Recent Results in Infinite Dimensional Analysis and Applications to Feynman Integrals

TL;DR

This thesis develops a comprehensive framework for infinite dimensional analysis, including non-Gaussian and Gaussian cases, and applies it to rigorously construct Feynman integrals, advancing mathematical understanding of quantum path integrals.

Contribution

It introduces a unified approach to infinite dimensional analysis, extending to non-Gaussian measures, and provides a rigorous construction of Feynman integrals using White Noise analysis.

Findings

Developed a general theory for non-Gaussian analysis including Poissonian measures.

Constructed Feynman integrals rigorously using White Noise distributions.

Proposed a generalized method for interacting potentials in quantum mechanics.

Abstract

The first part of this thesis proposes a general approach to infinite dimensional non-Gaussian analysis, including the Poissonian case. In particular distribution theory is developed. Using appropriate integral transformations, generalized and test functionals are characterized in terms of holomorphy. Furthermore differential operators, Wick product and change of measure are discussed. In the second part the Gaussian case (White Noise Analysis) is worked out in more detail. Furthermore operators on distribution spaces e.g. compositions with shifts and complex scaling are discussed. In the third part Feynman integrals are constructed using White Noise distributions as integrands. Its expectation yields the path integral. This rigorous approach is applied to the interacting case. A generalization of the Khandekar Streit method is proposed. The resulting class of admissible potentials covers signed measures. The Albeverio Hoegh-Krohn class, which consists of Fourier transforms of measures, is discussed. The third approach is based on complex scaling. The so-called Doss class allows analytic potentials which obey some growth condition. Using the White Noise calculus of differential operators, the functional form of the canonical commutation relation is derived. Finally Ehrenfest's theorem is proven.