni, A., Matthews, J., Meißner, U. G., Milstead, D., Mitchell, R. E., Mönig, K., Molaro, P., Moortgat, F., Moskovic, M., Murayama, H., Narain, M., Nason, P., Navas, S., Neubert, M., Nevski, P., Nir, Y., Olive, K. A., Pagan, G. S., Parsons, J., Patrignani, C., Peacock, J. A., Pennington, M., Petcov, S. T., Petrov, V. A., Pianori, E., Piepke, A., Pomarol, A., Quadt, A., Rademacker, J., Raffelt, G., Ratcliff, B. N., Richardson, P., Ringwald, A., Roesler, S., Rolli, S., Romaniouk, A., Rosenberg, L. J., Rosner, J. L., Rybka, G., Ryutin, R. A., Sachrajda, C. T., Sakai, Y., Salam, G. P., Sarkar, S., Sauli, F., Schneider, O., Scholberg, K., Schwartz, A. J., Scott, D., Sharma, V., Sharpe, S. R., Shutt, T., Silari, M., Sjöstrand, T., Skands, P., Skwarnicki, T., Smith, J. G., Smoot, G. F., Spanier, S., Spieler, H., Spiering, C., Stahl, A., Stone, S. L., Sumiyoshi, T., Syphers, M. J., Terashi, K., Terning, J., Thoma, U., Thorne, R. S., Tiator, L., Titov, M., Tkachenko, N. P., Törnqvist, N. A., Tovey, D. R., Valencia, G., Van de Water, R., Varelas, N., Venanzoni, G., Verde, L., Vincter, M. G., Vogel, P., Vogt, A., Wakely, S. P., Walkowiak, W., Walter, C. W., Wands, D., Ward, D. R., Wascko, M. O., Weiglein, G., Weinberg, D. H., Weinberg, E. J., White, M., Wiencke, L. R., Willocq, S., Wohl, C. G., Womersley, J., Woody, C. L., Workman, R. L., Yao, W. M., Zeller, G. P., Zenin, O. V., Zhu, R. Y., Zhu, S. L., Zimmermann, F., Zyla, P. A., Anderson, J., Fuller, L., Lugovsky, V. S., and Schaffner, P. (Particle Data Group) 2018. Physical Review D, 98(3):030001. https://link.aps.org/doi/10.1103/PhysRevD.98.030001 49

96

4. PROBLEMS IN ELECTROSTATICS AND ELECTRODYNAMICS

[2] Land, M. 1996. Foundations of Physics, 27:19. 50 [3] Pierce, P. O. 1899. A Short Table of Integrals, Ginn and Company, New York. 52, 88 [4] Hestenes, D. 1966. Space-Time Algebra, Documents on modern physics, Gordon and Breach. https://books.google.co.il/books?id=OoRmatRYcs4C DOI: 10.1007/9783-319-18413-5. 55 [5] Land, M. 2013. Journal of Physics: Conference Series, 437. https://doi.org/10.1088%2F 1742-6596%2F437%2F1%2F012012 58 [6] Land, M. and Horwitz, L. 1991. Foundations on Physics Letters, 4:61. 62 [7] Jackson, J. 1975. Classical Electrodynamics, 9:391, Wiley, New York. DOI: 10.1063/1.3057859. 65 [8] Land, M. 2019. Journal of Physics: Conference Series, 1239:012005. https://doi.org/10. 1088%2F1742-6596%2F1239%2F1%2F012005 65 [9] Land, M. 2017. Journal of Physics: Conference Series, 845:012024. http://stacks.iop.o rg/1742-6596/845/i=1/a=012024 69, 94 [10] Anderson, C. D. 1932. Physical Review, 41:405. 73 [11] Bethe, H. A. and Heitler, W. 1934. Proc. Royal Society of London, A(146):83. 73 [12] Horwitz, L. P. 2017 Journal of Physics: Conference Series, 845:012026. http://stacks.i op.org/1742-6596/845/i=1/a=012026 89 [13] Horwitz, L. P. and Arshansky, R. I. 2018. Relativistic Many-Body Theory and Statistical Mechanics, 2053–2571, Morgan & Claypool Publishers. http://dx.doi.org/10.1088/ 978-1-6817-4948-8 DOI: 10.1088/978-1-6817-4948-8. 89

97

CHAPTER

5

Advanced Topics 5.1

ELECTRODYNAMICS FROM COMMUTATION RELATIONS

In (2.1) we introduced an unconstrained 8D phase space .x  ; p  / along with Poisson brackets for which @x  @p  @x  @p  D g  .x/ fx  ; p  g D @p @x  @x  @p in curved spacetime. In 1990, Dyson [1] published a 1948 attempt by Feynman to derive the force law and homogeneous Maxwell equations starting from Euclidean relations ˚ i Lorentz x ; p j D ı ij on 6D phase space. Several authors noted that the derived equations have only Galilean symmetry, and so are not actually the Maxwell theory, leading to a number of interesting theoretical developments. Tanimura [2] generalized Feynman’s derivation to Lorentz covariant form and obtained expressions similar to Maxwell theory, but including a fifth electromagnetic potential, a scalar evolution parameter that cannot be identified with proper time, absence of reparameterization invariance, and violations of the mass-shell constraint. His result can be identified with SHP electrodynamics. Significantly, Hojman and Shepley [3] proved that the existence of quantum commutation relations is a strong assumption, sufficient to determine a corresponding classical action, from which this system can be derived. We generalize Tanimura’s result to curved spacetime and show that this approach to SHP provides the final step in Feynman’s program. Using the technique of Hojman and Shepley, we show that SHP electrodynamics follows as the most general interacting system consistent with the unconstrained commutation relations we have assumed [4]. We begin with the commutation relations among the quantum operators Œx  ; x   D 0

for ;  D 0; 1;    ; D

mŒx  ; xP   D i„g  .x/

(5.1)

1, and suppose equations of motion mxR  D F  .; x; x/: P

We regard these quantities as operators in a Heisenberg picture, so that the field equations and the Lorentz force may be interpreted, in the Ehrenfest sense, as relations among the expectation values which correspond to relations among classical quantities. It follows that ŒxP  ; q.x/ D

i„ @q m @x

(5.2)

98

5. ADVANCED TOPICS

for any function q.x/. Differentiating (5.1) with respect to  we find mŒxP  ; xP   C mŒx  ; xR   D i„@ g  .x/xP 

and so define W  D W  by W  D

m2   ŒxP ; xP  : i„

(5.3)

From (5.1) and the Jacobi identity, Œ x  ; Œ xP  ; xP    C Œ xP  ; Œ xP  ; x    C Œ xP  ; Œ x  ; xP    D 0

we find that Œ x  ; W   D

  m2  Œ Œ x  ; xP  ; xP   C Œ xP  ; Œ x  ; xP    D i„ @ g  i„

Defining f  D f  by

D m .@ g 

f  D W 

E @ g  /xP  ;

