Index
285
Localized gauge transformations, 268
Local observables, 95
Local structure, 22, 95
Local Weyl algebra, 270
Long range dynamics, 225
Lower bound condition, 20
M
Mean ergodic limit, 106
Molecular field approximation, 231
N
Noether charge, 37
Noether current, 38
Noether second theorem, 253
Noether theorem, 36
Noether theorem and symmetry breaking, 37
Non-abelian Higgs mechanism, 257
absence of physical Goldstone bosons,
260
Non-Fock representation, 87
Non-linear elliptic problem, 59
Non-regular representations, 270
O
Observable, 70
P
Phase, 6, 21, 77
pure, 148
Physically realizable operations, 97
Plasmon energy gap, 241
Plasmon energy spectrum, 237
Plasmon excitations, 241
Plasmons as Goldstone excitations, 241
Q
QCD
U (1) problem, 263
Lagrangian, 263
Quantum particle on a circle, 279
Quasi-local algebra, 97, 226
S
Schwinger model, 279
Sector, 22
physical, 32, 60
Seizing of the vacuum, 230
Semi-classical instanton approximation, 266
Sine-Gordon equation, 44
Soliton, 45
Spectral condition, 101
Spin systems, 111
short range interaction, 111
Spontaneous symmetry breaking, 7
Stability group of a sector, 35
Stability under space translations, 31, 97
Stability under time evolution, 2, 97
State
mixed, 79
pure, 79
Swieca condition, 176
Symmetries of classical systems, 5
Symmetry, 5
internal, 13
local generation, 174
Symmetry breaking
vacuum seizing, 229
Symmetry breaking axial in QCD
examples, 45
non perturbative Ruelle strategy, 43
Symmetry breaking criterion
Bogoliubov non-perturbative strategy,
136
Goldstone, 133
Symmetry breaking in gauge theories, 203
Symmetry breaking in quantum systems,
124, 125
Symmetry breaking in superconductivity,
172
Symmetry breaking order parameter, 127
T
Temporal gauge, 268
physical states, 270
residual gauge group, 268
subsidiary condition, 270
Thermal states, 147
θ angle, 266
θ sectors, 277
θ vacuum, 267, 275, 277
Topological current, 272
Topological group, 275
Topological term, 266
Transfer matrix, 143
Translationally invariant state,uniqueness,
103
U
Unitary implementation of symmetries, 37
285
Localized gauge transformations, 268
Local observables, 95
Local structure, 22, 95
Local Weyl algebra, 270
Long range dynamics, 225
Lower bound condition, 20
M
Mean ergodic limit, 106
Molecular field approximation, 231
N
Noether charge, 37
Noether current, 38
Noether second theorem, 253
Noether theorem, 36
Noether theorem and symmetry breaking, 37
Non-abelian Higgs mechanism, 257
absence of physical Goldstone bosons,
260
Non-Fock representation, 87
Non-linear elliptic problem, 59
Non-regular representations, 270
O
Observable, 70
P
Phase, 6, 21, 77
pure, 148
Physically realizable operations, 97
Plasmon energy gap, 241
Plasmon energy spectrum, 237
Plasmon excitations, 241
Plasmons as Goldstone excitations, 241
Q
QCD
U (1) problem, 263
Lagrangian, 263
Quantum particle on a circle, 279
Quasi-local algebra, 97, 226
S
Schwinger model, 279
Sector, 22
physical, 32, 60
Seizing of the vacuum, 230
Semi-classical instanton approximation, 266
Sine-Gordon equation, 44
Soliton, 45
Spectral condition, 101
Spin systems, 111
short range interaction, 111
Spontaneous symmetry breaking, 7
Stability group of a sector, 35
Stability under space translations, 31, 97
Stability under time evolution, 2, 97
State
mixed, 79
pure, 79
Swieca condition, 176
Symmetries of classical systems, 5
Symmetry, 5
internal, 13
local generation, 174
Symmetry breaking
vacuum seizing, 229
Symmetry breaking axial in QCD
examples, 45
non perturbative Ruelle strategy, 43
Symmetry breaking criterion
Bogoliubov non-perturbative strategy,
136
Goldstone, 133
Symmetry breaking in gauge theories, 203
Symmetry breaking in quantum systems,
124, 125
Symmetry breaking in superconductivity,
172
Symmetry breaking order parameter, 127
T
Temporal gauge, 268
physical states, 270
residual gauge group, 268
subsidiary condition, 270
Thermal states, 147
θ angle, 266
θ sectors, 277
θ vacuum, 267, 275, 277
Topological current, 272
Topological group, 275
Topological term, 266
Transfer matrix, 143
Translationally invariant state,uniqueness,
103
U
Unitary implementation of symmetries, 37