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Direct Images, Fields of Hilbert Spaces, and Geometric Quantization

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Abstract

Geometric quantization often produces not one Hilbert space to represent the quantum states of a classical system but a whole family H s of Hilbert spaces, and the question arises if the spaces H s are canonically isomorphic. Axelrod et al. (J. Diff. Geo. 33:787–902, 1991) and Hitchin (Commun. Math. Phys. 131:347–380, 1990) suggest viewing H s as fibers of a Hilbert bundle H, introduce a connection on H, and use parallel transport to identify different fibers. Here we explore to what extent this can be done. First we introduce the notion of smooth and analytic fields of Hilbert spaces, and prove that if an analytic field over a simply connected base is flat, then it corresponds to a Hermitian Hilbert bundle with a flat connection and path independent parallel transport. Second we address a general direct image problem in complex geometry: pushing forward a Hermitian holomorphic vector bundle \({E \to Y}\) along a non–proper map \({Y \to S}\). We give criteria for the direct image to be a smooth field of Hilbert spaces. Third we consider quantizing an analytic Riemannian manifold M by endowing TM with the family of adapted Kähler structures from Lempert and Szőke (Bull. Lond. Math. Soc. 44:367–374, 2012). This leads to a direct image problem. When M is homogeneous, 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 means that in those cases quantization is unique.

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Correspondence to László Lempert.

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Communicated by S. Zelditch

Róbert Szőke dedicates this paper to his son, Bálint

Research partially supported by NSF grant DMS0700281 and OTKA grants NK81203, K72537. Both authors are grateful to the Mathematics Department of Purdue University and to the Mittag–Leffler Institute, where they collaborated on this project. L.L. also acknowledges a Clay Senior Fellowship during his stay at the Mittag–Leffler Institute, and R.Sz. a fellowship at the Rényi Mathematical Institute, Budapest.

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Lempert, L., Szőke, R. Direct Images, Fields of Hilbert Spaces, and Geometric Quantization. Commun. Math. Phys. 327, 49–99 (2014). https://doi.org/10.1007/s00220-014-1899-y

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