  We discuss the role of the multiplicative anomaly for a complex scalar field
at finite temperature and density. It is argued that physical considerations
must be applied to determine which of the many possible expressions for the
effective action obtained by the functional integral method is correct. This is
done by first studying the non-relativistic field where the thermodynamic
potential is well-known. The relativistic case is also considered. We emphasize
that the role of the multiplicative anomaly is not to lead to new physics, but
rather to preserve the equality among the various expressions for the effective
action.
