From Classical Mechanics to Quantum Field Theory

240 From Classical Mechanics to Quantum Field Theory. A Tutorial

time, 220, 221
transformations, 222
Euler-Lagrange, 193, 201
Euler-MacLaurin formula, 230
evolution, 3, 7, 14, 19, 53
kernel, 39
operator, 14, 167
expectation value, 9, 31, 32, 73, 122, 146

factor, 150, 152
fermions, 11, 38, 44, 46
Feynman, 3, 39
Fock
basis, 12, 27, 30, 33
space, 37, 38, 212, 214, 220
Fourier
modes, 208, 230
symplectic transform, 55
transform, 10, 51, 81, 99, 236
free particle, 14, 42, 49, 53

G ̊arding space, 172
Gateaux derivative, 205
gauge group, 155
Grassmann variables, 38, 39, 47
gravitation, 228

Hamiltonian, 14, 19, 76, 103, 116, 167,
192–197, 202–206, 212, 214, 215, 217,
235
harmonic oscillator, 15, 36, 43, 46, 54, 60,
103
heat kernel, 207, 208, 210
Heisenberg, 190
law, 218
representation, 196, 218
Heisenberg-Weyl
algebra, 26
group, 27, 29, 36, 180
Hermite polynomials, 196, 197, 212
Hermitean
form, 4, 19, 20
product, 17, 23, 72, 85
Hilbert space, 6, 13, 85, 193, 200, 205,
211, 214, 233
projective, 21

Hilbertian basis, 87

infrared divergences, 204, 207

irreducibility, 135

Jordan bracket, 24

K ̈ahler manifold, 4, 21–23

Kadison symmetry, 159

Klein-Gordon equation, 199

lattice, 129

commuting elements in a, 136

homomorphism, 129

orthogonal elements in a, 136

Lie

algebra, 6, 26, 171

bracket, 24

derivative, 18, 52

group, 163

Lie-Jordan algebra, 25

local causality, 201, 219, 220

Lorentz

invariance, 218

symmetry, 209

transformations, 223

Maxwell equations, 199

measure

Borelian probability, 233

Gaussian, 211, 212, 215, 225, 227, 231

joint spectral, 119

Lebesgue, 211

positive operator valued, 16, 180

process, 7, 13

projection operator valued, 108

projection valued, 17

Riemann-Stieltjes, 227

Minkowski

metric, 223

space-time, 198, 201, 219

Moyal

bracket, 63

product, 61

multiplier, 162