Nonperturbative methods for calculating the heat kernel

We study the low-energy approximation for calculation of the heat kernel which is determined by the strong slowly varying background fields in strongly curved quasi-homogeneous manifolds. A new covariant algebraic approach, based on taking into account a finite number of low-order covariant derivatives of the background fields and neglecting all covariant derivatives of higher orders, is proposed. It is shown that a set of covariant differential operators together with the background fields and their low-order derivatives generates a finite dimensional Lie algebra. This algebraic structure can be used to present the heat semigroup operator in the form of an average over the corresponding Lie group. Closed covariant formulas for the heat kernel diagonal are obtained. These formulas serve, in particular, as the generating functions for the whole sequence of the Hadamard-\-Minakshisundaram-\-De~Witt-\-Seeley coefficients in all symmetric spaces.Comment: 15 pages, Plain TeX, 42 KB, no figures; Talk given at the Int. Conf. `Global Analysis, Differential Geometry and Lie Algebras', Thessaloniki, Greece, December, 1994; to be published in `Algebras, Groups and Geometries

Heat Kernel Asymptotics on Symmetric Spaces

We develop a new method for the calculation of the heat trace asymptotics of the Laplacian on symmetric spaces that is based on a representation of the heat semigroup in form of an average over the Lie group of isometries and obtain a generating function for the whole sequence of all heat invariants.Comment: 20 pages, Proceedings of the Conference "Midwest Geometry Conference 2006", University of Oklahoma, Norman, OK, May 5-7, 2006; References are update

Covariant approximation schemes for calculation of the heat kernel in quantum field theory

This paper is an overview on our recent results in the calculation of the heat kernel in quantum field theory and quantum gravity. We introduce a deformation of the background fields (including the metric of a curved spacetime manifold) and study various asymptotic expansions of the heat kernel diagonal associated with this deformation. Especial attention is payed to the low-energy approximation corresponding to the strong slowly varying background fields. We develop a new covariant purely algebraic approach for calculating the heat kernel diagonal in low-energy approximation by taking into account a finite number of low-order covariant derivatives of the background fields, and neglecting all covariant derivatives of higher orders. Then there exist a set of covariant differential operators that together with the background fields and their low-order derivatives generate a finite dimensional Lie algebra. In the zeroth order of the low-energy perturbation theory, determined by covariantly constant background, we use this algebraic structure to present the heat operator in the form of an average over the corresponding Lie group. This simplifies considerably the calculations and allows to obtain closed explicitly covariant formulas for the heat kernel diagonal. These formulas serve as the generating functions for the whole sequence of the Hadamard-Minakshisundaram- De Witt-Seeley coefficients in the low-energy approximation.Comment: 19 pages, Plain TeX, 51 KB, no figure

A method for calculating the heat kernel for manifolds with boundary

The covariant technique for calculating the heat kernel asymptotic expansion for an elliptic differential second order operator is generalized to manifolds with boundary. The first boundary coefficients of the asymptotic expansion which are proportional to $t^{1/2}$ and $t^{3/2}$ are calculated. Our results coincide with completely independent results of previous authors.Comment: 11 pages, Plain TeX, 29 KB, no figures, published in Yadernaya Fizika, 56 (1993) 245, transl. in: Sov. J. of Nucl. Phy

Heat Trace and Functional Determinant in One Dimension

We study the spectral properties of the Laplace type operator on the circle. We discuss various approximations for the heat trace, the zeta function and the zeta-regularized determinant. We obtain a differential equation for the heat kernel diagonal and a recursive system for the diagonal heat kernel coefficients, which enables us to find closed approximate formulas for the heat trace and the functional determinant which become exact in the limit of infinite radius. The relation to the generalized KdV hierarchy is discussed as well.Comment: 28 page

The heat kernel approach for calculating the effective action in quantum field theory and quantum gravity

