Projective Limits of State Spaces II. Quantum Formalism

TL;DR

This paper explores the quantum formalism within the projective framework, analyzing how quantum state spaces relate to inductive limits and tensor products, extending previous results to more complex configuration spaces.

Contribution

It extends the projective quantum state space formalism to Lie group configuration spaces and holomorphic quantization, building on prior linear space results.

Findings

Relation between projective quantum states and inductive limit Hilbert spaces clarified.

Extension of quantization methods to Lie group configuration spaces.

Inclusion of holomorphic quantization in the projective framework.

Abstract

In this series of papers, we investigate the projective framework initiated by Jerzy Kijowski and Andrzej Oko{\l}\'ow, which describes the states of a quantum theory as projective families of density matrices. After discussing the formalism at the classical level in a first paper, the present second paper is devoted to the quantum theory. In particular, we inspect in detail how such quantum projective state spaces relate to inductive limit Hilbert spaces and to infinite tensor product constructions. Regarding the quantization of classical projective structures into quantum ones, we extend the results by Oko{\l}\'ow [Oko{\l}\'ow 2013, arXiv:1304.6330], that were set up in the context of linear configuration spaces, to configuration spaces given by simply-connected Lie groups, and to holomorphic quantization of complex phase spaces.