What does Quantum Logic Explain?

In 1981, Takeuti introduced quantum set theory as the quantum counterpart of Boolean valued models of set theory by constructing a model of set theory based on quantum logic represented by the lattice of closed subspaces in a Hilbert space and showed that appropriate quantum counterparts of ZFC axioms hold in the model. Here, Takeuti's formulation is extended to construct a model of set theory based on the logic represented by the lattice of projections in an arbitrary von Neumann algebra. A transfer principle is established that enables us to transfer theorems of ZFC to their quantum counterparts holding in the model. The set of real numbers in the model is shown to be in one-to-one correspondence with the set of self-adjoint operators affiliated with the von Neumann algebra generated by the logic. Despite the difficulty pointed out by Takeuti that equality axioms do not generally hold in quantum set theory, it is shown that equality axioms hold for any real numbers in the model. It is also shown that any observational proposition in quantum mechanics can be represented by a corresponding statement for real numbers in the model with the truth value consistent with the standard formulation of quantum mechanics, and that the equality relation between two real numbers in the model is equivalent with the notion of perfect correlation between corresponding observables (self-adjoint operators) in quantum mechanics. The paper is concluded with some remarks on the relevance to quantum set theory of the choice of the implication connective in quantum logic.

Quantum information technology mainly includes quantum communication, quantum radar, quantum computer these three aspects, in recent years, rapid development, many countries invested heavily in development. The awarding of the Nobel Prize in Physics in 2022 to three scientists for their work in quantum informatics has further stimulated people's interest in quantum theory, which has been studied and discussed. This paper not only reviews the historical situation, but also thinks and innovates in theory. The main contents of this paper are as follows: It is pointed out that there are fundamental contradictions between relativity and quantum mechanics; The wave function, quantum statistics and uncertainty principle are discussed in detail. The hidden variable theory is reviewed and the Bell inequality is discussed. The Aspect two-photon experiment was analyzed. The rationality of Bohm experiment scheme is discussed. The development of Bell type experiment is discussed. It is pointed out that the entangled state is not acting at a distance but propagating faster than the speed of light. Reviews the Copenhagen interpretation of quantum mechanics; Quantum communication and Wootters theorem are discussed. etc. This paper holds that quantum mechanics has been finalized from 1926 to 1928, and its basic content has not changed much. But for some of the accusations, it is necessary to answer them theoretically. For example, we believe that the Copenhagen interpretation fundamentally changes our understanding of nature and marks a profound revolution in physics. And no other theory has since emerged that has such a profound understanding and wide application of microscopic phenomena as this interpretation. Another example is Bohm's two-particle correlation spin scheme in quantum entanglement experiments, which has been proved to be effective in a series of experiments, which is an important contribution in Bohm's life. This paper holds that the so-called quantum field theory is a failure, and the original quantum mechanics should still be advocated today. Current theoretical research should pay great attention to quantum entangled states, because its nature is still unclear. Understanding this "first mystery of the physical world" not only has scientific significance, but also has great philosophical significance for understanding the universe.

Friedman and Putnam have argued (Friedman and Putnam 1978) that the quantum logical interpretation of quantum mechanics gives us an explanation of interference that the Copenhagen interpretation cannot supply without invoking an additional ad hoc principle, the projection postulate. I show that it is possible to define a notion of equivalence of experimental arrangements relative to a pure state φ, or (correspondingly) equivalence of Boolean subalgebras in the partial Boolean algebra of projection operators of a system, which plays a role in the Copenhagen explanation of interference analogous to the role played by the material equivalence, given φ, of certain propositions in the Friedman-Putnam quantum logical analysis. I also show that the quantum logical interpretation and the Copenhagen interpretation are equally capable of avoiding the paradoxical conclusion of the Einstein-Podolsky-Rosen argument (Einstein, Podolsky, and Rosen 1935). Thus, neither interference phenomena nor the correlations between separated systems provide a test case for distinguishing between the relative acceptability of the Copenhagen interpretation and the quantum logical interpretation as explanations of quantum effects.

