Geometric quantization, complex structures and the coherent state transform

Florentino, Carlos; Matias, Pedro; Mourão, José M.; Nunes, João P.

doi:10.1016/j.jfa.2004.10.021

Cited by 34 publications

(57 citation statements)

References 13 publications

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“…In [Hal02, FMMN05,FMMN06], it was shown that the CST can be understood in terms of the geometric quantizations of T * K arising from the vertical and Kähler polarizations. Indeed, one has the isomorphism of Hilbert spaces…”

Section: Preliminariesmentioning

confidence: 99%

“…Hall computed this BKS pairing, with the following result. In [FMMN05,FMMN06], the authors considered a one-real-parameter family of polarizations connecting the vertical polarization with the Kähler polarization on T * K. The corresponding BKS pairing maps were shown to be unitary. Degenerating one of the Kähler polarizations to the vertical polarization, one recovers Hall's result relating the BKS pairing with the CST.…”

Section: Preliminariesmentioning

confidence: 99%

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Complex time evolution in geometric quantization and generalized coherent state transforms

Kirwin

Mourão²,

Nunes³

2013

Journal of Functional Analysis

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For the cotangent bundle T * K of a compact Lie group K, we study the complextime evolution of the vertical tangent bundle and the associated geometric quantization Hilbert space L 2 (K) under an infinite-dimensional family of Hamiltonian flows. For each such flow, we construct a generalized coherent state transform (CST), which is a unitary isomorphism between L 2 (K) and a certain weighted L 2 -space of holomorphic functions. For a particular set of choices, we show that this isomorphism is naturally decomposed as a product of a Heisenberg-type evolution (for complex time −τ ) within L 2 (K), followed by a polarization-changing geometric quantization evolution (for complex time +τ ). In this case, our construction yields the usual generalized Segal-Bargmann transform of Hall. We show that the infinite-dimensional family of Hamiltonian flows can also be understood in terms of Thiemann's "complexifier" method (which generalizes the construction of adapted complex structures). We will also investigate some properties of the generalized CSTs, and discuss how their existence can be understood in terms of Mackey's generalization of the Stone-von Neumann theorem.

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Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

Complex time evolution in geometric quantization and generalized coherent state transforms

Kirwin

Mourão²,

Nunes³

2013

Journal of Functional Analysis

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“…In [FMMN05,FMMN06], Florentino-Mattias-Mourão-Nunes looked at the results of [Hal02a] in terms of the one-parameter family of complex structures described above, namely those obtained from the adapted complex structure by scaling in the fibers. Using ideas similar to the ones in [ADW91], these authors consider a parallel transport in the Hilbert bundle associated to the family of complex structures.…”

Section: Connections To Geometric Quantizationmentioning

confidence: 99%

“…For example, this way of thinking provides a simple explanation for the formula expressing the adapted complex structure in terms of Jacobi fields (Section 3.3), and sheds light on the results of [FMMN05,FMMN06] (Section 3.4). We also hope that, as suggested by Thiemann, the use of other imaginary-time Hamiltonian flows will lead to the construction of new complex structures.…”

Section: Introductionmentioning

confidence: 98%

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Adapted complex structures and the geodesic flow

Hall

Kirwin²

2010

Math. Ann.

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In this paper, we give a new construction of the adapted complex structure on a neighborhood of the zero section in the tangent bundle of a compact, real-analytic Riemannian manifold. Motivated by the "complexifier" approach of T. Thiemann as well as certain formulas of V. Guillemin and M. Stenzel, we obtain the polarization associated to the adapted complex structure by applying the "imaginary-time geodesic flow" to the vertical polarization. Meanwhile, at the level of functions, we show that every holomorphic function is obtained from a function that is constant along the fibers by "composition with the imaginary-time geodesic flow." We give several equivalent interpretations of this composition, including a convergent power series in the vector field generating the geodesic flow.

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Coherent States for Compact Lie Groups and Their Large-N Limits

Hall¹

2018

Springer Proceedings in Physics

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The first two parts of this article surveys results related to the heatkernel coherent states for a compact Lie group K. I begin by reviewing the definition of the coherent states, their resolution of the identity, and the associated Segal-Bargmann transform. I then describe related results including connections to geometric quantization and (1 + 1)-dimensional Yang-Mills theory, the associated coherent states on spheres, and applications to quantum gravity.The third part of this article summarizes recent work of mine with Driver and Kemp on the large-N limit of the Segal-Bargmann transform for the unitary group U (N ). A key result is the identification of the leading-order large-N behavior of the Laplacian on "trace polynomials."

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Geometric quantization, complex structures and the coherent state transform

Cited by 34 publications

References 13 publications

Complex time evolution in geometric quantization and generalized coherent state transforms

Complex time evolution in geometric quantization and generalized coherent state transforms

Adapted complex structures and the geodesic flow

Coherent States for Compact Lie Groups and Their Large-N Limits

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