Quantum Electrodynamics (eBook)

Volume 4

V B Berestetskii, E.M. Lifshitz, L. P. Pitaevskii (Autoren)

eBook Download: PDF | EPUB

2012 | 2. Auflage
667 Seiten
Elsevier Science (Verlag)
978-0-08-050346-2 (ISBN)

Infos

Several significant additions have been made to the second edition, including the operator method of calculating the bremsstrahlung cross-section, the calcualtion of the probabilities of photon-induced pair production and photon decay in a magnetic field, the asymptotic form of the scattering amplitudes at high energies, inelastic scattering of electrons by hadrons, and the transformation of electron-positron pairs into hadrons.

Inhalt

Front Cover 1
Quantum Electrodynamics: Course of Theoretical Physics 
4 
Copyright Page 5
PREFACE TO THE SECOND EDITION 6
FROM THE PREFACE TO THE FIRST EDITION 8
TABLE OF CONTENTS 10
NOTATION 14
INTRODUCTION 18
1. The uncertainty principle in the relativistic case 18
CHAPTER I. PHOTONS 22
2. Quantization of the free electromagnetic field 22
3. Photons 27
4. Gauge invariance 29
5. The electromagnetic field in quantum theory 31
6. The angular momentum and parity of the photon 33
7. Spherical waves of photons 36
8. The polarization of the photon 41
9. A two-photon system 46
CHAPTER II. BOSONS 50
10. The wave equation for particles with spin zero 50
11. Particles and antiparticles 54
12. Strictly neutral particles 58
13. The transformations C, P and T 61
14. The wave equation for a particle with spin one 67
15. The wave equation for particles with higher integral spins 70
16. Helicity states of a particle 72
CHAPTER III. FERMIONS 79
17. Four-dimensional spinors 79
18. The relation between spinors and 4-vectors 81
19. Inversion of spinors 85
20. Dirac's equation in the spinor representation 90
21. The symmetrical form of Dirac's equation 92
22. Algebra of Dirac matrices 97
23. Plane waves 101
24. Spherical waves 105
25. The relation between the spin and the statistics 108
26. Charge conjugation and time reversal of spinors 111
27. Internal symmetry of particles and antiparticles 116
28. Bilinear forms 118
29. The polarization density matrix 123
30. Neutrinos 128
31. The wave equation for a particle with spin 3/2 132
CHAPTER IV. PARTICLES IN AN EXTERNAL FIELD 135
32. Dirac's equation for an electron in an external field 135
33: Expansion in powers of 1/c 139
34. Fine structure of levels of the hydrogen atom 143
35. Motion in a centrally symmetric field 145
36. Motion in a Coulomb field 150
37. Scattering in a centrally symmetric field 157
38. Scattering in the ultra-relativistic case 159
39. The continuous-spectrum wave functions for scattering in a Coulomb field 161
40. An electron in the field of an electromagnetic plane wave 165
41. Motion of spin in an external field 168
42. Neutron scattering in an electric field 174
CHAPTER V. RADIATION 176
43. The electromagnetic interaction operator 176
44. Emission and absorption 178
45. Dipole radiation 181
46. Electric multipole radiation 183
47. Magnetic multipole radiation 188
48. Angular distribution and polarization of the radiation 190
49. Radiation from atoms: the electric type 198
50. Radiation from atoms: the magnetic type 203
51. Radiation from atoms: the Zeeman and Stark effects 206
52. Radiation from atoms: the hydrogen atom 209
53. Radiation from diatomic molecules: electronic spectra 214
54. Radiation from diatomic molecules: vibrational and rotational spectra 220
55. Radiation from nuclei 222
56. The photoelectric effect: non-relativistic case 224
57. The photoelectric effect: relativistic case 229
58. Photodisintegration of the deuteron 233
CHAPTER VI. SCATTERING OF RADIATION 238
59. The scattering tensor 238
60. Scattering by freely oriented systems 248
61. Scattering by molecules 254
62. Natural width of spectral lines 257
63. Resonance fluorescence 261
CHAPTER VII. THE SCATTERING MATRIX 264
64. The scattering amplitude 264
65. Reactions involving polarized particles 269
66. Kinematic invariants 273
67. Physical regions 275
68. Expansion in partial amplitudes 281
69. Symmetry of helicity scattering amplitudes 285
70. Invariant amplitudes 291
