Electromagnetic Processes 9780691215846

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Electromagnetic Processes

PRINCETON SERIES IN ASTROPHYSICS EDITED BY DAVID N. SPERGEL

Theory of Rotating Stars, by Jean-Louis Tassoul Theory of Stellar Pulsation, by John P. Cox Galactic Dynamics, by James Binney and Scott Tremaine Dynamical Evolution of Globular Clusters, by Lyman Spitzer, Jr. Supernovae and Nucleosynthesis: An Investigation of the History of Matter, from the Big Bang to the Present, by David Arnett Unsolved Problems in Astrophysics, edited by John Bahcall and Jeremiah P. Ostriker Galactic Astronomy, by James Binney and Michael Merrifield Active Galactic Nuclei: From the Central Black Hole to the Galactic Environment, by Julian H. Krolik Plasma Physics for Astrophysics, by Russell M. Kulsrud Electromagnetic Processes, by Robert J. Gould

Electromagnetic Processes ROBERTJ. GOULD

§ PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD

Copyright © 2006 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 3 Market Place, Woodstock, Oxfordshire OX20 1SY All Rights Reserved Library of Congress Cataloging-in-Publication Data Gould, Robert J. (Robert Joseph), 1935Electromagnetic processes / Robert J. Gould p. cm.—(Princeton series in astrophysics) Includes bibliographical references and index. ISBN-13: 978-0-691-12443-8 (acid-free paper) ISBN-10: 0-691-12443-4 (acid-free paper) ISBN-13: 978-0-691-12444-5 (pbk. : acid-free paper) ISBN-10: 0-691-12444-2 (pbk. : acid-free paper) 1. Electromagnetic theory. 2. Quantum electrodynamics. 3. Nonrelativistic quantum mechanics. 4. Scattering (Physics) I. Title. II. Series. QC670.G616 2006 530.14'33—dc22

2005040562

British Library Cataloging-in-Publication Data is available Printed on acid-free paper, oo pup.princeton.edu Printed in the United States of America 10 9 8 7 6 5 4 3 2 1

Contents

Preface

ix

Chapter 1. Some Fundamental Principles

1

1.1 1.2

1.3

1.4 1.5

Units and Characteristic Lengths, Times, Energies, Etc. Relativistic Covariance and Relativistic Invariants 1.2.1 Spacetime Transformation 1.2.2 Other Four-Vectors and Tensors—Covariance 1.2.3 Some Useful and Important Invariants 1.2.4 Covariant Mechanics and Electrodynamics Kinematic Effects 1.3.1 Threshold Energies in Non-Relativistic and Relativistic Processes 1.3.2 Transformations of Angular Distributions Binary Collision Rates Phase-Space Factors 1.5.1 Introduction 1.5.2 Simple Examples 1.5.3 General Theorems—Formulation 1.5.4 General Formulas—Evaluation of Multiple Integrals 1.5.5 One-Particle Distributions 1.5.6 Invariant Phase Space

Chapter 2. Classical Electrodynamics

2.1

2.2

2.3 2.4

2.5 2.6

Retarded Potentials 2.1.1 Fields, Potentials, and Gauges 2.1.2 Retarded Potentials in the Lorentz Gauge Multipole Expansion of the Radiation Field 2.2.1 Vector Potential and Retardation Expansion 2.2.2 Multipole Radiated Power Fourier Spectra Fields of a Charge in Relativistic Motion 2.4.1 Lienard-Wiechert Potentials 2.4.2 Charge in Uniform Motion 2.4.3 Fields of an Accelerated Charge Radiation from a Relativistic Charge Radiation Reaction 2.6.1 Non-Relativistic Limit 2.6.2 Relativistic Theory: Lorentz-Dirac Equation

1 5 5 8 10 13 15 15 17 18 21 21 23 26 28 32 34 37

37 37 39 41 41 43 46 49 49 51 53 54 57 57 60

VI

CONTENTS

2.7

2.8

2.9

Soft-Photon Emission 2.7.1 Multipole Formulation 2.7.2 Dipole Formula 2.7.3 Emission from Relativistic Particles Weizsacker-Williams Method 2.8.1 Fields of a Moving Charge 2.8.2 Equivalent Photon Fluxes Absorption and Stimulated Emission 2.9.1 Relation to Spontaneous Emission 2.9.2 General Multiphoton Formula 2.9.3 Stimulated Scattering

