Two new possibilities in the axiomatic foundations of quantum mechanics are examined: first, the possibility of introducing a non-symmetric transition probability between pure states, and second, showing that the concept of orthocomplementation in the logic of events is unnecessary and of secondary importance. An axiomatic scheme is presented which does not involve the concept of orthocomplementation and yet has all the advantages of the well-known quantum logic axiomatics, because the generalised logic of events admits an extension, which is a complete orthocomplemented orthomodular lattice with the covering law holding in it. Thus, the approach to quantum axiomatics may be seen as answering both the old questions of the quantum logic approach (e.g. the questions of the complete lattice structure of the logic, atomicity, the validity of the covering law) and the question concerning the necessity of the orthocomplementation in the logic of events, recently raised by Mielnik (1976).
Two new possibilities in the axiomatic foundations of quantum mechanics are examined: first, the possibility of introducing a non-symmetric transition probability between pure states, and second, showing that the concept of orthocomplementation in the logic of events is unnecessary and of secondary importance. An axiomatic scheme is presented which does not involve the concept of orthocomplementation and yet has all the advantages of the well-known quantum logic axiomatics, because the generalised logic of events admits an extension, which is a complete orthocomplemented orthomodular lattice with the covering law holding in it. Thus, the approach to quantum axiomatics may be seen as answering both the old questions of the quantum logic approach (e.g. the questions of the complete lattice structure of the logic, atomicity, the validity of the covering law) and the question concerning the necessity of the orthocomplementation in the logic of events, recently raised by Mielnik (1976).
The logic approach to axiomatic quantum mechanics via orthocomplemented partial ordered sets of yes–no measurements, which constitute the observing part of a concretely realizable experiment on microworld, has been criticized from the empirical point of view by Mielnik, which on the contrary privileges the convex scheme linked to the preparing part. In this work we do assume that a description of quantum phenomenology must take into account both these two parts in which every elementary experiment can be decomposed. According to this predecision, we develop an axiomatic approach based on indistinguishability principles of a quantum information system. The very general concept of yes–no measurement or ‘‘question’’ is accepted, and then the set of all questions is classified according to the behavior with respect to a phenomenological orthogonality relation. In particular, we single out the set F of fuzzy events or effects and the set E⊆F of exact events. The Mielnik critique is then refused since it regards the order structure of E using counterexamples which pertain to F/E. The notions of physical property and noperty are then introduced and an axiomatic foundation of quantum mechanics based on a pre-Hilbert space is discussed.
Abstract The paper, summarizing the results of recent papers of the author, is devoted to a detailed analysis of postulates of nonrelativistic quantum theory appearing within two classical axiomatic frameworks the “quantum logic” and the “algebraic approach” respectively, which are presently the two main alternatives for quantum axiomatics. The first part of the paper concerns the structure of two important sets, the set of questions (propositional logic) and the set of pure states. It is shown how the quantum logic axiomatic scheme can be modified and improved in order to overcome the well known troubles connected with the lack of satisfactory physical justification for some postulates of this approach. In particular, an axiom system is developed, related closely to the quantum logic axiomatic scheme, in which the questions of the complete lattice structure, the atomisticity, and the validity of the covering law in the propositional logic do not appear so problematic. The particular attention is directed to the covering law, which is here obtained as a consequence of physically clear properties of the experimental procedures (“filters”) associated with the quantum‐mechanical propositions. In Section 1.5 we formulate an axiom system for quantum mechanics based exclusively on the concepts of pure state, transition probability, and pure filter. In the second part of the paper the axiomatic scheme presented in Part 1 is developed with the aim to analyse the algebraic structure of some subspaces of the set of simple observables. The quantum logic axiomatic framework is here shown closely connected with the Jordan‐Banach algebraic scheme. In the third part of the paper, similarly as it was done in Parts 1 and 2, the correspondence between the propositions and the experimental (measuring) procedures verifying these propositions is put as the basic assumption of the axiomatization of quantum mechanics. As a consequence of this assumption, there is established the structure of a real Jordan‐Banach algebra in the set Ob of the bounded observables associated with a physical system, so that we can apply the GNS representation theorem proved recently for such algebras by Alfsen et al. (1978) to obtain the Hilbert space (or, to be more precise, the C*‐algebraic) representation for Ob.