71. The unitarity condition 295
CHAPTER VIII. INVARIANT PERTURBATION THEORY 300
72. The chronological product 300
73. Feynman diagrams for electron scattering 303
74. Feynman diagrams for photon scattering 309
75. The electron propagator 312
76. The photon propagator 317
77. General rules of the diagram technique 321
78. Crossing invariance 328
79. Virtual particles 329
CHAPTER IX. INTERACTION OF ELECTRONS 334
80. Scattering of an electron in an external field 334
81. Scattering of electrons and positrons by an electron 338
82. Ionization losses of fast particles 347
83. Breit's equation 353
84. Positronium 360
85. The interaction of atoms at large distances 364
CHAPTER X. INTERACTION OF ELECTRONS WITH PHOTONS 371
86. Scattering of a photon by an electron 371
87. Scattering of a photon by an electron. Polarization effects 376
88. Two-photon annihilation of an electron pair 385
89. Annihilation of positronium 388
90. Synchrotron radiation 393
91. Pair production by a photon in a magnetic field 403
92. Electron–nucleus bremsstrahlung. The non-relativistic case 406
93. Electron–nucleus bremsstrahlung. The relativistic case 417
94. Pair production by a photon in the field of a nucleus 427
95. Exact theory of pair production in the ultra-relativistic case 430
96. Exact theory of bremsstrahlung in the ultra-relativistic case 436
97. Electron–electron bremsstrahlung in the ultra-relativistic case 443
98. Emission of soft photons in collisions 448
99. The method of equivalent photons 455
100. Pair production in collisions between particles 461
101. Emission of a photon by an electron in the field of a strong electromagnetic wave 466
CHAPTER XI. EXACT PROPAGATORS AND VERTEX PARTS 473
102. Field operators in the Heisenberg representation 473
103. The exact photon propagator 476
104. The self-energy function of the photon 482
105. The exact electron propagator 485
106. Vertex parts 489
107. Dyson's equations 493
108. Ward's identity 495
109. Electron propagators in an external field 498
110. Physical conditions for renormalization 504
111. Analytical properties of photon propagators 510
112. Regularization of Feynman integrals 513
CHAPTER XII. RADIATIVE CORRECTIONS 518
113. Calculation of the polarization operator 518
114. Radiative corrections to Coulomb's law 521
115. Calculation of the imaginary part of the polarization operator from the Feynman integral 525
116. Electromagnetic form factors of the electron 530
117. Calculation of electron form factors 534
118. Anomalous magnetic moment of the electron 538
119. Calculation of the mass operator 541
120. Emission of soft photons with non-zero mass 546
121. Electron scattering in an external field in the second Born approximation 551
122. Radiative corrections to electron scattering in an external field 557
123. Radiative shift of atomic levels 561
124. Radiative shift of mesic-atom levels 568
125. The relativistic equation for bound states 569
126. The double dispersion relation 576
127. Photon–photon scattering 583
128. Coherent, scattering of a photon in the field of a nucleus 590
129. Radiative corrections to the electromagnetic field equations 592
130. Photon splitting in a magnetic field 602
131. Calculation of integrals over four-dimensional regions 609
CHAPTER XIII. ASYMPTOTIC FORMULAE OF QUANTUM ELECTRODYNAMICS 614
132. Asymptotic form of the photon propagator for large momenta 614
133. The relation between unrenormalized and actual charges 618
134. Asymptotic form of the scattering amplitudes at high energies 620
135. Separation of the double-logarithmic terms in the vertex operator 625
136. Double-logarithmic asymptotic form of the vertex operator 631
137. Double-logarithmic asymptotic form of the electron–muon scattering amplitude 633
CHAPTER XIV. ELECTRODYNAMICS OF HADRONS 641
138. Electromagnetic form factors of hadrons 641
139. Electron–hadron scattering 646
140. The low-energy theorem for bremsstrahlung 649
141. The low-energy theorem for photon–hadron scattering 652
142. Multipole moments of hadrons 655
143. Inelastic electron–hadron scattering 660
144. Hadron formation from an electron-positron pair 662
INDEX 666