Chapter 3. Quantum Electrodynamics

3.1 3.2 3.3

3.4

3.5

3.6

Brief Historical Sketch Relationship with Classical Electrodynamics Non-Relativistic Formulation 3.3.1 Introductory Remarks 3.3.2 Classical Interaction Hamiltonian 3.3.3 Quantum-Mechanical Interaction Hamiltonian 3.3.4 Perturbation Theory 3.3.5 Processes, Vertices, and Diagrams Relativistic Theory 3.4.1 Modifications of the Non-Covariant Formulation 3.4.2 Photon Interactions with Charges without Spin 3.4.3 Spin- \ Interactions 3.4.4 Invariant Transition Rate Soft-Photon Emission 3.5.1 Non-Relativistic Limit 3.5.2 Emission from Spin Transitions 3.5.3 Relativistic Particles without Spin 3.5.4 Relativistic Spin-± Particles Special Features of Electromagnetic Processes 3.6.1 "Order" of a Process 3.6.2 Radiative Corrections and Renormalization 3.6.3 Kinematic Invariants 3.6.4 Crossing Symmetry

Chapter 4. Elastic Scattering of Charged Particles

4.1

4.2

Classical Coulomb Scattering 4.1.1 Small-Angle Scattering 4.1.2 General Case 4.1.3 Two-Body Problem—Relative Motion 4.1.4 Validity of the Classical Limit Non-Relativistic Born Approximation and Exact Treatment 4.2.1 Perturbation-Theory Formulation 4.2.2 Sketch of Exact Theory 4.2.3 Two-Body Problem 4.2.4 Scattering of Identical Particles 4.2.5 Validity of the Born Approximation

61 61 62 63 65 66 68 70 71 72 73 75

76 78 80 80 80 83 84 88 94 94 97 103 107 109 109 11 3 11 6 119 123 1 23 127 130 1 32 135

135 135 138 139 141 142 142 145 148 150 1 54

CONTENTS

4.3

4.4

Scattering of Relativistic Particles of Zero Spin 4.3.1 Coulomb Scattering 4.3.2 Scattering of Two Distinguishable Charges 4.3.3 Two Identical Charges 4.3.4 Scattering of Charged Antiparticles Scattering of Relativistic Spin- \ Particles 4.4.1 Spin Sums, Projection Operators, and Trace Theorems 4.4.2 Coulomb Scattering 4.4.3 M0ller and Bhabha Scattering

Chapter 5. Compton Scattering

5.1

5.2

5.3 5.4 5.5

5.6 5.7

Classical Limit 5.1.1 Kinematics of the Scattering 5.1.2 Derivation of the Thomson Cross Section 5.1.3 Validity of the Classical Limit Quantum-Mechanical Derivation: Non-Relativistic Limit 5.2.1 Interactions and Diagrams 5.2.2 Calculation of the Cross Section Scattering by a Magnetic Moment Relativistic Spin-0 Case Relativistic Spin- \ Problem: Klein-Nishina Formula 5.5.1 Formulation 5.5.2 Evaluation of the Cross Section 5.5.3 Invariant Forms 5.5.4 Limiting Forms and Comparisons Relationship to Pair Annihilation and Production Double Compton Scattering 5.7.1 Non-Relativistic Case. Soft-Photon Limit 5.7.2 Non-Relativistic Case. Arbitrary Energy 5.7.3 Extreme Relativistic Limit

Chapter 6. Bremsstrahlung

6.1

6.2

6.3

6.4

Classical Limit 6.1.1 Soft-Photon Limit 6.1.2 General Case: Definition of the Gaunt Factor Non-Relativistic Born Limit 6.2.1 General Formulation for Single-Particle Bremsstrahlung 6.2.2 Coulomb (and Screened-Coulomb) Bremsstrahlung 6.2.3 Born Correction: Sommerfeld-Elwert Factor 6.2.4 Electron-Positron Bremsstrahlung Electron-Electron Bremsstrahlung. Non-Relativistic 6.3.1 Direct Born Amplitude 6.3.2 Photon-Emission Probability (without Exchange) 6.3.3 Cross Section (with Exchange) Intermediate Energies 6.4.1 General Result. Gaunt Factor 6.4.2 Soft-Photon Limit

VII

1 56 156 1 58 162 163 166 166 170 171 177

177 17 7 178 181 182 182 184 186 188 191 191 1 93 194 195 197 199 199 202 207 211

