Physical Properties of Quantum Field Theory Measures

TL;DR

This paper employs measure theory to analyze physical properties of measures in quantum field theory, comparing configurations of scalar fields with different masses and examining the ergodic and discontinuous nature of quantum connections under the Ashtekar-Lewandowski measure.

Contribution

It introduces new measure-theoretic methods to study quantum field theory measures and proves ergodicity and discontinuity properties of the Ashtekar-Lewandowski measure on quantum connection spaces.

Findings

Differences between free scalar field configurations with varying masses.

Ergodic action of the diffeomorphism group on the AL measure.

Discontinuity of quantum connections along curves under the AL measure.

Abstract

Well known methods of measure theory on infinite dimensional spaces are used to study physical properties of measures relevant to quantum field theory. The difference of typical configurations of free massive scalar field theories with different masses is studied. We apply the same methods to study the Ashtekar-Lewandowski (AL) measure on spaces of connections. We prove that the diffeomorphism group acts ergodically, with respect to the AL measure, on the Ashtekar-Isham space of quantum connections modulo gauge transformations. We also prove that a typical, with respect to the AL measure, quantum connection restricted to a (piecewise analytic) curve leads to a parallel transport discontinuous at every point of the curve.