A
Algebra, multilinear over Hilbert space, 305-316
Amplitudes, representation of, 43-44 Analyticity, A2, derivation of, 76-82 Annihilation operator, 117, 118, 309 Ansatz
Pekar's product, 285-288 variational, 298-301 Asymptotic condition, 67-96
bound states, 67-74 Β Banach space, 306
Bargmann, Hall, Wightman theorem, 130-132
BCS « single particle » excitation, 277 Beta decay
axial vector coupling term, 93-94 global symmetry, 223-227
interaction Hamiltonian, 224-225 time reversal invariance, 194-195 Beth-Uhlenbeck expression of second
virial coefficient, 261-262 Blatt-Jackson straight line, 111 Bose statistics, 1
Boson system, collective behavior, 267-273
Branching equations, 123
« Breit» system, momenta in, 77 Brueckner
expression for ground state energy, 264
reaction matrix, 264 C
Cauchy formula for operators, 41 Causality, 2
Charge
distribution, 24 renormalization, 48
state, scattering amplitudes in, 97
Charges, generalized, 20 Chew
and Low effective range approxi
mation, 108-110 proposal, 98, 101 Collision matrix, 44
Collisions, binary, contribution to the shielded potential, 259-265 Commutation
free field, 7
relations, spin zero, 3
Commutator, multiple, 165-166, 169- 171
Configuration space method, 113-122
nuclear form factors in, 211-215 Coupling constant
determination with dispersion rela
tions, 97-111 ratio, 226-227
renormalization, 57-59 Creation operator, 309
D D'Alembert equation, 314
DAN AD representation, reduction of parameters in, 154-157 Decay
beta
axial vector coupling term, 93-94 global symmetry, 223-227 interaction Hamiltonian, 224-225 time reversal invariance, 194-195 exponential, 41-43
integration contour, 42
products, scattering amplitudes, 59-62
of unstable particles, 219-220 Degeneracy, 176-179
Dipole moment, magnetic, 22-23 Dirac theory of holes, 1, 20 Dispersion relations
and the asymptotic condition, 67-96 cases which have been proven, 82
SUBJECT INDEX derivation, 76-82
determination of coupling con
stants, 97-111
physical interpretation, 86-90 possible causes of limitations, 82-85 for proton-neutron scattering, 104 Distributive law, restricted, 205 Divergences, ultraviolet, renormaliza
tion of, 123-124 Domain
analyticity of vacuum expectation values, 147-162
of holomorphy, 152-153 permuted, 132-137
RN, real regularity points, 132-137 BN9 structure, 140-143
Doublet
Pais, 183-187
Dyson's theorem, high energy beha
vior, 90-94
£
Electrodynamics, gauge transforma
tion, 178
Electron-positron field, 312-314 Energy
of temperature dependent elemen
tary excitations, 283
total of nucleon-antinucleon system, 100
Energies, phonon excitation, 268 Exchange graph, 99
Exterior product, 115 F Fermi
gas
of Ν-particles, Hamiltonian, 229 perturbation theory difficulties,
237-240 interactions, 225 momentum, 230 statistics, 1
Fermion distribution, 282 Feynman
diagram, vacuum - vacuum type, 248 ft
graph, 99
Field
electron-positron, 312-314 equations, 121-122 free, 2, 305-316
commutation, 7 Hamiltonian, 7 hermitian, 2
Heisenberg, differential equation, 28 Majorana, 2
non-interacting, 2 photon, 314-316 quantities, 121-122 scalar neutral, 311-312 spin zero, quantization of, 2-3 theory of two interacting, 28 Forces, nuclear, time reversal in,
189-197 Form factor
isotropic, 212
nuclear in configuration space, 211- 215
spectral representation, 212 threshold limit, 214 Fourier transform, 70, 72 Functional
basic, 27
generating, 28-30 G Gauge transformation
electrodynamics, 178 generalized, 32-34 Gibbs potential, 248-251
transformation of expansion, 251- 255
Goldstone
determination of ground state en
ergy, 230-233
expansion of ground state energy, 254
Grand partition function expansion, 242-245 free particle system, 243
second method of expanding, 255- 257
Grassman
interior multiplication, 117 product, 115
Ground state energy
Brueckner expression, 264 Goldstone expression, 230-233,
254
of many-body system, 230-233 shift due to the interaction V, 232 in weak coupling limit, 281 perturbed, 230
