Quantum Fields in Hyperbolic Space-Times with Finite Spatial Volume

Abstract

The one-loop effective action for a massive self-interacting scalar field is investigated in $4$-dimensional ultrastatic space-time $R\times H^3/\Gamma$, $H^3/\Gamma$ being a non-compact hyperbolic manifold with finite volume. Making use of the Selberg trace formula, the $\zeta$-function related to the small disturbance operator is constructed. For an arbitrary gravitational coupling, it is found that $\zeta (s)$ has a simple pole at $s=0$. The one-loop effective action is analysed by means of proper-time regularisations and the one-loop divergences are explicitly found. It is pointed out that, in this special case, also $\zeta$-function regularisation requires a divergent counterterm, which however is not necessary in the free massless conformal invariant coupling case. Finite temperature effects are studied and the high-temperature expansion is presented. A possible application to the problem of the divergences of the entanglement entropy for a free massless scalar field in a Rindler-like space-time is briefly discussed.

13 pages, LaTex. The contribution of hyperbolic elements has been added. Other minor corrections and references

References in corpus (6)

• On Geometric Entropy
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• Heat-kernels and functional determinants on the generalized cone
• Black Hole Entropy: Off-Shell vs On-Shell
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• Effective Finite Temperature Partition Function for Fields on Non-Commutative Flat Manifolds
• Effective action for scalar fields and generalised zeta-function regularisation
• One-loop effective potential for scalar and vector fields on higher dimensional noncommutative flat manifolds
• Generalized Zeta Function Regularization and the Multiplicative Anomaly
• Aspects and applications of quantum field theory on spaces with conical singularities