Quantum Mechanics in Infinite Symplectic Volume

TL;DR

This paper develops a method to quantize classical phase spaces with infinite symplectic volume by embedding them into an infinite-dimensional projective space, enabling a consistent quantum description.

Contribution

It introduces a novel approach to quantize infinite-volume phase spaces via holomorphic embedding into complex projective space, preserving metric structures.

Findings

Successful embedding of infinite-volume phase space into CP(H)

Construction of Hilbert-space bundle as a pullback

Preservation of metric structures during quantization

Abstract

We quantise complex, infinite-dimensional projective space CP(H). We apply the result to quantise a complex, finite-dimensional, classical phase space C whose symplectic volume is infinite, by holomorphically embedding it into CP(H). The embedding is univocally determined by requiring it to be an isometry between the Bergman metric on C and the Fubini-Study metric on CP(H). Then the Hilbert-space bundle over C is the pullback, by the embedding, of the Hilbert-space bundle over CP(H).