136 research outputs found
Covariant algebraic calculation of the one-loop effective potential in non-Abelian gauge theory and a new approach to stability problem
We use our recently proposed algebraic approach for calculating the heat
kernel associated with the Laplace operator to calculate the one-loop effective
action in the non-Abelian gauge theory. We consider the most general case of
arbitrary space-time dimension, arbitrary compact simple gauge group and
arbitrary matter and assume a covariantly constant gauge field strength of the
most general form, having many independent color and space-time invariants
(Savvidy type chromomagnetic vacuum) and covariantly constant scalar fields as
a background. The explicit formulas for all the needed heat kernels and
zeta-functions are obtained. We propose a new method to study the vacuum
stability and show that the background field configurations with covariantly
constant chromomagnetic fields can be stable only in the case when more than
one independent field invariants are present and the values of these invariants
differ not greatly from each other. The role of space-time dimension is
analyzed in this connection and it is shown that this is possible only in
space-times with dimensions not less than five .Comment: 14 pages, LATeX, University of Greifswald (1994
Heat Kernel Asymptotics on Homogeneous Bundles
We consider Laplacians acting on sections of homogeneous vector bundles over
symmetric spaces. By using an integral representation of the heat semi-group we
find a formal solution for the heat kernel diagonal that gives a generating
function for the whole sequence of heat invariants. We argue that the obtained
formal solution correctly reproduces the exact heat kernel diagonal after a
suitable regularization and analytical continuation.Comment: 29 pages, Proceedings of the 2007 Midwest Geometry Conference in
Honor of Thomas P. Branso
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.Comment: 31 pages, LaTeX, no figures, Invited Lecture at the University of
Iowa, Iowa City, April, 199
Noncommutative Einstein Equations
We study a noncommutative deformation of general relativity where the
gravitational field is described by a matrix-valued symmetric two-tensor field.
The equations of motion are derived in the framework of this new theory by
varying a diffeomorphisms and gauge invariant action constructed by using a
matrix-valued scalar curvature. Interestingly the genuine noncommutative part
of the dynamical equations is described only in terms of a particular tensor
density that vanishes identically in the commutative limit. A noncommutative
generalization of the energy-momentum tensor for the matter field is studied as
well.Comment: 17 Pages, LaTeX, reference adde
Covariant Algebraic Method for Calculation of the Low-Energy Heat Kernel
Using our recently proposed covariant algebraic approach the heat kernel for
a Laplace-like differential operator in low-energy approximation is studied.
Neglecting all the covariant derivatives of the gauge field strength
(Yang-Mills curvature) and the covariant derivatives of the potential term of
third order and higher a closed formula for the heat kernel as well as its
diagonal is obtained. Explicit formulas for the coefficients of the asymptotic
expansion of the heat kernel diagonal in terms of the Yang-Mills curvature, the
potential term and its first two covariant derivatives are obtained.Comment: 19 pages, Plain TeX, 44 KB, no figure
Lack of strong ellipticity in Euclidean quantum gravity
Recent work in Euclidean quantum gravity has studied boundary conditions
which are completely invariant under infinitesimal diffeomorphisms on metric
perturbations. On using the de Donder gauge-averaging functional, this scheme
leads to both normal and tangential derivatives in the boundary conditions. In
the present paper, it is proved that the corresponding boundary value problem
fails to be strongly elliptic. The result raises deep interpretative issues for
Euclidean quantum gravity on manifolds with boundary.Comment: 14 pages, Plain Tex, 33 KB, no figure
Low-Energy Effective Action in Non-Perturbative Electrodynamics in Curved Spacetime
We study the heat kernel for the Laplace type partial differential operator
acting on smooth sections of a complex spin-tensor bundle over a generic
-dimensional Riemannian manifold. Assuming that the curvature of the U(1)
connection (that we call the electromagnetic field) is constant we compute the
first two coefficients of the non-perturbative asymptotic expansion of the heat
kernel which are of zero and the first order in Riemannian curvature and of
arbitrary order in the electromagnetic field. We apply these results to the
study of the effective action in non-perturbative electrodynamics in four
dimensions and derive a generalization of the Schwinger's result for the
creation of scalar and spinor particles in electromagnetic field induced by the
gravitational field. We discover a new infrared divergence in the imaginary
part of the effective action due to the gravitational corrections, which seems
to be a new physical effect.Comment: LaTeX, 42 page
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