The notion of a joint distribution in $\sigma$ -finite measures of observables of a quantum logic defined on some system of $\sigma$ -independent Boolean sub- $\sigma$ -algebras of a Boolean $\sigma$ -algebra is studied. In the present first part of the paper the author studies a joint distribution of compatible observables. It is shown that it may exists, although a joint obsevable of compatible observables need not exist.The notion of a joint distribution in $\sigma$-finite measures of observables of a quantum logic defined on some system of $\sigma$-independent Boolean sub-$\sigma$-algebras of a Boolean $\sigma$-algebra is studied. In the present first part of the paper the author studies a joint distribution of compatible observables. It is shown that it may exists, although a joint obsevable of compatible observables need not exist.

Applications of MathematicsJan 1, 19872 citations

Foundations of Quantum Logic

Andrei Galiautdinov

Jan 1, 20170 citations

On joint distribution in quantum logics. II. Noncompatible observables

Anatolij Dvurečenskij

This paper i a continuation of the first part under the same title. The author studies a joint distribution in $\sigma$ -finite measures for noncompatible observables of a quantum logic defined on some system of $\sigma$ -independent Boolean sub- $\sigma$ -algebras of a Boolean $\sigma$ -algebra. We present some necessary and sufficient conditions fot the existence of a joint distribution. In particular, it is shown that an arbitrary system of obsevables has a joint distribution in a measure iff it may be embedded into a system of compatible observables of some quantum logic. The methods used are different from those developed for finite measures. Finally, the author deals with the connection between the existence of a joint distribution and the existence of a commutator of observables, and the quantum logic of a nonseparable Hilbert space is mentioned.This paper i a continuation of the first part under the same title. The author studies a joint distribution in $\sigma$-finite measures for noncompatible observables of a quantum logic defined on some system of $\sigma$-independent Boolean sub-$\sigma$-algebras of a Boolean $\sigma$-algebra. We present some necessary and sufficient conditions fot the existence of a joint distribution. In particular, it is shown that an arbitrary system of obsevables has a joint distribution in a measure iff it may be embedded into a system of compatible observables of some quantum logic. The methods used are different from those developed for finite measures. Finally, the author deals with the connection between the existence of a joint distribution and the existence of a commutator of observables, and the quantum logic of a nonseparable Hilbert space is mentioned.

Applications of MathematicsJan 1, 19871 citation

Effects in quantum logic of observables

Vladimir L. Vasyukov

Логические исследованияJan 1, 20030 citations

The notion of a joint distribution in

\sigma

-finite measures of observables of a quantum logic defined on some system of

\sigma

-independent Boolean sub-

\sigma

-algebras of a Boolean

\sigma

-algebra is studied. In the present first part of the paper the author studies a joint distribution of compatible observables. It is shown that it may exists, although a joint obsevable of compatible observables need not exist.

This paper i a continuation of the first part under the same title. The author studies a joint distribution in

\sigma

-finite measures for noncompatible observables of a quantum logic defined on some system of

\sigma

-independent Boolean sub-

\sigma

-algebras of a Boolean

\sigma

-algebra. We present some necessary and sufficient conditions fot the existence of a joint distribution. In particular, it is shown that an arbitrary system of obsevables has a joint distribution in a measure iff it may be embedded into a system of compatible observables of some quantum logic. The methods used are different from those developed for finite measures. Finally, the author deals with the connection between the existence of a joint distribution and the existence of a commutator of observables, and the quantum logic of a nonseparable Hilbert space is mentioned.