 @ g  :

(5.4)

where the brackets h:::i represent Weyl ordering, we find Œx  ; f   D 0;

which shows that f  is independent of xP . When lowering indices, we define xP  D hg .x/xP  i

and from



E ˝  hD ˛i xP  ; xP  D g xP  ; g xP 

we may show that f D g g f  D

m2 ŒxP  ; xP   i„

leading to the Bianchi relation @ f C @ f C @ f D 0:

Rearranging Equation (5.1) and using (5.3) and (5.4), we see that mŒx  ; xR   D

where €  D

i„  f C 2i „h€  xP  i; m

1   .@ g C @ g  2

@ g  /

(5.5)

5.1. ELECTRODYNAMICS FROM COMMUTATION RELATIONS

is the Levi–Civita connection. We now define g  through the equation F  D mxR  D g  .x; x; P  / C hf  xP  i

mh€  xP  xP  i

and it follows that Œx  ; g   D Œx  ; f   f  Œx  ; xP   C m €  Œx  ; xP   xP  C €  xP  Œx  ; xP   D E i„ D E i„  D f C 2 i „ €  xP  C f  ı  i„ €  ı  xP  C €  xP  ı  m m D0

so that g  is also independent of xP . We may write the force as Ei h D D xP  g  C hf  xP  i D m xR  C €  xP  xP  D m D and since mxR  D m

d  hg xP  i ; d

we lower the index of g  to find g D g f 

hg f  xP  i C mhg €  xP  xP  i:

We write the first term on the right-hand side as g f  D m hg xR  i D m g

d  hg xP  i D m xR  C m hg @ g  xP  xP  i: d

Since the indices  and  of @ g  occur in symmetric combination, we may write 1   .@ g C @ g  / D 2

so that

€  C

1   @ g 2

1 g D m xR  C m h@ g  xP  xP  i hf g  xP  i: 2 Using (5.2) and (5.5) we obtain  1 i„ ŒxP  ; g  D mŒxP  ; xR   C i„@ @ g  xP  xP  @ g  .f xP  C xP  f / 2 2m  i„ i„   @ .f g / xP  C 2 f g f m  m 1 i„ D mŒxP  ; xR   C i„@ @ g  xP  xP  .@ g  /f xP  2 m  i„ i„ i„    .@ g /f xP  @ f g xP  C 2 f g f : m m m

99

100

5. ADVANCED TOPICS

Finally, antisymmetrization with respect to the indices  and  gives E i„ D ŒxP  ; g  ŒxP  ; g  D mŒxP  ; xR   ŒxP  ; xR   .@ f @ f /g  xP  m E d i„ D D m ŒxP  ; xP   .@ f C @ f /xP  d m ˛ i„ d i„ ˝ D f .@ f C @ f /xP  m d m ˝ ˛  i„ D .@ f C @ f C @ f /xP  C @ f m and so using the Bianchi identity for f , @ g

@ g C

@f D 0: @

Regarding these equations in the Ehrenfest sense, we may summarize the classical theory as m

D xP  D mŒxR  C   xP  xP   D f  xP  C g  D @ f C @ f C @ f D 0 @ g

@ g C

@f D 0: @

Introducing the definitions xD D 

@ D @D

fD D

fD D g :

We may then combine the inhomogeneous field equations as @˛ fˇ C @ˇ f ˛ C @ f˛ˇ D 0

(5.6)

(for ˛; ˇ; = 0;    ; D ), which shows that the two form f is closed on the formal (D C 1)dimensional manifold .; x/. Hence, if this manifold is contractable, then f is an exact form which can be obtained as the derivative of some potential with the form f D da. The Lorentz force equation becomes m

D xP  D mŒxR  C €  xP  xP   D f  .; x/xP  C g  .; x/ D f  ˇ .; x/xP ˇ : D

(5.7)

Following Dyson and Feynman, we observe that given Equation (5.6), the two-form f ˛ˇ is determined if we know functions  and j  such that

D˛ f ˛ D j 

D˛ f d˛ D ;

5.1. ELECTRODYNAMICS FROM COMMUTATION RELATIONS

101

where D˛ is the covariant derivative. By denoting  D j , these equations can be written compactly as D˛ f ˇ ˛ D j ˇ ; (5.8) d

where, due to the antisymmetry of f ˇ ˛ , we see that j ˇ is conserved as D˛ j ˛ D 0.

D˛ j ˛ D 0 : By comparing the Lorentz force (5.7) with (3.6), and the field Equations (5.6) and (5.8) with (3.19) and (3.17), we see conclude that the assumption of unconstrained commutation relations leads to a field theory equivalent to classical SHP electrodynamics. In Sections 3.2 and 3.3 we found the Lorentz force and field equations from an action principle. Hojman and Shepley [3] set out to prove that the assumed commutation relations are sufficient to establish the existence of a unique Lagrangian of electromagnetic form. To accomplish this goal, they demonstrate a new connection between the commutation relations and well-established results from the inverse problem in the calculus of variations, a theory which concerns the conditions under which a system of differential equations may be derived from a variational principle. We consider a set of ordinary second-order differential equations of the form Fk .; q; q; P q/ R D0

qP j D

dq j d

qR j D

d 2qj d2

j; k D 1;    ; n:

Under variations of the path q. / q. P / q./ R

! q./ C dq. /

d dq. / d d2 ! q. R / C d q. R / D q. R /C dq. / d2 ! q./ P C d q. P / D q. P /C

the function Fk .; q; q; P q/ R admits the variational one-form defined by dFk D

@Fk j @Fk @Fk dq C j d qP j C j d qR j j @q @qP @qR

and the variational two-form dq k dFk D

@Fk k @Fk @Fk dq ^ dq j C j dq k ^ d qP j C j dq k ^ d qR j ; j @q @qP @qR

where the 3n path variations .dq k ; d qP k ; d qR k / for k D 1;    ; n are understood to be linearly independent. The system of differential equations Fk .; q; q; P q/ R is called self-adjoint if there exists a two-form 2 .dq; d q/ P such that for all admissible variations of the path, dq k dFk .dq/ D

d 2 .dq; d q/: P d

102

5. ADVANCED TOPICS

Through integration by parts, one may show [5] that such a two-form exists and is unique up to an additive constant, if and only if @Fi @qR k

@Fi @Fk C i k @qP @qP @Fi @q k

@Fk @q i

D D D

@Fk @qR i   @Fk d @Fi C i d  @qR k @qR   1 d @Fi @Fk ; 2 d  @qP k @qP i

(5.9) (5.10) (5.11)

known as the Helmholtz conditions [6, 7]. Introducing the notation ı D dqˇk

@ @qˇk

qˇk D



d d

it follows that ı 2 D dqˇk ^ dq˛l



qk

@2 @qˇk @q˛l

ˇ D 0; 1; 2;

