Does A Special Relativistic Liouville Equation Exist?

TL;DR

This paper investigates the existence of a relativistic Liouville equation and finds that it does not exist in 8N phase space due to Hamiltonian constraints related to parametrization symmetry.

Contribution

It demonstrates that a special relativistic Liouville equation cannot be formulated in 8N phase space because the canonical Hamiltonian is zero due to symmetry constraints.

Findings

No relativistic Liouville equation exists in 8N phase space.

The zero Hamiltonian results from parametrization symmetry and constraints.

The situation is analogous to diffeomorphism invariance in general relativity.

Abstract

The Liouville Equation, the starting point of non-relativistic, non-equilibrium classical statistical mechanics, is problematic in special relativity because of two problems. A relativistic Hamiltonian is claimed not to exist for interacting particles and the problem of what time to use since the particles will all have their own time as part of their space-time coordinate. In this paper, I look at this problem and surprisingly found that there is no special relativistic Liouville equation in 8N phase space, where N is the number of particles, because the canonical Hamiltonian is zero for both non-interacting and interacting particles. This is due to the parametrization symmetry in defining a single time for the Liouville equation evolution, which results in a constraint.This is similar to the fact that in general relativity, diffeomorphism invariance of the theory always give a zero Hamiltonian because of a constraint.