Relativistic Quantum Mechanics. Wave Equations

Cover......Page 1
Half-title......Page 3
Series-title......Page 4
Title......Page 5
Copyright......Page 6
Contents......Page 7
Foreword......Page 13
Introduction......Page 15
References......Page 20
1.1.1 Invariant scattering amplitude and cross section......Page 22
1.1.3 Singularities......Page 23
1.2 Mandelstam variables for two-particle scattering......Page 24
1.2.1 The Mandelstam plane......Page 25
1.2.2 Threshold singularities on the Mandelstam plane......Page 26
1.3 Partial wave expansion and unitarity......Page 27
1.3.2 Singularities of Im A on the Mandelstam plane (Karplus curve)......Page 29
1.4 The Froissart theorem......Page 31
1.5 The Pomeranchuk theorem......Page 33
2 Physics of the t-channel and complex angular momenta......Page 36
2.1 Analytical continuation of the t-channel unitarity condition......Page 37
2.1.1 The Mandelstam representation......Page 41
2.1.2 Inconsistency of the ‘black disk ’model of diffraction......Page 42
2.2 Complex angular momenta......Page 43
2.3 Partial wave expansion and Sommerfeld–Watson representation......Page 44
2.4.1 Non-relativistic quantum mechanics......Page 47
2.4.2 Relativistic theory......Page 48
2.5 Gribov–Froissart projection......Page 49
2.6 t-Channel partial waves and the black disk model......Page 52
3 Singularities of partial waves and unitarity......Page 53
3.1 Continuation of partial waves with complex  l to t < 0......Page 54
3.1.1 Threshold singularity and partial waves.........Page 55
3.1.2 ...(t ) At t < 0 and its discontinuity......Page 56
3.2 The unitarity condition for partial waves with complex l......Page 57
3.3.1 Left cut in non-relativistic theory......Page 58
3.3.3 Moving singularities......Page 60
3.4 Moving poles and resonances......Page 62
4.1 Resonances......Page 65
4.3 Elementary particle or bound state?......Page 67
4.3.2 Regge pole exchange and particle exchange (t > 0)......Page 68
4.3.3 Regge exchange and elementary particles (t < 0)......Page 70
4.3.5 Asymptotics of s-channel amplitudes and reggeization......Page 71
4.4 Factorization......Page 72
5.1 t-Channel dominance......Page 75
5.2.1 Quantum numbers of the pomeron......Page 78
5.2.2 Slope of the pomeron trajectory......Page 79
5.3.1 s-Channel partial waves in the impact parameter space......Page 80
5.4 Relation between total cross sections......Page 83
6 Scattering of particles with spin......Page 85
6.1 Vector particle exchange......Page 86
6.2.1 Reggeon quantum numbers…......Page 89
6.2.2 Vacuum pole in Pi N and NN scattering......Page 90
6.3 Conspiracy......Page 91
7 Fermion Regge poles......Page 93
7.1 Backward scattering as a relativistic effect......Page 95
7.2 Pion–nucleon scattering......Page 96
7.2.1 Parity in the u-channel......Page 98
7.2.2 Fermion poles with definite parity and singularity at u = 0......Page 99
7.2.3 Oscillations in the fermion pole amplitude......Page 101
7.3 Reggeization of a neutron......Page 103
8.2 Scalar field theory…......Page 104
8.2.1 …Theory in the Duffin–Kemmer formalism......Page 105
8.2.2 Analytic properties of the amplitudes......Page 107
8.2.3 Order g ln s......Page 111
8.2.4 Order g ln s......Page 115
8.2.5 Ladder diagrams in all orders......Page 118
8.2.6 Non-ladder diagrams......Page 119
8.3 Interaction with vector mesons......Page 120
9 Reggeization of an electron......Page 126
9.1 Electron exchange in O (g) Compton scattering amplitude......Page 127
9.2.1 Conspiracy in perturbation theory......Page 129
9.2.2 Reggeization in QED (with massless photon)......Page 131
9.3 Electron reggeization from the cross-channel point of view: nonsense states......Page 132
10 Vector field theory......Page 135
10.1 Rõle of spin effects in reggeization......Page 136
10.1.1 Nonsense states in the unitarity condition......Page 137
10.1.3 Nonsense states from the s- and t-channel points of view......Page 139
