On the covariant formulation of classical relativistic statistical mechanics

Abstract

By developing Dirac's ideas, it is shown that a clear, covariant formulation of relativistic statistical mechanics can be constructed explicity. The basis lies in the duality between the observer's physical space-time which has the well-known Minkowski structure, and the dynamical system's phase space which has a completely different geometrical structure. The Lorentz transformations in physical space - expressed by the usual tensiorial formalism - are mapped in phase space by corresponding canonical transformations which form a representation of the Lorentz group. The generators of this representation are explicitly constructed in the case of a gas of charged particles interacting through the electromagnetic field. The latter is treated explicitly as a dynamical system. The equation for the Lorentz transformation of the distribution function in phase space is written explicitly; it is closely analogous to the Liouville equation. It can be solved exactly both for the free particles and for the free field, but not in the presence of interactions. In the latter case, it poses a problem which is quite analogous, and of the same degree of difficulty, as the solution of the ordinary Liouville equation. The whole theory has been presented in such a form as to exhibit the possibility of adaptation of the powerful modern many-body techniques to the problem of explicit Lorentz transformation of the dynamical quantities.