The book gives an overview of path integral quantization and stochastic quantization of classical mechanics and field theory. The non-Abelian gauge field, the gravitational field, and the path integral representation of quantum statistical mechanics and stochastic quantization are described.

Klappentext

In this book, we discuss the path integral quantization and the stochastic quantization of classical mechanics and classical field theory. For the description of the classical theory, we have two methods, one based on the Lagrangian formalism and the other based on the Hamiltonian formal ism. The Harniltonian formalisni is derived from the Lagrangian formalism. In the standard formalism of quantum mechanics, we usually make use of the Hamiltonian formalism. This fact originates from the following circumstance which dates back to the birth of quantum mechanics. The first formalism of quantum mechanics is Schrodinger's wave mechan ics. In this approach, we regard the Hamilton Jacobi equation of analytical mechanics as the Eikonal equation of "geometrical mechanics". Bsed on the optical analogy, we obtain the Schrodinger equation as a result of the inverse of the Eikonal approximation to the Hamilton Jacobi equation, and thus we arrive at "wave mechanics" . The second formalism of quantum mechanics is Heisenberg's "matrix me chanics". In this approach, we arrive at the Heisenberg equation of motion frorn consideration of the consistency of the Ritz combination principle, the Bohr quantization condition and the Fourier analysis of a physical quantity. These two forrnalisrns make up the Hamiltonian formalism of quantum me chanics.

Inhalt

Path Integral Representation of Quantum Mechanics.- Path Integral Representation of Quantum Field Theory.- Path Integral Quantization of Gauge Field.- Path Integral Representation of Quantum Statistical Mechanics.- Stochastic Quantization.