Products of Logics

Abstract

As pointed out by Gudder,¹ the problem of providing a definition of tensor product for general quantum logics seems to be unavoidable if a theory of quantum measurement is addressed and developed in the context of quantum logics. More specifically, suppose we have two physical systems Σ and $\tilde \Sigma$ with corresponding logics L and $\tilde L$. For instance Σ could be the physical system under study and $\tilde \Sigma$ a measure-merit apparatus. If one wants to study the compound physical system $\Sigma \; + \;\tilde \Sigma$ with corresponding logic L, one has to provide a definition of (tensor) product of L with $\tilde L$ in order to obtain L. A solution of the general problem should contain as special cases the following standard situations. Suppose first that both Σ and E are classical systems with phase space A and à respectively and associated logics L = P(A), L = P(Ã) the power set of their phase spaces. In this case L is provided by the power set of the cartesian product of the phase spaces, namely by L = P(A × Ã).