Casimir effects on Riemann surfaces
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 dimensional unitary representation \(\chi\) of \(\Gamma\) can be written as a weighted average of the logarithm of the Selberg zeta-function \(Z_\Gamma\): \[ E_{\text{Casimir}} [M,\chi] = \frac{1}{4\pi} \int_1^\infty \frac{dt}{\sqrt{1-t^{-2}}} \log Z_\Gamma \biggl(\frac{t+1}{2},\chi \biggr). \] It turns out that for \(\chi =1\), \(E_{\text{Casimir}} [M,1]<0\). As properly acknowledged by the authors, an alternative proof of the main theorem in the paper can be obtained using an integral representation formula for the modified Bessel functions of the third kind, \(K_\nu (z)\), as it was nicely computed in the book by \textit{E. Elizalde, S. D. Odintsov, A. Romeo, A. A. Bytsenko} and \textit{S. Zerbini} [Zeta regularization techniques with applications, World Scientific, Singapore (1994; Zbl 1050.81500)]. In any case, the paper does a very rigorous job in stablishing the basis of the general definition of a Casimir energy on Riemann surfaces, and it has to be viewed as a first step towards an extension to more general hyperbolic cases, which could have very interesting cosmological applications.
scientific article; zbMATH DE number 1893630