v 1 2 6 M ay 1 99 8 preprint - Local ζ - function techniques vs point - splitting proce - dure : a few rigorous results

Abstract: Some general properties of local ζ-function procedures to regularize (and renormalize) some quantities in D-dimensional (Euclidean) Quantum Field Theory in curved background are rigorously discussed. A comparison is carried out with respect the more known point-splitting procedure. The results are given in general closed manifolds and a few comments are given for nonclosed manifolds too. Previous results, concerning the relation between these two different approaches to renormalize the effective Lagrangian in D = 4 for fields operators −∆+m2, are generalized to any integer D > 1 for positive operators −∆+V (x). This generalized comparison is carried out concerning the averaged squared field fluctuations of a real scalar quantum field. For this task the (Euclidean) Hadamard structure of the D-dimensional “Green functions” is carefully analyzed. It is proven that, for D > 1, the local ζ-function and point-splitting approaches lead essentially to the same results apart from the choice of the coefficient w0(x, x) in the Hadamard expansion which is fixed by the local ζ-function approach, whereas it remains arbitrary in the point-splitting procedure. Also the presence of null modes, which is studied in all considered cases, produces some differences between the two analysed approaches. Finally, a formal identity concerning the field fluctuations, usually supposed true by physicists, is discussed and proven within the local ζ-function approach. This is done also to reply to part of a recent criticism against ζ function techniques.