211 211 214 217 217 222 223 226 228 228 232 234 236 236 239

viii 6.5

6.6

Index

CONTENTS Relativistic Coulomb Bremsstrahlung 6.5.1 Spin-0 Problem 6.5.2 Spin- \: Bethe-Heitler Formula 6.5.3 Relativistic Electron-Electron Bremsstrahlung 6.5.4 Weizsacker-Williams Method Electron-Atom Bremsstrahlung 6.6.1 Low Energies 6.6.2 Born Limit—Non-Relativistic 6.6.3 Intermediate Energies—Non-Relativistic 6.6.4 Relativistic Energies—Formulation 6.6.5 Relativistic Energies—Results and Discussion

240 241 244 248 251 254 254 256 257 259 264 269

Preface

The aim of this book is to provide an understanding of processes that are electromagnetic in nature. That is, they take place as a result of the interaction of a particle's charge or magnetic moment with the electromagnetic field. It is the subject of Quantum Electrodynamics (QED), but the more general designation "Electromagnetic Processes" is adopted for a title. The reason for doing this is that for some processes there is a limit where a classical treatment is valid in the sense that both the charge motion and the electromagnetic field can be treated classically. There is also the limit where the charge motion must be described quantum mechanically but a non-relativistic Born-approximation treatment is adequate. In fact, usually calculations in this domain are simpler than those in a purely classical treatment. For a full understanding of the subject the more general relativistic covariant treatment is necessary, but a detailed systematic formulation of this topic requires a lengthy formal development and this is done in a number of fine textbooks. The covariant theory is what we generally refer to as QED, and it would include the nonrelativistic (NR) limit. For problems involving scattering in a Coulomb field the classical domain corresponds to low energy and there is a gap between the classical and non-relativistic Born domains. That is, the two domains do not overlap, but for some processes there are ways of bridging the gap in approximate treatments. However, non-relativistic, non-covariant QED is a valid limit as is, even, the classical domain. We shall try to make an intuitive transition from the NR theory to the covariant treatment, instead of giving a systematic formulation of the latter. Actually, for some processes the covariant calculations are very difficult and a general formula for, say, a cross section cannot be computed from a relativistic expression from which the NR limit would be taken. In some such cases the correct analytic expressions for cross sections can be obtained through a non-relativistic treatment. The NR theory can introduce perturbation "diagrams" just like Feynman diagrams are used in the covariant theory. While these diagrams are not necessary in the calculations, they are useful as guides to outline the computation of the effects of perturbations that cause processes to take place. Although the diagrams in the NR theory do not have the same exact meamng as in the covariant theory, they are helpful in making the transition from the NR theory to the covariant formulation. The covariant theory is much simpler for spin-0 charged particles than it is for the more important case of spin-^, and for the processes considered it is treated first in the relativistic calculations. In the interest of brevity and simplicity, the scope of this book is limited to only a few processes, although some phenomena that are closely related to the ones considered are discussed. For example, in the chapter on Compton scattering,

X

CONTENTS

pair production in photon-photon collisions and pair annihilation to two photons are brought in. For the processes considered the treatment follows the same path. We go from the classical theory to NR QED to the relativistic covariant QED. Moreover, in the latter we first consider the case of spin-0 before going on to the spin- \ calculation. The first chapter of this book gives a summary of some important principles, including a review of special relativity. In particular, some emphasis is given to the method or device of inferring a correct relativistic covariant formula from its nonrelativistic limiting expression. Also, certain kinematic effects are discussed, such as the use of invariant momentum-space volume and the formula for the binary collision rate for distributions of relativistic particles. The chapter also gives a fairly complete account of "phase-space effects," showing how in some cases the phase-space integral should involve only energy conservation and not momentum conservation. Some simple examples are given for cases where a combination of zero-mass and NR particles are outgoing. Although we do not make extensive use of the results on phase space integrals, the treatment of the topic could be useful to people working in other areas, for example, in problems of multiparticle production in high-energy and cosmic-ray physics. Chapter 2 gives a treatment of some features of classical electrodynamics (CED), especially for its application in radiation problems. The third chapter is on QED, treating both the non-relativistic and covariant theories. In an attempt to make the book as self-contained as possible, certain basic developments are formulated from first principles. For example, time-dependent perturbation theory is treated, including the derivation of the Fermi Golden Rule. The Dirac equation is introduced, as well, with its modern covariant notation. However, parts of some calculations for spin- j particles are not given in detail, such as the calculation of the corresponding "trace" for a particular process. Reference to textbooks is given, instead, since the task is sometimes lengthy (but not difficult). Processes involving bound states are not treated extensively. The various types of radiative transitions in atoms, molecules, and nuclei constitute an enormous subject. Nevertheless, at least the foundations are developed in the simple treatment of nonrelativistic QED, which yields the necessary forms for the interaction Hamiltonian for couplings to charges and magnetic moments and for the two-photon coupling. I am very grateful to the staff at Princeton University Press for their help and encouragement in seeing the project of this book come to completeness. In particular, my sincere thanks go to Ingrid Gnerlich and Carmel Lyons, without whom it would not have been accomplished. The work by Ginny Dunn and Mark Bellis was also indespensible. I am grateful, as well, to two referees, who read the original manuscript carefully and made useful suggestions. La Jolla, December, 2004

