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Czechoslovak Mathematical Journal

Quantum logics representable as kernels of measures

Mirko Navara

Published Jan 1, 1996• 6 citations
Publisher

Abstract

Tlic classical Kohnogorov's model of probability assumes that every pair of ev(Tits is siinult.aneouslyobservable.This principle is violated in several applications including quantum mechanics, artificial intelligence, psychology, sociology etc.In these areas iioneoinpatible events are encountered.These are events which can be observed separately, but not simultaneously, so they are not contained in a Boolean subalgebra i = classical subsystem) of the event structure describing the system in question.Various attempts have been made to generalize the probability theory to a more general structure admitting noneonipatibility.Among them, classes of subsets (more generally, concrete logics) wore studied for many years (see e.g.[15, 18]).Although some results were successfully generalized (see e.g.[4. 10, 19]).the theory proceeded slowly and with serious difficulties.Hero we introduce a more special-but still reasonably general-structure, a kernel logic.As it is described in terms of Boolean algebras using measure-theoretic notions, we believe that there is a greater chance to generalize classical results for Boolean algebras to kernel logics.Kernel logics seem to be interesting also from the algebraic point of view as a new const ruction technique for concrete logics.Its usefulness was proved by solutions of several quite nontrivial problems.Besides this, it scorns desirable to describe kernels of measures on Boolean algebras.

DOI
10.21136/cmj.1996.127321

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Tlic classical Kohnogorov's model of probability assumes that every pair of ev(Tits is siinult.aneouslyobservable.This principle is violated in several applications including quantum mechanics, artificial intelligence, psychology, sociology etc.In these areas iioneoinpatible events are encountered.These are events which can be observed separately, but not simultaneously, so they are not contained in a Boolean subalgebra i = classical subsystem) of the event structure describing the system in question.Various attempts have been made to generalize the probability theory to a more general structure admitting noneonipatibility.Among them, classes of subsets (more generally, concrete logics) wore studied for many years (see e.g.[15, 18]).Although some results were successfully generalized (see e.g.[4. 10, 19]).the theory proceeded slowly and with serious difficulties.Hero we introduce a more special-but still reasonably general-structure, a kernel logic.As it is described in terms of Boolean algebras using measure-theoretic notions, we believe that there is a greater chance to generalize classical results for Boolean algebras to kernel logics.Kernel logics seem to be interesting also from the algebraic point of view as a new const ruction technique for concrete logics.Its usefulness was proved by solutions of several quite nontrivial problems.Besides this, it scorns desirable to describe kernels of measures on Boolean algebras.
Since 1933, when Kolmogorov laid the foundations for probability and statistics as we know them today [1], it has been recognized that propositions asserting that such and such an event occurred as a consequence of the execution of a particular random experiment tend to band together and form a Boolean algebra. In 1936, Birkhoff and von Neumann [2] suggested that the so-called logic of quantum mechanics should not be a Boolean algebra, but rather should form what is now called a modular ortholattice [3]. Presumably, the departure from Boolean algebras encountered in quantum mechanics can be attributed to the fact that in quantum mechanics, one must consider more than one physical experiment, e.g., an experiment measuring position, an experiment measuring charge, an experiment measuring momentum, etc., and, because of the uncertainty principle, these experiments need not admit a common refinement in terms of which the Kolmogorov theory is directly applicable. Mackey's Axioms I–VI for quantum mechanics [4] imply that the logic of quantum mechanics should be a σ-orthocomplete orthomodular poset [5]. Most contemporary practitioners of quantum logic seem to agree that a quantum logic is (at least) an orthomodular poset [6], [7], [8], [9], [10] or some variation thereof [11]. P. D. Finch [12] has shown that every completely orthomodular poset is the logic arising from sets of Boolean logics, where these sets have a structure similar to the structures generally given to quantum logic. In all of these versions of quantum logic, a fundamental relation, the relation of compatibility or commutativity, plays a decisive role.
In classical probability theory one assumes that all the events concerning a statistical experiment constitute a Boolean σ-algebra and defines a probability measure as a completely additive non-negative function which assigns the value unity for the identity element of the σ-algebra. Invariably, the σ-algebra is the Borel σ-algebra B \mathcal{B} B X (i.e., the smallest σ-algebra generated by the open subsets) of a nice topological space X. Under very general conditions it turns out that all probability measures on B \mathcal{B} B X constitute a convex set whose extreme points are degenerate probability measures.
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.
Abstract With many physical processes in which quantum mechanical phenomena can occur, it is essential to take into account a decision mechanism based on measurement data. This can be achieved by means of so-called numerical events, which are specified as follows: Let S be a set of states of a physical system and p(s) the probability of the occurrence of an event when the system is in state s∈Ss\in Ss∈S s ∈ S . A function p:S→[0,1]p:S\rightarrow [0,1]p:S→[0,1] p : S → [ 0 , 1 ] is called a numerical event or alternatively, an S-probability. If a set P of S-probabilities is ordered by the order of real functions, it becomes a poset which can be considered as a quantum logic. In case the logic P is a Boolean algebra, this will indicate that the underlying physical system is a classical one. The goal of this paper is to study sets of S-probabilities which are not far from being Boolean algebras by means of the addition and comparison of functions that occur in these sets. In particular, certain classes of so-called Boolean posets of S-probabilities are characterized and related to each other and descriptions based on sets of states are derived.
Sentences Associated with a Quantum System and Their Truth Values In a sense this chapter is marginal to the content of this volume; the reader can skip it without prejudice to the comprehension of chapters to follow. Quantum logic is a discipline that branched off from the 1936 paper of Birkhoff and von Neumann and has the orthomodular-poset structures encountered in previous chapters as basic mathematical carriers: that is why we think it worthwhile to devote some pages to the subject. But the main interests quantum logic calls into play are in the territory of logicians and philosophers: that is why we (who are neither logicians nor philosophers) shall confine ourselves to a very simplified introduction to the subject. Roughly, the starting question is whether the propositions of a quantum system can be associated with, or can be interpreted as, sentences of a language (or propositional calculus) and which rules this language inherits from the ordered structure of propositions. In raising this question one has in mind the fact that when the physical system is classical its propositions form a Boolean algebra, and Boolean algebras are the algebraic models of the calculus of classical logic. Thus the question above can also be phrased as follows: when a Boolean algebra is relaxed into an orthomodular nondistributive lattice, which logic is it the model of? “Quantum logic” is the name that designates the answer, but there are several views about the content of this name. Here we sketch one approach to a quantum logic.