Local $ζ$-function techniques vs point-splitting procedure: a few rigorous results

Details Local $ζ$-function techniques vs point-splitting procedure: a few rigorous results Local $ζ$-function techniques vs point-splitting procedure: a few rigorous results

1998-05-26

arxiv.org/abs/gr-qc/9805091v3

Commun.Math.Phys. 201 (1999) 327-363

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

40 pages, latex, no figures, shortened version, some previous Comments and minor errors corrected, final version accepted for

Scientific paper

10.1007/s002200050558

Some general properties of local $\zeta$-function procedures to renormalize some quantities in $D$-dimensional (Euclidean) Quantum Field Theory in curved background are rigorously discussed for positive scalar operators $-\Delta + V(x)$ in general closed $D$-manifolds, and a few comments are given for nonclosed manifolds too. A general comparison is carried out with respect to the more known point-splitting procedure concerning the effective Lagrangian and the field fluctuations. It is proven that, for $D>1$, the local $\zeta$-function and point-splitting approaches lead essentially to the same results apart from some differences in the subtraction procedure of the Hadamard divergences. It is found that the $\zeta$ function procedure picks out a particular term $w_0(x,y)$ in the Hadamard expansion. Also the presence of an untrivial kernel of the operator $-\Delta +V(x)$ may produce some differences between the two analyzed approaches. Finally, a formal identity concerning the field fluctuations, used by physicists, is discussed and proven within the local $\zeta$-function approach. This is done also to reply to recent criticism against $\zeta$-function techniques.

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