Hot and Dense Gauge Theories with External Fields

In the introductory part of this thesis, I briefly review quantum statistichal mechanics; quantum field theory with external fields; the real and imaginary time formalisms in thermal field theory; and the hard thermal loop resummation.<p /> In the second part, consisting of here reprinted research articles, we apply this to quantum electrodynamics and quantum chromodynamics with a background (chromo-) magnetic field in the presence of a heat and charge reservoir.<p /> The effective Lagrangian for a background static uniform (chromo-) magnetic field in scalar and spinor QED as well as QCD at finite temperature and density, is calculated. The spinor QED result contains a term that had been missed in the earlier attempts to calculate the effective Lagrangian. At low temperatures this term is shown to be responsible for the relativistic de Haas--van Alphen oscillations, not earlier known. The vacuum magnetization is shown to dominate the ordinary matter magnetization in the limit of high fields. We resolve the ostensible discrepancy between the renormalized paramagnetic vacuum of QED found here, and the paramagnetic behavior of the vacuum of bare regularized QCD, used to explain asymptotic freedom in a pedagogigal manner. In hot QCD, the effective charge here obtained is found to be decreasing with the temperature. This indicates the formation of a quark--gluon plasma at high temperatures, in for example heavy-ion collisions. The electron self-energy in a magnetic field at finite temperature and density, is calculated. In the high temperature limit, particle as well as hole solutions to the self-consistent dispersion relation are found, with interesting chiral properties. The matter contribution to the electron anomalous magnetic moment is obtained from the self-energy. The result differs from the earlier known, obtained through the vertex correction, using ordinary perturbation theory and plane wave external states. Since the true external states are Landau levels, it is our result that has to be correct. The anomalous magnetic moment will become very large at high densities. This could affect the properties of for example neutron stars. Furthermore, also the electron self-energy is shown to exhibit de Haas--van Alphen oscillations. It is shown that using the general form of the spatial part of the electron wave-functions in a magnetic field it is possible to calculate a general self energy matrix, and find a self-consistent dispersion relation also this dispersion relation is solved, but only resulting in a mass-shift. The condensation of a magnetized relativistic Bose gas is considered. This had earlier been treated, but using the external field, instead of the true mean field, as the acting field in the gas, which led to erroneous conclusions about the Meissner--Ochsenfeldt effect. Furthermore, we find the relativistic generalization of the magnetization law governing the true Meissner--Ochsenfeldt effect