Path Integrals in Physics: Volume I Stochastic Processes and Quantum Mechanics

by Chaichian; M

• ISBN13: 9780750308014
• ISBN10: 075030801X
• Edition: 1st
• Format: Hardcover
• Copyright: 2001-07-01
• Publisher: CRC Press

Summary

Comprehensive, detailed, and systematic, this volume presents the fundamentals of path integrals, both the Wiener and Feynman type, and their many applications in physics. Accessible to a broad community of theoretical physicists, the book deals with systems possessing a infinite number of degrees in freedom. It discusses the general physical background and concepts of the path integral approach used, followed by a detailed presentation of the most typical and important applications as well as problems with either their solutions or hints how to solve them. The book describes in detail various applications, including systems with Grassmann variables.

Table of Contents

Preface	ix
Introduction	1	(8)
Notational conventions	9	(3)
Path integrals in classical theory	12	(110)
Brownian motion: introduction to the concept of path integration	12	(44)
Brownian motion of a free particle, diffusion equation and Markov chain	12	(10)
Wiener's treatment of Brownian motion: Wiener path integrals	22	(7)
Wiener's theorem and the integration of functionals	29	(7)
Methods and examples for the calculation of path integrals	36	(9)
Change of variables in path integrals	45	(4)
Problems	49	(7)
Wiener path integrals and stochastic processes	56	(66)
A short excursion into the theory of stochastic processes	56	(7)
Brownian particles in the field of an external force: treatment by functional change of variables in the path integral	63	(3)
Brownian particles with interactions	66	(3)
Brownian particles with inertia: a Wiener path integral with constraint and in the space of velocities	69	(3)
Brownian motion with absorption and in the field of an external deterministic force: the Bloch equation and Feynman-Kac formula	72	(6)
Variational methods of path-integral calculations: semiclassical and quadratic approximations and the method of hopping paths	78	(16)
More technicalities for path-integral calculations: finite-difference calculus and Fourier decomposition	94	(7)
Generating (or characteristic) functionals for Wiener integrals	101	(7)
Physics of macromolecules: an application of path integration	108	(3)
Problems	111	(11)
Path integrals in quantum mechanics	122	(196)
Feynman path integrals	123	(30)
Some basic facts about quantum mechanics and the Schrodinger equation	123	(14)
Feynman-Kac formula in quantum mechanics	137	(4)
Properties of Hamiltonian operators from the Feynman-Kac formula	141	(3)
Bohr-Sommerfeld (semiclassical) quantization condition from path integrals	144	(5)
Problems	149	(4)
Path integrals in the Hamiltonian formalism	153	(47)
Derivation of path integrals from operator formalism in quantum mechanics	154	(7)
Calculation of path integrals for the simplest quantum-mechanical systems: a free particle and a harmonic oscillator	161	(8)
Semiclassical (WKB) approximation in quantum mechanics and the stationary-phase method	169	(7)
Derivation of the Bohr-Sommerfeld condition via the phase-space path integral, periodic orbit theory and quantization of systems with chaotic classical dynamics	176	(7)
Particles in a magnetic field: the Ito integral, midpoint prescription and gauge invariance	183	(4)
Applications of path integrals to optical problems based on a formal analogy with quantum mechanics	187	(3)
Problems	190	(10)
Quantization, the operator ordering problem and path integrals	200	(30)
Symbols of operators and quantization	200	(9)
General concept of path integrals over trajectories in pahse space	209	(7)
Normal symbol for the evolution operator, coherent-state path integrals, perturbation expansion and scattering operator	216	(10)
Problems	226	(4)
Path integrals and quantization in spaces with topological constraints	230	(15)
Point particles in a box and on a half-line	231	(7)
Point particles on a circle and with a torus-shaped phase space	238	(5)
Problems	243	(2)
Path integrals in curved spaces, spacetime transformations and the Coulomb problem	245	(41)
Path integrals in curved spaces and the ordering problem	245	(6)
Spacet ne transformations of Hamiltonians	251	(7)
Path integrals in polar coordinates	258	(8)
Path integral for the hydrogen atom: the Coulomb problem	266	(6)
Path integrals on group manifolds	272	(10)
Problems	282	(4)
Path integrals over anticommuting variables for fermions and generalizations	286	(32)
Path integrals over anticommuting (Grassmann) variables for fermionic systems	286	(12)
Path integrals with generalized Grassmann variables	298	(6)
Localization techniques for the calculation of a certain class of path integrals	304	(11)
Problems	315	(3)
Appendices	318	(10)
A General pattern of different ways of construction and applications of path integrals	318	(1)
B Proof of the inequality used for the study of the spectra of Hamiltonians	318	(4)
C Proof of lemma 2.1 used to derive the Bohr-Sommerfeld quantization condition	322	(4)
D Tauberian theorem	326	(2)
Bibliography	328	(5)
Index	333