Casimir effects on Riemann surfaces
type: Item
instance of: scholarly article
title: Casimir effects on Riemann surfaces
zbMATH Open document ID: 1027.81026
DOI: 10.1016/S0019-3577(02)90006-6
author: Nobushige Kurokawa
author: Masato Wakayama
published in: Indagationes Mathematicae. New Series
publication date: 2003-12-21T00:00:00Z
review text: The purpose of this paper is to introduce an analogue of the Casimir energy for an arbitrary compact Riemann surface, \(M\), of genus \(g\) (\(\geq 2\)), and to express it as a special value of the Mellin transform of a theta series arising from the heat kernel of the Laplacian operator on \(M\), and also as a certain weighted integral average of the logarithm of the Selberg zeta-function. The main idea for a satisfactory definition of the Casimir energy in this context is based, as the authors beautifully describe, in considering the sum of the eigenvalues of the Laplacian on \(M\), which is obtained from the Casimir element of the group PSL\(_2(\mathbb{R})\), consisting of all line fractional transformations of the upper half plane. Further, the idea is to use the Selberg trace formula as a non-commutative counterpart of the Poisson summation formula. The playground is the complex upper half plane, \(H\), and \(M\) is regarded as a domain realized in \(H\), so that the Casimir energy for \(M\) is defined by the authors as \[ E_{\text{Casimir}} [M] = \{\text{Zero-point energy for } M \} - \{\text{Zero-point energy for } H \} \] More technically, a fundamental domain for a Fuchsian group in the complex upper half plane, \(H\), is taken as a bounded domain, \(M\), and the Casimir operator on \(M\) (that is, the hyperbolic Laplacian) plays the role of field (differential) operator. The authors are interested in Fuchsian groups of the first kind, that is, discrete subgroups \(\Gamma \subset \text{PSL}_2(\mathbb{R})\) of motions of \(H\) for which there exists a fundamental domain of finite volume (it is compact iff there is no parabolic element in \(\Gamma\)). \(M\) can be realized as a form \(M=\Gamma \backslash H\), with \(\Gamma\) a strictly hyperbolic subgroup of PSL\(_2(\mathbb{R})\). The main result in the paper can be expressed as follows. The Casimir energy \(E_{\text{Casimir}} [M,\chi]\) for a domain \(M=\Gamma \backslash H\) associated with a finite dimension…
reviewed by: Emilio Elizalde
Mathematics Subject Classification ID: 81T20
Mathematics Subject Classification ID: 58J35
Mathematics Subject Classification ID: 11F72
Mathematics Subject Classification ID: 11M36
Mathematics Subject Classification ID: 30F99
zbMATH DE Number: 1893630
zbMATH Keywords: Casimir energy
zbMATH Keywords: Riemann surfaces
zbMATH Keywords: zeta functions
zbMATH Keywords: Mellin transform of a theta series
zbMATH Keywords: heat kernel
zbMATH Keywords: Selberg zeta function
zbMATH Keywords: Selberg trace formula
zbMATH Keywords: Fuchsian groups of the first kind
MaRDI profile type: Publication
cites work: Q5787728
cites work: On determinants of Laplacians on Riemann surfaces
cites work: Q5820721
cites work: Q4894948
cites work: Asymptotic behaviour of spectra of compact quotients of certain symmetric spaces
cites work: Zeta functions of Selberg's type for compact space forms of symmetric spaces of rank one
cites work: Q4733924
cites work: Q4516332
cites work: Determinants of Laplacians
cites work: Q3237356
cites work: Q4111790
cites work: A local index theorem for families of \({\bar \partial}\)-operators on punctured Riemann surfaces and a new Kähler metric on their moduli spaces
cites work: Zeta functions of Selberg's type associated with homogeneous vector bundles
description: scientific article; zbMATH DE number 1893630