Derivation of Local Gauge Freedom from a Measurement Principle

TL;DR

This paper links the gauge principle to quantum measurements by deriving local gauge freedom from the structure of operator manifolds and spectral measures, showing the gauge group as U(m).

Contribution

It introduces operator manifolds with spectral measures to mathematically connect quantum measurement to local gauge freedom, deriving the gauge group as U(m).

Findings

Vectors can be represented as functions on the manifold.

Local gauge freedom arises from the arbitrariness of this representation.

The gauge group is restricted to U(m).

Abstract

We define operator manifolds as manifolds on which a spectral measure on a Hilbert space is given as additional structure. The spectral measure mathematically describes space as a quantum mechanical observable. We show that the vectors of the Hilbert space can be represented as functions on the manifold. The arbitrariness of this representation is interpreted as local gauge freedom. In this way, the physical gauge principle is linked with quantum mechanical measurements of the position variable. We derive the restriction for the local gauge group to be U(m), where m is the number of components of the wave functions.