The aim of this book is to present and explain the concept of the path integral which is intensively used nowadays in almost all the branches of theoretical physics. The notion of path integral (sometimes also called functional integral or integral over trajectories or integral over histories or continuous integral) was introduced, for the first time, in the 1920s by Norbert Wiener (1921, 1923, 1924, 1930) as a method to solve problems in the theory of diffusion and Brownian motion. This integral, which is now also called the Wiener integral, has played a central role in the further development of the subject of path integration.
The aim of this book is to present and explain the concept of the path integral which is intensively used nowadays in almost all the branches of theoretical physics. The notion of path integral (sometimes also called functional integral or integral over trajectories or integral over histories or continuous integral) was introduced, for the first time, in the 1920s by Norbert Wiener (1921, 1923, 1924, 1930) as a method to solve problems in the theory of diffusion and Brownian motion. This integral, which is now also called the Wiener integral, has played a central role in the further development of the subject of path integration.
Introduction Preliminary remarks The path integral formalism is one of the most important tools in modern theoretical physics. Integration in functional spaces was considered for the first time by Norbert Wiener in his studies of diffusion processes [60, 61]. The corresponding integration measure is now known as the Wiener measure, i.e. the so-called Gaussian path integrals were studied. In quantum mechanics, functional integrals were introduced by R. Feynmann [62, 63] who was inspired by important observations due to P. A. M. Dirac concerning a relation between wave function and classical action [64, 65]. In modern physics the path integral formalism is applied not only to developing a quantum description for systems with infinitely many degrees of freedom as, for example, in quantum field theories or condensed matter physics, but also to ordinary quantum mechanical problems. Moreover there is a tendency to reformulate standard quantum mechanical problems (such as a particle in a “box” or potential well [66, 67], or a particle in the Coulomb field [68]) within the framework of the path integral method. The reason for doing this is rather obvious: physicists develop a new mathematical tool, a new language and, in order to comprehend all of its subtleties, it is necessary to investigate as many concrete examples as possible. Use of the path integral formalism is widespread in quantum field theory because it is the most adequate mathematical tool for a wide range of problems which allows, not only for elegant formulation of problems, but also for a great deal of simplification of their theoretical analysis.
The aim of this chapter is to present and to discuss the general concept and mathematical structure of path integrals, introduced for the first time by N Wiener (1921, 1923, 1924, 1930), as a tool for solving problems in the theory of classical systems subject to random influences from the surrounding medium. The most famous and basic example of such a system is a particle performing the so-called Brownian motion. This phenomenon was discovered in 1828 by the British botanist R Brown, who investigated the pollen of different plants dispersed in water. Later, scientists realized that small fractions of any kind of substance exhibit the same behaviour, as a result of random fluctuations driven by the medium. The theory of Brownian motion emerged in the beginning of the last century as a result of an interplay between physics and mathematics and at present it has a wide range of applications in different areas, e.g., diffusion in stellar dynamics, colloid chemistry, polymer physics, quantum mechanics.
One usually refers the concept of Feynman path integral to the work of Norbert Wiener on Brownian motion in the early 1920s. This view is not false and we show in this article that Wiener used the first path integral of the history of physics to describe the Brownian motion. That said, Wiener, as he pointed out, was inspired by the work of some French mathematicians, particularly Gateaux and Levy. Moreover, although Richard Feynman has independently found this notion, we show that in the course of the 1930s, while searching a kind of geometrization of quantum mechanics, another French mathematician, Adolphe Buhl, noticed by the philosopher Gaston Bachelard, had himself been close to forge such a notion. This reminder does not detract from the remarkable discovery of Feynman, which must undeniably be attributed to him. We also show, however, that the difficulties of this notion had to wait many years before being resolved, and it was only recently that the path integral could be rigorously established from a mathematical point of view.
This book is a self-contained and concise introduction to the techniques and applications of path integral quantization and functional techniques, aimed at students and practitioners. The first half of the text focuses on quantum mechanics, including a review of the action formulation of classical mechanics and quantum mechanics in the Dirac operator and state formalism, and further examination of the path integral. The second part examines relativistic field theories, reviewing special relativity, as well as derivation of the path integral representation of the vacuum transition element for quantized scalar, spinor, and vector fields from the coherent state representation of the respective field theories
Hilbert-space techniques are widely used not only for quantum theory, but also for classical physics. Two important examples are the Koopman-von Neumann (KvN) formulation and the method of ``classical'' wave functions. As this paper explains, these two approaches are conceptually distinct. In particular, the method of classical wave functions was not due to Bernard Koopman and John von Neumann, but was developed independently by a number of later researchers, perhaps first by Mario Schönberg, with key contributions from Angelo Loinger, Giacomo Della Riccia, Norbert Wiener, and E. C. George Sudarshan. The primary goals of this paper are to explain these two approaches, describe the relevant history in detail, and give credit where credit is due.