A short informal overview about recent progress in the calculation of the effective action in quantum gravity is given. I describe briefly the standard heat kernel approach to the calculation of the effective action and discuss the applicability of the Schwinger - De Witt asymptotic expansion in the case of strong background fields. I propose a new ansatz for the heat kernel that generalizes the Schwinger - De Witt one and is always valid. Then I discuss the general structure of the asymptotic expansion and put forward some approximate explicitly covariant methods for calculating the heat kernel, namely, the high-energy approximation as well as the low-energy one. In both cases the explicit formulae for the heat kernel are given.Comment: 21 pages, Plain TeX, 52 KB, no figure

Non-Laplace type Operators on Manifolds with Boundary

We study second-order elliptic partial differential operators acting on sections of vector bundles over a compact manifold with boundary with a non-scalar positive definite leading symbol. Such operators, called non-Laplace type operators, appear, in particular, in gauge field theories, string theory as well as models of non-commutative gravity theories, when instead of a Riemannian metric there is a matrix valued self-adjoint symmetric two-tensor that plays the role of a ``non-commutative'' metric. It is well known that there is a small-time asymptotic expansion of the trace of the corresponding heat kernel in half-integer powers of time. We initiate the development of a systematic approach for the explicit calculation of these coefficients, construct the corresponding parametrix of the heat equation and compute explicitly the first two heat trace coefficients.Comment: Introduction and bibliography significantly expanded. 39 pages, no figures, to be published in: Proceedings of the Workshop ``Krzysztof Wojciechowski 50 years - Analysis and Geometry of Boundary Value Problems'', Roskilde, Denmark, 20-22 May, 2005; Minor cosmetic changes in version

Dynamical Systems on Leibniz Algebroids

In this paper we study the differential systems on Leibniz algebroids. We introduce a class of almost metriplectic manifolds as a special case of Leibniz manifolds. Also, the notion of almost metriplectic algebroid is introduced. These types of algebroids are used in the presentation of associated differential systems. We give some interesting examples of differential systems on algebroids and the orbits of the solutions of corresponding systems are established.Comment: 14 pages, 6 figures, the paper will be presented at The 5-th Conference of Balkan Society of Geometers, August 29-September 2, 2005 Mangalia, Romani

On the Gravitationally Induced Schwinger Mechanism

In this paper we will present very recent results obtained in the ambit of quantum electrodynamics in curved spacetime. We utilize a newly developed non-perturbative heat kernel asymptotic expansion on homogeneous Abelian bundles over Riemannian manifolds in order to compute the one-loop effective action for scalar and spinor fields in curved spacetime under the influence of a strong covariantly constant electromagnetic field. In this framework we derived, in particular, the gravitational corrections, up to linear terms in Riemannian curvature, to Schwinger's result for the creation of particles in a strong electric field.Comment: 7 pages, LaTeX file. To appear in the proceedings of the Conference "QFEXT09: Quantum Field Theory Under the Influence of External Conditions", Norman OK, September 21 - September 25, 200

Investigation of Relationship Between High-Energy X-ray Sources and Photospheric and Helioseismic Impacts of X1.8 Solar Flare of October 23, 2012

The X-class solar flare of October 23, 2012, generated continuum photospheric emission and a strong helioseismic wave ("sunquake") that points to an intensive energy release in the dense part of the solar atmosphere. We study properties of the energy release with high temporal and spatial resolutions, using photospheric data from the Helioseismic Magnetic Imager (HMI) onboard Solar Dynamics Observatory (SDO), and hard X-ray observations made by the Ramaty High-Energy Solar Spectroscopic Imager (RHESSI). For this analysis we use level-1 HMI data (filtergrams), obtained by scanning the Fe I line (6731~\AA) with the time cadence of $\sim 3.6$ s and spatial resolution of $\sim 0.5^{\prime\prime}$ per pixel. It is found that the photospheric disturbances caused by the flare spatially coincide with the region of hard X-ray emission, but are delayed by $\lesssim 4$ seconds. This delay is consistent with predictions of the flare hydrodynamics RADYN models. However, the models fail to explain the magnitude of variations observed by the HMI. The data indicate that the photospheric impact and helioseismic wave might be caused by the electron energy flux substantially higher than that in the current flare radiative hydrodynamic models.Comment: 23 pages, 7 figure