theory of many-particle system, 229-240
Η Haag's theorem, 137
Hadamard concept of partie-finie in
tegral, 126 Half-life integral, 58 Hamiltonian
for Fermi gas of Ν-particles, 229 free field, 7
interaction, for beta-decay, 224-225 Lee model total, 50
polaron model, 285 renormalization, 49-50 spectrum of, 38, 39
for system of interacting electrons and photons, 242
Heisenberg
fields, differential equations for, 28 representation, 18-19
type theory, 173 ff Hermitian
adjoint of anti-linear operator, 11 fields, 2
Hilbert space, 67, 68, 71
multilinear algebra over, 305-316 transformations in, 203
Holes, Dirac theory of, 1, 20 Holomorphy, domain of, 152-153 Hypersurface, analytic, 153, 154
I Interaction
direct, term, 103
Hamiltonian for β-decay, 224-225 Interactions
Fermi, 225
strangeness, violating, 183 strong, 184 ff, 219-220
weak and Pais doublets, 183-187 Isofermions, 220
J
JLD representation, 76, 78, 79 Jost
points, 168
proof of TCP theorem, 19 Κ
Kernel part, 217 Klein
-Gordon equation, 177, 309 transformation, 8
K-meson theory, symmetrical, 220 Kronecker
symbol, 61 weighted, 232
L Lee model,
properties of scattering amplitudes, 49-54
total Hamiltonian, 50 Lehmann
definition of propagator, 54, 59 weight function, 63, 64
Lifetime, definition from resonance, 48-49
Linear independence, 205 Locality, 2
condition, 68 Lorentz
condition, 314
group, inhomogeneous, 3 group L, homogeneous, 128-130
Μ
JKZ boundary
proof of a non-analytic hyper
surface, 157-159
reduction of parameters in DA Ν AD representation, 154-157
SUBJECT INDEX Macrocausality, 85, 86-90
Majorana fields, 2 gauge, 173 ff spinor, 173, 179
Many-body problem at non-zero tem
peratures, 241-265
Many-body system, ground state en
ergy, 230-233 Mass
definition from resonance, 48-49 and degeneracy, 176-179 renormalization, 48, 57-59 theorem, 175-176
uncertainty, 47 Matrix
Brueckner reaction, 264 collision, 44
element
diagram analysis, 231-232, 234, 239
of resolvant, 40 unitarity, 55-56 T, scattering, 107
Momenta in the « B r e i t » system, 77 Momentum, Fermi, 230
Motion, operator of, 234 Ν
Nuclei, unoriented, correlation func
tion for double cascade, 195 Nucleon-antinucleon system, total en
ergy, 100
Nucleons in nuclear matter, 241
« Nullteiler » components, 179 Ο
Observables, transformation of, 204- 205
Operator
annihilation, 117, 118, 309 anti-hermitian, 11
anti-linear, 10-12 hermitian adjoint, 11 an ti-unitary, 11 Cauchy formula for, 41 creation, 117, 118, 309
linear, 11
of AR, 117-119 of motion, 234 one-particle, 119-120
resolvent, singularities, 38-40 8-, 68
0, properties, 12-15 two-particle, 120-121 weighted, 232
Ρ Pais doublets, 183-187 Parity
of K+ and K°, 184 violation, 218-219 Particles
and antiparticles, 20-26
« degenerate », 174, 176-179 electromagnetic properties, 22-24 elementary,
diatomic-molecular model, 217 ff global symmetry, 217-228
beta-decay, 223-227
fundamental assumptions, 217- 223
many-, systems, 229-240 physical V-, 51-52
simple with mass =£0, 173-174 stable, 20-22
axiomatic approach, 37 unstable, 24-26
complex poles of the propagators, 37-45
decay of, 219-220
description in quantum field theory, 47-66
time-graph of, 65
Partie-finie integral, Hadamard con
cept, 126 Pauli principle, 115
Pekar's product ansatz, 285-288 Perturbation
theoretical treatment of propaga
tors, 40
theory, difficulties for the Fermi gas, 237-240
Phonon excitation energies, 268 Photon field, 314-316
Photoproduction of charged pion, 103 K+ 104
Pion,
charged, photoproduction, 103 Polarization measurements, 193-194 Polaron
Hamiltonian, 285 model, 285-304 Pole, existence, 57-59 Poles
complex of the propagators, 37-45 spurious of the propagators, 62-65 Potential
Gibbs, 248-251 scattering, 43-44
one-dimensional, 89 shielded, 259-265 Propagator
analytical continuation, 55-57 in more general field theory, 59-65 complex poles of, 37-45
definition of unstable V-particle, 54-59
