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

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.