D 0;

which permits the equivalence of a set of self-adjointness differential equations to a Lagrangian formulation to be easily demonstrated [8]. Varying the Lagrangian L,     @L d @L @L d @L k @L k d k k C k dq C dq D Fk dq k C ıL D k dq C k d qP D 1 k k d  @qP d  @qP d @q @qP @q so that ı 2 D 0 H)

dq k ıFk C

d ı1 D d

dq k ıFk C

d 2 D 0 d

which demonstrates self-adjointness. Conversely, self-adjoint of Fk requires that dq k ıFk d  D 0 and since ı 2 D 0, d 2 d d 2 D ı 1 : d d Therefore, k

0 D dq ıFk

 d 2 D ı dq k Fk d

d 1 d



D ıL

by variation of L under  -integration, one obtains the differential equations Fk D 0. For the second-order equations considered here, it follows [5] from self-adjointness that the most general form of Fk is Fk .; q; q; P q/ R D Akj .; q; q/ P qR j C Bk .; q; q/: P

(5.12)

5.1. ELECTRODYNAMICS FROM COMMUTATION RELATIONS 3 i

103

3

To see this, notice that Fk is independent of d q =dt , so that the right-hand side of (5.10) must be independent of qR i . Inserting (5.12) into (5.9)–(5.11), one finds the Helmholtz conditions on Akj and Bk @Akj @Aij D k @qP i @qP   @ k @ D 2 C qP Aij @ @q k   1 @ @Bi k @ D C qP k 2 @ @qP j @q

Aij D Aj i @Bi @Bj C i j @qP @qP @Bi @q j

@Bj @q i

@Bj @qP i



along with the useful identity @Akj 1 @ D i @q 2 @qP j

@Aij @q k



@Bk @qP i

@Bi @qP k



(5.13)

:

In the domain of invertibilty of the Aj k , one can write (5.12) as Fk .; q; q; P q/ R D Akj .; q; q/ΠP qR j

f j

f j .; q; q/ P D

.A

1 jk

/ Bk

and the Helmholtz conditions become @Akj @Aij D k @qP i " @qP # k 1 @f @f k Aik j C Aj k i 2 @qP @qP

Aij D Aj i

1 D 2 D

"

Aik

@f k @qP j

where

D Aij D D # @f k @f k Aj k i D Ai k j @qP @q

Aj k

(5.14) (5.15)

@f k ; @q i

D @ @ @ D C qP k k C f k k D @ @q @qP

is the total time derivative subject to the constraint qR k

f k .; q; q/ P D0:

(5.16)

The identity (5.13) becomes @Aij @q k

@Akj D @q i

1 @ 2 @qP j



@ .Ain f n / @qP k

 @ n .Ak n f / : @qP i

(5.17)

Within the domain of applicability of the inverse function theorem, (5.16) is equivalent to (5.12), and the Helmholtz conditions become the necessary and sufficient conditions for the existence

104

5. ADVANCED TOPICS

of an integrating factor Aj k such that Fk D Akj .; q; q/ΠP qR

j

d f D d j



@L @qP k



@L : @q k

(5.18)

Employing this apparatus, Hojman and Shepley prove that given the quantum mechanical commutation relations Œx i ./; xP j . / D i„G ij ; the classical function g ij D lim G ij „!0

has an inverse !ij D .g

1

/ij

which satisfies the Helmholtz conditions. Following Santilli, we take the function A D g .x/ to be a Riemannian metric independent of xP , so that Equation (5.14) is satisfied automatically. Since g does not depend on xP  , Equation (5.15) becomes   D 1 @f @f @ g D xP   g D (5.19) D @x 2 @xP  @xP  and Equation (5.17) becomes 1 @ 2 @xP 



@f @xP 

 @f @g D  @xP @x 

@g  : @x 

Acting on (5.19) with @=@xP  and exchanging ( $  ), we obtain   1 @2 f @2 f g; D C ; 2 @xP  @xP  @xP  @xP 

(5.20)

(5.21)

where g; D @g =@x  . Combining (5.20) and (5.21), we find 1 @2 f D 2 @xP  @xP 

1 .g; C g; 2

g ; / D

€ ;

where € is the Levi-Civita connection. Thus, the most general expression for f .; x; x/ P is f D

€ xP  xP 

 .; x/xP 

 .; x/:

Now from (5.19) we find xP 

 @g 1 D 2€ xP  C 2€ xP  C  C   @x 2

(5.22)

5.1. ELECTRODYNAMICS FROM COMMUTATION RELATIONS

105

and using .€ C € /xP  D g; xP 

we find that all terms except for those in  cancel, so that 0 D  C  :

We now apply Equation (5.16) which becomes   1 D @f  @f  @f  @f  g  g  D g  g  2 D @xP @xP  @x @x @f 1 D @f D f; f; g; f  C g; f  : 2 D @xP  @xP 

(5.23)

using (5.22) to expand the left-hand side,    @f 1 D 1 D @f @  D € xP  xP  C  .; x/xP  2 D @xP  @xP  2 D @xP    C  .; x/ . $ /   1 D  D 2.€ € /xP C   2D h i @  @  @  D C xP C f .g g / x P C   ; ; @ @x  @xP  D .g; g; /f  ; (5.24) xP  xP  .g; g; / C xP  ; ; where ; D @ =@ , and we have used 2.€

€ /xP 

D xP  . g; C g; C g; C g; D 2xP  .g; g; / :

g;

g; /

Again using (5.22) we have f;

D D

     € xP xP C  .; x/xP C  .; x/ ;      €; xP xP C ; xP C ;

so that the right-hand side of (5.23) is f;

f;

g; f  C g; f  D

D

.€;

€; /xP  xP  C .;

.g;

g; /f  :

; /xP  C ;

;



106

5. ADVANCED TOPICS

Now canceling common terms, we are left with @ C xP  ; D xP  .; @

; / C ;

;

which, because the xP  are arbitrary, is equivalent the two expressions @ @ D @ @x  Therefore, we may identify

@ @x 

f D



@  C @  C @  D 0: and

f5 D



showing that SHP electrodynamics is the most general interaction consistent with the unconstrained commutation relations. Moreover, these commutation relations are sufficient to establish the existence of an equivalent Lagrangian for the classical problem associated with the quantum commutators. We observe that in flat space (5.18) implies   d @L @L  ŒM xR  f   D d  @xP  @x  @2 L @2 L @2 L @L D   xR  C   xP  C  @xP @xP @xP @x @xP @ @x  so that the solution @2 L @2 L @L @2 L  f  D   xP  C  M  D   @xP @xP @xP @x @xP @ @x  is unique. Therefore, we see that L may consist of the quadratic term integrated from the first expression, plus terms at most linear in xP  . Thus, we may write the Lagrangian 1 e ec5 M xP  xP  C a .; x/xP  C a5 2 c c which is the SHP event Lagrangian (3.3) in flat space. This demonstrates that SHP electrodynamics represents the conditions on the most general velocity dependent forces that may be obtained from a variational principle. LD

