Coherent state transforms and the Mackey-Stone-Von Neumann theorem

TL;DR

This paper explores the relationship between Mackey's theorem and the unitary equivalence of geometric quantizations on cotangent bundles of compact Lie groups, using coherent state transforms and asymptotic analysis.

Contribution

It demonstrates how Mackey's theorem underpins the unitary equivalence of quantizations for a family of polarizations, connecting geometric quantization with coherent state transforms.

Findings

In the semiclassical and large imaginary time limits, the Mackey transform approximates the composition of geometric evolutions.

For quadratic Hamiltonians, the asymptotic equivalence becomes exact, linking heat operators with coherent state transforms.

The results unify representation theory, geometric quantization, and coherent state analysis for compact Lie groups.

Abstract

Mackey showed that for a compact Lie group $K$, the pair $(K,C^{0}(K))$ has a unique non-trivial irreducible covariant pair of representations. We study the relevance of this result to the unitary equivalence of quantizations for an infinite-dimensional family of $K\times K$ invariant polarizations on $T^{\ast}K$. The K\"{a}hler polarizations in the family are generated by (complex) time-$\tau$ Hamiltonian flows applied to the (Schr\"{o}dinger) vertical real polarization. The unitary equivalence of the corresponding quantizations of $T^{\ast}K$ is then studied by considering covariant pairs of representations of $K$ defined by geometric prequantization and of representations of $C^0(K)$ defined via Heisenberg time-$(-\tau)$ evolution followed by time-$(+\tau)$ geometric-quantization-induced evolution. We show that in the semiclassical and large imaginary time limits, the unitary transform whose existence is guaranteed by Mackey's theorem can be approximated by composition of the time-$(+\tau)$ geometric-quantization-induced evolution with the time-$(-\tau)$ evolution associated with the momentum space [W. D. Kirwin and S. Wu, Momentum space for compact Lie groups and the Peter-Weyl theorem, to appear] quantization of the Hamiltonian function generating the flow. In the case of quadratic Hamiltonians, this asymptotic result is exact and unitary equivalence between quantizations is achieved by identifying the Heisenberg imaginary time evolution with heat operator evolution, in accordance with the coherent state transform of Hall.