Quantum Logic and Related Systems

George Svetlichny

R I G Hughes

According to quantum mechanics, the pure states of a microsystem are represented by vectors in a Hilbert Space. Sentences of the form, "x є L" (where x is the state vector for a system, L a subspace of the appropriate Hilbert space), may be called Q-propositions: such sentences serve to summarise our information about the results of possible experiments on the system. Quantum logic investigates the relations which hold among the Q-propositions about a given physical sys tem. These logical relations correspond to algebraic relations among the subspaces of Hilbert space. The algebra of this set of subspaces is non-Boolean, and may be regarded either as an orthomodular lattice or as a partial Boolean algebra. With each type of structure we can associate a logic. A general approach to the semantics for such a logic is provided in terms of interpretations of a formal language within an algebraic structure; an interpretation maps sentences of the language homomorphically onto elements of the structure. When the structure in question is a Boolean algera, the resulting logic is classical; here we develop a semantics for the logic associated with partial Boolean algebras. Two systems of proof, based on the natural deduction systems of Gentzen, are shown for this logic. With respect to the given sematics, these calculi are sound and weakly complete. Strong completeness is conjectured. Quantum logic deals with the logical relations between sentences, and so is properly called a logic. However, it is the logic appropriate to a limited class of sentences: proposals that it should replace classical logic wherever the latter is used should be viewed with suspicion.

J.A.J. van Leunen

This is part of a slide presentation. It treats logic systems that relate to the foundation of physics. Classical logic is well known. Traditional quantum logic has a weaker structure and is often assigned as the base of quantum physics. Hilbert logic restricts quantum logic, such that it fits better with Hilbert spaces.

M. P. Vaughan

A projective quantum logic in terms of relative states is developed, emphasizing the importance of information transfer between a system under study and its environment. The need for accounting for the historical evolution of system is highlighted and it is found that the conjunction of observations involving conjugate variables can be consistently defined but is found to be non-commutative. It is shown that the Birkhoff and von Neumann approach to quantum logic is unable to deal with such conjunctions. It is found that whilst the proposed scheme is still not distributive in general, the discrepancy is directly related to interference effects that may disappear when information is transferred from the system to its environment. It is argued that the probabilities associated with projections be mapped to an orthocomplemented ternary logic, in which it is shown that the law of the excluded middle still holds.