Fluctuations of quantum fields via zeta function regularization

Explicit expressions for the expectation values and the variances of some observables, which are bilinear quantities in the quantum fields on a D-dimensional manifold, are derived making use of zeta function regularization. It is found that the variance, related to the second functional variation of the effective action, requires a further regularization and that the relative regularized variance turns out to be $2/N,$ where N is the number of the fields, thus being independent of the dimension D. Some illustrating examples are worked through. The issue of the stress tensor is also briefly addressed.