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arXiv:2004.02472 · International Journal of Geometric Methods in Modern Physics

Schwinger’s picture of quantum mechanics IV: Composition and independence

Florio M. Ciaglia, Fabio Di Cosmo, A. Ibort, G. Marmo

Published Feb 11, 2020• 19 citations

Abstract

The groupoid description of Schwinger’s picture of quantum mechanics is continued by discussing the closely related notions of composition of systems, subsystems, and their independence. Physical subsystems have a neat algebraic description as subgroupoids of the Schwinger’s groupoid of the system. The groupoid picture offers two natural notions of composition of systems: Direct and free products of groupoids, that will be analyzed in depth as well as their universal character. Finally, the notion of independence of subsystems will be reviewed, finding that the usual notion of independence, as well as the notion of free independence, find a natural realm in the groupoid formalism. The ideas described in this paper will be illustrated by using the EPRB experiment. It will be observed that, in addition to the notion of the non-separability provided by the entangled state of the system, there is an intrinsic “non-separability” associated to the impossibility of identifying the entangled particles as subsystems of the total system.

DOI: 10.1142/s0219887820500589

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A new picture of Quantum Mechanics based on the theory of groupoids is presented. This picture provides the mathematical background for Schwinger’s algebra of selective measurements and helps to understand its scope and eventual applications. In this first paper, the kinematical background is described using elementary notions from category theory, in particular the notion of 2-groupoids as well as their representations. Some basic results are presented, and the relation with the standard Dirac–Schrödinger and Born–Jordan–Heisenberg pictures are succinctly discussed.

A new picture of Quantum Mechanics based on the theory of groupoids is presented. This picture provides the mathematical background for Schwinger's algebra of selective measurements and helps to understand its scope and eventual applications. In this first paper, the kinematical background is described using elementary notions from category theory, in particular the notion of 2-groupoids as well as their representations. Some basic results are presented, and the relation with the standard Dirac-Schr\"odinger and Born-Jordan-Heisenberg pictures are succinctly discussed.

The challenges posed by the development of field theories, both classical and quantum, force us to question their most basic and foundational ideas like the role and origin of space-time, the meaning of physical states, etc. Among them the notion of ``field'' itself is notoriously difficult to address. These notes aim to analyze such notion from the perspective offered by the groupoid description of quantum mechanics inspired by Schwinger's picture of quantum mechanics. Then, a natural interpretation of the notion of physical fields as functors among appropriate groupoids will emerge. The domain of a field in this new picture is a groupoid that describes ``test particles'', and its codomain is a groupoid that describes the intrinsic nature of the system being probed. Such a space of functors carries some natural structures, which are best described in a categorical language. Some illustrative examples will be presented that could help clarify the various abstract notions discussed in the text.

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.