A manifestly covariant relativistic Boltzmann equation for the evolution of a system of events

A quantum mechanical derivation of a manifestly covariant relativistic Boltzmann equation is given in a framework in which the fundamental dynamical constituents of the system are a family of N events with motion in space-time parametrized by an invariant “historical time” τ. The relativistic analog of the BBGKY hierarchy is generated. Approximating the effect of correlations of second and higher order by two event collision terms, one obtains a manifestly covariant Boltzmann equation. The Boltzmann equation is used to prove the H-theorem for evolution in τ. For ensembles containing only positive energy (or only negative energy) states, a precise H-theorem is also proved for increasing t. In the nonrelativistic limit, the usual H-theorem is recovered. It is shown that a covariant form of the Maxwell-Boltzmann distribution is obtained in the equilibrium limit; an examination of the energy momentum tensor for the free gas yields the ideal gas law in this limit. It is shown that in local equilibrium the internal energy is defined by a Lorentz transformation to a local rest frame. We study the conserved quantities in this theory. There is a new nontrivially conserved quantity corresponding to the particle mass. We obtain continuity equations for these quantities.