Symmetry groups, density-matrix equations and covariant Wigner functions

A representation theory for Lie groups is developed taking the Hilbert spac e, say H-w, of the w*-algebra standard representation as the representation space. In this context the states describing physical systems are amplitud e wave functions but closely connected with the notion of the density matri x. Then, based on symmetry properties, a general physical interpretation fo r the dual variables of thermal theories, in particular the thermofield dyn amics (TFD) formalism, is introduced. The kinematic symmetries, Galilei and Poincare, are studied and (density) amplitude matrix equations are derived for both of these cases. In the same context of group theory, the notion o f phase space in quantum theory is analysed. Thus, in the non-relativistic situation, the concept of density amplitude is introduced, and as an exampl e, a spin-half system is algebraically studied; Wigner function representat ions for the amplitude density matrices are derived and the connection of T FD and the usual Wigner-function methods are analysed. For the Poincare sym metries the relativistic density matrix equations are studied for the scala r and spinorial fields. The relativistic phase space is built following the lines of the non-relativistic case. So, for the scalar field, the kinetic theory is introduced via the Klein-Gordon density-matrix equation, and a de rivation of the Juttiner distribution is presented as an example, thus maki ng it possible to compare with the standard approaches. The analysis of the phase space for the Dirac field is carried out in connection with the dual spinor structure induced by the Dirac-field density-matrix equation, with the physical content relying on the symmetry groups. Gauge invariance is co nsidered and, as a basic result, it is shown that the Heinz density operato r (which has been used to develope a gauge covariant kinetic theory) is a p articular solution for the (Klein-Gordon and Dirac) density-matrix equation . (C) 2000 Elsevier Science B.V. All rights reserved.