Relational Quantum Geometry

TL;DR

This paper unifies gauge theory, quantum theory, and quantum reference frames through a mathematical framework of non-commutative geometry, providing new insights into quantum orbifolds and relational interpretations.

Contribution

It introduces a rigorous quantum geometric framework that unifies extended phase space, crossed products, and quantum reference frames, including a novel quantum orbifold concept.

Findings

Extended phase space as a classical principal bundle with a Poisson base

Crossed product as a trivial quantum principal bundle

Quantum orbifold as a unified structure for multiple QRFs

Abstract

A common feature of the extended phase space of gauge theory, the crossed product of quantum theory, and quantum reference frames (QRFs) is the adjoining of degrees of freedom followed by a constraining procedure for the resulting total system. Building on previous work, we identify non-commutative or quantum geometry as a mathematical framework which unifies these three objects. We first provide a rigorous account of the extended phase space, and demonstrate that it can be regarded as a classical principal bundle with a Poisson manifold base. We then show that the crossed product is a trivial quantum principal bundle which both substantiates a conjecture on the quantization of the extended phase space and facilitates a relational interpretation. Combining several crossed products with possibly distinct structure groups into a single object, we arrive at a novel definition of a quantum orbifold. We demonstrate that change of frame maps within the quantum orbifold correspond to quantum gauge transformations, which are QRF preserving maps between crossed product algebras. Finally, we conclude that the quantum orbifold is equivalent to the G-framed algebra proposed in prior work, thereby placing systems containing multiple QRFs squarely in the context of quantum geometry.