Multiplicative anomaly and finite charge density

In field theory we often have to deal with functional determinants of differential operators. These, as formal products of infinite eigenvalues, are divergent objects (UV divergence) and a regularization scheme is therefore necessary. One of the most successful and powerful ones is the zeta-function regularization method [1–5]. It permits us to give a meaning to the ill defined quantity ln detA, where A is a second order elliptic differential operator, through the zeta function ζ(s|A) = TrA−s, which is well defined for a sufficiently large real part of s and can be analytically continued to a function meromorphic in all the plane and analytic at s = 0. As such its derivative to respect to s at zero is well defined and the logarithm of the zeta-function regularized functional determinant will then be defined by