Geometric Quantization and Quantum Mechanics

Śniatycki, Jędrzej

doi:10.1007/978-1-4612-6066-0

Cited by 290 publications

(388 citation statements)

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“…As our physical Hilbert space we take a certain space of square-integrable holomorphic functions which arises by Kähler quantization [51,56]. Through an analogue of the Peter-Weyl theorem, this space is related with the physical Hilbert space arising by ordinary Schrödinger quantization on K .…”

Section: The Quantum Picturementioning

confidence: 99%

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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Section: The Quantum Picturementioning

confidence: 99%

A Gauge Model for Quantum Mechanics on a Stratified Space

Huebschmann

Rudolph

Schmidt

2008

Commun. Math. Phys.

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“…According to (7.20) of [3], together with the similarity transform (14) adjustment and the adjustment in the BKS pairing described in (19),…”

Section: Bks Pairing Casementioning

confidence: 99%

“…We will denote by Q(X) the quantization of the phase space X, suppressing in our notation the choices of polarizations and pre-quantization line bundles etc, via the standard procedure of geometric quantization [3,4]. Suppose we choose the vertical polarization on T * N so that the quantization Q(T * N ) of T * N gives L 2 (N ).…”

Section: Introductionmentioning

confidence: 99%

“…Then J −1 (O µ )/G has a canonical symplectic structure given by the Marsden-Weinstein reduction [1]. This reduced phase space is the appropriate phase space of a particle in a background Yang-Mills field α of charge µ [2].We will denote by Q(X) the quantization of the phase space X, suppressing in our notation the choices of polarizations and pre-quantization line bundles etc, via the standard procedure of geometric quantization [3,4]. Suppose we choose the vertical polarization on T * N so that the quantization Q(T * N ) of T * N gives L 2 (N ).…”

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Quantization of a particle in a background Yang–Mills field

1998

Journal of Mathematical Physics

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Two classes of observables defined on the phase space of a particle are quantized, and the effects of the Yang-Mills field are discussed in the context of geometric quantization.PACS: 03.65.Bz, 02.40.Vh Short title: Particle in background Yang-Mills field. Let Q be a Riemannian manifold considered as the configuration space of a particle, the purpose of this paper is to discuss the quantization of the observables on the phase space T * Q of this particle when it is moving under the influence of a background Yang-Mills field, the Yang-Mills potential is a connection α on a principal bundle N over Q.The free G−action on N can be lifted to a Hamiltonian G−action on T * N with an equivariant moment map J : T * N → g * . Let µ ∈ g * , and denote by O µ the coadjoint orbit through µ. Then J −1 (O µ )/G has a canonical symplectic structure given by the Marsden-Weinstein reduction [1]. This reduced phase space is the appropriate phase space of a particle in a background Yang-Mills field α of charge µ [2].We will denote by Q(X) the quantization of the phase space X, suppressing in our notation the choices of polarizations and pre-quantization line bundles etc, via the standard procedure of geometric quantization [3,4]. Suppose we choose the vertical polarization on T * N so that the quantization Q(T * N ) of T * N gives L 2 (N ). Moreover, suppose the co-adjoint orbit O µ is integral, so that the quantization of this coadjoint orbit gives a irreducible representation space H µ of G [5] [6]. A theorem of Guillemin-Sternberg [7] (see also [8]) then suggests that the quantization ofAnd this result holds independent of whether there is a Yang-Mills field present in the backgroud. Thus when some technical assumptions are made so that the procedure of geometric quantization can be carried out smoothly, the Yang-Mills field plays no role in the quantization of the phase spaceWe will discuss the effect of the Yang-Mills field in quantizing observables that are lifted from functions on T * Q. We will show that the resulting quantum operators are expressed in terms of the covariant derivatives, which is defined by the connection α. In particular, we will show that the quantum operators for f of the form

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“…The reader is referred to Woodhouse [45], Sniatycki [39] or Puta [36] for comprehensive expositions.…”

Section: Geometric Quantizationmentioning

confidence: 99%

Geometric quantization of reduced contangent bundles

Robson¹

1996

Journal of Geometry and Physics

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The method of geometric quantization is applied to a particle moving on an arbitrary Riemannian manifold Q in an external gauge field, that is a connection on a principal H-bundle N over Q. The phase space of the particle is a Marsden-Weinstein reduction of T * N , hence this space can also be considered to be the reduced phase space of a particular type of constrained mechanical system. An explicit map is found from a subalgebra of the classical observables to the corresponding quantum operators. These operators are found to be the generators of a representation of the semi-direct product group, Aut N ⋉ C ∞ c (Q). A generalised Aharanov-Bohm effect is shown to be a natural consequence of the quantization procedure. In particular the rôle of the connection in the quantum mechanical system is made clear. The quantization of the Hamiltonian is also considered.Additionally, our approach allows the related quantization procedures proposed by Mackey and by Isham to be fully understood.

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Geometric Quantization and Quantum Mechanics

Cited by 290 publications

References 0 publications

A Gauge Model for Quantum Mechanics on a Stratified Space

A Gauge Model for Quantum Mechanics on a Stratified Space

Quantization of a particle in a background Yang–Mills field

Geometric quantization of reduced contangent bundles

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