Noncommutative Dynamics of Random Operators

Details Noncommutative Dynamics of Random Operators Noncommutative Dynamics of Random Operators

2004-09-16

arxiv.org/abs/gr-qc/0409063v1

Int.J.Theor.Phys. 44 (2005) 619-628

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

13 pages, LaTex

Scientific paper

10.1007/s10773-005-3992-7

We continue our program of unifying general relativity and quantum mechanics in terms of a noncommutative algebra ${\cal A}$ on a transformation groupoid $\Gamma = E \times G$ where $E$ is the total space of a principal fibre bundle over spacetime, and $G$ a suitable group acting on $\Gamma $. We show that every $a \in {\cal A}$ 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 ${\cal A}$ which can be used to define a state dependent dynamics; i.e., the pair $({\cal A}, \phi)$, where $\phi $ is a state on ${\cal A}$, is a ``dynamic object''. Only if certain additional conditions are satisfied, the Connes-Nikodym-Radon theorem can be applied and the dependence on $\phi $ disappears. In these cases, the usual unitary quantum mechanical evolution is recovered. We also notice that the same pair $({\cal A}, \phi)$ defines the so-called free probability calculus, as developed by Voiculescu and others, with the state $\phi $ 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.

Affiliated with

Also associated with

No associations

Rating

Noncommutative Dynamics of Random Operators does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Noncommutative Dynamics of Random Operators, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Noncommutative Dynamics of Random Operators will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-199955