Functional integrals for QCD at nonzero chemical potential and zero density.

In a Euclidean space functional integral treatment of the free energy of QCD, a chemical potential enters only through the functional determinant of the Dirac operator which for any flavor is /D+m-mu(f)gamma(0) (where mu(f) is the chemical potential for the given flavor). Any nonzero mu alters all of the eigenvalues of the Dirac operator relative to the mu=0 value, leading to a naive expectation that the determinant is altered and which thereby alters the free energy. Phenomenologically, this does not occur at T=0 for sufficiently small mu, in contradiction to this naive expectation. The problem of how to understand this phenomenological behavior in terms of functional integrals is solved for the case of an isospin chemical through the study of the spectrum of the operator gamma(0)(/D+m). The case of the baryon chemical potential is briefly discussed.