Boolean representations of fuzzy quantum spaces

Anatolij Dvurečenskij

Roberto Leporini, Cesarino Bertini, Filippo Carone Fabiani

Jarosław Pykacz

V. P. Belavkin

We give an axiomatic formulation of quantum structures like semilogics and quasilogics which generalize the boolean semirings of events and fuzzy logics. The notions of distributions, states, representations observables and semiobservables are introduced and their Hilbert space realizations are found. The closed and open structures in semilogics are introduced and the regular distributions on the semilogics are studied.

Anatolij Dvurečenskij, Ferdinand Chovanec

S. Twareque Ali, Eduard Prugovečki

The problem of expressing quantum mechanical expectation values as averages with respect to nonnegative density functions on phase space, by analogy with classical mechanics, is reexamined in the light of some earlier work on fuzzy phase spaces. It is shown that such phase space representations are possible if ordinary phase space is replaced by a so-called fuzzy phase space, on which the usual marginal distribution functions are redefined to conform to the fact that arbitrarily precise simultaneous measurements on position and momentum are not compatible with quantum mechanics. In the process a generalization of Wigner’s theorem on the nonexistence of phase space representations of quantum mechanics, which also satisfy the standard (classical) marginality conditions in position and momentum, is obtained. It is shown that a (continuous) representation of quantum mechanics exists on a given fuzzy phase space if an only if the corresponding confidence functions for position and momentum measurements satisfy the Heisenberg uncertainty relations.