Diffusion in multiscale spacetimes

Calcagni, Gianluca

doi:10.1103/physreve.87.012123

Cited by 54 publications

(145 citation statements)

References 162 publications

(380 reference statements)

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“…(15) and (19), distinct diffusion equations with different underlying stochastic processes (as well as different stochastic processes sharing the same diffusion equation) can lead to nearly identical profiles d S (σ). As discussed in [18,57,60], multifractional spacetimes can reproduce the same profiles, too, even in versions violating ordinary Lorentz symmetries. Thus, the spectral dimension constitutes only a very rough characterization of the quantum spacetime.…”

Section: Discussionmentioning

confidence: 96%

“…5. Smooth profiles of multiscale geometries can be found in [18,57]. To find a smooth Ansatz, the simplest way is to replace σ β in Eq.…”

Section: Multiscale Diffusion Processesmentioning

confidence: 99%

“…Within analytical approaches to quantum gravity, it has been proposed that quantum effects in spacetime, foremost a dynamical dimensional change, can be captured by a modification of the Laplacian operator appearing in the classical diffusion equation [6][7][8][9][10][11][12][13]18]. We will call this new operator "generalized Laplacian" or, as done in probability theory, spatial generator.…”

Section: Introductionmentioning

confidence: 99%

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Probing the quantum nature of spacetime by diffusion

2013

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Many approaches to quantum gravity have resorted to diffusion processes to characterize the spectral properties of the resulting quantum spacetimes. We critically discuss these quantum-improved diffusion equations and point out that a crucial property, namely positivity of their solutions, is not preserved automatically. We then construct a novel set of diffusion equations with positive semidefinite probability densities, applicable to asymptotically safe gravity, Hořava-Lifshitz gravity and loop quantum gravity. These recover all previous results on the spectral dimension and shed further light on the structure of the quantum spacetimes by assessing the underlying stochastic processes. Pointing out that manifestly different diffusion processes lead to the same spectral dimension, we propose the probability distribution of the diffusion process as a refined probe of quantum spacetime.

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Section: Discussionmentioning

confidence: 96%

“…5. Smooth profiles of multiscale geometries can be found in [18,57]. To find a smooth Ansatz, the simplest way is to replace σ β in Eq.…”

Section: Multiscale Diffusion Processesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Probing the quantum nature of spacetime by diffusion

2013

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“…In this context, we focus on theories of multiscale spacetimes [4,[30][31][32][33][34][35][36][37][38][39][40][41][42][43][44]. These have been proposed either as stand-alone models of exotic geometry [31,32,40,43] or as an effective means to study, in a controlled manner, the change of dimensionality with the probed scale (known as dimensional flow 1 of these models and of their status).…”

Section: A Dimensional Flow and Multiscale Theoriesmentioning

confidence: 99%

Standard Model in multiscale theories and observational constraints

2016

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We construct and analyze the Standard Model of electroweak and strong interactions in multiscale spacetimes with (i) weighted derivatives and (ii) q-derivatives. Both theories can be formulated in two different frames, called fractional and integer picture. By definition, the fractional picture is where physical predictions should be made. (i) In the theory with weighted derivatives, it is shown that gauge invariance and the requirement of having constant masses in all reference frames make the Standard Model in the integer picture indistinguishable from the ordinary one. Experiments involving only weak and strong forces are insensitive to a change of spacetime dimensionality also in the fractional picture, and only the electromagnetic and gravitational sectors can break the degeneracy. For the simplest multiscale measures with only one characteristic time, length and energy scale t Ã , l Ã and E Ã , we compute the Lamb shift in the hydrogen atom and constrain the multiscale correction to the ordinary result, getting the absolute upper bound t Ã < 10 −23 s. For the natural choice α 0 ¼ 1=2 of the fractional exponent in the measure, this bound is strengthened to t Ã < 10 −29 s, corresponding to l Ã < 10 −20 m and E Ã > 28 TeV. Stronger bounds are obtained from the measurement of the fine-structure constant.(ii) In the theory with q-derivatives, considering the muon decay rate and the Lamb shift in light atoms, we obtain the independent absolute upper bounds t Ã < 10 −13 s and E Ã > 35 MeV. For α 0 ¼ 1=2, the Lamb shift alone yields t Ã < 10

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“…As I demonstrate in section 3.3, the rate of decay for σ > σ cl sets the scale of the large scale curvature, which is also the scale dS that characterizes the central accumulation. 5 If one possessed a model of the spectral dimension's behavior for σ < σ cl , of which there are several [15,18,19,21,22,23,29,32,34,37,38], then one could presumably also extract a scale qm from the rate of increase. Once again, as expected, based on numerical measurements alone, these three scales are characterized by dimensionless numbers.…”

Section: Spectral Dimensionmentioning

confidence: 99%

On a renormalization group scheme for causal dynamical triangulations

Cooperman

2016

Gen Relativ Gravit

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The causal dynamical triangulations approach aims to construct a quantum theory of gravity as the continuum limit of a lattice-regularized model of dynamical geometry. A renormalization group schemein concert with finite size scaling analysis-is essential to this aim. Formulating and implementing such a scheme in the present context raises novel and notable conceptual and technical problems. I explored these problems, and, building on standard techniques, suggested potential solutions in the first paper of this two-part series. As an application of these solutions, I now propose a renormalization group scheme for causal dynamical triangulations. This scheme differs significantly from that studied recently by Ambjørn, Görlich, Jurkiewicz, Kreienbuehl, and Loll.

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Diffusion in multiscale spacetimes

Cited by 54 publications

References 162 publications

Probing the quantum nature of spacetime by diffusion

Probing the quantum nature of spacetime by diffusion

Standard Model in multiscale theories and observational constraints

On a renormalization group scheme for causal dynamical triangulations

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