On the physical and mathematical foundations of quantum physics via functional integrals

TL;DR

This paper develops a rigorous mathematical framework for quantum field theory using functional integrals, Banach spaces, and the Henstock-Kurzweil integral to better understand the foundations of quantum physics.

Contribution

It introduces a new Hilbert space and measure construction that rigorously defines Feynman path integrals and incorporates time as a universal parameter called historical time.

Findings

Constructed a measure on separable Hilbert spaces for path integrals.

Defined a universal historical time parameter for quantum evolution.

Extended Feynman calculus to include time-dependent groups and semigroups.

Abstract

In order to preserve the leading role of the action principle in formulating all field theories one needs quantum field theory, with the associated BRST symmetry, and Feynman-DeWitt-Faddeev-Popov ghost fields. Such fields result from the fibre-bundle structure of the space of histories, but the physics-oriented literature used them formally because a rigorous theory of measure and integration was lacking. Motivated by this framework, this paper exploits previous work of Gill and Zachary, where the use of Banach spaces for the Feynman integral was proposed. The Henstock-Kurzweil integral is first introduced, because it makes it possible to integrate functions like the exponential of (i times x squared). The Lebesgue measure on R(infinity) is then built and used to define the measure on every separable Hilbert space. The subsequent step is the construction of a new Hilbert space KS2, which contains square-integrable functions on R**n as a continuous dense embedding, and contains both the test functions and their dual, the Schwartz space of distributions, as continuous embeddings. This space allows us to construct the Feynman path integral in a manner that maintains its intuitive and computational advantages. We also extend this space to KS2[H], where H is any separable Banach space. Last, the existence of a unique universal definition of time, tau(h), that we call historical time, is proved. We use tau(h) as the order parameter for our construction of Feynman's time ordered operator calculus, which in turn is used to extend the path integral in order to include all time dependent groups and semigroups with a kernel.