5.2

CLASSICAL NON-ABELIAN GAUGE THEORY

A classical non-Abelian gauge theory was given by Wong [9] possessing the following structure: mR IP F   @ F C gA  F A D Aa I a F

D gF  I. / P  D gA  I P  D @ A @ A C gA  A D j D Fa I a ŒI a ; I b  D i„"abc Ic ;

5.2. CLASSICAL NON-ABELIAN GAUGE THEORY 

107

a

where  ./ is the particle worldline and the I . / are an operator representation of the generators of a non-Abelian gauge group. From the form of the field f , one has the inhomogeneous equation D F C D F C D F D 0 with covariant derivative .D F /a D @ Fa

"abc Ab Fc :

Lee [10] followed Feynman’s method, supplementing the phase space commutation relations with ŒI a ; I b  D i„"abc Ic Œxi ; I a .t / D 0 IP C gAi  I xP i D 0 for i D 1; 2; 3, and arrived at the Wong’s equations in Newtonian form. Tanimura [2] generalized Lee’s derivation to D -dimensional flat Minkowski space and a general gauge group satisfying ŒI a ; I b  D i„ fc ab I c IPa D Fcab Ab .x/ xP  I c (5.25) for  -independent fields. We now extend the presentation of Section 5.1 by generalizing the Helmholtz conditions to take account of classical non-Abelian gauge fields according to Wong’s formulation. To achieve this, we associate with variations dq of the path q. /, a variation dI a of the generators I a , which may be understood as the variation of the orientation of the tangent space under q./ ! q. / C dq./. The explicit form of this variation follows from (5.25): for small d  , dI a D fc ab ŒAb .; x/ dx  C b .; x/d  I c ;

(5.26)

where we have allowed an explicit  -dependence for the gauge field, and have included a Lorentz scalar gauge field a , in analogy with the Abelian case. The quantity M D Ma I a undergoes the variation of the path .; x/ ! . C d ; x C dx/ according to d M D .dMa /I a C Ma .dI a /   @Ma @Ma  @Ma  @Ma  a D d C dx C d x P C d x R I @ @x  @xP  @xR  CMa Œfc ab Ab dx  C b d  I c     @Ma @Ma bc a bc fa b Mc I d  C fa Ab Mc I a dx  D @ @x  @Ma @Ma a  C  I a d xP  C I d xR @xP @xR  @M @M D D Md  C D Mdx  C  d xP  C  d xR  @xP @xR

108

5. ADVANCED TOPICS

in which the spacetime part of the covariant derivative D has the form .D F /a D @ Fa

fabc ab Fc 

and a similar covariant derivative for the  component appears which contains a . Now, the entire structure of self-adjoint equations follows with the replacements @ ! D @x  so that the Helmholtz conditions become

where

! D ; @A @A D  @ x P @xP    1 @f  @f  A  C A  2 @xP @xP

A D A  1 D @f  A  2 D @xP

@ @

D A D D  @f  A  D A D f  @xP

A D f  ;

(5.27) (5.28) (5.29)

@ D D D C xP  D C f   D @xP is the total  derivative subject to xR 

fa .; x; x/I P a D 0:

Since Hojman and Shepley’s argument relates only to the commutation relations among the coordinates, not to the structure of the forces, their result carries over unchanged. In flat spacetime, with A D g D  D constant, (5.27) is trivially satisfied and (5.28) becomes @f @2 f @f @2 f C D 0 H) C D 0: (5.30) @xP  @xP  @xP  @xP  @xP  @xP  Recalling the identity (5.17), we may also write (since the metric carries no group indices) @2 f @2 f D0 @xP  @xP  @xP  @xP  and so the most general form of fa is

!

@2 f D 0; @xP  @xP 

fa D fa .; x/xP  C ga .; x/;

(5.31)

where (5.30) requires that fa C fa D 0. Finally, applying (5.29) leads to   1 D @f @f D D f D f 2 D @xP  @xP  1 D Œfa 2 D

fa  D D fa xP  C D ga

.D C xP  D /fa D xP  .D fa

D fa xP  C D ga

D fa / C D ga

D ga

5.2. CLASSICAL NON-ABELIAN GAUGE THEORY 

and since xP is arbitrary, we obtain

D fa C D fa C D fa D 0 D fa C D ga D ga D 0

for the fields fa and ga . Now, in analogy to the Abelian case, we may write 1 M xP  xP  C Aa .; x/I a ./xP  C a .; x/I a ./ 2 and applying the Euler-Lagrange equations, we obtain LD

 d  @ mxP  C Aa I a D  ŒAa I a xP  C a I a  d @x @Aa a @Aa  a @Aa  a @a I C xP I C Aa IPa D xP I C  I a : @ @x  @x  @x Rearranging terms and using (5.26) to express IPa , we find    @Aa  @Aa  @a @Aa a a a P M xR  D xP xP I Aa I C  I a I   @x @x @x @    @Aa  @Aa  @Aa a @a a ab  c D x P x P I I A f .A x P C  /I C Ia a c b b @x  @x  @x @     @Aa @Aa @a @Aa bc  a bc D C fa Ab Ac xP I C C fa Ab c I a : @x  @x  @x  @ M xR  C

Comparing this with (5.31), we may express the field strengths in terms of the potentials as   @Aa @Aa bc f D C fa Ab Ac xP  I a @x  @x    @Aa @a bc a C f A  g D b c I ; a @x  @ from which it follows that the field equations are satisfied. Introducing the definitions xD D 

@ D @D

fD D

fD D g ;

the field equations and Lorentz force assume the form @˛ fˇ C @ˇ f ˛ C @ f˛ˇ D 0

where

 ˇ M xR  D fa xP  I a C ga I a D fa I a xP  C faD I a xP D D faˇ xP ;

 @Aaˇ @Aa˛ bc a C f A A b˛ cˇ I a @x ˛ @x ˇ recovers the usual relationship of the field strength tensor to the non-Abelian potential. f˛ˇ D



