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

Max Planck Institute for Mathematics in the Sciences, Inselstraße 22, 04103 Leipzig, Germany

Instituto de Ciencias Matemáticas, (CSIC – UAM – UC3M – UCM) ICMAT and Dipartimento de Matemáticas, Universitaris Carlos III de Madrid, Avda. de la Universidad 30, 28911 Leganés, Madrid, Spain

E-mail Address: fcosmo@math.uc3m.es

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A. Ibort

http://orcid.org/0000-0002-0580-5858

E-mail Address: albertoi@math.uc3m.es

Corresponding author.

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, and

G. Marmo

Dipartimento di Fisica E. Pancini dell Universitá, Federico II di Napoli and INFN-Sezione di Napoli, Complesso Universitario di Monte S. Angelo, via Cintia, 80126 Naples, Italy

E-mail Address: marmo@na.infn.it

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https://doi.org/10.1142/S0219887820500589Cited by:16 (Source: Crossref)

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.

Keywords:

AMSC: 22E46, 53C35, 57S20

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