Curving flat space-time by deformation quantization?

Much, Albert

doi:10.1063/1.4995820

Cited by 4 publications

(3 citation statements)

References 29 publications

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“…Non-commutative geometry has long been put forward as a candidate for a theory of quantum gravity. There, the general idea that spacetime should be somehow quantised at microscopic scales has been realised in many different concrete approaches: Connes' non-commutative spectral triples [1], Lorentzian spectral triples [2,3], Snyder and 𝜅-Minkowski spacetimes and their curvedspace generalisations [4][5][6][7][8], or strict deformation quantisation [9][10][11][12].1 A very popular approach is the DFR spacetime [14], where one promotes the Cartesian coordinates describing flat Minkowski spacetime to operators x𝜇 and postulates commutation relations…”

Section: Introductionmentioning

confidence: 99%

Non-commutative coordinates from quantum gravity

Fröb,

Much,

Papadopoulos

2023

Proceedings of Corfu Summer Institute 2022 "School and Workshops on Elementary Particle Physics and Gravity" — PoS(CORFU2022)

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Local observables in (perturbative) quantum gravity are notoriously hard to define, since the gauge symmetry of gravity -diffeomorphisms -moves points on the manifold. In particular, this is a problem for backgrounds of high symmetry such as Minkowski space or de Sitter spacetime which describes the early inflationary phase of our universe. Only recently this obstacle has been overcome, and a field-dependent coordinate system has been constructed to all orders in perturbation theory, in which observables are fully gauge-invariant. We show that these fielddependent coordinates are non-commutative, and compute their commutator to second order in the Planck length. This provides the first systematic derivation of non-commutativity that arises due to quantum gravity effects.

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Section: Introductionmentioning

confidence: 99%

Non-commutative coordinates from quantum gravity

Fröb,

Much,

Papadopoulos

2023

Proceedings of Corfu Summer Institute 2022 "School and Workshops on Elementary Particle Physics and Gravity" — PoS(CORFU2022)

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“…On the geometrical side, a prominent approach to quantisation is given by non-commutative geometry. In this framework, there are again many different approaches, for example Connes' non-commutative spectral triples [2], Lorentzian spectral triples [3,4], Snyder and κ-Minkowski spacetimes and their curved-space generalisations [5][6][7][8][9], or strict deformation quantisation [10][11][12][13]. [14] The overarching general idea is to obtain classical geometry from the limit of a non-commutative algebra, which is conceptually analogous to the quantisation of the classical phase space in the quantum mechanical setting.…”

Section: Introductionmentioning

confidence: 99%

Non-commutative Geometry from Perturbative Quantum Gravity

Fröb¹,

Much²,

Papadopoulos³

2022

Preprint

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Trying to connect a fundamentally non-commutative spacetime with the conservative perturbative approach to quantum gravity, we are led to the natural question: are non-commutative geometrical effects already present in the regime where perturbative quantum gravity provides a predictive framework? Moreover, is it necessary to introduce non-commutativity by hand, or does it arise through quantum-gravitational effects? We show that the first question can be answered in the affirmative, and the second one in the negative: perturbative quantum gravity predicts non-commutativity at the Planck scale, once one clarifies the structure of observables in the quantum theory.

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“…Nowadays, investigations prove the physical and/or mathematical necessity for spacetimes that carry a noncommutative structure, [DFR95], [And13], [Muc14] and [Muc17]. One of the most common studied models is the so called Moyal-Weyl spacetime, generated by self-adjoint coordinate operators x that fulfill,…”

Section: Introductionmentioning

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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Curving flat space-time by deformation quantization?

Cited by 4 publications

References 29 publications

Non-commutative coordinates from quantum gravity

Non-commutative coordinates from quantum gravity

Non-commutative Geometry from Perturbative Quantum Gravity

A Poincaré covariant noncommutative spacetime

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