Functional properties of quantum logics
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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 fori≠j there isb∈L 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.
KeywordsHilbert Space Field Theory Elementary Particle Quantum Field Theory Functional Property
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- Gleason, A. M. (1967).Journal of Rational Mechanics and Analysis,6, 885.Google Scholar
- Gudder, S. (1970). Axiomatic quantum mechanics and generalized probability theory, inProbabilistic Methods in Applied Mathematics, Vol. 2, pp. 53–129. Academic Press, New York.Google Scholar
- Kakutani, S. and Mackey, G. W. (1946).Bulletin of the American Mathematical Society,52, 727.Google Scholar
- Mackey, G. (1963).The Mathematical Foundations of Quantum Mechanics. Benjamin, New York.Google Scholar
- Mączyński, M. J. (1973a).International Journal of Theoretical Physics, Vol. 8, No. 5, p. 353.Google Scholar
- Mączyński, M. J. (1973b).Colloquium Mathematicum,27, 207.Google Scholar
- Meyer, P. D. (1970).Bulletin of the Australian Mathematical Society,3, 163.Google Scholar
- Varadarajan, V. S. (1968).Geometry of Quantum Theory, Vol. I. Van Nostrand, Princeton.Google Scholar