Michio Masujima Livres

All there is to know about functional analysis, integral equations and calculus of variations in a single volume. This advanced textbook starts with a short introduction to functional analysis, including a review of complex analysis, before continuing a systematic discussion of different types of integral equations. After a few remarks on the historical development, the second part gives an introduction to the calculus of variations and the relationship between integral equations and applications of the calculus of variations. It further covers applications of the calculus of variations developed in the second half of the 20th century in the fields of quantum mechanics, quantum statistical mechanics and quantum field theory.

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.