Relational Quantum Geometry

Understand

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.