Electromagnetic Processes

Chapter One Some Fundamental Principles

1.1 UNITS AND CHARACTERISTIC LENGTHS, TIMES, ENERGIES, ETC. In the measurement of quantities by laboratory instruments, both the c.g.s. and m.k.s. units are convenient. However, for the description of particle and atomic processes, the c.g.s. system is preferable in that equations and formulas are sometimes simpler in form; for this reason, the c.g.s. system will be employed throughout this book. At the same time, it is often useful to express quantities in dimensionless units in terms of certain "fundamental" values denned in terms of the fundamental physical constants. Different fundamental quantities—for example, a characteristic length— can be formed from different combinations of physical constants, and the particular choice appropriate for the description of some process is dictated by the nature of the process. Concerning the physical constants themselves, the most fundamental one is perhaps the "velocity of light" (c). The constant is of more general significance than the name given to it, since it is the characteristic parameter of spacetime, and its value is relevant to all dynamical processes in physics. Our fundamental theory of spacetime is special relativity, and we shall review certain basic features of the theory in the following section. Considerations of some general consequences of special relativity are extremely powerful, in particular, as a guide in formulating the fundamental equations of physics. After c, the most fundamental physical constant is probably Planck's constant (h). Loosely put, this constant might be designated as a "quantization parameter," but this is probably not a good description. Another try at description might be to call it the "fundamental indeterminacy parameter," but it is questionable whether the "uncertainty relations" deserve the title of principle, since they follow from the superposition principle (which really is a principle). Given that discrete particle motion is to be described in terms of an associated wave or propagation vector k and frequency co, Planck's constant is then the proportionality factor between k and the particle momentum: p = hk.

(1.1)

The uncertainty relations for an individual particle follow from this relation and the superposition principle. If momentum is to be regarded as a particle dynamical property and the wave propagation vector a kinematical variable, we might designate h more descriptively as a parameter of particle dynamics. However, we shall, as usual, refer to h simply as Planck's constant like everyone else.

2

CHAPTER 1

The third most fundamental physical constant may be the "electronic charge" (e), since it seems to be a fundamental unit common to the various charged elementary particles. That is, although there is a spectrum of masses for the particles, except for the fractionally charged "quarks," the particle charges are multiples of e. From the three physical constants c, ft, and e, it is not possible to construct a fundamental length by various combinations of products. From e and h it is possible to form a characteristic velocity v0 = e2/h,

(1.2)

and this velocity is of significance for particle processes. Combining the fundamental physical constants, a dimensionless number a = e2/hc « 1/137

(1.3)

can be formed that is of great importance, especially for electromagnetic processes. This number is called the "fine structure constant" because of its role in determining the magnitude of the small relativistic level shifts in atomic hydrogen; it can also be regarded as a dimensionless coupling constant for electromagnetic processes. Because of its small value, these processes can be calculated well by perturbation theory. The masses of the various elementary particles play a major role in particle processes. The electron (and positron) mass (m), being the smallest of all, is of great importance because the particle is easily perturbed by an electromagnetic field. In particular, a variety of radiative (photon-producing) processes are associated with the electron and its interactions. A description of these processes is the principal task of this book. Almost all of our knowledge about the world outside our solar system comes from the analysis of the spectral distribution of radiation from distant sources. Our understanding of the details of the microscopic photon-producing processes allows us to interpret these source spectra and learn something of the nature of the sources. Fortunately, the electromagnetic processes are very well understood, and they can be calculated to high accuracy by perturbation theory. The nucleon mass (M)—say, the mass of the proton, which is stable—is significant in that it is much larger (about 1836m) than that of the electron. Along with their corresponding antiparticles, the electron and proton are the only stable "particles." In fact, there is now good evidence, from inelastic scattering of very high energy electrons off protons, that the latter are not "elementary" or "fundamental" particles. Instead, protons are thought to be composites, built from quarks, and they have, as a consequence, structure. For example, protons have a characteristic size and charge distribution that can be measured. Pions (and also kaons) are also quark composites, and the pion is especially important as the least massive of the strongly interacting species. In the older theory of strong interactions, the pion was treated as a fundamental particle and its mass {mn) determined the characteristic range of the interaction. These ideas are still useful in understanding certain particle and nuclear processes. The masses of the elementary particles determine the various fundamental or characteristic lengths, all of which are inversely proportional to the mass value. There are different kinds of lengths, each having a different physical meaning and playing

