Geometric Quantization¶and the Generalized Segal--Bargmann Transform¶for Lie Groups of Compact Type

Hall, Brian C.

doi:10.1007/s002200200607

Cited by 79 publications

(155 citation statements)

References 51 publications

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“…Half-form Schrödinger quantization on T * K yields the Hilbert space L 2 (K , dx) of ordinary square-integrable functions on K [25] with scalar product…”

Section: Schrödinger Quantizationmentioning

confidence: 99%

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A Gauge Model for Quantum Mechanics on a Stratified Space

Huebschmann

Rudolph

Schmidt

2008

Commun. Math. Phys.

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Abstract:In the Hamiltonian approach on a single spatial plaquette, we construct a quantum (lattice) gauge theory which incorporates the classical singularities. The reduced phase space is a stratified Kähler space, and we make explicit the requisite singular holomorphic quantization procedure on this space. On the quantum level, this procedure yields a costratified Hilbert space, that is, a Hilbert space together with a system which consists of the subspaces associated with the strata of the reduced phase space and of the corresponding orthoprojectors. The costratified Hilbert space structure reflects the stratification of the reduced phase space. For the special case where the structure group is SU(2), we discuss the tunneling probabilities between the strata, determine the energy eigenstates and study the corresponding expectation values of the orthoprojectors onto the subspaces associated with the strata in the strong and weak coupling approximations.

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“…Half-form Schrödinger quantization on T * K yields the Hilbert space L 2 (K , dx) of ordinary square-integrable functions on K [25] with scalar product…”

Section: Schrödinger Quantizationmentioning

confidence: 99%

“…For details, see [25] (description in terms of the heat kernel on K ) or Theorem 6.5 in [35] (description in terms of representative functions).…”

Section: As λ Ranges Over the Highest Weights Yields A Unitary Isomomentioning

confidence: 99%

A Gauge Model for Quantum Mechanics on a Stratified Space

Huebschmann

Rudolph

Schmidt

2008

Commun. Math. Phys.

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“…Hall [9]. In that paper, the pairing map is shown to coincide, up to multiplication by a constant, with a version of the Segal-Bargmann coherent state transform developed, in turn, over Lie groups admitting a bi-invariant Riemannian metric, in a sequence of preceding papers [7]- [9]. The main technique in those papers is heat kernel harmonic analysis and, in fact, in [9], Hall derives the unitarity of the pairing map by identifying the measure on K C coming from the half-form bundle with an appropriate heat kernel measure which, in turn, he has shown in the preceding papers to furnish a unitary transform.…”

Section: Introductionmentioning

confidence: 99%

Kirillov’s character formula, the holomorphic Peter–Weyl theorem, and the Blattner–Kostant–Sternberg pairing

Huebschmann¹

2008

Journal of Geometry and Physics

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Abstract. Let K be a compact Lie group, endowed with a bi-invariant Riemannian metric, which we denote by κ. The complexification K C of K inherits a Kähler structure having twice the kinetic energy of the metric as its potential; let ε denote the symplectic volume form. Left and right translation turn the Hilbert space HL 2 (K C , e −κ/t ηε) of square-integrable holomorphic functions on K C relative to a suitable measure written as e −κ/t ηε into a unitary (K × K)-representation; here η is an additional term coming from the metaplectic correction, and t > 0 is a real parameter. In the physical interpretation, this parameter amounts to Planck 's constant .We establish the statement of the Peter-Weyl theorem for the Hilbert space HL 2 (K C , e −κ/t ηε) to the effect that (i) HL 2 (K C , e −κ/t ηε) contains the vector space of representative functions on K C as a dense subspace and that (ii) the assignment to a holomorphic function of its Fourier coefficients yields an isomorphism of Hilbert algebras from the convolution algebra HL 2 (K C , e −κ/t ηε) onto an algebra of the kind b ⊕End(V ). Here V ranges over the irreducible rational representations of K C and b ⊕ refers to a suitable completion of the direct sum algebra ⊕End(V ).Consequences are: (i) the existence of a uniquely determined unitary isomorphism between L 2 (K, dx) (where dx refers to Haar measure on K) and the Hilbert space HL 2 (K C , e −κ/t ηε), and (ii) a proof that this isomorphism coincides with the Blattner-Kostant-Sternberg pairing map from L 2 (K, dx) to HL 2 (K C , e −κ/t ηε), multiplied by (4πt) − dim(K)/4 . Among our crucial tools is Kirillov's character formula. Our methods are geometric, rely on the orbit method, and are independent of heat kernel harmonic analysis, which is used by B. C. Hall to obtain many of these results

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“…where ϑ is the tautological 1-form on T * K. An explicit calculation which establishes this fact may be found in [6] (but presumably it is a folk-lore observation). We note that, given Y 0 ∈ k, the assignment to (x, Y ) ∈ K × k of |Y − Y 0 | 2 yields a Kähler potential as well which, in turn, determines the same Kähler structure as κ.…”

Section: Stratified Kähler Spacesmentioning

confidence: 72%