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Philosophical Problems in Science (Zagadnienia Filozoficzne w Nauce)

Independent quantum systems and the associativity of the product of quantum observables

Klaus Fredenhagen

Published Jul 30, 2019• 0 citations
Publisher

Abstract

We start from the assumption that the real valued observables of a quantum system form a Jordan algebra which is equipped with a compatible Lie product characterizing infinitesimal symmetries, and ask whether two such systems can be considered as independent subsystems of a larger system. We show that this is possible if and only if the associator of the Jordan product is a fixed multiple of the associator of the Lie product. In this case it is known that the two products can be combined to an associative product in the Jordan algebra or its complexification, depending on the sign of the multiple.

DOI
10.59203/zfn.66.465

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This is a transcription of a conference proceedings from 1985. It reviews the Jordan algebra formulation of quantum mechanics. A possible novelty is the discussion of time evolution; the associator takes over the role of iii times the commutator in the standard density matrix formulation, and for the Jordan algebra of complex Hermitian matrices this implies a Hamiltonian of the form H=i[x,y]+λIH= i[x,y] + λ\mathbb{I}H=i[x,y]+λI for traceless Hermitian x,yx,yx,y and real number λλλ. Other possibilities for time evolution in the Jordan formulation are briefly considered.