109

110

5. ADVANCED TOPICS

5.3

EVOLUTION OF THE LOCAL METRIC IN CURVED SPACETIME

General relativity has been summarized as: “Space acts on matter, telling it how to move. In turn, matter reacts back on space, telling it how to curve.” [11] The action of space on matter is expressed in equations of motion describing geodesic evolution with respect to a local metric g .x/. Such equations were found from a Lagrangian in (3.6) and from canonical commutation relations in (5.7). They can also be described in a Hamiltonian formulation on the phase space of position and momentum, an approach amenable to the canonical quantum dynamics for general relativity developed in [12, 13]. To express the action of matter on space, we look to Einstein equations that relate the local metric to sources of mass and energy, which evolve dynamically with  . We therefore consider a  -dependent metric that may also evolve along with its sources. One possible approach, proposed by Pitts and Schieve [14, 15], is to develop general relativity on the 5D manifold .x  ; /, introducing an ADM-type foliation with  as a preferred time direction. In the approach followed here, we adhere to the restriction imposed in SHP electrodynamics, maintaining the role of  as external, non-dynamical parameter throughout. General relativity treats the interval between a pair of instantaneously displaced points in spacetime ıx 2 D g ıx  ıx  D .x2 x1 /2 as an invariance of the manifold. To transform geometry into dynamics, a particle trajectory maps an arbitrary parameter  to a continuous sequence of events x  ./ in the manifold. For any timelike path we may put  D s D proper time, and although the path consists of instantaneous displacements in a 4D block universe, “motion” is observed through changes in x 0 .s/ with proper time. Treating the sequence as a function, the invariant interval can be written ıx 2 D g ıx  ıx  D g

dx  dx  2 ıs ds ds

suggesting a dynamical description of the path by the action Z dx  dx  1 S D ds g 2 ds ds which removes the constraint xP 2 D c 2 associated with the usual square root form. A physical event x  . / in SHP theory occurs at time  and chronologically precedes events occurring at subsequent times. The physical picture that emerges in SHP electrodynamics can thus be understood as describing the evolution of a Maxwell–Einstein 4D block universe defined at time  to an infinitesimally close 4D block universe defined at  C d  . As c5 ! 0, evolution slows to zero, recovering Maxwell theory as an equilibrium limit. The form of the gauge fields draws our attention to idea that while geometric relations on spacetime, such as O(3,1) invariance, are defined within a given block universe, the dynamics operate through the

5.3. EVOLUTION OF THE LOCAL METRIC IN CURVED SPACETIME

111

 -dependent gauge interaction, and in this sense are defined in the transition from one 4D block manifold to another. We therefore consider the interval dx  D xN  . C ı /

x  . /

between an event x  ./ and an event xN  . C ı / occurring at a displaced spacetime location at a subsequent time, and expand as dx 2 D g ıx  ıx  C g5 ıx  ıx 5 C g55 ıx 5 ıx 5 D g˛ˇ .x;  / ıx ˛ ıx ˇ

referred to the coordinates of x . This interval contains both the geometrical distance ıx  between two neighboring points in one manifold, and the dynamical distance ıx 5 D c5 ı between events occurring at two sequential times. This leads to the Lagrangian LD

1 Mg˛ˇ .x;  /xP ˛ xP ˇ 2

and equations of motion 0D

D xP   ˛ ˇ D xR  C €˛ˇ xP xP D

0D

D xP 5 5 D xR 5 C €˛ˇ xP ˛ xP ˇ ; D

is the standard Christoffel symbol in 5D. But as in the electrodynamic Lagrangian, where €˛ˇ we do not treat x 5 ./  c5  as a dynamical variable, and take the 5-index to denote scalar quantities, not elements of a 5D tensor. This symmetry breaking of 5D ! 4D+1 is expressed through the prescription  D €5˛

1  g .@5 g˛ C @˛ g5 2

@ g˛5 /

5 €˛ˇ 0

(5.32)

which extends the geodesic Equations (3.6) and (5.7) to 5D. We define n.x;  / to be the number of events (non-thermodynamic dust) per spacetime volume, so that j ˛ .x;  / D .x;  /xP ˛ ./ D M n.x; /xP ˛ ./ is the 5-component event current, and r˛ j ˛ D

@j ˛ @ ˛ C j € ˛ D C r j  D 0 ˛ @x @

is the continuity equation. Generalizing the 4D stress-energy-momentum tensor to 5D, the mass-energy-momentum tensor [16, 17] is (  T D MnxP  xP  D xP  xP  ˛ˇ ˛ ˇ ˛ ˇ T D MnxP xP D xP xP ! T 5ˇ D xP 5 xP ˇ  D c5 j ˇ

112

5. ADVANCED TOPICS

combining T  with j ˛ , and is conserved by virtue of the continuity and geodesic equations. The Einstein equations are similarly extended to G˛ˇ D R˛ˇ

1 8G Rg˛ˇ D 4 T˛ˇ ; 2 c

where the Ricci tensor R˛ˇ and scalar R are obtained by contracting indices of the 5D curvature tensor R ı ˛ˇ . Since conservation of T ˛ˇ depends on prescription (5.32), we must similarly 5 when constructing the Ricci tensor to insure rˇ G ˛ˇ D 0. Working through the suppress €˛ˇ 4D algebra we find that R D R and obtain 1       @ € @ €5 C €5 € € €5 c5 1       €55 € @ €55 C €5 €5 D @ €5 c5

R5 D R55

as new components. The weak field approximation [11] is generalized to 5D as g˛ˇ D ˛ˇ C h˛ˇ

! @ g˛ˇ D @ h˛ˇ

h˛ˇ

2

0

leading to R˛ˇ '

1 @ˇ @ h ˛ C @˛ @ h ˇ 2

Defining hN ˛ˇ D h˛ˇ

1  h, 2 ˛ˇ

@ @ h˛ˇ

 @˛ @ˇ h

R ' ˛ˇ R˛ˇ

h ' ˛ˇ h˛ˇ :

the Einstein equations become

16G T˛ˇ D @ˇ @ hN ˛ C @˛ @ hN ˇ c4

@ @ hN ˛ˇ

@˛ @ˇ hN

which take the form of a wave equation 16G T˛ˇ D c4

@ @ hN ˛ˇ D

  55 2 N  @ @ C 2 @ h˛ˇ c5

after imposing the gauge condition @ hN ˛ D 0. Using the Green’s function GMaxwell from (3.24) for this equation leads to   jx x0 j 0 Z T t ; x ;  ˛ˇ c 4G hN ˛ˇ .x;  / D 4 d 3x0 c jx x0 j relating the field hN ˛ˇ .x;  / to the source T˛ˇ .x;  /. In analogy to the Coulomb problem, we take a point source X D .cT . /; 0/ in a co-moving frame, with T 00 D mc 2 TP 2 ı 3 .x/ ' .t

T . //

T ˛i D 0

T 55 D

c52 00 T ; c2

5.3. EVOLUTION OF THE LOCAL METRIC IN CURVED SPACETIME

where '. / is the smoothing function (3.15). Writing M D m ' .t 4GM hN 00 .x;  / D 2 TP 2 c R