Textauszug

INTRODUCTION

§ 1 The uncertainty principle in the relativistic case

The quantum theory described in Volume 3 (Quantum Mechanics) is essentially non-relativistic throughout, and is not applicable to phenomena involving motion at velocities comparable with that of light. At first sight, one might expect that the change to a relativistic theory is possible by a fairly direct generalization of the formalism of non-relativistic quantum mechanics. But further consideration shows that a logically complete relativistic theory cannot be constructed without invoking new physical principles.

Let us recall some of the physical concepts forming the basis of non-relativistic quantum mechanics (QM, §1). We saw that one fundamental concept is that of measurement, by which is meant the process of interaction between a quantum system and a classical object or apparatus, causing the quantum system to acquire definite values of some particular dynamical variables (coordinates, velocities, etc.). We saw also that quantum mechanics greatly restricts the possibility that an electron † simultaneously possesses values of different dynamical variables. For example, the uncertainties Δq and Δp in simultaneously existing values of the coordinate and the momentum are related by the expression ‡ ΔqΔp ˜ ħ; the greater the accuracy with which one of these quantities is measured, the less the accuracy with which the other can be measured at the same time.

It is important to note, however, that any of the dynamical variables of the electron can individually be measured with arbitrarily high accuracy, and in an arbitrarily short period of time. This fact is of fundamental importance throughout non-relativistic quantum mechanics. It is the only justification for using the concept of the wave function, which is a basic part of the formalism. The physical significance of the wave function ψ(q) is that the square of its modulus gives the probability of finding a particular value of the electron coordinate as the result of a measurement made at a given instant. The concept of such a probability clearly requires that the coordinate can in principle be measured with any specified accuracy and rapidity, since otherwise this concept would be purposeless and devoid of physical significance.

The existence of a limiting velocity (the velocity of light, dénoted by c) leads to new fundamental limitations on the possible measurements of various physical quantities (L. D. Landau and R. E. Peierls, 1930).

In QM, §44, the following relationship has been derived:

′−v)ΔpΔt∼h, (1.1)

(1.1)

relating the uncertainty Δp in the measurement of the electron momentum and the duration Δt of the measurement process itself; v and v′ are the velocities of the electron before and after the measurement. From this relationship it follows that a momentum measurement of high accuracy made during a short time (i.e. with Δp and At both small) can occur only if there is a large change in the velocity as a result of the measurement process itself. In the non-relativistic theory, this showed that the measurement of momentum cannot be repeated at short intervals of time, but it did not at all diminish the possibility, in principle, of making a single measurement of the momentum with arbitrarily high accuracy, since the difference v′ – v could take any value, no matter how large.

The existence of a limiting velocity, however, radically alters the situation. The difference v′ – v, like the velocities themselves, cannot now exceed c (or rather 2c). Replacing v′- v in (1.1) by c, we obtain

pΔt∼h/c, (1.2)

(1.2)

which determines the highest accuracy theoretically attainable when the momentum is measured by a process occupying a given time Δt. In the relativistic theory, therefore, it is in principle impossible to make an arbitrarily accurate and rapid measurement of the momentum. An exact measurement (Δp → 0) is possible only in the limit as the duration of the measurement tends to infinity.

