Zeta function regularization for a scalar field in a compact domain

Abstract

Abstract. We express the zeta function associated to the Laplacian operator on S 1 r × M in terms of the zeta function associated to the Laplacian on M , where M is a compact connected Riemannian manifold. This gives formulas for the partition function of the associated physical model at low and high temperature for any compact domain M . Furthermore, we provide an exact formula for the zeta function at any value of r when M is a D-dimensional box or a D-dimensional torus; this allows a rigorous calculation of the zeta invariants and the analysis of the main thermodynamic functions associated to the physical models at finite temperature.

Finite temperature Casimir energy in closed rectangular cavities: a rigorous derivation based on a zeta function technique

“…Hence if ζ(0) = −N , the Casimir energy turns out to be independent of µ or λ and can be calculated by using (2.11) in contrast to the usual prescription F = − 1 2β ζ ′ (0). The expression for log Z (2.9), with the presence of ω k = 0 terms has been obtained in [14]. However, in [14], the discrepancy between Z and the thermodynamic partition function Z was not emphasized.…”

“…Among the different geometries of space that have been under consideration, rectangular cavities of different dimensions are among the most extensively studied [15,17,18,19,20,21,22,23,24,25,26,27,28,29,30], partly due to the simple geometry and also the well-developed mathematical tools. Various aspects of the effect, such as the low and high temperature expansions of the Casimir energy or force [12,13,14,17,31,32], the attractive or repulsive nature of the Casimir force [15,17,18,24,26,33], the effect of extra dimension [14,15,17,24], etc, have been discussed.…”

Zeta invariants for Dirichlet series

“…The case c = 0 can be dealt with directly by using the Kronecker first limit formula [Lang 1987;Ortenzi and Spreafico 2004] or by taking the limit of the result obtained in Section 2, and this shows that Theorem 1 reduces continuously to the classical Kronecker formula. We get …”

“…In particular, the Poisson summation formula, namely the Fourier expansion of the theta function, has suitable generalizations to the multidimensional case (see [Chandrasekharan 1985, XI.2, 3], for example). Using these formulas and properties of special functions, it is possible to compute the main zeta invariants for multiple series of Epstein type, also called multiple Eisenstein zeta functions (see [Weil 1976;Ortenzi and Spreafico 2004;Elizalde et al 1994;Cassou-Noguès 1990] and references therein). Moving up to the zeta functions associated to series of Dirichlet type (namely when the sums are over ‫ގ‬ k 0 ), the main difficulty is precisely the lack of a formula of Poisson type.…”

Some developments of the Casimir effect in p-cavity of (D + 1)-dimensional space–time

“…[46][47][48][49][50][51][52][53][54][55][56] The zeta function technique, 46,47,50,51,53,54 which can be traced back to G. H. Hardy,57,58 is used to be regarded as an elegant and unique regularization method 59,60 different from other ones such as frequency cut-off method (see e.g. Ref.…”