113

T .// produces

c2 hN 55 .x;  / D 52 hN 00 c

hN ˛i .x;  / D 0

so using h˛ˇ D hN ˛ˇ 12 ˛ˇ hN and neglecting c52 =c 2  1, we see that h00 D hN 00 . Since g ˛ˇ hˇ ' ˛ˇ hˇ the non-zero Christoffel symbols are 1   @ h00 2 1 0 D  @ h00 2c5

1   @i h0 2 1  D  @ h55 2

 €00 D

 €0i D

 €50

 €55

so the equations of motion split into tR D .@ h00 / tP C xP  .rh00 / tP2

xR D

c2 .rh00 / tP2 : 2

In spherical coordinates, putting  D =2, the angular and radial equations are 2RP P C RR D 0

! P D

L MR2

! RR

L2 D M 2 R3

GM 2 P 2 tP T R2

and the t equation is tR D

2G@ M 4GM tP C 2 TP TR tP c2R c R

  2GM 2GM P P 2 ˛ ./ R T  1 C ˛P . / tP; R2 c 2 c2R 2

P  0 and @ '  0 (taking  large), and define where we neglect R=c   ˛ . / ˛ . / ˛P . / 2 P P P R T D1C ! T ' 1 C ˛ . / ! T T ' 1 C : 2 2 2

In the Newtonian case, ˛ D 0 ! tP D 1, but this t equation has the solution      2GM 1 2 1 2GM P2 TP 2 ' 1 C tP D exp ˛ C ˛ ! t 1 C ˛ c2R 4 2 c2R which, since 2GM=c 2 R  1, leads finally to the radial equation in the form    GM 1 dK GM d d 1 P2 1 l2 . / R C 1 C ˛ D D ˛ ./ : 2 2 d 2 2M R R 2 d 2R d  We recognize K on the LHS as the Hamiltonian of the particle moving in this local metric. The mass fluctuation of the point source is seen to induce a fluctuation in the mass of a distant particle through the field g˛ˇ .x;  /, producing a small modification of Newtonian gravity.

114

5. ADVANCED TOPICS

Interactions in SHP electrodynamics form an integrable system in which event evolution generates an instantaneous current defined over spacetime at  , and in turn, these currents induce  -dependent fields that act on other events at  . We expect that in a similar way, a fully developed SHP formulation of general relativity will describe how the instantaneous distribution of mass at  expressed through T˛ˇ .x;  / induces the local metric g˛ˇ .x;  /, which, in turn, determines geodesic equations of motion for any particular event at x  ./.

5.4

ZEEMAN AND STARK EFFECTS

As discussed in Section 2.4, reasonable solutions to the relativistic central force problem are obtained in a restricted Minkowski space (RMS) with fixed unit vector n ˚ RMS.n/ D x 2 x j Œx .x  n/n2  0 invariant under O(2,1) but not general Lorentz transformations. Because quantum states are classified by their symmetry representations, Horwitz and Arshansky [18–20] generalized their solutions to the quantum central force problem to an induced representation of O(3,1). Studying the Lorentz transformations on n and the RMS(n), they found the generators h of O(3,1) for the combined space, formed a maximally commuting set of operators, and solved for eigenstates of these operators. The energy levels of these degenerate quantum states split in a constant electromagnetic field—the Zeeman and Stark effects. To couple the electromagnetic field to these states, we construct a gauge theory for the induced representation in its classical form [21, 22]. We denote nı D .0; 0; 0; 1/ so that the parameterization (2.5) describes RMS.nı /. Given the Lorentz transformation nı D L.n/ n it follows that x 2 RMS.n /

and

y D L.n/ x

ı

y 2 RMS.n /

H)

and so we may characterize the full spacelike region x D LT .n/y by  D .n; y/. Since a Lorentz transformation ƒ acts as n ! n0 D ƒ n and x ! x 0 D ƒ x , it follows that x 0 D ƒ x D ƒL.n/T y D L.ƒn/T L.ƒn/ ƒ L.n/T y D L.n0 /T y 0 :

Thus, y transforms under the O(2,1) little group defined through y ! y0 D D

1

.ƒ; n/ y

D

1

.ƒ; n/ D L.ƒn/ ƒ L.n/T

and since D 1 .ƒ; n/nı D nı , the little group preserves RMS(nı ). We have taken L.n.// to be  -dependent, but one can show that since d=d  is Lorentz-invariant and commutes with ƒ .y/ P 0DD

1

.ƒ; n/yP C DP

1

.ƒ; n/ y D

d ŒD d

1

.ƒ; n/ y

is form-invariant. Representing the Lorentz transform ƒ W x ! x 0 as ƒD1C

1 ! M C o.! 2 / 2

.M / D  

  

5.4. ZEEMAN AND STARK EFFECTS

115

N W  D .n; y/ !  D .n ; y / can be represented as the Lorentz transform ƒ 0

0

0

N D 1 C 1 ! X  C o.! 2 / ƒ 2

and the generators are found as  x T M rx C nT M rn D

X D

 y T LM LT ry C nT M D ;

where we introduce S D L .@=@n / LT

D D .rn / C y T S ry :

It is easily shown that for a function of x alone (even as n varies with  ) D acts as a kind of covariant derivative with D f .n; y/ D D f .L.n/T y/  0. P we put As a classical Lagrangian on the phase space f ; g  1 L D M xP 2 C 2 nP 2 C e .xP  A.x/ C nP  .n// V .x 2 / 2 i h i 1 h D M .yP C nP  S y/2 C 2 nP 2 C e .yP C nP  S y/  A.n/ .y/ C nP  .n/ 2

V .x 2 /;

where  is a length scale required because n is a unit vector, A.n/ D LA transforms under the little group, and we used   xP D LT yP C LP T y D LT yP C LLP T y D LT .yP C nP  S y/ : This L is scalar and represents a generalized Maxwell electrodynamics including n as a new dynamical degree of freedom. The conjugate momenta are found to be   @L  .n/ D M y P C n P S y C eA    @yP   @L  D  D M 2 nP  y T S p C e @nP

p D

having used the antisymmetry of the matrices S . The Hamiltonian is obtained from the Lagrangian as K D yP  p C nP  

LD

1  p 2M

eA.n/

2

C

1 .P 2Md 2

e/2 C V;

where P D  C y T S p . Taking A.n/ D  D 0 and applying Noether’s theorem to the variation produced by a Lorentz transformation ı D 12 !  X  we obtain the conserved quantities   h D p  X y C   X n D y T L.n/M LT p C nT M P

116

5. ADVANCED TOPICS

˚ which satisfy Poisson brackets h ; K D 0. Now interpreting p and  as quantum operators, so that p D i„

@ @y 

 D i„

@ @n

P D i„D

the h are precisely the Lorentz generators found by Horwitz and Arshansky for   solutions .x;  / to the Stueckelberg–Schrodinger equation i@ D K and satisfy h ; K D 0. This system is invariant under U.1/ gauge transformations !e

i e‚./=„

A.n/ ! A.n/   C

@ ‚ @y 

 !  C D ‚:

For interactions cyclic in n, we may put nP D 0 for the classical system which remains within RM S.n/ with fixed n, so the classical and quantum dynamics reduce to L D L0 D

1 M yP 2 2

V

K D K0 D

„2 @ @ CV 2M @y  @y

and quantum wavefunctions satisfy D D 0. The Zeeman and Stark effects are thus obtained in perturbation theory by expressing a constant field strength F  as A .x/ D

1  F x 2

 .n/ D  A .n/ D

   F n 2 

and writing 1 .LF LT y/ 2 for the potential in RMS(nı ). To first order in e , the Hamiltonian is just  T A.n/  .y/ D L A .L y/ D

K D K0 C

e F X  4M

e e e so that the Zeeman effect follows from 4m F˛ˇ X ˛ˇ ! 4m Fij X ij D 2m B k Lk splitting the energy levels along the diagonal component Lk of angular momentum. For the Stark effect, we put e e e F X ˛ˇ ! 2m F0i X 0i D 2m E k Ak , where Ak is a boost, and to reproduce the phenomenol4m ˛ˇ ogy we must include an additional scalar potential V ! V C A5 , where A5 D e  x , hinting at the 5D gauge theory.

5.5

CLASSICAL MECHANICS AND QUANTUM FIELD THEORY

Although quantum field theory differs from classical mechanics in both methodology and results, classical SHP electrodynamics presents a number of interesting qualitative implications for QED.

5.5. CLASSICAL MECHANICS AND QUANTUM FIELD THEORY

117

As seen in Sections 1.3 and 3.1, the Stueckelberg–Schrodinger equation is first-order in  , and the Hamiltonian operator is a Lorentz scalar, so that manifest covariance is preserved throughout the second quantization procedure. In constructing canonical momenta, the kinetic term for the fields fˆ˛ˇ f˛ˇ formed from the cross derivatives of a˛ leads to momentum fields  D @ a but no 5 component, because @ a5 does not appear in f˛ˇ . In Dirac quantization for gauge theories [23], one inserts a momentum 5 conjugate to a5 and a Lagrange multiplier to enforce the primary constraint 5 D 0. The secondary constraint (that the primary constraint commutes with the Hamiltonian) leads to the Gauss law @ f 5 D .ec5 =c/j 5 . But because this system is first-order, one may apply the Jackiw quantization scheme [24], in which we first eliminate the constraint from the Lagrangian by solving the Gauss law, and then construct the Hamiltonian from the unconstrained degrees of freedom, which are the matter fields and the transverse electromagnetic modes. Since the momentum modes are not constrained to be lightlike, as we saw for plane waves in Section 4.4, there can be three transverse polarization modes. The resulting system is amenable to both canonical and path integral quantization. In canonical quantization, one finds the propagator G.x; / for the matter fields as the vacuum expectation value of  -ordered operator products (equivalent to a Fourier transform of the momentum representation with a Feynman contour). The propagator enforces  -retarded causality, with G.x; / D 0 for  < 0, so that SHP quantum field theory is free of matter loops. Extracting the propagator for a sharp mass eigenvalue recovers the Feynman propagator for the Klein–Gordon equation. As in classical mechanics, quantum systems evolve as  increases, with advance or retreat of x 0 treated on an equal footing. Perturbation theory is constructed in an interaction picture obtained by a unitary transformation constructed from the scalar interaction Hamiltonian and  . As a result, this method has been shown [25] to circumvent the Haag no-go theorem [26], summarized as, “Haag’s theorem is very inconvenient; it means that the interaction picture exists only if there is no interaction.” [27] As seen in Section 4.7, particles interacting through the electromagnetic field can exchange mass. The treatment of Moller scattering leads to a cross-section identical to the standard QED result for spinless particles when mass exchange is absent. When mass is exchanged, the usual pole in the cross-section at 0o splits into a zero and two poles close to but away from the forward beam axis, providing a small experimental signature (and one very difficult to observe). Because there are no matter loops in this theory, the problem of renormalization reduces to treatment of photon loops in the matter field (gauge and vertex factors become unity by the Ward identities). Mass renormalization can be absorbed into the first order mass term  i @ in the quantum Lagrangian. To remove singularities from the loop contributions to the matter propagator in standard QED, some regularization scheme is required. However in SHP QED, h i 1 the field interaction kernel (3.11) places a multiplicative factor 1 C ./2 in the photon propagator. This factor acts as a mass cut-off rendering the theory superrenormalizable. Unlike a momentum cut-off, this factor leaves the Lorentz and gauge symmetries of the original

118

5. ADVANCED TOPICS

theory unaffected, recalling Schwinger’s motivation for his “proper time method” discussed in Section 1.3.

5.6

BIBLIOGRAPHY

[1] Dyson, F. J. 1990. American Journal of Physics, 58:209–211. https://doi.org/10.1119/ 1.16188 97 [2] Tanimura, S. 1992. Annals of Physics, 220:229–247. http://www.sciencedirect.com/ science/article/pii/000349169290362P 97, 107 [3] Hojman, S. A. and Shepley, L. C. 1991. Journal of Mathematical Physics, 32:142–146. ht tps://doi.org/10.1063/1.529507 97, 101 [4] Land, M., Shnerb, N., and Horwitz, L. 1995. Journal of Mathematical Physics, 36:3263. 97 [5] Santilli, R. M. 1990. Foundations of Theoretical Mechanics I, Springer-Verlag. 102 [6] Helmholtz, H. 1887. Journal für die Reine Angewandte Mathematik, 100:137. 102 [7] Darboux, G. 1894. Leçons sur la Théory Générale des Surfaces, 3, Gauthier-Villars. 102 [8] Dedecker, P. 1950. Bulletin de l’Académie Royale des Sciences de Belgique Classe des Sciences, 36:63. 102 [9] Wong, S. K. 1970. Nuovo Cimento, 65A:689. 106 [10] Lee, C. R. 1950. Physics Letters, 148A:36. 107 [11] Misner, C. W., Thorne, K. S., and Wheeler, J. A. 1973. Gravitation, W.H. Freeman and Co., San Francisco, CA. 110, 112 [12] Horwitz, L. P. 2019. Journal of Physics: Conference Series, 1239. https://doi.org/10. 1088%2F1742-6596%2F1239%2F1%2F012014 110 [13] Horwitz, L. P. 2019. The European Physical Journal Plus, 134:313. https://doi.org/10. 1140/epjp/i2019-12689-7 110 [14] Pitts, J. B. and Schieve, W. C. 1998. Foundations of Physics, 28:1417–1424. https://do i.org/10.1023/A:1018801126703 110 [15] Pitts, J. B. and Schieve, W. C. 2001. Foundations of Physics, 31:1083–1104. https://do i.org/10.1023/A:1017578424131 110 [16] Saad, D., Horwitz, L., and Arshansky, R. 1989. Foundations of Physics, 19:1125–1149. 111

