Quantum Field Theory on Quantum Spacetime

Doplicher, Sergio

doi:10.1088/1742-6596/53/1/051

Cited by 14 publications

(17 citation statements)

References 16 publications

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“…Our conclusions, eq. (3.16), agree with the heuristic argument [Do01,Do06] which suggests to modify the Planck length in (3.2) by, as a rough approximation, the factor g −1/2 00 ; the minimal distance between two events [BDFP10] would then be modified accordingly. Such a rough argument points too to an infinite extension of non local effects near a singularity, where g 00 vanishes; so that, near the "Big Bang", thermal equilibrium would have been established globally.…”

Section: Backreaction On Quantum (Frw) Spacetimesupporting

confidence: 85%

“…We recall that the horizon problem of standard cosmology arises from the observation that there should be regions of the universe which were never in causal contact since the Big Bang, which seems to be in contrast with the homogeneity of the universe, as shown for instance by the Cosmic Microwave Background. Our result here, although obtained under oversimplifying assumptions, goes in the same direction as those drawn at a heuristic level from a full use of the Principle of Gravitational Stability against localization of events [Do01,Do06], which point to a background dependence of the effective Planck length, through which the acausal effects which are typical of QFT on noncommutative spacetime may be transmitted over large distances (cf comments at the end of Section 3.2).…”

Section: Introductionsupporting

confidence: 70%

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On Quantum Spacetime and the horizon problem

Doplicher¹,

Morsella²,

Pinamonti³

2013

Journal of Geometry and Physics

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In the special case of a spherically symmetric solution of Einstein equations coupled to a scalar massless field, we examine the consequences on the exact solution imposed by a semiclassical treatment of gravitational interaction when the scalar field is quantized. In agreement with Doplicher et al. (1995) [2], imposing the principle of gravitational stability against localization of events, we find that the region where an event is localized, or where initial conditions can be assigned, has a minimal extension, of the order of the Planck length. This conclusion, though limited to the case of spherical symmetry, is more general than that of [2] since it does not require the use of the notion of energy through the Heisenberg Principle, nor of any approximation as the linearized Einstein equations. We shall then describe the influence of this minimal length scale in a cosmological model, namely a simple universe filled with radiation, which is effectively described by a conformally coupled scalar field in a conformal KMS state. Solving the backreaction, a power law inflation scenario appears close to the initial singularity. Furthermore, the initial singularity becomes light like and thus the standard horizon problem is avoided in this simple model. This indication goes in the same direction as those drawn at a heuristic level from a full use of the principle of gravitational stability against localization of events, which point to a background dependence of the effective Planck length, through which a-causal effects may be transmitted. (C) 2013 Elsevier B.V. All rights reserved

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Section: Backreaction On Quantum (Frw) Spacetimesupporting

confidence: 85%

Section: Introductionsupporting

confidence: 70%

On Quantum Spacetime and the horizon problem

Doplicher¹,

Morsella²,

Pinamonti³

2013

Journal of Geometry and Physics

Self Cite

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“…We summarize in this section the analysis developed at length in [43]. The 2N -dimensional DFR model of quantum spacetime (see [22,23] as well as [46] for a recent review) is described by coordinate operators q µ , µ = 1, 2N , that satisfy the commutation relations [24] [q µ , q ν ] = iλ 2 P Θ µν I, (6.15) with Θ the matrix given in (2.2) and λ P the Planck length. It carries a representation of the Poincaré group G under which (6.15) is covariant (the left-hand side transforms under Ad G).…”

Section: Vi2 Quantum Length In the Dfr Modelmentioning

confidence: 99%

Noncommutative Geometry of the Moyal Plane: Translation Isometries, Connes’ Distance on Coherent States, Pythagoras Equality

Martinetti

Tomassini

2013

Commun. Math. Phys.

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We study the metric aspect of the Moyal plane from Connes' noncommutative geometry point of view. First, we compute Connes' spectral distance associated with the natural isometric action of R 2 on the algebra of the Moyal plane A. We show that the distance between any state of A and any of its translated is precisely the amplitude of the translation. As a consequence, we obtain the spectral distance between coherent states of the quantum harmonic oscillator as the Euclidean distance on the plane. We investigate the classical limit, showing that the set of coherent states equipped with Connes' spectral distance tends towards the Euclidean plane as the parameter of deformation goes to zero. The extension of these results to the action of the symplectic group is also discussed, with particular emphasize on the orbits of coherent states under rotations. Second, we compute the spectral distance in the double Moyal plane, intended as the product of (the minimal unitization of) A by C 2 . We show that on the set of states obtained by translation of an arbitrary state of A, this distance is given by Pythagoras theorem. On the way, we prove some Pythagoras inequalities for the product of arbitrary unital & non-degenerate spectral triples. Applied to the Doplicher-Fredenhagen-Roberts model of quantum spacetime [DFR], these two theorems show that Connes' spectral distance and the DFR quantum length coincide on the set of states of optimal localization.

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“…We shall not discuss here the first two condition (see [21, sec.3] as well as [22]). However, in order to have a satisfactory model of spacetime, symmetries must be taken into account: in the same way as C 0 (R 2N ) carries a natural representation of the 2N dimensional Poincaré group, one may ask for an action on the quantum space, that is…”

Section: Iii3 Quantum Pointsmentioning

confidence: 99%

Minimal Length in Quantum Space and Integrations of the Line Element in Noncommutative Geometry

2012

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We question the emergence of a minimal length in quantum spacetime, comparing two notions that appeared at various points in the literature: on the one side, the quantum length as the spectrum of an operator L in the Doplicher Fredenhagen Roberts (DFR) quantum spacetime, as well as in the canonical noncommutative spacetime (θ-Minkowski); on the other side, Connes' spectral distance in noncommutative geometry. Although on the Euclidean space the two notions merge into the one of geodesic distance, they yield distinct results in the noncommutative framework. In particular on the Moyal plane, the quantum length is bounded above from zero while the spectral distance can take any real positive value, including infinity. We show how to solve this discrepancy by doubling the spectral triple. This leads us to introduce a modified quantum length d L , which coincides exactly with the spectral distance d D on the set of states of optimal localization. On the set of eigenstates of the quantum harmonic oscillator -together with their translations -d L and d D coincide asymptotically, both in the high energy and large translation limits. At small energy, we interpret the discrepancy between d L and d D as two distinct ways of integrating the line element on a quantum space. This leads us to propose an equation for a geodesic on the Moyal plane.

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Quantum Field Theory on Quantum Spacetime

Abstract: Condensed account of the Lectures delivered at the Meeting onNoncommutative Geometry in Field and String Theory, Corfu, September 18 -20, 2005.

Cited by 14 publications

References 16 publications

On Quantum Spacetime and the horizon problem

On Quantum Spacetime and the horizon problem

Noncommutative Geometry of the Moyal Plane: Translation Isometries, Connes’ Distance on Coherent States, Pythagoras Equality

Minimal Length in Quantum Space and Integrations of the Line Element in Noncommutative Geometry

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