Noncommutative Dynamics of Random Operators

Abstract

We continue our program of unifying general relativity and quantum mechanics in terms of a noncommutative algebra $\mathcal{A}$ 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 $a\in \mathcal{A}$ 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 $\mathcal{A}$ which can be used to define a state dependent dynamics; i.e., the pair $(\mathcal{A},\varphi )$, where $\varphi$ is a state on $\mathcal{A}$, is a ``dynamic object''. Only if certain additional conditions are satisfied, the Connes-Nikodym-Radon theorem can be applied and the dependence on $\varphi$ disappears. In these cases, the usual unitary quantum mechanical evolution is recovered. We also notice that the same pair $(\mathcal{A},\varphi )$ defines the so-called free probability calculus, as developed by Voiculescu and others, with the state $\varphi$ 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.

13 pages, LaTex

References in corpus (5)

• Von Neumann Algebra Automorphisms and Time-Thermodynamics Relation in General Covariant Quantum Theories
• Towards Noncommutative Quantization of Gravity
• Noncommutative Unification of General Relativity and Quantum Mechanics. A Finite Model
• State Vector Reduction as a Shadow of a Noncommutative Dynamics
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Cited by in corpus (7)

• Noncommutative Unification of General Relativity and Quantum Mechanics
• Conceptual Unification of Gravity and Quanta
• Anatomy of Malicious Singularities
• Geometry and General Relativity in the Groupoid Model with a Finite Structure Group
• A Sheaf of von Neumann Algebras and Its Geometry
• General Relativity on Random Operators
• Creation of Matter in a Noncommutative Universe