Evolution of Classical and Quantum States in the Groupoid Picture of Quantum Mechanics

Abstract

The evolution of states of the composition of classical and quantum systems in the groupoid formalism for physical theories introduced recently is discussed. It is shown that the notion of a classical system, in the sense of Birkhoff and von Neumann, is equivalent, in the case of systems with a countable number of outputs, to a totally disconnected groupoid with Abelian von Neumann algebra. The impossibility of evolving a separable state of a composite system made up of a classical and a quantum one into an entangled state by means of a unitary evolution is proven in accordance with Raggio’s theorem, which is extended to include a new family of separable states corresponding to the composition of a system with a totally disconnected space of outcomes and a quantum one.

Feynman's Propagator in Schwinger's picture of Quantum Mechanics

“…As it was discussed in Ref. [27] , there is a natural classical system associated to the quantum system with groupoid of configurations K ⇒ Ω. Such system is determined by the Abelian algebra L ∞ (Ω), which is just the von Neumann algebra of the totally disconnected subgroupoid defined by the units 1 x of the groupoid K. Then, the restriction of the DFS state defined by ϕ to L ∞ (Ω × R), is the measure defined by the function p, that is, the restriction of the positive-type function ϕ , Eq.…”

Dynamics of System States in the Probability Representation of Quantum Mechanics

“…operators and the concepts of classical mechanics can be formulated in the language analogous to quantum mechanics language. New fundamental aspects of quantum mechanics based on groupoid approach are investigated in [25]. In [26] the evolution of states of system containing quantum and classical parts was studied.…”

Causality in Schwinger’s Picture of Quantum Mechanics

“…In such a case, the space of outputs would be an interval (or a interval in ). The groupoid , being a classical system, will be identified with the set itself as no nontrivial transitions exist on apart from the units (see [ 44 ] for details on dynamics of classical and quantum systems). Hence, the direct product of both systems will represent the standard composition of both systems whose transitions will have the form , so there is not an intrinsic notion of causal precedence build in the system.…”