Optical approach for the thermal partition function of photons

Abstract

The optical manifold method to compute the one-loop effective action in a static space-time is extended from the massless scalar field to the Maxwell field in any Feynman-like covariant gauge. The method is applied to the case of the Rindler space obtaining the same results as the point-splitting procedure. The result is free from Kabat's surface terms which instead affect the -function or heat-kernel approaches working directly in the static manifold containing conical singularities. The relation between the optical method and the direct -function approach on the Euclidean Rindler manifold is discussed both in the scalar and the photon cases. Problems with the thermodynamic self-consistency of the results obtained from the stress tensor in the case of the Rindler space are pointed out. ͓S0556-2821͑97͒01806-7͔PACS number͑s͒: 04.62.ϩv, 04.70.Dy Another important point is that the optical method produces thermodynamic quantities which agree with those obtained form the point-splitting procedure. This happens in the case of a massless scalar field conformally coupled in the Rindler wedge at least, but also, as we shall see, in the case of the photon field.In the first part of this paper we shall review the compu-*Electronic address: moretti@science.unitn.it † Electronic address: iellici@science.unitn.it PHYSICAL REVIEW D

Electromagnetic fields in Schwarzschild and Reissner-Nordström geometry: Quantum corrections to the black hole entropy

“…In this way we have shown that the first quantum correction to the black hole entropy due to the electromagnetic field, which we have computed by using the semi-classical approximation, is exactly twice the one which one has for a scalar field. This means that the anomalous contributions, which one has for the Rindler case [25][26][27], are not present in this context.…”

“…In any case, as suggested in Ref. [27], such terms are non physical and have to be discharged. All results of this section have a good limit for Q → 0 and so they are valid also for the Schwarzschild black hole, with the simple substitution r h = 2M , r − = 0.…”

“…In the Rindler space, the contribution to the entropy due to scalar and also higher spin fields have been considered in Refs. [25][26][27], where in particular it has been shown that, depending on the method of calculation used, the contribution of the electromagnetic field is not just twice the scalar one, but it contains some unexpected anomalous surface terms.…”

Local ζ-Function Techniques vs. Point-Splitting Procedure: A Few Rigorous Results

“…Formal proofs concerning the effective Lagrangian in more general cases can be found in [BD82]. Recently, the local approaches have proven to give untrivial results if compared with global approaches based on heat kernel procedures [ZCV96,IM96,MI97] whenever the manifold contains singularities physically relevant. Just as conjectured by Wald, methods based on local ζ function have been found out which are able to regularize and renormalize the one-loop averaged squared field fluctuations [IM98] and the one-loop stress tensor [Mo97a] directly (anyhow, a first remarkable formal attempt still appeared in [Ha77]).…”

“…Since the appearance of these papers, a large use of these techniques has been done by physicists, in particular, to compute one-loop partition functions within semiclassical approaches to the Quantum Gravity and also concerning other related areas [BCVZ96,EORBZ94,El95,Ca90]. After the fundamental works cited above, many efforts have been spent in studying the black hole entropy and related problems by these approaches (for recent results see [CKVZ,MI97,IM96]). The local ζ-function approach differs from integrated ζ function approaches because the former defines quantities which may depend on the point on the manifold and thus can be compared with analogue regularized and renormalized quantities produced from different more usual local approaches as the point-splitting one [BD82,Fu91].…”