arXiv subjects

W. Wachowski

Publications and source records attributed to W. Wachowski.

7 recordsLinked to original sources

arXiv2026

Multiple Mellin-Barnes integrals in Schwinger-DeWitt technique

We consider off-diagonal asymptotic series for integral kernels of functions of Laplace-type operators on curved backgrounds. These expansions are obtained by applying integral transforms to the DeWitt series for the heat kernel of the corresponding operator and thus represent a DeWitt-type series in the heat kernel coefficients with the coefficients of this expansion (which we call basis kernels) being some hypergeometric-type functions of the Synge world function. Basis kernels of a certain class of operator functions were found previously in terms of $N$-fold Mellin-Barnes integrals. In this paper we study series representations of the corresponding Mellin-Barnes integrals in both non-resonant and resonant cases and suggest a physical interpretation for the emerging series, which is related to the UV and IR properties of operator functions.

hep-th↗

arXiv2025

Pseudodifferential calculus in Schwinger--DeWitt formalism: UV and IR parts

We consider expansions for the kernels of operator functions of second-order minimal operators on a curved background. We show that the terms of these expansions originate in the ultraviolet or infrared regions. We propose a systematic approach to obtaining ultraviolet terms using term-by-term integration of the DeWitt expansion of the heat kernel. We discuss two methods for regularizing infrared divergences arising at intermediate computational steps -- using analytic continuation and introducing a mass term -- and the relationship between them.

hep-th↗

arXiv2023

Notes on conformal anomaly, nonlocal effective action and the metamorphosis of the running scale

We discuss the structure of nonlocal effective action generating the conformal anomaly in classically Weyl invariant theories in curved spacetime. By the procedure of conformal gauge fixing, selecting the metric representative on a conformal group orbit, we split the renormalized effective action into anomalous and Weyl invariant parts. A wide family of thus obtained anomalous actions is shown to include two special cases of Riegert--Fradkin--Tseytlin and Fradkin--Vilkovisky actions. Both actions are shown to be contained in the first three orders of the curvature expansion for a generic one-loop effective action obtained by covariant perturbation theory. The complementary Weyl invariant part of the action is given by the ``conformization'' of the full effective action -- restricting its argument to the conformally invariant representative of the orbit of the conformal group. This is likely to resolve a long-standing debate between the proponents of the Riegert action and adherents of the perturbation expansion for the effective action with typical nonlocal logarithmic form factors. We derive the relation between quantum stress tensors on conformally related metric backgrounds, which generalizes the known Brown-Cassidy equation to the case of nonzero Weyl tensor, and discuss applications of this relation in the cosmological model driven by conformal field theory. We also discuss the issue of renormalization group running for the cosmological and gravitational coupling constants and show that it exhibits a kind of a metamorphosis to the nonlocal form factors of the so-called partners of the cosmological and Einstein terms -- nonlocal curvature squared terms of the effective action.

hep-th↗

arXiv2019

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

We consider the heat kernel for higher-derivative and nonlocal operators in $d$-dimensional 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 Wentzel-Kramers-Brillouin (WKB) ansatz. Rather it is represented by the generalized exponential function (GEF) directly related to what is known in mathematics as the Fox-Wright $\varPsi$-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 noninteger order. Finally, we briefly discuss further applications of the GEF technique to generic higher-derivative and pseudodifferential operators in curved space-time, which might be critically important for applications of Horava-Lifshitz and other UV renormalizable quantum gravity models.

hep-th↗

arXiv2018

Heat kernel for higher-order differential operators in Euclidean space

We consider heat kernel for higher-order operators with constant coefficients in $d$-dimensio\-nal 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 $ \mathcal{E}_{\nu, \alpha}(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 non-integer order. Possible applications of the obtained results in quantum field theory and the connection with fractional calculus are discussed.

hep-th↗

arXiv2016

Differential calculus over N-graded commutative rings

The Chevalley-Eilenberg differential calculus and differential operators over N-graded commutative rings are constructed. This is a straightforward generalization of the differential calculus over commutative rings, and it is the most general case of the differential calculus over rings that is not the non-commutative geometry. Since any N-graded ring possesses the associated Z_2-graded structure, this also is the case of the graded differential calculus over Grassmann algebras and the supergeometry and field theory on graded manifolds.

math-ph↗

arXiv2014

SUSY gauge theory on graded manifolds

Lagrangian classical field theory of even and odd fields is adequately formulated in terms of fibre bundles and graded manifolds. In particular, conventional Yang-Mills gauge theory is theory of connections on smooth principal bundles, but its BRST extension involves odd ghost fields an antifields on graded manifolds. Here, we formulate Yang-Mills theory of Grassmann-graded gauge fields associated to Lie superalgebras on principal graded bundles. A problem lies in a geometric definition of odd gauge fields. Our goal is Yang--Mills theory of graded gauge fields and its BRST extension.

math-ph↗