Quantum mechanical effects from deformation theory

Much, Albert

doi:10.1063/1.4865459

Cited by 16 publications

(26 citation statements)

References 21 publications

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“…This fact is first of all supported by working in a relativistic second-quantized context. Secondly, in the one particle non-relativistic limit we obtain the well known Landau plane (see [Muc14]). Hence, the QFT -noncommutative plane seems to be an intermediate stepping stone from the non-relativistic one-particle Landau quantization to the second quantized relativistic one.…”

Section: Discussionmentioning

confidence: 99%

“…In the context of QM the deformation of the coordinate operator with the momentum operator gave as the quantum plane of the Landau-quantization, (see [Muc14,Lemma 4.3]). Since the commutator commutes with the generators of deformation, one can view the result as the deformed commutator of the coordinate operators, (see Definition 2.2).…”

Section: Qft-moyal-weyl From Deformationmentioning

confidence: 99%

“…Such a concept was developed for the quantum-mechanical (QM) case in [Muc14]. It was done in a rigorous mathematical fashion by using the deformation technique known under warped convolutions [BLS11].…”

Section: Introductionmentioning

confidence: 99%

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Relativistic corrections to the Moyal-Weyl spacetime

Much

2015

Journal of Mathematical Physics

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Section: Discussionmentioning

confidence: 99%

Section: Qft-moyal-weyl From Deformationmentioning

confidence: 99%

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Relativistic corrections to the Moyal-Weyl spacetime

Much

2015

Journal of Mathematical Physics

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“…a noncommutative spacetime, from a given set of coordinate operators. In [And13] and [Muc14] the noncommutative Moyal-Weyl space was obtained by deformation quantization of the ordinary quantum mechanical coordinate operator, by using as generators of the deformation the translation operators, i.e. the momentum operators.…”

Section: Extending Snyder-spacetime By Deformationmentioning

confidence: 99%

A Poincaré covariant noncommutative spacetime

Much

Vergara

2018

Int. J. Geom. Methods Mod. Phys.

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We interpret, in the realm of relativistic quantum field theory, the tangential operator given by Coleman, Mandula [CM67] (see also [MPS16]) as an appropriate coordinate operator. The investigation shows that the operator generates a Snyder-like noncommutative spacetime with a minimal length that is given by the mass. By using this operator to define a noncommutative spacetime, we obtain a Poincaré invariant noncommutative spacetime and in addition solve the soccer-ball problem. Moreover, from recent progress in deformation theory we extract the idea how to obtain, in a physical and mathematical well-defined manner, an emerging noncommutative spacetime. This is done by a strict deformation quantization known as Rieffel deformation (or warped convolutions). The result is a noncommutative spacetime combining a Snyder and a Moyal-Weyl type of noncommutativity that in addition behaves covariant under transformations of the whole Poincaré group. Discussion 16Conventions 0.1. We use d = n + 1, for n ∈ N and the Greek letters are split into µ, ν = 0, . . . , n. Moreover, we use Latin letters for the spatial components which run from 1, . . . , n and we choose the following convention for the Minkowski scalar product of d-dimensional vectors, a · b = a 0 b 0 + a k b k = a 0 b 0 − a · b. Furthermore, we use the common symbol S (R n,d ) for the Schwartz-space and H for a Hilbert space. * amuch@matmor.unam.mx † vergara@nucleares.unam.mx 2

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“…In restriction to the smooth subspace B ∞ m ⊂ B m (with its usual Fréchet topology), we thus find an action complying with Definition 4.1. This space occurs in the context of quantum fields satisfying polynomial energy bounds [11], see also [1,26] for recent related work in a quantum mechanics context. We show that this fits into our general framework.…”

Section: Deformations Of Unbounded Operatorsmentioning

confidence: 99%

Strict deformation quantization of locally convex algebras and modules

Lechner

Waldmann

2016

Journal of Geometry and Physics

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In this work various symbol spaces with values in a sequentially complete locally convex vector space are introduced and discussed. They are used to define vector-valued oscillatory integrals which allow to extend Rieffel's strict deformation quantization to the framework of sequentially complete locally convex algebras and modules with separately continuous products and module structures, making use of polynomially bounded actions of R n . Several well-known integral formulas for star products are shown to fit into this general setting, and a new class of examples involving compactly supported R n -actions on R n is constructed.

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Quantum mechanical effects from deformation theory

Cited by 16 publications

References 21 publications

Relativistic corrections to the Moyal-Weyl spacetime

Relativistic corrections to the Moyal-Weyl spacetime

A Poincaré covariant noncommutative spacetime

Strict deformation quantization of locally convex algebras and modules

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