SOME FUNDAMENTAL PRINCIPLES

3

a different role in determining the characteristic magnitude of importance of various processes. Along with h and e, the electron mass determines the characteristic atomic size a0 = H2/me2.

(1.4)

This is the Bohr radius, and it is one of the triumphs of quantum mechanics that the atomic radius (~ ao ~ 10~8 cm) is explained by physical principles. Classical physics had no explanation for the characteristic size of atoms as determined in the last century. The basic physical meaning of the characteristic length a0 can be indicated through considerations of atomic binding. The classical total energy of an electron of momentum p in the neighborhood of a proton is Eci = p2/2m - e2/r.

(1.5)

In a quantum-mechanical description, the spectra of position and momentum values are such that there is a spread in each, determined by the uncertainty relation. Setting pr ~ h as a constraint condition added to Equation (1.5), we see that Ec\ is minimized at a value (£ci)min ~ -e2/2ao = -Ry

(1.6)

rmin~ao-

(1-7)

for the r-value This little analysis shows, very simply, why atoms have a ground state or state of minimum energy. In a classical model with h —*• 0 the electron "orbit" size could be infinitesimally small and the energy would be infinitely negative. A characteristic length that does not involve h is the "classical electron radius" r 0 . If the electron mass is attributed to its electrostatic self-energy (~ e2/r0 ~ me2), the result is ro = e2/mc2.

(1.8)

13

This is a very small distance (~ 3 x 10~ cm), and the quantity really has no physical meaning, because the classical self-energy considerations are not valid. However, the combination e2/mc2 appears often to various powers in expressions for parameters for electromagnetic quantities. Thus, it is still designated ro and called by its original name. The erroneous nature of the classical model for electromagnetic self-energy is clear through considerations that introduce another characteristic length. If we attempt to localize an electron to a very small distance, of necessity we introduce a spectrum of momentum states extending to high values. For p ~ me, the energy values become large enough to produce e± pairs, which affect and limit the localization. The uncertainty relation then suggests a minimum localization distance loc ~ ft/me = A.

(1.9)

Again for historical reasons, the quantity A is called the electron Compton wavelength. It appears often as a factor in formulas for cross sections for electromagnetic processes and, in general, in many equations describing phenomena involving electrons.

4

CHAPTER 1

The three lengths ao, ro, and A are related through a linear equation with the fine-structure constant as a proportionality factor: r0 =

a

A = Qt2fl0-

(1.10)

Although the three lengths are connected by means of the factor a, only ao and A have a useful physical meaning, and most formulas given throughout this work will not be expressed in terms of ro. It might be noted that each of the lengths ao, A, and ro is inversely proportional to the electron mass. For some problems it is convenient to consider corresponding lengths involving masses of other particles. While a0 determines the characteristic (electron) atomic unit of length, and Eo = e 2 / fl o(= 2Ry) the atomic unit of energy, the electron mass can be replaced by the nucleon (proton) mass M to introduce a "nucleon atomic unit" of distance aM = (m/M)a0

(1.11)

and a characteristic "nucleon Rydberg energy" RyM = (M/m)Ry.