5.6. BIBLIOGRAPHY

119

[17] Land, M. 2019. Journal of Physics: Conference Series, 1239. https://doi.org/10.1088% 2F1742-6596%2F1239%2F1%2F012005 111 [18] Arshansky, R. and Horwitz, L. 1989. Journal of Mathematical Physics, 30:66. 114 [19] Arshansky, R. and Horwitz, L. 1989. Journal of Mathematical Physics, 30:380. [20] Horwitz, L. P. 2015. Relativistic Quantum Mechanics, Springer, Dordrecht, Netherlands. 114 [21] Land, M. and Horwitz, L. 1995. Jounal of Physics A: Mathematical and General, 28:3289– 3304. 114 [22] Land, M. and Horwitz, L. 2001. Foundations of Physics, 31:967–991. 114 [23] Dirac, P. 1964. Lectures on Quantum Mechanics, Yeshiva University, New York. 117 [24] Jackiw, R. 1993. https://arxiv.org/pdf/hep-th/9306075.pdf 117 [25] Seidewitz, E. 2017. Foundations of Physics, 47:355–374. 117 [26] Haag, R. 1955. Kong. Dan. Vid. Sel. Philosophical Magazine Series, 746, 376. 117

Mat.

Fys.

Med.,

29N12:1–37.

[27] Streater, R. F. and Wightman, A. S. 1964. PCT, Spin, Statistics, and All That, Princeton University Press. 117

121

Authors’ Biographies MARTIN LAND Martin Land was born in Brooklyn in 1953. He grew up in the New York City area, strongly influenced by his mother, a social worker who worked with Holocaust survivors, and his father, a second-generation engineer in small manufacturing businesses associated with the garment industry. In his school years he cleaned swimming pools and stables, worked as a carpenter on a construction site, and expedited orders in the garment center. In 1972, he entered Reed College in Portland, Oregon, where he received a Kroll Fellowship for original research which permitted him to devote an extra year to extensive study in the humanities along with his specialization in physics. After completing his BA in 1977, he returned to New York City where he received an M.S. in electrical engineering from Columbia University in 1979 as a member of the Eta Kappa Nu engineering honor society. He joined Bell Laboratories, developing specialized hardware for fiber optic communication with application in computer networks and video transmission. In 1982, he worked as a telecommunications engineer at a major Wall Street bank. Returning to theoretical physics at Hebrew University in Jerusalem, he worked with Eliezer Rabinovicci on supersymmetric quantum mechanics to receive a second M.S. in 1986. In 1985, he married Janet Baumgold, a feminist therapist and co-founder of the Counseling Center for Women. Following a year devoted to full-time fatherhood and another in compulsory national service, he began working toward a Ph.D. in high energy physics with Lawrence Horwitz at Tel Aviv University in 1988. He elaborated many aspects of the classical and quantum theories known as Stueckelberg-Horwitz-Piron (SHP) theory, producing a dissertation developing the SHP quantum field theory. Concurrently with his doctoral work, he was on the research faculty of the Computer Science Department at Hebrew University, developing specialized hardware for parallel computing. After submitting his dissertation in 1995, he taught communications engineering for three years at the Holon Institute of Technology, before joining the Department of Computer Science at Hadassah College in Jerusalem, teaching computer architecture, microprocessors, embedded systems, and computer networking. He was a founding member of the International Association for Relativistic Dynamics (IARD) in 1998 and has served as IARD president since 2006. In parallel to his activities in physics and computer science, he has

122

AUTHORS’ BIOGRAPHIES

enjoyed a long collaboration with Jonathan Boyarin of Cornell University in various areas of the humanities, critical theory, and Jewish studies. This collaboration has allowed him to communicate contemporary thinking in physics, especially notions of time associated with SHP theory, to scholars in other fields as modern context for philosophical consideration of temporality.

AUTHORS’ BIOGRAPHIES

123

LAWRENCE P. HORWITZ Lawrence Paul Horwitz was born in New York City on October 14, 1930. He lived in Westchester County until 1934, then went to London where his father founded and managed a chain of womens wear shops, called the Richard Shops, and then returned to the United States in 1936. After a few years in Brooklyn, NY, his family moved to Forest Hills in Queens, NY, where he learned tennis and attended Forest Hills High School, a school dedicated to teaching students how to think, where he came to love physics. He then went to the College of Engineering, New York University, where he studied Engineering Physics and graduated summa cum laude with a Tau Beta Pi key and the S.F.B. Morse medal for physics. He met a young lady, Ruth Abeles, who arrived from Germany in the U.S. in 1939 and became his wife before moving on to Harvard University in 1952 with a National Science Foundation Fellowship. He received his doctorate at Harvard working under the supervision of Julian Schwinger in 1957. He then worked at the IBM Watson Research Laboratory where he met Herman Goldstine, a former assistant to John von Neumann and, among other things, explored with him octononic and quaternionic Hilbert spaces from both physical and mathematical points of view. He then moved on to the University of Geneva in 1964, becoming involved in scattering theory as well as continuing his studies of hypercomplex systems with L. C. Biedenharn and becoming involved in particle physics with Yuval Neeman at CERN. He became full professor at the University of Denver in 1966–1972; he then accepted a full professorship at Tel Aviv University. After stopping for a year to work with C. Piron at the University of Geneva on the way to Israel, he has been at Tel Aviv University since 1973, with visits at University of Texas at Austin, Ilya Prigogine Center for Statistical Mechanics and Complex Systems in Brussels, and at CERN, ETH (Honggerberg, Zurich), University of Connecticut (Storrs, CT), IHES (Bures-sur-Yvette, Paris), and Institute for Advanced Study (Princeton, NJ), where he was a Member in Natural Sciences, 1993, 1996, 1999, 2003 with short visits in August 1990, and January 1991, working primarily with S. L. Adler. He is now Professor Emeritus at Tel Aviv University, Bar Ilan University, and Ariel University. His major interests are in particle physics, statistical mechanics, mathematical physics, theory of unstable systems, classical and quantum chaos, relativistic quantum mechanics, relativistic many body theory, quantum field theory, general relativity, representations of quantum theory on hypercomplex Hilbert modules, group theory and functional analysis, theories of irreversible quantum evolution, geometrical approach to the study of the stability of classical Hamiltonian systems, and to the dark matter problem, and classical and quantum chaos. He is a member of the American Physical Society (Particle Physics), Swiss Physical Society, European Physical Society, International Association for Mathematical Physics, Israel Physical Society, Israel Mathematics Union, European Mathematical Society, International Quantum Structures As-

124

AUTHORS’ BIOGRAPHIES

sociation, Association of Members of the Institute for Advanced Study, and the International Association for Relativistic Dynamics.