Zeta function regularization for a scalar field in a compact domain

Abstract

We express the zeta function associated to the Laplacian operator on $S_r^1\times M$ 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.

19 pages, no figures, to appear in J. Phys. A

References in corpus (5)

• Heat-kernel coefficients of the Laplace operator on the D-dimensional ball
• Casimir energy in the MIT bag model
• Zeta function determinant of the Laplace operator on the $D$-dimensional ball
• The thermal energy of a scalar field in a one-dimensional compact space
• Finite temperature effective potential on hyperbolic spacetimes

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• Some Developments of the Casimir Effect in $p$-Cavity of $(D+1)$-Dimensional Spacetime
• Remarks on the thermodynamics and the vacuum energy of a quantum Maxwell gas on compact and closed manifolds
• Aspects of Higher-Abelian Gauge Theories at zero and finite temperature: Topological Casimir effect, duality and Polyakov loops
• Equivalence of zeta function technique and Abel-Plana formula in regularizing the Casimir energy of hyper-rectangular cavities