general properties, 54-55 Lehmann definition, 54, 59 in quantum electrodynamics, 27-35 perturbation theoretical treatment,
40
relation between scattering ampli
tudes, 59-62 spurious poles, 62-65
Q Quantization
second, 113-122 of spin-zero field, 2-3 Quantum
electrodynamics, gauge transforma
tion of propagators, 27-35 field theory, description of unstable
particles, 47-66
mechanical system, fixing, 200-201 mechanics, symmetry operations in,
199-209 R
Radiations, successive, correlations in, 195-197
Kays
defining, 201-202 unit, 201-202
Kay transformations, 203
Reactions, nuclear, no polarization, 191-193
Reduction technique, 75-76 Eeflection
R, 174-175
significance, 181-182 strong, 16-17
Regularization, 123-126
[Regularity points, real, of domain RN, 132-137
Renormalization, 123-126 of charge, 48
of coupling constant, 57-59 Hamiltonian, 49-50
of mass, 48, 57-59
of ultraviolet divergences, 123-124 V-field, 50
Kepresentation up to a factor, 209 Resolvent
operator
matrix elements, 40 singularities, 38-40 technique, 233-237 r-function, 75
connection with Wightman func
tion, 163-172
Rotation generator, infinitesimal, 174 .R-products, 75
Ruelle's theorem, 134 S Scattering
amplitudes
in a given charge state, 97 of decay products, 59-62 forward, 52
properties of 7i(z), 53-54
properties in Lee model, 49-54 singlet at 90°, 108-110
elastic two-particle, 88
forward, extrapolation procedure, 105-107
matrix T, 107 potential, 43-44
one-dimensional, 89
SUBJECT INDEX proton-neutron, forward dispersion
relation, 104 states, 52-53 Schur's lemma, 173 if Screwon, 217
« Shape » parameter P, 108 Schrodinger equation, second quan
tized, 310-311 Slater determinant, 114 Space
configuration, wave function, 113- 114
dual, 116-117 Hilbert, 67, 68, 71
mapping, 10
multilinear algebra over, 305-316 transformations, 203
Spin
connection with statistics, 1-9, 137- 139
conservation, isobaric, 184
invariance under sign change of field operators, 7-8
zero
commutation relations, 3 field quantization, 2-3 Spinor
complex, 181 Majorana, 173, 179 States
bound, 67-74
axiomatic method, 69-72 semiaxiomatic method, 72-74
^-matrix method, 67-69 many-fold of, 21
State vector, time dependence, 15 Statistics
Bose, 1
connection with spin, 1-9, 137-139 Fermi, 1
Strangeness, 21
-violating interactions, 183 Strong coupling
limits, variational method, 296-298 Bogoliubov-Tiablikov, 290-296 Pekar's product ansatz, 285-288 Subspaces, superselection, 205 Sum, weighted, 232
Superconductivity
ground state vector, 276
temperature dependence, 281-282 theory, 275-284
Symmetry global, 185
of elementary particles, 217-228 operations in quantum mechanics,
199-209 Τ
TCP theorem, 1-26, 137-139 proof, 15-19
Temperature
dependence of superconductivity, 281-282
dependent excitations, energy, 283 Tensor rank, 18
/-function, 75 Time
dependence of state vector, 15 graph of an unstable particle, 65 reversal invariance
beta-decay, 194-195
correlations in successive radia
tions, 195-197
nuclear reactions, no polariza
tion, 191-193
polarization measurements, 193- 194
reversal in nuclear forces, 189-197 T-process, 223-224
T-products, 75 Transformation
of the expansion of Gibbs potential, 251-255
gauge, of propagators in quantum electrodynamics, 27-35 in Hilbert space, 203 Klein, 8
of observables, 204-205 ray, 203
TCP, 12-13, 14
0-0, properties of, 203-207 Translational invariance, 288
U Unitarity
^-matrix, 55-56
#-operator, 68
V Vacuum
expectation values, 4
analyticity domain, 147-162 physical, 13
Vectors, state, 201-202
algebraic representation, 114- 116
V-field renormalization, 50 Virial coefficient, second, 261-262 V-particle
condition for existence, 51 physical, 51-52
unstable, 54-59
W Ward's identity, 21
Wave functions, configuration space, 113-114
Weight function, Lehmann, 63, 64 Wick
product, 17
theorem, generalized, 245-248 Wightman functions, 127-145
connection between r-functions, 163-172
Y Y-process, 223-224