(1.12)

These units are convenient, for example, in the treatment of proton-proton scattering; in that problem, in which the nuclear and Coulomb forces contribute, the Coulomb force plays the major role (except at very high energy). Another important characteristic distance is the particle Compton wavelength associated with the least massive of the strongly interacting particles (i.e., the pion). The quantity An = h/mnc

(1.13)

determines the range of the strong interaction and the magnitude of characteristic cross sections associated purely with this interaction. The cross section is as ~ A\ ~ 20 mb,

(1.14)

where 1 mb = 10~3 b, the barn (b), defined as 10~24cm2, being a cross section unit common in nuclear and strong-interaction physics (1 barn is a large cross section for nuclear processes: "as big as a barn"). The choice of units for the description of some particular phenomenon is dictated not just by considerations of characteristic numerical values of relevant quantities. Depending on the type of units chosen, the equations describing a process take on slightly different form. When formulated in terms of the "most natural" units, the equations are more transparent in exhibiting the nature of the physics involved. In problems of atomic structure or in the description of the scattering of non-relativistic electrons by atomic systems or by a pure Coulomb field, the so-called atomic or "hartree" units are natural. In these units e, h, and m are each set equal to unity, and lengths are in units of the Bohr radius ao, cross sections are in units of a\, and energies are in units of e 2 /ao = 2Ry. The atomic units are, however, not as convenient in problems involving relativistic particles; then the more useful choice is H = c — 1 for which e2 = a is fixed by the dimensionless fine-structure constant [Equation (1.3)]. These units are particularly useful in describing electromagnetic

SOME FUNDAMENTAL PRINCIPLES

5

phenomena. Further, if the process involves electrons or positrons, the rest energy m(c2) is a natural characteristic energy. Throughout this book, certain important results will often be expressed in forms that exhibit dimensions clearly by collecting products of factors that are dimensionless ratios. For example, if a cross section for some electromagnetic process at energy E is expressed in terms of a factor A2, a function of E/mc1, and a factor a", we immediately identify n as the "order" of the process. Higher order electromagnetic processes have cross sections down by powers of a. Equations expressed in this manner are preferable to those in which numerical values of physical constants are substituted in.

1.2 RELATIVISTIC COVARIANCE AND RELATIVISTIC INVARIANTS Ideas of covariance are extremely powerful as a guide in formulating basic physical laws and in the derivation of results in mathematical descriptions of certain physical processes. Considerations of covariance can even provide a path to the discovery of new fundamental laws and then to the development of these new areas of physics. In the description of physical processes, it is often possible to simplify derivations by imposing conditions of relativistic covariance as a trick to arrive at formulas of general validity. We shall often make use of this kind of device. 1.2.1 Spacetime Transformation The basic laws of physics are generally expressed as differential equations with space and time coordinates as independent variables. The spacetime coordinates refer, in some cases, to "events" such as the position (or possible position) of a particle or of a particle process. Further, the properties of spacetime are described in terms of its "structure" or its transformation properties, and this is the theory of special relativity. For spacetime reference frames K and K' whose spatial coordinate axes are moving with constant relative velocity, the relationship between the coordinates of events in the two frames is the Lorentz transformation ^ = Evxv-

(1-15)

V

Here,x v , with v = 0,1, 2, 3, represents the time (v = 0)andspace(v = 1, 2, 3)coordinates. Because of the fundamental isotropy of space, it is convenient to choose Cartesian coordinates (xi, xi, x$ = x,y,z) for the spatial coordinate description. These are thus "natural" or "preferred" coordinates for formulating the basic equations of physics. It is, in one sense, convenient to choose an imaginary component xo = ict for the time variable. This is because the fundamental property of spacetime can then be described by, in addition to the property (1.15), the equation ds2 = J2 dx'^ dx'^ = J2 dxv dxv — invariant (inv),

(1.16)

in which dx^ are the differential coordinate separations between two spacetime events.

6

CHAPTER 1

Because of the choice of an imaginary time component, it has not been necessary to introduce a "metric" or metric tensor. The spacetime is essentially fourdimensional cartesian, and the metric tensor {g^v) is identical to the Kronecker 5-function

