Manifold topology, observables, and gauge group

Abstract

The relation between the manifold topology, observables, and gauge group is clarified on the basis of the classification of the representations of the algebra of observables associated with 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, topologically trivial, regions, (ii) diffeomorphism covariance, which guarantees the intrinsic character of the analysis, and (iii) the exclusion of additional local degrees of freedom with respect to the Schrödinger 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 M̃ of M and on the decomposition of L2(M̃) 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.