Manifold Topology, Observables and Gauge Group

TL;DR

This paper clarifies how manifold topology influences the structure of quantum observables and gauge groups, classifying representations based on fundamental group properties and their implications for particle statistics.

Contribution

It introduces a classification of observable algebra representations using fundamental group representations, linking topology, gauge groups, and observable structure.

Findings

Locally normal representations are classified by fundamental group representations.

All representations are obtainable if the fundamental group is amenable.

Implications for the observability of permutation groups in particle statistics.

Abstract

The relation between manifold topology, observables and gauge group is clarified on the basis of the classification of the representations of the algebra of observables associated to positions and displacements on the manifold. The guiding, physically motivated, principles are i) locality, i.e. the generating role of the algebras localized in small, topological trivial, regions, ii) diffeomorphism covariance, which guarantees the intrinsic character of the analysis, iii) the exclusion of additional local degrees of freedom with respect to the Schroedinger representation. The locally normal representations of the resulting observable algebra are classified by unitary representations of the fundamental group of the manifold, which actually generate an observable, "topological", subalgebra. The result is confronted with the standard approach based on the introduction of the universal covering ${\tilde{\cal M}}$ of $\cal{M}$ and on the decomposition of $L^2({\tilde{\cal M}})$ according to the spectrum of the fundamental group, which plays the role of a gauge group. It is shown that in this way one obtains all the representations of the observables iff the fundamental group is amenable. The implications on the observability of the Permutation Group in Particle Statistics are discussed.