From Classical Mechanics to Quantum Field Theory

242 From Classical Mechanics to Quantum Field Theory. A Tutorial

point spectrum, 105
residual spectrum, 105
standard model, 191, 209
states, 4
algebraic, 182
entangled, 13
minimal uncertainty relations, 31
mixed, 8, 10, 144
pure, 7, 9, 144
quantum - as measure, 140
quasi-classical, 31
separable, 13
space of, 7
stationary, 14
strongly continuous
one-parameter unitary group, 164
unitary representation of a Lie group,
171
superposition, 8, 22, 144
superselection
charges, 152
rules, 152
sector, 153
Symanzik, 225
symplectic
form, 6, 20, 22, 27, 52
manifold,6,7,47
transformation, 53

tangent space/bundle, 17
theorem
Bargmann criterion, 163
Bell-Kochen-Speker theorem, 143
closed graph theorem, 96
Coleman-Mermin-Wagner theorem,
224
double commutant theorem, 147
Gleason theorem, 142
GNS theorem, 184
Goldstone theorem, 224
Hellinger-Toepliz theorem, 98
Kadison theorem, 159
Nelson criterion for essentially
selfadjointness, 101

Nelson theorem, 175

Noether quantum theorem, 168

Pauli theorem, 180

spectral representation theorem for

selfadjoint operators, 118

spectral theorem for selfadjoint

operators, 114

Stone - von Neumann - Mackey

theorem, 179

Stone theorem, 166

Wick theorem, 216, 227, 232, 233

Wigner theorem, 159

topology

strong operator, 125

uniform operator, 125

weak operator, 125

two-level system, 8, 11

unitarity, 200

vacuum, 11

energy, 192, 206–210, 216, 217, 230,

231

quantum, 195, 206

vector field, 18, 19, 26

constant, 18

dilation, 18

gradient, 25

Hamiltonian, 25

von Neumann algebra, 148

center of, 152

of observables, 150

Weyl

algebra, 5

map/system, 48, 50, 52

quantization, 47

Wick theorem, 215

Wightman functions, 220, 225

Wigner, 199

map, 57, 58, 61

symmetry, 159

Wilson’s renormalization group, 190, 229