Heat kernel for flat generalized Laplacians with anisotropic scaling

Mamiya, Arthur A.; Pinzul, A.

doi:10.1063/1.4882157

Cited by 10 publications

(14 citation statements)

References 27 publications

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“…We explicitly verified that the isotropic limit z = 1 of our heat-kernel coefficients correctly reproduces the expansion (2.18) written in terms of ADM fields. Moreover, our formulas agree with special cases for z = 2 and d = 1 studied in [34], the results for the conformal anomaly in z = 2, d = 2 obtained in [35] and the scaling analysis for anisotropic Laplace operators in flat space [36]. This match provides a highly non-trivial test for the results obtained in this work.…”

Section: Terms Containing Spatial Derivatives Of the Lapse Functionsupporting

confidence: 85%

Covariant computation of effective actions in Hořava-Lifshitz gravity

D'Odorico

Goossens

Saueressig

2015

J. High Energ. Phys.

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We initiate the systematic computation of the heat-kernel coefficients for Laplacian operators obeying anisotropic dispersion relations in curved spacetime. Our results correctly reproduce the limit where isotropy is restored and special anisotropic cases considered previously in the literature. Subsequently, the heat kernel is used to derive the scalar-induced one-loop effective action and beta functions of Hořava-Lifshitz gravity. We identify the Gaussian fixed point which is supposed to provide the UV completion of the theory. In the present setting, this fixed point acts as an infrared attractor for the renormalization group flow of Newton's constant and the high-energy phase of the theory is screened by a Landau pole. We comment on the consequences of these findings for the renormalizability of the theory.2 For a recent interesting analysis including mixed derivative terms see [22,23].

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Section: Terms Containing Spatial Derivatives Of the Lapse Functionsupporting

confidence: 85%

Covariant computation of effective actions in Hořava-Lifshitz gravity

D'Odorico

Goossens

Saueressig

2015

J. High Energ. Phys.

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“…However, in this paper the parameter ν is bounded by the interval (1/2, 1) and no expressions are given in terms of the Fox-Wright Ψ -functions. Relatively recently, analogous expressions for the case d = 3 applicable to models of the Hoȓava-Lifshitz type were obtained in the work [56].…”

Section: Generalized Exponential Functions and Their Propertiesmentioning

confidence: 76%

Heat kernel for higher-order differential operators and generalized exponential functions

Barvinsky¹,

Pronin²,

Wachowski³

2019

Phys. Rev. D

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We consider the heat kernel for higher-derivative and nonlocal operators in ddimensional Euclidean space-time and its asymptotic behavior. As a building block for operators of such type, we consider the heat kernel of the minimal operator generic power of the Laplacian and show that it is given by the expression essentially different from the conventional exponential WKB ansatz. Rather it is represented by the generalized exponential function (GEF) directly related to what is known in mathematics as the Fox-Wright Ψ -functions and Fox H-functions. The structure of its essential singularity in the proper time parameter is different from that of the usual exponential ansatz, which invalidated previous attempts to directly generalize the Schwinger-DeWitt heat kernel technique to higher-derivative operators. In particular, contrary to the conventional exponential decay of the heat kernel in space, we show the oscillatory behavior of GEF for higher-derivative operators. We give several integral representations for the generalized exponential function, find its asymptotics and semiclassical expansion, which turns out to be essentially different for local operators and nonlocal operators of non-integer order. Finally, we briefly discuss further applications of the GEF technique to generic higher-derivative and pseudo-differential operators in curved space-time, which might be critically important for applications of Hoȓava-Lifshitz and other UV renormalizable quantum gravity models.

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“…Relatively recently, analogous expressions for the case d = 3 applicable to models of the Hoȓava-Lifshitz type were obtained in the work [24].…”

Section: Operators Of the Formmentioning

confidence: 76%

“…The heat kernel method also can be used in the investigation of higher order operators. This is important for regularization by means higher covariant derivatives, as well as for theories with higher derivatives that have attracted much interest in recent years, namely, R 2 -gravity [20], nonlocal and superrenormalizable theories [21,22] and Hoȓava-Lifshitz type theories [23,24]. One of the possible extensions of the standard heat kernel method was proposed in [12].…”

Section: Introductionmentioning

confidence: 99%

Heat kernel for higher-order differential operators in Euclidean space

Wachowski,

Pronin

2018

Preprint

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We consider heat kernel for higher-order operators with constant coefficients in ddimensional Euclidean space and its asymptotic behavior. For arbitrary operators which are invariant with respect to O(d)-rotations we obtain exact analytical expressions for the heat kernel and Green functions in the form of infinite series in Fox-Wright psi functions and Fox H-functions. We investigate integro-differential relations and the asymptotic behavior of the functions E ν,α (z), in terms of which the heat kernel of O(d)invariant operators are expressed. It is shown that the obtained expressions are well defined for non-integer values of space dimension d, as well as for operators of noninteger order. Possible applications of the obtained results in quantum field theory and the connection with fractional calculus are discussed.

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Heat kernel for flat generalized Laplacians with anisotropic scaling

Cited by 10 publications

References 27 publications

Covariant computation of effective actions in Hořava-Lifshitz gravity

Covariant computation of effective actions in Hořava-Lifshitz gravity

Heat kernel for higher-order differential operators and generalized exponential functions

Heat kernel for higher-order differential operators in Euclidean space

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