Local $\hat{I}\P$-function techniques vs point-splitting procedure: a few rigorous results

Abstract

Some general properties of local $\hat{I}\P$-function procedures to renormalize some quantities in $D$-dimensional (Euclidean) Quantum Field Theory in curved background are rigorously discussed for positive scalar operators $-\hat{I}”+V(x)$ 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 $\hat{I}\P$-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 $\hat{I}\P$ 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 $-\hat{I}”+V(x)$ 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 $\hat{I}\P$-function approach. This is done also to reply to recent criticism against $\hat{I}\P$-function techniques.

40 pages, latex, no figures, shortened version, some previous Comments and minor errors corrected, final version accepted for publication in Commun. Math. Phys