International Journal of Theoretical Physics

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

Functional properties of quantum logics

  • M. J. Mączyński


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.


Hilbert Space Field Theory Elementary Particle Quantum Field Theory Functional Property 
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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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