From Classical Mechanics to Quantum Field Theory

Index 241

number operator, 12, 16, 28, 30, 35, 104

observables
classical, 4
maximal set of compatible, 151
quantum, 4, 64
space of, 7, 9
operator, 19, 24, 25, 40
adjoint, 92
bounded, 89
closable, 94
closed, 94
closure, 94
creation/annihilation, 11, 12, 26, 29,
33, 35, 37, 38, 104
density matrix, 8
essentially selfadjoint, 97
Fourier-Plancherel, 100
Hermitean, 97
momentum, 5, 10, 15, 102
normal, 97
position, 5, 10, 15, 102
projection, 8
range of, 88
selfadjoint, 97
standard domains of composition, 88
symmetric, 97
trace of, 137
trace-class, 137
unitary, 97
orthomodularity, 135
orthonormal set of vectors, 87
Osterwalder-Schrader, 223
positivity, 224, 226

partition function, 36, 43, 46, 47, 60
path integral, 39
in coordinate space, 42
in imaginary time, 43
with coherent states, 44
Pauli matrices, 11
Peierls brackets, 220, 234–236
phase space, 4, 6, 47, 55
Poincar ́e
group, 197, 199, 200
invariance, 201, 204

symmetry, 223, 227

Poisson

bracket, 5, 24, 194, 202, 203, 205, 235,

236

tensor, 23, 62

probability, 8, 16, 30

amplitude, 144

density, 9, 17, 40

measure, 7

transition, 144

projective representation, 48, 162

quantization, 4, 6, 202, 215

canonical, 5, 15, 201

covariant, 220

geometric, 6

path integral, 39

second, 2

quantum field theory, 191, 195, 200, 208,

211, 214, 215, 221

reflection positivity, 222

relativity, 193, 197, 199

resolution of the identity, 33, 37–40, 45,

108

Riemann, 192

Riemannian

form, 20

metric, 22

tensor, 23

scalar field, 201, 208, 224

Schr ̈odinger, 2

equation, 14, 19, 167

lagrangian, 46

representation, 196, 205

Schwinger, 190

formula, 227

function, 224, 225

selfadjoint generator, 166, 171

separability, 135

spectral

condition, 200

decomposition, 4, 9, 17, 114

spectrum, 105

continuous spectrum, 105