arXiv subjects

Ivan G. Avramidi

Publications and source records attributed to Ivan G. Avramidi.

50 records · Page 3Linked to original sources

arXiv1997

Covariant techniques for computation of the heat kernel

The heat kernel associated with an elliptic second-order partial differential operator of Laplace type acting on smooth sections of a vector bundle over a Riemannian manifold, is studied. A general manifestly covariant method for computation of the coefficients of the heat kernel asymptotic expansion is developed. The technique enables one to compute explicitly the diagonal values of the heat kernel coefficients, so called Hadamard-Minackshisundaram-De Witt-Seeley coefficients, as well as their derivatives. The elaborated technique is applicable for a manifold of arbitrary dimension and for a generic Riemannian metric of arbitrary signature. It is very algorithmic, and well suited to automated computation. The fourth heat kernel coefficient is computed explicitly for the first time. The general structure of the heat kernel coefficients is investigated in detail. On the one hand, the leading derivative terms in all heat kernel coefficients are computed. On the other hand, the generating functions in closed covariant form for the covariantly constant terms and some low-derivative terms in the heat kernel coefficients are constructed by means of purely algebraic methods. This gives, in particular, the whole sequence of heat kernel coefficients for an arbitrary locally symmetric space.

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arXiv1997

Singularities of Green functions of the products of the Laplace type operators

The structure of diagonal singularities of Green functions of partial differential operators of even order acting on smooth sections of a vector bundle over a Riemannian man ifold is studied. A special class of operators formed by the products of second-order operators of Laplace type defined with the help of a unique Riemannian metric and a unique bundle connection but with different potential terms is investigated. Explicit simple formulas for singularities of Green functions of such operators in terms of the usual heat kernel coefficients are obtained.

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arXiv1997

Universal Functions in Euclidean Quantum Gravity

A key problem in the attempt to quantize the gravitational field is the choice of boundary conditions. These are mixed, in that spatial and normal components of metric perturbations obey different sets of boundary conditions. In the covariant quantization scheme this leads to a boundary operator involving both normal and tangential derivatives of metric perturbations. On studying the corresponding heat-kernel asymptotics, one finds that universal, tensorial, nonpolynomial structures contribute through the integrals over the boundary of linear combinations of all geometric invariants of the problem. These universal functions are independent of conformal rescalings of the background metric, and they might lead to a deep revolution in the current understanding of quantum gravity.

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arXiv1996

Axial Gauge in Euclidean Quantum Gravity

The axial gauge is applied to the analysis of Euclidean quantum gravity on manifolds with boundary. A set of boundary conditions which are completely invariant under infinitesimal diffeomorphisms require that spatial components of metric perturbations should vanish at the boundary, jointly with all components of the ghost one-form and of the gauge-averaging functional. If the latter is taken to be of the axial type, all components of metric perturbations obey Dirichlet conditions, and all ghost modes are forced to vanish identically. The one-loop divergence coincides with the contribution resulting from three-dimensional transverse-traceless perturbations.

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arXiv1996

Boundary Operators in Euclidean Quantum Gravity

Gauge-invariant boundary conditions in Euclidean quantum gravity can be obtained by setting to zero at the boundary the spatial components of metric perturbations, and a suitable class of gauge-averaging functionals. This paper shows that, on choosing the de Donder functional, the resulting boundary operator involves projection operators jointly with a nilpotent operator. Moreover, the elliptic operator acting on metric perturbations is symmetric. Other choices of mixed boundary conditions, for which the normal components of metric perturbations can be set to zero at the boundary, are then analyzed in detail. Last, the evaluation of the 1-loop divergence in the axial gauge for gravity is obtained. Interestingly, such a divergence turns out to coincide with the one resulting from transverse-traceless perturbations.

