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International Journal of Theoretical Physics

, Volume 11, Issue 3, pp 149–156 | Cite as

Functional properties of quantum logics

  • M. J. Mączyński
Article

Abstract

A quantum logic is defined as a setL of functions from the set of all statesS into [0, 1] satisfying the orthogonality postulate: for any sequencea1,a2, ... of members ofL satisfyingai+aj≤1 forij there isbL such thatb+a1+a2+...=1. Every logicL is in a natural way an orthomodular σ-orthocomplemented partially ordered set (L, ≤, ′) with members ofS inducing a full set of measures onL. It is shown that a logicL is quite full if and only if (L,≤,′) is isomorphic to an orthocomplemented set lattice of subsets ofS. Sufficient conditions are given in order that a quite full logic be representable in the set of projection quadratic formsf(u)=(Pu, u) on a complex Hilbert space, or in the set of trace functionsf(A)=Trace (AP) generated by projectionsP, where the domain off is the set of non-negative self-adjoint trace operators of trace 1 in a complex Hilbert space.

Keywords

Hilbert Space Field Theory Elementary Particle Quantum Field Theory Functional Property 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Plenum Publishing Company Limited 1974

Authors and Affiliations

  • M. J. Mączyński
    • 1
  1. 1.Institute of MathematicsTechnical University of WarsawWarszawaPoland

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