Characteristic classes of star products on Marsden–Weinstein reduced symplectic manifolds

Reichert, T.

doi:10.1007/s11005-016-0921-z

Cited by 9 publications

(12 citation statements)

References 30 publications

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“…In the case of a regular G-Hamiltonian system Bordemann-Herbig-Waldmann [16] constructed a star product on the reduced symplectic space via homological perturbation of the classical homological reductioná la Batalin-Fradkin-Vilkoviski; see also [35,20]. In [65], Reichert relates the characteristic classes of the unreduced with the reduced star product and thus shows that, under reasonable assumptions on the initial data of the Hamiltonian system, deformation quantization commutes with homological reduction. The method from [16] was generalized by Herbig [37] and Bordemann-Herbig-Pflaum [13] to the singular case under the condition that the zero level set is a complete intersection and that its vanishing ideal is generated by the components of the moment map.…”

Section: The Methodsmentioning

confidence: 99%

Deformation Quantization and Homological Reduction of a Lattice Gauge Model

Pflaum

Rudolph

Schmidt

2021

Commun. Math. Phys.

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For a compact Lie group G we consider a lattice gauge model given by the G-Hamiltonian system which consists of the cotangent bundle of a power of G with its canonical symplectic structure and standard moment map. We explicitly construct a Fedosov quantization of the underlying symplectic manifold using the Levi-Civita connection of the Killing metric on G. We then explain and refine quantized homological reduction for the construction of a star product on the symplectically reduced space in the singular case. Afterwards we show that for G = SU(2) the main hypotheses ensuring the method of quantized homological reduction to be applicable hold in the case of our lattice gauge model. For that case, this implies that the-in general singular-symplectically reduced phase space of the corresponding lattice gauge model carries a star product.

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Section: The Methodsmentioning

confidence: 99%

Deformation Quantization and Homological Reduction of a Lattice Gauge Model

Pflaum

Rudolph

Schmidt

2021

Commun. Math. Phys.

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“…As already mentioned in the former section, the purpose of the kinematic Hilbert space is to implement the adjointness and commutation relations among the operators acting on H kin . The question we want to address is then how to formulate the physical inner product, defined through the group averaging procedure (35) in terms of a phase space integration of an appropriate Wigner distribution as depicted in (17). First, one only needs to consider the phase space function corresponding to the non-diagonal density operatorρ(φ, φ ′ ) for the kinematic quantum states φ, φ ′ ∈ D kin ⊂ H kin , that is, a self-adjoint and positive semi-definite operator written asρ…”

Section: The Physical Wigner Distributionmentioning

confidence: 99%

“…for each of the constraints operatorsĈ I = Φ(C I ) ∈ (Ĉ I ) I∈I , related to the classical constraints C I (p, q) by means of the Weyl quantization map (10). Using the integral property of the Wigner distribution (17), and the cyclic phase space behavior of the star-product (24), the physical inner product obtained in (35) can be expressed as…”

Section: The Physical Wigner Distributionmentioning

confidence: 99%

Deformation quantization of constrained systems: a group averaging approach

Berra─Montiel

Molgado

2020

Class. Quantum Grav.

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Motivated by certain concepts introduced by the Refined Algebraic Quantization formalism for constrained systems which has been successfully applied within the context of Loop Quantum Gravity, in this paper we propose a phase space implementation of the Dirac quantization formalism to appropriately include systems with constraints. In particular, we propose a physically prescribed Wigner distribution which allows the definition of a well-defined inner product by judiciously introducing a star version of the group averaging of the constraints. This star group averaging procedure is obtained by considering the star-exponential of the constraints and then integrating with respect to a suitable measure. In this manner, the proposed physical Wigner distribution explicitly solves the constraints by including generalized functions on phase space. Finally, in order to illustrate our approach, our proposal is tested in a couple of simple examples which recover standard results found in the literature.

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“…The map J is called quantum momentum map and the pairs (⋆, J ) are called equivariant star products, see also [28][29][30] for a classification in the symplectic setting and [11] for the more general Poisson case.…”

Section: Introductionmentioning

confidence: 99%

BRST Reduction of Quantum Algebras with $^*$-Involutions

Esposito,

Kraft,

Waldmann

2019

Preprint

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In this paper we investigate the compatibility of the BRST reduction procedure with the Hermiticity of star products. First, we introduce the generalized notion of abstract BRST algebras with corresponding involutions. In this setting we define adjoint BRST differentials and as a consequence one gets new BRST quotients. Passing to the quantum BRST setting we show that for compact Lie groups the new quantum BRST quotient and the quantum BRST cohomology are isomorphic in zero degree implying that reduction is compatible with Hermiticity.

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Characteristic classes of star products on Marsden–Weinstein reduced symplectic manifolds

Cited by 9 publications

References 30 publications

Deformation Quantization and Homological Reduction of a Lattice Gauge Model

Deformation Quantization and Homological Reduction of a Lattice Gauge Model

Deformation quantization of constrained systems: a group averaging approach

BRST Reduction of Quantum Algebras with $^*$-Involutions

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