Contents 1 Notation and mathematical preliminaries 5 1.1 Generalized functions(distributions) . 5 1.2 Functional differentiation . 5 1.3 Gaussian integration . 5 1.4 Groups and their representations .
5 1.5 Exercises . 5 2 Basic notions of the scalar field theory 7 2.1 Classical field theory. Lagrange equations and the Noether theorem . 7 2.2 Classical scalar free field . 7 2.
3 Quantization of the scalar field . 7 2.4 The Poincare group and its representations . 7 2.5 Functional representation of quantum fields . 7 2.6 Exercises . 7 3 Interacting fields and scattering amplitudes 9 3.
1 Interaction picture:correlation functions . 10 3.2 Gell-Mann-Low formula . 10 3.3 The integral kernel of an operator . 10 3.4 Momentum representation . 10 3.
5 Coupling constant renormalization . 10 3.6 Euclidean correlation functions . 10 3.7 A dimensional regularization . 10 3.8 Generating functional: a perturbative formula . 10 3.
9 The Euclidean quantum field theory: Osterwalder-Schrader formulation . 10 3.10 Heisenberg picture: the asymptotic fields . 10 3.11 Reduction formulas . 10 3.12 Exercises . 10 4 Thermal states and quantum scalar field on a curved manifold 11 4.
1 Fields at finite temperature . 11 4.2 Scalar free field on a globally hyperbolic manifold . 11 4.3 Exercises . 11 5 The functional integral 13 5.1 Trotter product formula and the Feynman integral . 13 5.
2 Evolution for time-dependent Hamiltonians . 13 5.3 The Wiener integral and Wiener-Feynman integral . 13 5.4 The stochastic integral: the Feynman integral for a particle in an electromagnetic field . 13 5.5 Solution of stochastic equations . 13 5.
6 Exercises . 13 6 Feynman integral in terms of the Wiener integral 15 6.1 Feynman-Wiener integral for polynomial potentials . 15 6.2 Feynman-Wiener integral for potentials which are FourierLaplace transforms of a measure . 15 6.3 Functional integration in terms of oscillatory paths in QFT 15 6.4 Wiener-Feynman integration in two-dimensional QFT .
15 6.5 Exercises . 15 7 Application of the Feynman integral for approximate calculations 17 7.1 Semi-classical expansion:the stationary phase method . 18 7.2 Stationary phase for an anharmonic oscillator . 18 7.3 The loop expansion in QFT .
18 7.4 The saddle point method:the loop expansion in Euclidean field theory . 18 7.5 Effective action . 18 7.6 Determinants of differential operators . 18 7.7 The functional integral for Euclidean fields at finite temperature .
18 7.8 Exercises . 18 8 Feynman path integral in terms of expanding pathsâ 19 8.1 Expansion around a particular solution . 20 8.2 An example:the upside-down oscillator . 20 8.3 Solution in the Heisenberg picture .
20 8.4 Quantum mechanics at an imaginary time . 20 8.5 Paths at imaginary time as Euclidean fields . 20 8.6 Free field on a static manifold . 20 8.7 Time-dependent Gaussian state in quantum field theory .
20 8.8 Free field in an expanding universe . 20 8.9 Free field in De Sitter space . 20 8.10 Interference of classical and quantum waves . 20 8.11 Exercises .
20 9 An interaction with a quantum electromagnetic field 21 9.1 Functional integral quantization of the electromagnetic field 22 9.2 The Abelian Higgs model . 22 9.3 Euclidean version:the polymer representation . 22 9.4 One-loop determinant in the Abelian Higgs model:a nonperturbative method . 22 9.
5 Non-relativistic QED:a charged particle interacting with quantum electromagnetic field . 22 9.6 Heisenberg equations of motion for a particle in QED environment . 22 9.7 Squeezed states in QED . 22 9.8 Noise in the squeezed state . 23 9.
9 Feynman formula in QED with an axion . 23 9.10 Exercises . 23 10 Particle interaction with gravitonsâ 25 10.1 Quantum geodesic deviation . 25 10.2 Heisenberg equations of motion . 25 10.
3 Stochastic deviation equations in the thermal environment . 25 10.4 Exercises . 25 11 Quantization of non-Abelian gauge fields 27 11.1 Non-Abelian gauge theories . 27 11.2 The Non-Abelian Higgs model:symmetry breaking and mass generation . 27 11.
3 The effective scalar field action in non-Abelian gauge field . 27 11.4 Fadeev-Popov procedure . 27 11.5 The background field method . 27 11.6 The effective action in non-Abelian gauge theories . 27 11.
7 Exercises . 27 12 Lattice approximation 29 12.1 Lattice approximation in Euclidean scalar field theory . 29 12.2 Lattice approximation in gauge theories . 29 12.3 Exercises . 29 13 Bibliography 31 14 The index 33.