Emergence of Time

In the groupoid approach to noncommutative quantization of gravity, gravitational field is quantized in terms of a 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 quantum gravitational regime the concepts of space and time are meaningless. We study the “emergence of time” in the transitions process from the noncommutative regime to the standard spacetime geometry. Precise conditions are specified under which modular groups of the von Neumann algebra generated by 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 quantum gravitational dynamics can be expressed in terms of modular groups. ∗Correspondence address: ul. Powstańców Warszawy 13/94, 33-110 Tarnów, Poland. E-mail: [email protected]