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 $\mathrm{D̸}+m-{\mu }_f{\gamma }_0$ (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 ${\gamma }_0(\mathrm{D̸}+m)$. The case of the baryon chemical potential is briefly discussed.