Relational Quantum Geometry
Shadi Ali Ahmad, Wissam Chemissany, Marc S. Klinger, Robert G. Leigh
TL;DR
The paper tackles constrained quantum systems with gauge symmetry by unifying extended phase spaces, crossed products, and quantum reference frames under a noncommutative geometric lens. It shows that the classical extended phase space is a principal bundle with a Poisson base and that its quantization yields a trivial quantum principal bundle, which is equivalent to a von Neumann crossed product via Landstad duality. By introducing quantum gauge transformations, it demonstrates that QRF changes are inner conjugations that preserve the underlying crossed-product structure, and it extends these ideas to locally trivial bundles, culminating in the quantum orbifold, which organizes multiple QRFs and their overlaps as a global geometric object. This framework clarifies how different QRFs relate, provides algebraic tools for relational reasoning in constrained systems, and points to rich directions in noncommutative geometry and quantum gravity, including possible quantum cohomology invariants and curvature concepts that are frame-independent.
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.