Trace dynamics and a noncommutative special relativity

Abstract

Trace Dynamics is a classical dynamical theory of noncommuting matrices in which cyclic permutation inside a trace is used to define the derivative with respect to an operator. We use the methods of Trace Dynamics to construct a noncommutative special relativity. We define a line-element using the Trace over spacetime coordinates which are assumed to be operators. The line-element is shown to be invariant under standard Lorentz transformations, and is used to construct a noncommutative relativistic dynamics. The eventual motivation for constructing such a noncommutative relativity is to relate the statistical thermodynamics of this classical theory to quantum mechanics.