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arXiv1996

A model of stable chromomagnetic vacuum in higher-dimensional Yang-Mills theory

We study the effective action in Euclidean Yang-Mills theory with a compact simple gauge group in one-loop approximation assuming a covariantly constant gauge field strength as a background. For groups of higher rank and spacetimes of higher dimensions such field configurations have many independent color components taking values in Cartan subalgebra and many ``magnetic fields'' in each color component. In our previous investigation it was shown that such background is stable in dimensions higher than four provided the amplitudes of ``magnetic fields'' do not differ much from each other. In present paper we calculate exactly the relevant zeta-functions in the case of equal amplitudes of ``magnetic fields''. For the case of two ``magnetic fields'' with equal amplitudes the behavior of the effective action is studied in detail. It is shown that in dimensions $d=4,5,6,7$ $({\rm mod}\, 8)$, the perturbative vacuum is metastable, i.e., it is stable in perturbation theory but the effective action is not bounded from below, whereas in dimensions $d=9,10,11$ $({\rm mod}\, 8)$ the perturbative vacuum is absolutely stable. In dimensions $d=8$ $({\rm mod}\, 8)$ the perturbative vacuum is stable for small values of coupling constant but becomes unstable for large coupling constant leading to the formation of a non-perturbative stable vacuum with nonvanishing ``magnetic fields''. The critical value of the coupling constant and the amplitudes of the vacuum ``magnetic fields'' are evaluated exactly.

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arXiv1996

Covariant methods for the calculation of the effective action in quantum field theory and investigation of higher-derivative quantum gravity

The main results are: 1. A manifestly covariant technique for the calculation of De Witt coefficients is elaborated; 2. The coefficients $a_3$ and $a_4$ are calculated; 3. Covariant methods for the study of the nonlocal structure of the effective action are developed. The terms of first and second order in background fields in De Witt coefficients are calculated. The summation of these terms is carried out and nonlocal covariant expression for the Green function, the heat kernel and the effective action are obtained. It is shown that in the conform-invariant case the Green function is finite. A finite effective action in the conform-invariant case of massless scalar field in two-dimensional space is obtained; 4. The off-shell one-loop divergences of the effective action in arbitrary covariant gauge as well as those of the `unique' effective action in higher-derivative quantum gravity are calculated; The ultraviolet asymptotics of coupling constants are found. It is shown that in the `physical' region of coupling constants (no tachyons!) the conformal sector has `zero-charge' behavior contrary to previous authors. This means that the theory goes beyond the limits of weak conformal coupling at higher energies. In other words, the condition of conformal stability of flat background is incompatible with the asymptotic freedom in the conformal sector. There is a stable non-flat ground state but only in the case of positive definite Euclidean action. In this case the theory is asymptotically free both in tensor and conformal sectors. The off-shell one-loop effective action in arbitrary covariant gauge and the `unique' effective action on De Sitter background are calculated.

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arXiv1996

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.

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arXiv1995

Algorithms for the calculation of the heat kernel coefficients

We present a brief overview of several approaches for calculating the local asymptotic expansion of the heat kernel for Laplace-type operators. The different methods developed in the papers of both authors some time ago are described in more detail.

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arXiv1995

A new algebraic approach for calculating the heat kernel in quantum gravity

It is shown that the heat kernel operator for the Laplace operator on any covariantly constant curved background, i.e. in symmetric spaces, may be presented in form of an averaging over the Lie group of isometries with some nontrivial measure. Using this representation the heat kernel diagonal, i.e. the heat kernel in coinciding points is obtained. Related topics concerning the structure of symmetric spaces and the calculation of the effective action are discussed.

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arXiv1995

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.

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arXiv1995

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.

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arXiv1995

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.

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arXiv1995

The heat kernel on symmetric spaces via integrating over the group of isometries

A new algebraic approach for calculating the heat kernel for the Laplace operator on any Riemannian manifold with covariantly constant curvature is proposed. It is shown that the heat kernel operator can be obtained by an averaging over the Lie group of isometries. The heat kernel diagonal is obtained in form of an integral over the isotropy subgroup.

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