Ergodic Theory and the Structure of Non-commutative Space-Time

Abstract

We develop further our fibre bundle construct of non-commutative space-time on a Minkowski base space. We assume space-time is non-commutative due to the existence of additional non-commutative algebraic structure at each point x of space-time, forming a quantum operator 'fibre algebra' A(x). This structure then corresponds to the single fibre of a fibre bundle. A gauge group acts on each fibre algebra locally, while a 'section' through this bundle is then a quantum field of the form {A(x); x in M} with M the underlying space-time manifold. In addition, we assume a local algebra O(D) corresponding to the algebra of sections of such a principal fibre bundle with base space a finite and bounded subset of space-time, D. The algebraic operations of addition and multiplication are assumed defined fibrewise for this algebra of sections. We characterise 'ergodic' extremal quantum states of the fibre algebra invariant under the subgroup T of local translations of space-time of the Poincare group P in terms of a non-commutative extension of entropy applied to the subgroup T. We also characterise the existence of T- invariant states by generalizing to the non-commutative case Kakutani's work on wandering projections. This leads on to a classification of the structure of the local algebra O(D) by using a 'T-Twisted' equivalence relation, including a full analysis of the T-type III case. In particular we show that O(D) is T-type III if and only if the crossed product algebra O(D)xT is type III in the sense of Murray-von Neumann.