Trace Dynamics and a non-commutative special relativity

Trace Dynamics is a classical dynamical theory of non-commuting 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 non-commutative special relativity. We define a line-element using the Trace over space–time 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 non-commutative relativistic dynamics. The eventual motivation for constructing such a non-commutative relativity is to relate the statistical thermodynamics of this classical theory to quantum mechanics.

► Classical time is external to quantum mechanics.
► This implies need for a formulation of quantum theory without classical time.
► A starting point could be a non-commutative special relativity.
► Such a relativity is developed here using the theory of Trace Dynamics.
► A line-element is defined using the Trace over non-commuting space–time operators.