10.1.4 The j = ½ pole in the perturbative nonsense–nonsense amplitude......Page 143
10.2 QED processes with photons in the t-channel......Page 145
10.2.1 The vacuum channel in QED......Page 146
10.2.2 The problem of the photon reggeization......Page 148
11.1 The pole l = -1 and restriction on the amplitude fall-off......Page 151
11.2 Contradiction with unitarity......Page 155
11.3 Poles condensing at l = -1......Page 156
11.4 Particles with spin: failure of the Regge pole picture......Page 157
12.1.1 Redefinition of partial wave amplitudes......Page 159
12.1.2 Particles with spin in the unitarity condition......Page 160
12.2 Particle scattering via a two-particle intermediate state......Page 162
12.3 Two-reggeon exchange and production vertices......Page 164
12.4 Asymptotics of two-reggeon exchange amplitude......Page 167
12.5 Two-reggeon branching and l = -1......Page 168
12.6 Movement of the branching in the t and j planes......Page 170
12.7 Signature of the two-reggeon branching......Page 171
13.1.1 Branchings in the j plane......Page 173
13.1.2 Branch singularity in the unitarity condition......Page 174
13.2 Branchings in the vacuum channel......Page 175
13.2.2 The Mandelstam representation in the presence of branchings......Page 176
13.3 Vacuum–non-vacuum pole branchings......Page 177
13.4 Experimental verification of branching singularities......Page 179
13.4.1 Branchings and conspiracy......Page 180
14 Reggeon diagrams......Page 182
14.1.1 Structure of the vertex......Page 186
14.1.2 Analytic properties of the vertex......Page 187
14.1.3 Factorization......Page 188
14.2 Partial wave amplitude of the Mandelstam branching......Page 189
15 Interacting reggeons......Page 197
16 Reggeon field theory......Page 207
16.1 Enhanced reggeon diagrams......Page 209
16.2 Effective field theory of interacting reggeons......Page 214
16.3 Equation for the Green function G......Page 215
16.4 Equation for the vertex function Gamma2......Page 216
16.5 Weak and strong coupling regimes......Page 218
16.6 Pomeron Green function and reggeon unitarity condition......Page 219
17.1.1 The Green function......Page 222
17.1.2 P …PP vertex......Page 223
17.1.3 Induced multi-reggeon vertices......Page 225
17.1.4 Vanishing of multi-reggeon couplings......Page 227
17.2 Problems of the strong coupling regime......Page 229
Introduction......Page 230
A.1 Wave function of the hadron. Orthogonality and normalization......Page 235
A.2 Distribution of the partons in space and momentum......Page 237
A.3 Deep inelastic scattering......Page 241
A.4 Strong interactions of hadrons......Page 245
A.5 Elastic and quasi-elastic processes......Page 249
References......Page 252
Introduction......Page 254
B.1 The absorptive parts of reggeon diagrams in the s-channel. Classification of inelastic processes......Page 258
B.2 Relations among the absorptive parts of reggeon diagrams......Page 261
B.3 Inclusive cross sections......Page 266
B.4 Main corrections to the inclusive cross sections in the central region......Page 269
B.5 Fluctuations in the distribution of the density of produced particles......Page 273
Appendix......Page 278
References......Page 280
C.1 Introduction......Page 281
C.2 Non-enhanced cuts at Alpha = 0......Page 283
C.3 Estimation of enhanced cuts at Alpha = 0......Page 285
C.4 Structure of the transition amplitude of one pomeron to two......Page 287
C.5 The Green function and the vertex part at Alpha = 0......Page 289
C.6 Properties of high energy processes in the theory with Alpha = 0......Page 296
C.6.1 Two-particle processes, total cross sections......Page 297
C.6.3 Correlation, multiplicity distribution......Page 298
C.6.4 Probability of fluctuations in individual events: the inclusive spectrum in the three-pomeron limit......Page 300
C.7 The case…......Page 305
C.8 The contribution of cuts at small Alpha......Page 306
Appendix......Page 308
References......Page 309
Index......Page 310