Universität zu Köln

Kirwin, William D., Mourao, Jose M. and Nunes, Joao P. (2014). Coherent state transforms and the Mackey-Stone-Von Neumann theorem. J. Math. Phys., 55 (10). MELVILLE: AMER INST PHYSICS. ISSN 1089-7658

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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 x K invariant polarizations on T*K. The Kahler polarizations in the family are generated by (complex) time-tau Hamiltonian flows applied to the (Schrodinger) 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-(-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with the momentum space [W. D. Kirwin and S. Wu, Momentum space for compact Lie groups and the Peter-Weyl theorem (unpublished)] quantization of the Hamiltonian function generating the flow.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. (C) 2014 AIP Publishing LLC.

Item Type:	Journal Article
Creators:	Creators Email ORCID ORCID Put Code Kirwin, William D. UNSPECIFIED UNSPECIFIED UNSPECIFIED Mourao, Jose M. UNSPECIFIED UNSPECIFIED UNSPECIFIED Nunes, Joao P. UNSPECIFIED UNSPECIFIED UNSPECIFIED
URN:	urn:nbn:de:hbz:38-426426
DOI:	10.1063/1.4896816
Journal or Publication Title:	J. Math. Phys.
Volume:	55
Number:	10
Date:	2014
Publisher:	AMER INST PHYSICS
Place of Publication:	MELVILLE
ISSN:	1089-7658
Language:	English
Faculty:	Unspecified
Divisions:	Unspecified
Subjects:	no entry
Uncontrolled Keywords:	Keywords Language GEOMETRIC-QUANTIZATION; COMPLEX STRUCTURES; FIELD-THEORY Multiple languages Physics, Mathematical Multiple languages
URI:	http://kups.ub.uni-koeln.de/id/eprint/42642