Hilbert field and hermitian Hilbert bundle

TL;DR

This paper presents an example of an analytic Hilbert field that does not induce a Hilbert bundle, challenging previous assumptions about the relationship between flat analytic Hilbert fields and Hilbert bundles.

Contribution

It provides a counterexample demonstrating that not all flat analytic Hilbert fields induce Hilbert bundles, refining the understanding of their relationship.

Findings

Counterexample of a flat analytic Hilbert field not inducing a Hilbert bundle

Clarification of limitations in the correspondence between Hilbert fields and bundles

Enhancement of the theoretical framework in geometric quantization

Abstract

In the paper "Direct Images, Fields of Hilbert Spaces, and Geometric Quantization", Lempert and Sz\H{o}ke proved that any flat analytic Hilbert field will induce a hermitian Hilbert bundle and gave an example of a flat Hilbert field that does not induce any Hilbert bundle. In this paper, we will provide an example of an analytic Hilbert field that does not induce any Hilbert bundle.