There is reason to suppose that the concept of measurability of the electron coordinate itself must also undergo modification. In the mathematical formalism of the theory, this situation is shown by the fact that an accurate measurement of the coordinate is incompatible with the assertion that the energy of a free particle is positive. It will be seen later that the complete set of eigenfunctions of the relativistic wave equation of a free particle includes, as well as solutions having the “correct” time dependence, also solutions having a “negative frequency”. These functions will in general appear in the expansion of the wave packet corresponding to an electron localized in a small region of space.

It will be shown that the wave functions having a “negative frequency” correspond to the existence of antiparticles (positrons). The appearance of these functions in the expansion of the wave packet expresses the (in general) inevitable production of electron–positron pairs in the process of measuring the coordinates of an electron. This formation of new particles in a way which cannot be detected by the process itself renders meaningless the measurement of the electron coordinates.

In the rest frame of the electron, the least possible error in the measurement of its coordinates is

q∼h/mc. (1.3)

(1.3)

This value (which purely dimensional arguments show to be the only possible one)corresponds to a momentum uncertainty Δp ˜ mc, which in turn corresponds to the threshold energy for pair production.

In a frame of reference in which the electron is moving with energy ε, (1.3) becomes

q∼cℏ/ɛ. (1.4)

(1.4)

In particular, in the limiting ultra-relativistic case the energy is related to the momentum by ε ˜ cp, and

q∼h/p, (1.5)

(1.5)

i.e. the error Δq is the same as the de Broglie wavelength of the particle.†

For photons, the ultra-relativistic case always applies, and the expression (1.5) is therefore valid. This means that the coordinates of a photon are meaningful only in cases where the characteristic dimensions of the problem are large in comparison with the wavelength. This is just the “classical” limit, corresponding to geometrical optics, in which radiation can be said to be propagated along definite paths or rays. In the quantum case, however, where the wavelength cannot be regarded as small, the concept of coordinates of the photon has no meaning. We shall see later (§4) that, in the mathematical formalism of the theory, the fact that the photon coordinates cannot be measured is evident because the photon wave function cannot be used to construct a quantity which might serve as a probability density satisfying the necessary conditions of relativistic invariance.

The foregoing discussion suggests that the theory will not consider the time dependence of particle interaction processes. It will show that in these processes there are no characteristics precisely definable (even within the usual limitations of quantum mechanics); the description of such a process as occurring in the course of time is therefore just as unreal as the classical paths are in non-relativistic quantum mechanics. The only observable quantities are the properties (momenta, polarizations) of free particles: the initial particles which come into interaction, and the final particles which result from the process (L. D. Landau and R. E. Peierls, 1930).

A typical problem as formulated in relativistic quantum theory is to determine the probability amplitudes of transitions between specified initial and final states (t → ∞) of a system of particles. The set of such amplitudes between all possible states constitutes the scattering matrix or S-matrix. This matrix will embody all the information about particle interaction processes that has an observable physical meaning (W. Heisenberg, 1938).

There is as yet no logically consistent and complete relativistic quantum theory. We shall see that the existing theory introduces new physical features into the nature of the description of particle states, which acquires some of the features of field theory (see §10). The theory is, however, largely constructed on the pattern of ordinary quantum mechanics. This structure of the theory has yielded good results in quantum electrodynamics. The lack of complete logical consistency in this theory is shown by the occurrence of divergent expressions when the mathematical formalism is directly applied, although there are quite well-defined ways of eliminating these divergences. Nevertheless, such methods remain, to a considerable extent, semiempirical rules, and our...

Produkt-Details

• Erscheint lt. Verlag: 2.12.2012
• Sprache: englisch
• Themenwelt: Geisteswissenschaften ► Psychologie ► Allgemeine Psychologie
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Elektrodynamik
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Quantenphysik
• Themenwelt: Technik
• ISBN-10: 0-08-050346-2 / 0080503462
• ISBN-13: 978-0-08-050346-2 / 9780080503462