Functional Integrals in Quantum Field Theory and Statistical Physics

Functional integration is one of the most powerful methods of contempo rary theoretical physics, enabling us to simplify, accelerate, and make clearer the process of the theoretician's analytical work. Interest in this method and the endeavour to master it creatively grows incessantly. This book presents a study of the application of functional integration methods to a wide range of contemporary theoretical physics problems. The concept of a functional integral is introduced as a method of quantizing finite-dimensional mechanical systems, as an alternative to ordinary quantum mechanics. The problems of systems quantization with constraints and the manifolds quantization are presented here for the first time in a monograph. The application of the functional integration methods to systems with an infinite number of degrees of freedom allows one to uniquely introduce and formulate the diagram perturbation theory in quantum field theory and statistical physics. This approach is significantly simpler than the widely accepted method using an operator approach.

1. Functional Integrals and Quantum Mechanics. - 1. Introduction. - 2. Functional Integrals in Quantum Mechanics. - 3. Quantization of Systems with Constraints. - 4. Functional Integrals and Quantization on Manifolds. - 2. Functional Integrals in Quantum Field Theory and Statistical Physics. - 5. Functional Integrals and Perturbation Theory in Quantum Field Theory. - 6. Functional Integrals and the Temperature Diagram Technique in Statistical Physics. - 3. Gauge Fields. - 7. Quantization of Gauge Fields. - 8. Quantum Electrodynamics. - 9. Yang-Mills Fields. - 10. Quantization of a Gravitational Field. - 11. Attempts to Construct a Gauge-Invariant Theory of Electromagnetic and Weak Interactions. - 4. Infrared Asymptotics of Green s Functions. - 12. Method of Successive Integration Over `Rapid and `Slow Fields. - 13. Infrared Asymptotic Behaviour of Green s Functions in Quantum Electrodynamics. - 5. Scattering of High-Energy Particles. - 14. Double Logarithmic Asymptotics in Quantum Electro-dynamics. - 15. Eikonal Approximation. - 6. Superfluidity. - 16. Perturbation Theory for Superfluid Bose Systems. - 17. Bose Gas of Small Density. - 18. Application of Functional Integrals to the Derivation of Low-Energy Asymptotic Behaviour of the Green s Function. - 19. Hydrodynamical Lagrangian of Nonideal Bose Gas. - 20. Superfluidity of Two-Dimensional and One-Dimensional Bose Systems. - 21. Quantum Vortices in a Bose Gas. - 7. Superconductivity. - 22. Perturbation Theory of Superconducting Fermi Systems. - 23. Superconductivity of the Second Kind. - 24. Bose Spectrum of Superfluid Fermi Gas. - 25. A System of the He3 Type. - 8. Plasma Theory. - 26. Hydrodynamical Action in Plasma Theory. - 27. Damping of Plasma Oscillations. - 9. The Ising Model. - 28. The Statistical Sum of the Ising Model as aFunctional Integral. - 29. The Correlation Function of the Ising Model. - 10. Phase Transitions. - 30. Special Role of Dimension d = 4. - 31. Calculation of Critical Indices and the Wilson Expansion. - 11. Vortex-Like Excitations in Relativistic Field Theory. - 32. Vortices in the Relativistic Goldstone Model. - 33. On Vortex-Like Solutions in Quantum Field Theory. - References.