Coherent state transforms and the MackeyStoneVon Neumann theorem
Abstract
Mackey showed that for a compact Lie group $K$, the pair $(K,C^{0}(K))$ has a unique nontrivial irreducible covariant pair of representations. We study the relevance of this result to the unitary equivalence of quantizations for an infinitedimensional family of $K\times K$ invariant polarizations on $T^{\ast}K$. The Kähler polarizations in the family are generated by (complex) time$\tau$ Hamiltonian flows applied to the (Schrödinger) 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$(+\tau)$ geometricquantizationinduced 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$(+\tau)$ geometricquantizationinduced 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 PeterWeyl 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.
 Publication:

arXiv eprints
 Pub Date:
 November 2012
 DOI:
 10.48550/arXiv.1211.2145
 arXiv:
 arXiv:1211.2145
 Bibcode:
 2012arXiv1211.2145K
 Keywords:

 Mathematics  Differential Geometry;
 High Energy Physics  Theory;
 Mathematical Physics;
 Mathematics  Symplectic Geometry;
 81S10;
 53D50;
 22E30