Time Flow in a Noncommutative Regime

Abstract

We develop an approach to dynamical and probabilistic properties of the model unifying general relativity and quantum mechanics, initiated in the paper (Heller et al. (2005) International Journal Theoretical Physics 44, 671). We construct the von Neumann algebra ${\cal M}$ of random operators on a groupoid, which now is not related to a finite group G, but is the pair groupoid of the Lorentzian frame bundle E over spacetime M. We consider the time flow on ${\cal M}$ depending on the state $\phi$. The state $\phi$ defining the noncommutative dynamics is assumed to be normal and faithful. Then the pair $({\cal M}, \phi)$ is a noncommutative probabilistic space and $\phi$ can also be interpreted as an equilibrium thermal state, satisfying the Kubo-Martin-Schwinger condition. We argue that both the “time flow” and thermodynamics have their common roots in the noncommutative unification of dynamics and probability.