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Title: Heat-kernel expansion on noncompact domains and a generalized zeta-function regularization procedure

Abstract

Heat-kernel expansion and zeta function regularization are discussed for Laplace-type operators with discrete spectrum in noncompact domains. Since a general theory is lacking, the heat-kernel expansion is investigated by means of several examples. It is pointed out that for a class of exponential (analytic) interactions, generically the noncompactness of the domain gives rise to logarithmic terms in the heat-kernel expansion. Then, a meromorphic continuation of the associated zeta function is investigated. A simple model is considered, for which the analytic continuation of the zeta function is not regular at the origin, displaying a pole of higher order. For a physically meaningful evaluation of the related functional determinant, a generalized zeta function regularization procedure is proposed.

Authors:
; ;  [1];  [2];  [3]
  1. Dipartimento di Fisica, Universita di Trento and Istituto Nazionale di Fisica Nucleare, Gruppo Collegato di Trento, Trento (Italy)
  2. (ICE/CSIC), Campus UAB, Facultat de Ciencies, Torre C5-Parell-2a Planta, 08193 Bellaterra, Barcelona, Spain and Institut d'Estudis Espacials de Catalunya (IEEC), Edifici Nexus, Gran Capita 2-4, 08034 Barcelona (Spain)
  3. (Italy)
Publication Date:
OSTI Identifier:
20860772
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Mathematical Physics; Journal Volume: 47; Journal Issue: 8; Other Information: DOI: 10.1063/1.2259580; (c) 2006 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; EVALUATION; EXPANSION; FUNCTIONS; HEAT; KERNELS; MATHEMATICAL OPERATORS; QUANTUM FIELD THEORY

Citation Formats

Cognola, Guido, Elizalde, Emilio, Zerbini, Sergio, Consejo Superior de Investigaciones Cientificas, Instituto de Ciencias del Espacio, and Dipartimento di Fisica, Universita di Trento and Istituto Nazionale di Fisica Nucleare, Gruppo Collegato di Trento, Trento. Heat-kernel expansion on noncompact domains and a generalized zeta-function regularization procedure. United States: N. p., 2006. Web. doi:10.1063/1.2259580.
Cognola, Guido, Elizalde, Emilio, Zerbini, Sergio, Consejo Superior de Investigaciones Cientificas, Instituto de Ciencias del Espacio, & Dipartimento di Fisica, Universita di Trento and Istituto Nazionale di Fisica Nucleare, Gruppo Collegato di Trento, Trento. Heat-kernel expansion on noncompact domains and a generalized zeta-function regularization procedure. United States. doi:10.1063/1.2259580.
Cognola, Guido, Elizalde, Emilio, Zerbini, Sergio, Consejo Superior de Investigaciones Cientificas, Instituto de Ciencias del Espacio, and Dipartimento di Fisica, Universita di Trento and Istituto Nazionale di Fisica Nucleare, Gruppo Collegato di Trento, Trento. Tue . "Heat-kernel expansion on noncompact domains and a generalized zeta-function regularization procedure". United States. doi:10.1063/1.2259580.
@article{osti_20860772,
title = {Heat-kernel expansion on noncompact domains and a generalized zeta-function regularization procedure},
author = {Cognola, Guido and Elizalde, Emilio and Zerbini, Sergio and Consejo Superior de Investigaciones Cientificas, Instituto de Ciencias del Espacio and Dipartimento di Fisica, Universita di Trento and Istituto Nazionale di Fisica Nucleare, Gruppo Collegato di Trento, Trento},
abstractNote = {Heat-kernel expansion and zeta function regularization are discussed for Laplace-type operators with discrete spectrum in noncompact domains. Since a general theory is lacking, the heat-kernel expansion is investigated by means of several examples. It is pointed out that for a class of exponential (analytic) interactions, generically the noncompactness of the domain gives rise to logarithmic terms in the heat-kernel expansion. Then, a meromorphic continuation of the associated zeta function is investigated. A simple model is considered, for which the analytic continuation of the zeta function is not regular at the origin, displaying a pole of higher order. For a physically meaningful evaluation of the related functional determinant, a generalized zeta function regularization procedure is proposed.},
doi = {10.1063/1.2259580},
journal = {Journal of Mathematical Physics},
number = 8,
volume = 47,
place = {United States},
year = {Tue Aug 15 00:00:00 EDT 2006},
month = {Tue Aug 15 00:00:00 EDT 2006}
}
  • The authors combine a zeta-function definition and heat-kernal series to derive Casimir energy expansions parametrizing the UV divergences in the presence of arbitrarily shaped smooth boundaries. Their terms, in the form of a geometrical object times a divergence, allow for drawing conclusions on the scale dependence and on the finiteness of the vacuum energy when limiting surfaces have been introduced. Different behaviors are found depending, among other factors, on the even or odd character of the space dimension. A number of controversial points are cleared up and some misstatements in the literature are properly rigorized.
  • A generalized zeta-function regularization is considered for computation of the superdeterminant of a matrix of differential operators in the case of a continuous spectrum. It is shown that in the general case the method makes use of a complete set of eigenfunctions of a certain system of integrodifferential equations.
  • The generalized {zeta}-function techniques will be utilized to investigate the Casimir energy for the transverse oscillations of a piecewise uniform closed string. We find that the {zeta}-function regularization method can lead straightforwardly to a correct result.
  • We propose regularization with the generalized zeta function in order to calculate one-loop corrections to the original action in quantum field theory. The Schwinger proper-time formalism is used to define the zeta function. The method is illustrated by calculations of the one-loop corrections to the Maxwell Lagrangian in QED with anomalous moments.
  • The {zeta}-function regularization method is used to evaluate the renormalized effective action for massless conformally coupling scalar fields propagating in a closed Friedmann spacetime perturbed by a small rotation. To the second order of the rotational parameter in the model spacetime the analytic form of the effective action is obtained with the help of the Schwinger perturbation formula. After investigating the time evolution of the rotational parameter we find that the quantum field effect can produce an effect which damps the cosmological rotation in the early universe. {copyright} 1997 Academic Press, Inc.