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

Joint Distributions of Observables

Stanley Gudder

Indiana University Mathematics JournalJan 1, 196827 citations

On joint distribution in quantum logics. I. Compatible observables

Anatolij Dvurečenskij

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

On the Joint Distribution of Observables

Beloslav Riečan

Foundations of PhysicsOct 1, 20001 citation

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.

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.