Quantum mechanics is usually described in the terminology of probability theory even though the properties of the probability spaces associated with it are fundamentally different from the standard ones of probability theory. For example, Kolmogorov's axioms are not general enough to encompass the non-commutative situations that arise in quantum theory. There have been many attempts to generalise these axioms to meet the needs of quantum mechanics. The focus of these attempts has been the observation, first made by Birkhoff and von Neumann (1936), that the propositions associated with a quantum-mechanical system do not form a Boolean σ-algebra. There is almost universal agreement that the probability space associated with a quantum-mechanical system is given by the set of subspaces of a separable Hilbert space, but there is disagreement over the algebraic structure that this set represents. In the most popular model for the probability space of quantum mechanics the propositions are assumed to form an orthocomplemented lattice (Mackey (1963), Jauch (1968)). The fundamental concept here is that of a partial order, that is a binary relation that is reflexive and transitive but not symmetric. The partial order is interpreted as embodying the logical concept of implication in the set of propositions associated with the physical system. Although this model provides an acceptable mathematical expression of the probabilistic structure of quantum mechanics in that the subspaces of a separable Hilbert space give a representation of an ortho-complemented lattice, it has several deficiencies which will be discussed later.
The purpose of this paper is to prove that the construction of a quantum logic is a task of specific interest for non-classical logic, but that it is relevant neither to the methodology of quantum mechanics nor to the formalisation of the alternative yes-no process. The main argument is that quantum structures (for example, orthocomplemented partially ordered sets, orthomodular lattices and partial algebras) are not systems of logic proper, but rather are perfectly formalizable structures in classical logic. In § 1, the orthodox formulation of Jauch and Piron are analyzed. I affirm that the objections made by these authors against “quantum logic” are not eliminated by the introduction of a conditional in the lattice structures, since the true reason why quantum “logic” is not really logic lies in its formulation, which is not that of a propositional system, but rather that of a specific theory whose model is these lattices. It is argued that the collection of the properties satisfied by certain objects of a structure does not justify their presentation as purely logical properties, nor does the presence of certain operations of the least upper bound and greatest lower bound permit us to call symbols introduced by analogy in propositionaI logic “connectives”. In any case, if that formalization is indeed possible, it does not imply the adoption of a new logic as a formal instrument for quantum mechanics. In § 2, aspects of quantum theories which are especially semantic in nature are analyzed. I agree with van Fraassen’s statement in the sense that that semantic formulation makes specific concepts of the theory clearer, but it does not attempt to construct its own logic. I deduce that it is non-classical logic which can actually take advantage of quantum semantics, as a model which, eventually, could throw light on the metalogical properties of certain special propositional systems. Finally, by using Benioff’s profound set theory analysis, I advance the hypothesis that classical logic is possibly the only one that can account for the richness of the meanings that appear in quantum mechanics.[Summary by Carlos Lungarzo]
Since Birkhoff and von Neumann [2] a new area of logical investigation has grown up under the name of quantum logic. However it seems to me that many authors have been inclined to discuss algebraic semantics as such (mainly lattices of a certain kind) almost directly without presenting any axiomatic system, far from developing any proof theory of quantum logic. See, e.g., Gunson [9], Jauch [10], Varadarajan [15], Zeirler [16], etc. In this sense many works presented under the name of quantum logic are algebraic in essence rather than genuinely logical, though it is absurd to doubt the close relationship between algebraic and logical study on quantum mechanics. The main purpose of this paper is to alter this situation by presenting an axiomatization of quantum logic as natural and as elegant as possible, which further proof-theoretical study is to be based on. It is true that several axiomatizations of quantum logic are present now. Several authors have investigated the so-called material implication α → β ( = ¬α∨(α ∧ β)) very closely with due regard to its importance. See, e.g., Finch [5], Piziak [11], etc. Indeed material implication plays a predominant role in any axiomatization of a logic in Hilbert-style. Clark [4] has presented an axiomatization of quantum logic with negation ¬ and material implication → as primitive connectives. In this paper we do not follow this approach. First of all, this approach is greatly complicated because orthomodular lattices are only locally distributive.
Several attempts have been made to establish the logical structure of quantum mechanics on an axiomatic foundation. It has been shown that, whereas the logical structure of the propositional calculus of classical mechanics is that of a Boolean lattice, the logical structure of the propositional calculus of quantum mechanics is non-Boolean. Moreover, a probability model of a quantum system, namely, the quantum probability model, has a mathematical structure completely different from that of the Kolmogorovian probability model, which is the standard probability model of a classical system. These results, which are still controversial, are analysed and put in critical perspective.