36,008 research outputs found
Mathematical Tools for Calculation of the Effective Action in Quantum Gravity
We review the status of covariant methods in quantum field theory and quantum
gravity, in particular, some recent progress in the calculation of the
effective action via the heat kernel method. We study 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 without boundary. We develop a manifestly covariant method for
computation of the heat kernel asymptotic expansion as well as new algebraic
methods for calculation of the heat kernel for covariantly constant background,
in particular, on homogeneous bundles over symmetric spaces, which enables one
to compute the low-energy non-perturbative effective action.Comment: 71 pages, 2 figures, submitted for publication in the Springer book
(in preparation) "Quantum Gravity", edited by B. Booss-Bavnbek, G. Esposito
and M. Lesc
Distribution Functions for Random Variables for Ensembles of positive Hermitian Matrices
Distribution functions for random variables that depend on a parameter are
computed asymptotically for ensembles of positive Hermitian matrices. The
inverse Fourier transform of the distribution is shown to be a Fredholm
determinant of a certain operator that is an analogue of a Wiener-Hopf
operator. The asymptotic formula shows that up to the terms of order ,
the distributions are Gaussian
Hypoellipticity: Geometrization and Speculation
To any finite collection of smooth real vector fields in we
associate a metric in the phase space . The relation between the
asymptotic behavior of this metric and hypoellipticity of , in the
smooth, real analytic, and Gevrey categories, is explored
Gravitational Waves in Open de Sitter Space
We compute the spectrum of primordial gravitational wave perturbations in
open de Sitter spacetime. The background spacetime is taken to be the
continuation of an O(5) symmetric instanton saddle point of the Euclidean no
boundary path integral. The two-point tensor fluctuations are computed directly
from the Euclidean path integral. The Euclidean correlator is then analytically
continued into the Lorentzian region where it describes the quantum mechanical
vacuum fluctuations of the graviton field. Unlike the results of earlier work,
the correlator is shown to be unique and well behaved in the infrared. We show
that the infrared divergence found in previous calculations is due to the
contribution of a discrete gauge mode inadvertently included in the spectrum.Comment: 17 pages, compressed and RevTex file, including one postscript figure
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