Noncommutative Dynamics of Random Operators |
| |
Authors: | Email author" target="_blank">Michael?HellerEmail author Leszek?Pysiak Wies?w?Sasin |
| |
Institution: | (1) Vatican Observatory, Vatican City State, V-00120;(2) Warsaw University of Technology, Plac Politechniki 1, Warsaw, 00-661, Poland;(3) ul. Powstańców Warszawy 13/94, Tarnóow, 33-110, Poland |
| |
Abstract: | We continue our program of unifying general relativity and quantum mechanics in terms of a noncommutative algebra А on a transformation groupoid Γ = E × G where E is the total space of a principal fibre bundle over spacetime, and G a suitable group acting on Γ . We show that every a ∊ А defines a random operator, and we study the dynamics of such operators. In the noncommutative regime, there is no usual time but, on the strength of the Tomita–Takesaki theorem, there exists a one-parameter group of automorphisms of the algebra А which can be used to define a state dependent dynamics; i.e., the pair (А, ϕ), where ϕ is a state on А, is a “dynamic object.” Only if certain additional conditions are satisfied, the Connes–Nikodym–Radon theorem can be applied and the dependence on ϕ disappears. In these cases, the usual unitary quantum mechanical evolution is recovered. We also notice that the same pair (А, ϕ) defines the so-called free probability calculus, as developed by Voiculescu and others, with the state ϕ playing the role of the noncommutative probability measure. This shows that in the noncommutative regime dynamics and probability are unified. This also explains probabilistic properties of the usual quantum mechanics. |
| |
Keywords: | general relativity quantum mechanics unification theory noncommutative dynamics random operators free probability |
本文献已被 SpringerLink 等数据库收录! |
|