English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

Geometric quantization often produces not one Hilbert space to represent thequantum states of a classical system but a whole family $H_s$ of Hilbertspaces, and the question arises if the spaces $H_s$ are canonically isomorphic.[ADW] and [Hi] suggest to view $H_s$ as fibers of a Hilbert bundle $H$,introduce a connection on $H$, and use parallel transport to identify differentfibers. Here we explore to what extent this can be done. First we introduce thenotion of smooth and analytic fields of Hilbert spaces, and prove that if ananalytic field over a simply connected base is flat, then it corresponds to aHermitian Hilbert bundle with a flat connection and path independent paralleltransport. Second we address a general direct image problem in complexgeometry: pushing forward a Hermitian holomorphic vector bundle $E-->Y$ along anon-proper map $Y-->S$. We give criteria for the direct image to be a smoothfield of Hilbert spaces. Third we consider quantizing an analytic Riemannianmanifold $M$ by endowing $TM$ with the family of adapted K\"ahler structuresfrom arxiv:1004.4069 [LSz]. This leads to a direct image problem. When $M$ ishomogeneous, we prove the direct image is an analytic field of Hilbert spaces.For certain such $M$---but not all---the direct image is even flat; which meansthat in those cases quantization is unique.

Direct images, fields of Hilbert spaces, and geometric quantization