Lorentz transformations in phase space and in physical space

Abstract

The relation between “tensorial” Lorentz transformations in physical space and “canonical” Lorentz transformations in phase space is examined in detail. It is shown in particular that the Lorentz-Liouville equation transforms the phase-space distribution function in such a way that the relevant average values computed with it, transform precisely as vectors or tensors. This has been proved explicitly in the case of the current-density vector and the energy-momentum tensor. An important fact which comes out of the theory is that the tensorial variance is not attached intrinsically to a given function but is determined by the dynamical nature of the system For instance, the average identified with the energy-momentum tensor for free particles, transforms as a tensor in a free particle system, but not in a system of interacting particles.