Asymptotic and Exact Expansions of Heat Traces

Eckstein, Michał; Zając, Artur

doi:10.1007/s11040-015-9197-2

Cited by 10 publications

(12 citation statements)

References 69 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In particular, the analytic techniques involved are based on the derivation of the terms of the asymptotic expansion of the spectral action for Robertson-Walker metrics discussed in the previous sections, based on the Feynman-Kac formula and Brownian bridge integrals as in [5]. In the case of Robertson-Walker metrics that are round 4-spheres, the result we obtain can also be obtained using the technique of [1], based on the results of [28] counting the contributions of the different levels in the fractal structure, and on results on the heat kernel on fractals, [13], [14], [15].…”

Section: 1mentioning

confidence: 84%

Bell Polynomials and Brownian Bridge in Spectral Gravity Models on Multifractal Robertson–Walker Cosmologies

Fathizadeh

Kafkoulis

Marcolli

2020

Ann. Henri Poincaré

View full text Add to dashboard Cite

We obtain an explicit formula for the full expansion of the spectral action on Robertson-Walker spacetimes, expressed in terms of Bell polynomials, using Brownian bridge integrals and the Feynman-Kac formula. We then apply this result to the case of multifractal Packed Swiss Cheese Cosmology models obtained from an arrangement of Robertson-Walker spacetimes along an Apollonian sphere packing. Using Mellin transforms, we show that the asymptotic expansion of the spectral action contains the same terms as in the case of a single Robertson-Walker spacetime, but with zeta-regularized coefficients, given by values at integers of the zeta function of the fractal string of the radii of the sphere packing, as well as additional log-periodic correction terms arising from the poles (off the real line) of this zeta function. Contents

show abstract

Section: 1mentioning

confidence: 84%

Bell Polynomials and Brownian Bridge in Spectral Gravity Models on Multifractal Robertson–Walker Cosmologies

Fathizadeh

Kafkoulis

Marcolli

2020

Ann. Henri Poincaré

View full text Add to dashboard Cite

show abstract

“…So, it would be very interesting to compare these two approaches on the level of matter dynamics. Another related issue is the following one: The spectral action is a residue coming from a heat kernel expansion [26][27][28], but can be obtained also from cancellation of anomalies [29][30][31][32] or a ζ-function regularization [33]. The presence of spaces, such as the one described here, with a built-in cutoff, alter profoundly the field theory, and in particular the UV dynamics of bosons [34,35].…”

Section: Discussionmentioning

confidence: 99%

Dimensional deception from noncommutative tori: An alternative to the Horava-Lifshitz model

Lizzi

Pinzul

2017

Phys. Rev. D

View full text Add to dashboard Cite

We study the dimensional aspect of the geometry of quantum spaces. Introducing a physically motivated notion of the scaling dimension, we study in detail the model based on a fuzzy torus. We show that for a natural choice of a deformed Laplace operator, this model demonstrates quite non-trivial behaviour: the scaling dimension flows from 2 in IR to 1 in UV. Unlike another model with the similar property, the socalled Horava-Lifshitz model, our construction does not have any preferred direction. The dimension flow is rather achieved by a rearrangement of the degrees of freedom.

show abstract

“…It is also worth noting (see [33,59] for the full story) that if a positive operator T admits an expansion of the form (12), then the associated spectral zeta function ζ T defined as…”

Section: B Dimension Spectrum From Asymptotic Expansionmentioning

confidence: 99%

Spectral dimensions and dimension spectra of quantum spacetimes

Eckstein

Trześniewski

2020

Phys. Rev. D

Self Cite

View full text Add to dashboard Cite

Different approaches to quantum gravity generally predict that the dimension of spacetime at the fundamental level is not 4. The principal tool to measure how the dimension changes between the IR and UV scales of the theory is the spectral dimension. On the other hand, the noncommutative-geometric perspective suggests that quantum spacetimes ought to be characterized by a discrete complex set-the dimension spectrum. We show that these two notions complement each other and the dimension spectrum is very useful in unraveling the UV behavior of the spectral dimension. We perform an extended analysis highlighting the trouble spots and illustrate the general results with two concrete examples: the quantum sphere and the κ-Minkowski spacetime, for a few different Laplacians. In particular, we find that the spectral dimensions of the former exhibit log-periodic oscillations, the amplitude of which decays rapidly as the deformation parameter tends to the classical value. In contrast, no such oscillations occur for either of the three considered Laplacians on the κ-Minkowski spacetime.

show abstract

Asymptotic and Exact Expansions of Heat Traces

Cited by 10 publications

References 69 publications

Bell Polynomials and Brownian Bridge in Spectral Gravity Models on Multifractal Robertson–Walker Cosmologies

Bell Polynomials and Brownian Bridge in Spectral Gravity Models on Multifractal Robertson–Walker Cosmologies

Dimensional deception from noncommutative tori: An alternative to the Horava-Lifshitz model

Spectral dimensions and dimension spectra of quantum spacetimes

Contact Info

Product

Resources

About