Recent Developments in Deformation Quantization

Waldmann, Stefan

doi:10.1007/978-3-319-26902-3_18

Cited by 11 publications

(7 citation statements)

References 81 publications

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“…All their results stand as interesting concepts in the theory of noncommutative geometry [11], further justified by many follow-ups and continuations. A particular class of noncommutative Cartan calculi is given by twisted Cartan calculi in the overlap of deformation quantization [7,32] and quantum groups [18,24]. Drinfel'd twists [13] are tools to deform Hopf algebras as well as the representation theory of the Hopf algebra in a compatible way.…”

Section: Introductionmentioning

confidence: 99%

Braided Cartan Calculi and Submanifold Algebras

Weber

2019

Preprint

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It is a necessity of derivation based Cartan calculi on noncommutative algebras to employ central bimodules [15,16]. In analogy to differential geometry we construct a noncommutative Cartan calculus for any braided commutative algebra in the symmetric braided monoidal category of equivariant braided symmetric bimodules. In particular, bimodules are considered over the full underlying algebra. Braided versions of the Lie derivative, the insertion and de Rham differential are related by braided commutators, also incorporating the braided Schouten-Nijenhuis bracket, generalizing the classical situation and twisted Cartan calculi. We further prove that Drinfel'd twist deformation corresponds to gauge equivalences of these noncommutative calculi. Then, braided covariant derivatives and metrics on braided commutative algebras are discussed. In particular, we show the existence of a braided Levi-Civita covariant derivative for a fixed braided metric and that braided covariant derivatives are compatible with twist deformation. Furthermore, we project braided Cartan calculi to submanifold algebras and prove that this process commutes with twist deformation if the Hopf algebra action respects the submanifold ideal.

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Section: Introductionmentioning

confidence: 99%

Braided Cartan Calculi and Submanifold Algebras

Weber

2019

Preprint

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“…Again, within our context, the unitary property is readily obtained from the Wigner functions (34) and (43), as both are related by a quantum canonical transformation of the form (18), where the unitary operator U T is in fact given by the integral transformation (47). Following this reasoning, since the time evolution of the pair of harmonic oscillators is a solution of Moyal's equation [16] …”

Section: Equal Frequency Limitmentioning

confidence: 99%

“…We refer the reader to [13], where results on the explicit construction of maps between classical and quantum observables are explained in detail, to Refs. [14,15], where conditions on the existence of the star product are exposed, and to the reviews [16,17,18] for general aspects of deformation quantization, as well as for more recent developments.…”

Section: Introductionmentioning

confidence: 99%

Deformation quantization of the Pais–Uhlenbeck fourth order oscillator

Berra─Montiel¹,

Molgado²,

Rojas³

2015

Annals of Physics

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We analyze the quantization of the Pais-Uhlenbeck fourth order oscillator within the framework of deformation quantization. Our approach exploits the Noether symmetries of the system by proposing integrals of motion as the variables to obtain a solution to the ⋆-genvalue equation, namely the Wigner function. We also obtain, by means of a quantum canonical transformation the wave function associated to the Schrödinger equation of the system. We show that unitary evolution of the system is guaranteed by means of the quantum canonical transformation and via the properties of the constructed Wigner function, even in the so called equal frequency limit of the model, in agreement with recent results.

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“…The next step is to find the phase-space function associated to the rigging map η constructed through the group averaging proposal (34). Since the star-product obtained in (18) defines a homomorphism between the classical observables, C ∞ (R 2n ), and linear operators L(H kin ), this means that the symbol corresponding to the formal operator…”

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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Recent Developments in Deformation Quantization

Cited by 11 publications

References 81 publications

Braided Cartan Calculi and Submanifold Algebras

Braided Cartan Calculi and Submanifold Algebras

Deformation quantization of the Pais–Uhlenbeck fourth order oscillator

Deformation quantization of constrained systems: a group averaging approach

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