相干态变换与Mackey-Stone-Von Neumann定理
Coherent state transforms and the Mackey-Stone-Von Neumann theorem
摘要：
麦基表明，对于一个紧凑的李群$K$，对$(K,C^{0}(K))$有一个独特的非平凡的不可约协变对表示。我们研究了这个结果的相关性，一个无限维家庭的$K\times K$不变的极化$T^{\ast}K$量子化的酉等价。Kähler极化是由（复数）时间- $τ$哈密顿流应用于（薛定谔）垂直实极化而产生的。然后，通过考虑由几何预量化定义的$K$的表示和通过海森堡时间-$(+τ)$几何量化诱导的演化之后的$(-τ)$演化定义的$C^0(K)$的表示的协变对，研究了$T^{\ast}K$的相应量化的酉等价性。我们证明了在半经典和大的虚时间极限中，由Mackey定理保证其存在性的幺正变换可以近似为时间-$(+τ)$几何量子化诱导的演化与与动量空间相关的时间- $(-τ)$演化的合成[W. D. Kirwin和S.吴，紧李群的动量空间和彼得-外尔定理，出现]量子化的哈密顿函数产生的流动。在二次哈密顿量情形下，该渐近结果是精确的，量子化之间的么正等价性是通过将Heisenberg虚时间演化与热算符演化相结合，并与Hall的相干态变换相一致来实现的.
英文摘要：
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 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 $T^{\ast}K$ 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-$(-τ)$ 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, 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.
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