ζ-function regularization and one-loop renormalization of field fluctuations in curved space-time

Abstract

Abstract:A method to regularize and renormalize the fluctuations of a quantum field in a curved background in the ζ-function approach is presented. The method produces finite quantities directly and finite scale-parametrized counterterms at most. These finite couterterms are related to the presence of a particular pole of the effective-action ζ function as well as to the heat kernel coefficients. The method is checked in several examples obtaining known or reasonable results. Finally, comments are given for as it concerns the recent proposal by Frolov et al. to get the finite Bekenstein-Hawking entropy from Sakharov's induced gravity theory. 04.62.+v, 04.70.Dy PACS number(s):

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

“…One is therefore forced to remove by hand these divergences, this nothing but the main idea of the point-splitting procedure. It is worth remarking that the definitions given in terms of ζ function and heat kernel [Ha77,Wa79,BD82,Mo97a,IM98] of the formal quantity in left hand sides of (5), (6), (7) contain an implicit infinite renormalization procedure in the sense that these are finally free from divergences.…”

“…The local ζ-function approach concerning the field fluctuations has produced also a few new results, e.g., the general form for renormalized trace of the one-loop stress tensor in the generally nonanomalous case, and several applications in symmetric spaces for general values of the parameter ξ which fixes the coupling of the field with the curvature (see [IM98]). …”

“…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]). In all examined cases, agreement with the point-splitting technique has arisen as well as, sometimes, differences with integrated heat kernel approaches.…”

Quantum scalar field on the massless(2+1)-dimensional black hole background

“…1 The coefficient 1/2ξD which appears in (13) of [27] is missprinted, and has to be corrected into 1/(4ξD − 1).…”

“…For the topological non trivial part T µν (x) Γ , it is convenient to proceed as follows. Making use of the of the eigenvalues equation for the scalar eigenfunctions 27) and the background metric form, a standard calculation for the stress tensor evaluated on the modes (5.15) leads to…”