fl for^-v °nv = i n . (1-17) t, 10 otherwise. It is somewhat a matter of taste whether this notation procedure is adopted. Actually, the trend in physics seems to be away from the use of the imaginary zero component of spacetime and other four-vectors, at the expense of the introduction of a metric tensor. There is then also the necessity of the introduction of "covariant" and "contravariant" vectors and tensors, etc. with component indices appearing as superscripts and subscripts. Perhaps the resurgence of interest in general relativity during the past forty years has led to this fashion, and it is a necessity in that subject. However, for the treatment of physical processes occurring in a localized region, even if a strong gravitational field is present, coordinates can be chosen corresponding to a "flat" or Minkowskian spacetime. We can then do without a metric and reduce the notation complexity by employing XQ = ict and indices that only appear as subscripts. The summation convention notation henceforth is adopted, in which summation over an index is always implied if it appears twice either on a single symbol or in a product of factors with subscripts. For example, in Equations (1.15) and (1.16), the summation sign could simply be left off. This procedure saves space and is employed extensively with no confusion or difficulty. It is a fundamental assumption that all inertial systems are equally good for formulating a mathematical description of physical phenomena, and in terms of coordinates and other quantities (momenta, fields, etc.), the equations must have the same form whether expressed in terms of "unprimed" or "primed" quantities. Ideas like this—a Principle of Relativity—really go back to Newton's time. The principle would imply that, if we transform from the primed to the unprimed coordinates, the transformation in the inverse direction with an inverse matrix should have the same form as Equation (1.15): xv = a v > ; -

d-18)

2

The invariance of the quantity ds implies the existence of sets of orthogonality relations for the transformation coefficients: = avpavi = x yields, by Equation (1.82), a condition on the relative velocity vr: l

ltivf

> x-

(1.83)

For the case, say, v2 = 0 (one particle initially at rest in the lab frame), we have a condition on the required kinetic energy of the incident particle:

d-84)

\j

But ii < mi, and we see that the kinetic energy required is always greater than the excitation energy x needed in K'. The physical reason for this result is very simple; to satisfy momentum conservation, the products of the collision process must, at threshold, all be moving in the direction of the incident particle at a velocity V = (ffii/Af)»i. The kinetic energy (in K) of the outgoing M is just EM = {m\/M)E\ at threshold, and the difference (E\ — EM) equals xThe kinematics of relativistic particle collisions can be handled with about the same degree of simplicity as the non-relativistic case. Instead of transforming the particle energy and momentum between the lab frame (K) and the cm. frame (K')by means of the four-vector Lorentz transformation, it is more convenient to employ the invariant (1.39). Also, in manipulations involving relativistic kinematics, to simplify the algebra somewhat, we can set c = 1; if necessary, in final formulas the factors of c to various powers can be reinserted through considerations of dimensionality. The invariant E2-

p2 = inv

(1.85)

SOME FUNDAMENTAL PRINCIPLES

17

can refer to either the components of the total energy and momentum or those of an individual particle, for which E2 — p\ = m\. When E and p refer to the total energy and momentum of a two-particle system, since the individual energy and momentum are related by px — E\V\, application of the invariant (1.85) yields ' m\ + m\ + 2ElE2{\ - i>i • v2) = E'2

(1.86)

where E' is the total energy in the cm. frame. This simple and general formula has many important applications. In the calculation of binary-collision processes, generally the cross section is most conveniently represented in terms of the energy E', and the expression (1.86) provides the explicit relationship to the lab-frame quantities.12 Equation (1.86) can also be applied to determine (lab-frame) threshold energies as in the non-relativistic limit. If mi = m2 = m and v2 = 0, for a process corresponding to a cm. threshold energy £" = X, the lab-frame kinetic energy required is Ti = Ei-m

= x2/2m - 2m.

(1.87)

For example, in producing particle-antiparticle pairs of the same type, the cm. threshold energy would be x = Am and the lab-frame value would be 7\ = 6m. As in our non-relativistic problem, because of the necessity of satisfying momentum conservation, much more lab-frame energy is required than in the cm. frame. In the case of photon-photon collisions or for collisions of highly relativistic particles, the relation (1.86) is further simplified: 2EiE2(\ - cos0i2) = E'2,

(1.88)

where #12 is the angle between the (massless) particle directions of motion. For example, in the process of pair production in photon-photon collisions, £" could be set equal to 2E'e, where E'e is the cm. energy of the electron or positron. For a head-on photon-photon collision (#12 = n), the threshold condition would be, very simply, E\E2 = m2. 1.3.2 Transformations of Angular Distributions In the evaluation of particle processes, it is often necessary to consider the relationship between the direction of particle motion in different reference frames. For relativistic particle processes, there is an especially important and interesting phenomenon associated with the directions of motion of other particles (with which it interacts) in the lab frame and in the rest frame of the relativisitic particle. Consider the motion of a particle (Pv) of velocity v in the lab frame (K). Let the axes of K be such that v = v(l, 0, O);thatis, D is along the x-axis. Now consider the motion of another particle (Pu) of velocity u; let this velocity lie in A"'s x-v-plane so that u = M(COS 6, sin 6,0). In Pv's rest frame (Kr), Pu will be moving at an angle 0' with respect to the x- and x'-axes. From the velocity transformation, tan0' = u'Ju'x = uy/y(ux 12

- v) = sin0/y(cos# - v/u),

(1.89)

Often it is necessary to integrate over a spectrum of lab-frame energies and directions of motion.

