Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
1996-05-29
Class.Quant.Grav. 14 (1997) 615-627
Physics
High Energy Physics
High Energy Physics - Theory
13 pages, LaTex. The contribution of hyperbolic elements has been added. Other minor corrections and references
Scientific paper
10.1088/0264-9381/14/3/008
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.
Bytsenko Andrei
Cognola Guido
Zerbini Sergio
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