When physics must be based on an axiomatic foundation then the law set of traditional quantum logic is a valid candidate. However, at first sight, these axioms do not treat physical fields and they do not treat dynamics. It only prescribes the static relations that exist between quantum logical propositions that treat static subjects. These subjects are considered to be physical subjects or their properties. Amongst these propositions statements exist that describe everything that can be said about the static condition of a given physical item. Such propositions represent that item.
When physics must be based on an axiomatic foundation then the law set of traditional quantum logic is a valid candidate. However, at first sight, these axioms do not treat physical fields and they do not treat dynamics. It only prescribes the static relations that exist between quantum logical propositions that treat static subjects. These subjects are considered to be physical subjects or their properties. Amongst these propositions statements exist that describe everything that can be said about the static condition of a given physical item. Such propositions represent that item.
Knowledge of the classical and nonclassical theories of logical propositions is not required for an understanding of the analysis of the nonseparability of systems. Some very elementary notions about that matter may help to place the general problem in appropriate perspective. Classically, physical propositions bear upon physical systems. A proposition a bearing on a system S is a qualitative statement about S. The quantum calculus of propositions has been successfully used as an alternative and more general basis for the foundations of quantum mechanics. It is a truism that the advent of the modern physical theories—relativity, quantum mechanics, quantum electrodynamics, S matrix theory, and so on—has induced us to abandon many familiar intuitive concepts. All the successful theories mentioned above are built on elaborate sets of axioms that are justified only a posteriori, that is, by agreement between some of their consequences and observed facts.
The term proposition usually denotes in quantum mechanics (QM) an element of (standard) quantum logic (QL). Within the orthodox interpretation of QM the propositions of QL cannot be associated with sentences of a language stating properties of individual samples of a physical system, since properties are nonobjective in QM. This makes the interpretation of propositions problematical. The difficulty can be removed by adopting the objective interpretation of QM proposed by one of the authors (semantic realism, or SR, interpretation). In this case, a unified perspective can be adopted for QM and classical mechanics (CM), and a simple first order predicate calculus L(x) with Tarskian semantics can be constructed such that one can associate a physical proposition (i.e., a set of physical states) with every sentence of L(x). The set Pf of all physical propositions is partially ordered and contains a subset PTf of testable physical propositions whose order structure depends on the criteria of testability established by the physical theory. In particular, PTf turns out to be a Boolean lattice in CM, while it can be identified with QL in QM. Hence the propositions of QL can be associated with sentences of L(x), or also with the sentences of a suitable quantum language LTQ(x), and the structure of QL characterizes the notion of testability in QM. One can then show that the notion of quantum truth does not conflict with the classical notion of truth within this perspective. Furthermore, the interpretation of QL propounded here proves to be equivalent to a previous pragmatic interpretation worked out by one of the authors, and can be embodied within a more general perspective which considers states as first order predicates of a broader language with a Kripkean semantics.
Why should anyone think that what is so deeply puzzling about the quantum world is really a matter of logic? Why should anyone think that studying the logic of quantum mechanics should be a route to a proper understanding of quantum mechanics? There is a tradition in the philosophy of quantum mechanics, a tradition stronger among philosophical commentators than among physicists, which sees the subject as the business of understanding quantum mechanical language and the way that language relates to the world. It is an idea more strongly held by philosophers – it comes much more naturally to them – because the analytic tradition, in which most Anglo-American philosophers of science are raised, is generally much exercised about words and how they hook on to things. The analytic tradition in the philosophy of quantum mechanics is not unreasonable. Everyone admits that we cannot picture the microphysical world, that a graphic or iconic representation of quantum systems is impossible. Therefore understanding quantum mechanics must be a matter of understanding the logic of the words and the mathematics of quantum mechanics. If this seems implausible, it is because of an ambiguity in our use of the verb ‘to understand’. We understand the physical world, and we understand physics. But physics is not the physical world. It is something of an entirely different sort, a human product, a way of representing the world.
The objects that occur in nature can be categorized in several levels. In this collection every level except the first level is built from lower level objects. This collection represents a simple model of nature. The model exploits the possibilities that mathematical concepts provide. Also typical physical ingredients will be used. The paper splits the hierarchy of objects in a logic model and a geometric model. These two hierarchies partly overlap. The underlying Hilbert Book Model is a simple self-consistent model of physics that is strictly based on quantum logic. This paper refines quantum logic to Hilbert logic such that it more directly resembles its lattice isomorphic companion, which is a separable Hilbert space. The HBM extends these static sub-models into a dynamic model that consists of an ordered sequence of the static sub-models. The paper is founded on three starting points: • A sub-model in the form of traditional quantum logic that represents a static status quo. • A correlation vehicle that establishes cohesion between subsequent members of a sequence of such sub-models. • The cosmological principle. Further it uses a small set of hypotheses. It turns out that the cosmological principle is already a corollary of the first two points. The paper explains or indicates the explanation of all features of fundamental physics that are encountered in the discussed hierarchy which ranges from propositions about physical objects until composites of elementary particles. Amongst them are the cosmological principle, the existence of quantum physics, the existence of a maximum speed of information transfer, the existence of physical fields, the origin of curvature, the origin of inertia, the dynamics of gravity, the existence of elementary particles, the existence of generations of elementary particles and the existence of the Pauli principle.
It is possible to completely explain all aspects of quantum mechanics by expressing the relations between physical properties in terms of complex conditional probabilities (Phys. Rev. A 89, 042115(2014)). These fully deterministic probabilities replace the classical notions of phase space geometries and continuous trajectories with a more accurate description that takes into account the role of dynamics in the emergence of reality. We can then understand why so many previous attempts to find a detailed explanation of quantum phenomena have failed: the assumption of a static reality breaks down when the interaction needed to obtain a real effect exceeds Planck`s constant. Beyond that limit, complex conditional probabilities are the only valid expression of the fundamental laws of physics.
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.