Some Comments on Rigorous Finite-Volume Euclidean Quantum Field Path Integrals in the Analytical Regularization Scheme

Luiz C L Botelho

doi:10.1155/2011/257916

Abstract

Through the systematic use of the Minlos theorem on the support of cylindrical measures on , we produce several mathematically rigorous finite-volume euclidean path integrals in interacting euclidean quantum fields with Gaussian free measures defined by generalized powers of finite-volume Laplacian operator.

Highlights

Feynman on representing the initial value solution of Schrodinger Equation by means of an analytically time continued integration on a infinite-dimensional space of functions, the subject of Euclidean Functional Integrals representations for Quantum Systems has became the mathematical-operational framework to analyze Quantum Phenomena and stochastic systems as showed in the previous decades of research on Theoretical Physics 1–3
The purpose of this paper is to present the formulation of Euclidean Quantum Field theories as Functional Fourier Transforms by means of the Bochner-Martin-Kolmogorov theorem for Topological Vector Spaces 4, 5, Theorem 4.35 and suitable to define and analyze rigorously Functional Integrals by means of the well-known Minlos theorem 5, Theorem 4.312 and 6, part 2 which is presented in full in Appendix A
2.39 with a function f x ∈ L2 RN and any given ε > 0, even if originally all fields configurations entering into the path integral were elements of the Schwartz Tempered Distribution Spaces S RN certainly very “rough” mathematical objects to characterize from a rigorous geometrical point of view

Summary

Introduction

It is worth calling the reader attention that due to the infrared regularization introduced on 2.1a , the domain of the Gaussian measure 4, 6 is given by the space of square integrable functions on R2 by the Minlos theorem of Appendix A, since for α > 1, the operator defines a trace class operator on L2 R2 , namely, Tr 1 L2 R2 Another important rigorously defined functional integral is to consider the following α-power Klein Gordon operator on Euclidean space-time with f ∈ L2 RN

Objectives

Results

Conclusion