Self-Similar Random Processes and Infinite-Dimensional Configuration Spaces

TL;DR

This paper explores infinite-dimensional configuration spaces with measures invariant under diffeomorphisms, constructed from self-similar random processes, with applications in quantum physics and extended objects.

Contribution

It introduces new measures on configuration spaces derived from self-similar processes, enabling unitary representations of diffeomorphism groups in quantum physics.

Findings

Construction of measures on configuration spaces from self-similar processes

Analysis of the measurable structure and topology of these spaces

Application to quantum statistical physics and extended objects

Abstract

We discuss various infinite-dimensional configuration spaces that carry measures quasiinvariant under compactly-supported diffeomorphisms of a manifold M corresponding to a physical space. Such measures allow the construction of unitary representations of the diffeomorphism group, which are important to nonrelativistic quantum statistical physics and to the quantum theory of extended objects in d-dimensional Euclidean space. Special attention is given to measurable structure and topology underlying measures on generalized configuration spaces obtained from self-similar random processes (both for d = 1 and d > 1), which describe infinite point configurations having accumulation points.