Emergence of time

Abstract

In the groupoid approach [Heller et al., J. Math. Phys. 38 (1997) 5840] a gravitational system is quantized in terms of a noncommutative C ∗- algebra A of complex valued functions on a groupoid G = E × Γ, where E is a suitable space and Γ a group of fundamental symmetries. In the noncommutative regime the concepts of space and time are meaningless. We study the “emergence of time” in the transition process from the noncommutative regime to the standard space-time geometry. Precise conditions are specified under which modular groups of the von Neumann algebra generated by the algebra A can be defined. These groups are interpreted as representing a state depending time flow of a quantum gravitational system. If the above conditions are further refined one obtains a state independent time flow. We show that the dynamics of the proposed model can be expressed in terms of modular groups.