Lectures on Quantum Field Theory and Functional Integration

ISBN-13: 9783031307119 / Angielski

Zbigniew Haba

Spis treści

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 paths∗ 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 gravitons∗ 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

O autorze

Born in 1951, Zbigniew Haba obtained his Ph.D. degree at the University of Wroclaw in 1976, where he then became an assistant professor. Since 1995, he holds a full professorship.

He has spent long periods visiting the Department of Physics of Bielefeld University, and also had research stays in the Department of Mathematics, Bochum University,1988, and the Max Planck Institute, Munich. In 1993, he was the Gulbenkian fellow at Lisbon University, and the visiting professor in Freie Universitaet Berlin in 2000.

He is the author of a monograph "Feynman integral and random dynamics in quantum physics. A probabilistic approach to quantum dynamics", Kluwer/Springer, 1999.

Blurb

This book offers a concise introduction to quantum field theory and functional integration for students of physics and mathematics. Its aim is to explain mathematical methods developed in the 1970s and 1980s and apply these methods to standard models of quantum field theory. In contrast to other textbooks on quantum field theory, this book treats functional integration as a rigorous mathematical tool. More emphasis is placed on the mathematical framework as opposed to applications to particle physics. It is stressed that the functional integral approach, unlike the operator framework, is suitable for numerical simulations. The book arose from the author's teaching in Wroclaw and preserves the form of his lectures. So some topics are treated as an introduction to the problem rather than a complete solution with all details. Some of the mathematical methods described in the book resulted from the author's own research.