18

CHAPTER 1

where y = (1 — v2/c2)~1/2. The transformation from K' to K is, of course, of the same form except that the sign of vx (= v) is reversed. For the case where Pu is nonrelativistic and v/u » cos9, tan#' ->• —(u/yv) sinO, for which JT/2 < 0' < 7r. In fact, if v -» c (y 3> 1), 9' will be very close to jr. This means that in Pv's frame (X"') every other particle is moving toward it head on. Even if the Pu 's are photons (u —>• c), this will be true as long as 9 > y" 1 . A highly relativistic particle moving through a gas "sees" particles incident head on (like a beam). We could also consider some process involving a highly relativistic Pv in which, in K', there is some outgoing particle Pu (for example, a scattered photon) moving at an angle 9'. Then in the lab frame (K) the corresponding angle would be 9 with tan (u/cy)sin9 (1-112) / o a n d P = 0 in the c m . frame. The condition (1.112) also follows from the evaluation of H'j0 in the position representation, employing plane-wave states for the particle wave functions in the initial and final states (see Chapter 3). Aside from spin states, the number of momentum (p) or wave-vector (k) states available in a spatial volume L 3 and momentum-space volume d3p is 1 4 dN = L3d3p/(27Thf = L3d3k/(27t)\

(1.113) 3

In the description of particle processes the factors involving L always cancel, since L 3 appears in inverse powers in the matrix element H'f0 as normalization factors in the individual one-particle wave functions. As remarked earlier in this chapter, although L3d3p is, strictly speaking, what is known as phase space, in the description of particle processes we often refer to the momentum-space part as the phase-space factor. If N particles are being described, the associated phase-space volume, with both energy and momentum-space conservation, is then, in the cm. frame, j

Z

j

)

E

j

)

(1.114)

For many processes this factor determines the magnitude of the transition rate or, for example, the cross section for the process. Of special importance is the dependence of $ W on the available energy E, which would determine the behavior of, say, a cross section in the neighborhood of the threshold energy for the process. This 14

See Equation (1.54) and any book on quantum mechanics treating the elementary "particle-in-abox" problem.

22

CHAPTER 1

dependence can be derived very readily, the result being that 0 form

m

has a power-law

£«, (1.115) for certain energy domains [outgoing particles non-relativistic (NR) or extreme relativistic (ER)], with q increasing with N. We shall derive these results, obtaining general formulas for w. They are fundamental for general processes in which particles are produced, and often the energies involved are relativistic. Unfortunately, the problem for (general) outgoing particle energies Ej ~ nijC2 is complex except for small N and, in fact, in this case the invariant expression

4>{N) = / • " / ft (d3p/2E)j S^(j2(pu)j

- />„)

(1.116)

is sometimes more useful. Considerations of particle processes in terms of invariant phase-space factors are employed extensively in high-energy physics. However, a covariant description15 is not always most appropriate, especially if the outgoing particles are non-relativistic. This is usually the case in applications to problems involving nuclear reactions (£o ~ MeV energies), and always in the area of chemical kinetics where the reaction products are atoms, molecules, ions, and electrons, at ~ eV energies. In some processes there might be outgoing particles both NR and ER, the latter type being photons, for example, as in some nuclear reactions; thus, it is useful to have expresssions for the phase-space factor in these "mixed" cases. To describe the cases we shall employ the notation N and N' for the total number of NR and ER particles, respectively, and the indices j and k to designate the particles among the N and N'. We shall see that it is possible to obtain exact formulas for 0 for arbitrary N and N' in this limit, where [Ej^p^llmj (y = ltoAO, J J \ J , (1.117) [Ek = pkc (k=ltoN). There is an important theorem involving the phase-space volume (1.114) for the case where one of the outgoing particles (say, m^) has a large mass. In this limit its energy would have to be non-relativistic and the arguments of the 5-functions in can be written N-l

N

y=l

J

J j=\

Y,Ej-E = p1N/2mN+t(iV> can then be eliminated by integrating over d3pN using the argument (1.118), and we have the result mjylarge

where V is a phase-space volume like