{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T11:27:44Z","timestamp":1777462064910,"version":"3.51.4"},"reference-count":28,"publisher":"AIP Publishing","issue":"10","content-domain":{"domain":["pubs.aip.org"],"crossmark-restriction":true},"short-container-title":[],"published-print":{"date-parts":[[2014,10,1]]},"abstract":"<jats:p>Mackey showed that for a compact Lie group K, the pair (K, C\u20090(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 \u00d7 K invariant polarizations on T*K. The K\u00e4hler polarizations in the family are generated by (complex) time-\u03c4 Hamiltonian flows applied to the (Schr\u00f6dinger) 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\u20090(K) defined via Heisenberg time-(\u2212\u03c4) evolution followed by time-(+\u03c4) 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-(+\u03c4) geometric-quantization-induced evolution with the time-(\u2212\u03c4) evolution associated with the momentum space [W. D. Kirwin and S. Wu, \u201cMomentum space for compact Lie groups and the Peter-Weyl theorem\u201d (unpublished)] quantization of the Hamiltonian function generating the flow.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.<\/jats:p>","DOI":"10.1063\/1.4896816","type":"journal-article","created":{"date-parts":[[2014,10,7]],"date-time":"2014-10-07T11:39:09Z","timestamp":1412681949000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":10,"title":["Coherent state transforms and the Mackey-Stone-Von Neumann theorem"],"prefix":"10.1063","volume":"55","author":[{"given":"William D.","family":"Kirwin","sequence":"first","affiliation":[{"name":"University of Cologne 1 Mathematics Institute, , Weyertal 86 - 90, 50931 Cologne, Germany"}]},{"given":"Jos\u00e9 M.","family":"Mour\u00e3o","sequence":"additional","affiliation":[{"name":"Universidade de Lisboa 2 Center for Mathematical Analysis, Geometry and Dynamical Systems and Department of Mathematics, Instituto Superior T\u00e9cnico, , Av. Rovisco Pais, 1049-001 Lisbon, Portugal"}]},{"given":"Jo\u00e3o P.","family":"Nunes","sequence":"additional","affiliation":[{"name":"Universidade de Lisboa 2 Center for Mathematical Analysis, Geometry and Dynamical Systems and Department of Mathematics, Instituto Superior T\u00e9cnico, , Av. Rovisco Pais, 1049-001 Lisbon, Portugal"}]}],"member":"317","published-online":{"date-parts":[[2014,10,6]]},"reference":[{"issue":"3","key":"2023062722322363700_c1","first-page":"787","article-title":"Geometric quantization of Chern-Simons gauge theory","volume":"33","year":"1991","journal-title":"J. Differ. Geom."},{"key":"2023062722322363700_c2","doi-asserted-by":"publisher","first-page":"3388","DOI":"10.1016\/j.jfa.2010.01.023","article-title":"Quantization of abelian varieties: distributional sections and the transition from K\u00e4hler to real polarizations","volume":"258","year":"2010","journal-title":"J. Funct. 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