Gauge theories at finite temperature and chemical potential

The relativistic partition function is studied respecting the internal symmetries. We consider generally the SU(N) gauge theories, and, in particular, the SU(3) symmetry relating to the quark-gluon plasma. The implications of the complex chemical potentials are analyzed and discussed in relation to the lattice gauge theories. It is explicitly shown that the physical partition function is real, which can be obtained from a complex Euclidean partition function by imposing Gauss’s law.