Mathematical Feynman Path Integrals and Their Applications

The following sections are included:

• Preface to the first edition
• Preface to the second edition
• About the Author
• Contents

One of the most challenging problems of modern physics is the connection between the macroscopic and the microscopic world, that is between classical and quantum mechanics. In principle a macroscopic system should be described as a collection of microscopic ones, so that classical mechanics should be deduced from quantum theory by means of suitable approximations. At a first glance the solution of the problem is not straightforward; indeed there are deep differences between the classical and the quantum description of the physical world…

This chapter presents some basic notions of measure and integration theory. Besides the classical results, we present some notions that will play an important role in the development of Feynman integration theory such as, e.g., the theory of complex measures and their Fourier transform. For additional details we refer to, e.g., [57, 56, 70, 96, 293, 66].

In the present chapter we describe the main topics of infinite dimensional measure and integration theory that are relevant in the mathematical definition of Feynman path integrals…

As explained in Section 3.5 it is impossible to construct a reasonable Feynman measure. In particular, the integrals appearing in the piecewise linear approximations of Feynman formula

In the theory of infinite dimensional Fresnel integrals on real separable Hilbert spaces ℋ described in Section 4.5, one of the main issues is the restriction on the class of integrable functions, in other words the domain of the functional. Indeed, the infinite dimensional Fresnel integral of a mapping f : ℋ→ ℂ

The present chapter is devoted to the application of infinite dimensional oscillatory integrals to the construction of the Feynman path integral representation of the solution of the Schrödinger equation with different potentials.

The present chapter is devoted to the description of an infinite dimensional version of the classical stationary phase method in the framework of infinite dimensional oscillatory integrals. In particular, Section 7.4 presents the application of these techniques to the study of the semiclassical asymptotic behaviour of the solution of Schrödinger equation in the limit ħ ↓ 0.

The following sections are included:

• Feynman path integrals and open quantum systems
• The Feynman-Vernon influence functional
• The stochastic Schrödinger equation

Since the first introduction of Feynman path integrals [138], several approaches for their mathematical definition and several applications have been proposed, both in the mathematical and in the physical literature. The present chapter is a brief survey of this topic, without any claim of completeness.

The following sections are included:

• Bibliography
• Index