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Published Online: 06 October 2014
Accepted: September 2014

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

Journal of Mathematical Physics 55, 102101 (2014); https://doi.org/10.1063/1.4896816
William D. Kirwin1, a), José M. Mourão2, b), and João P. Nunes2, c)
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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 × K invariant polarizations on T*K. The Kähler polarizations in the family are generated by (complex) time-τ Hamiltonian flows applied to the (Schrödinger) vertical real polarization. The unitary equivalence of the corresponding quantizations of T*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-(−τ) evolution followed by time-(+τ) 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-(+τ) geometric-quantization-induced evolution with the time-(−τ) evolution associated with the momentum space [W. D. Kirwin and S. Wu, “Momentum space for compact Lie groups and the Peter-Weyl theorem” (unpublished)] quantization of the Hamiltonian function generating the flow.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.

ACKNOWLEDGMENTS

We thank Brian Hall for discussions. This work was supported in part by the European Science Foundation (ESF) grant “Interactions of Low-Dimensional Topology and Geometry with Mathematical Physics (ITGP).” The last two authors were supported by CAMGSD-LARSys through FCT Program PEst-OE/EEI/LA0009/2013 and by the FCT Project No. PTDC/MAT/119689/2010.

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