Besides the physicomathematical controversy concerning the ‘phenomenological justification’ and the specific formal structure of the calculus known as quantum logic (QL), there is a philosophical controversy concerning whether this ‘calculus of experimental propositions’ is properly speaking a logic rather than simply an algebraic structure only analogous to logic properly so called. Directly associated with this controversy is the issue of whether logic is an empirical science on the par with, e.g., physical geometry. This article is not concerned, however, with the general question of the empirical character of logic, either classical or quantal; it is rather concerned only with one seemingly decisive objection against regarding QL as logic, namely, the objection advanced by Jauch and Piron (1970, p. 176), who argue that since QL lacks an essential feature of logic — viz., a deduction scheme — it is `very questionable whether we may properly call the lattice of general quantum mechanics a logic.’ Their argument is based on what they regard to be the general failure of the lattice of QL propositions to admit a material implication connective or conditional operation by means of which the modus ponens deduction scheme can be incorporated into QL. The view of Jauch and Piron is also supported by Greechie and Gudder (1971, 1973), who examine a number of results which suggest that no reasonable material conditional exists in QL.

CONCEPTUAL FOUNDATIONS OF QUANTUM MECHANICS: THE ROLE OF EVIDENCE THEORY, QUANTUM SETS, AND MODAL LOGIC

A non-symmetric transition probability in quantum mechanics

If a quantum logic16, 18, 21 admits hidden variables in a certain sense, then the resulting structure is called a “generalized measurable space”. Such a structure is more general than a traditional measurable space and is distinct from the conventional Hilbert space quantum mechanics. A generalized measurable space consists of a set Ω together with a collection of subsets C which form a α-orthocomplete orthomodular poset2 under set theoretic inclusion and complementation. We may think of (Ω, C) as giving a quantum set theory. Besides being of interest in quantum mechanics, there are possible applications of this theory to investigations in “elementary lengths”, pattern recognition, and computer science.1, 3, 5, 9, 12, 15, 22, 23, 24

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 $P^{f}$ of all physical propositions is partially ordered and contains a subset $P^{f}_{T}$ of testable physical propositions whose order structure depends on the criteria of testability established by the physical theory. In particular, $P^{f}_{T}$ 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 $L_{TQ}(x)$, 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.

Science sets itself apart from other paths to truth by recognizing that even its greate practitioners sometimes err.

This paper examines the three central theses of the Copenhagen interpretation of quantum mechanics. (1) Quantum mechanics must make use of concepts of classical physics; (2) descriptions which in classical physis are mutually exclusive are complementary descriptions of the same objects in quantum mechanics; (3) the uncertainty relations are natural laws which determine the limits of the application of classical concepts to micro-physical objects. Usually, these central theses of the Copenhagen interpretation are interpreted in a realistic-ontological manner. This paper criticizes this position, and, by means of a comparative investigation of the function of models in classical physics and quantum mechanics, proposes a model-theoretic interpretation. This interpretation lays special stress on the fictional status of models in quantum mechanics.

In the present work, I discuss the interpretation of quantum mechanics (QM) from the viewpoint of quantum logic (QL). I regard the objects of QL to be the possible (accidental) properties that can be ascribed to a quantum system SYS. The basic idea is that, at any given moment t, SYS actualizes some properties, but not others. The micro-state of SYS at time t is identified with the set of all properties that SYS actualizes at time t.One thing an interpretation of QM is supposed to do, I believe, is delineate the admissible quantum micro-states. Since the characterization of quantum micro-states is intimately related to the characterization of quantum logical consistency, the interpretation of QM is intimately tied to the interpretation of QL.Two kinds of interpretations are discussed. Strict interpretations are based on the assumption that the properties of a system are individuated by the projection operators on the associated Hilbert space.

The mathematical treatment of the concepts of vagueness and approximation is of increasing importance in artificial intelligence and related research. The theory of fuzzy sets was created by Zadeh [Z] to allow representation and mathematical manipulation of situations of partial truth, and proceeding from this a large amount of theoretical and applied development of this concept has occurred. The aim of this paper is to develop a natural logic and set theory that is a candidate for the formalisation of the theory of fuzzy sets. In these theories the underlying logic of properties and sets is intuitionistic, but there is a subset of formulae that are ‘crisp’, classical and two-valued, which represent the certain information. Quantum logic or logics weaker than intuitionistic can also be adopted as the basis, as described in [L]. The relationship of this theory to the intensional set theory MZF of [Gd] and the global intuitionistic set theory GIZF of Takeuti and Titani [TT] is also treated.

In this paper we shall describe a formal system which enables us to prove theorems within propositional calculus, logic of quantifiers and first order theories, including theorems containing programs. Its main feature relies on generating an additional set of assumptions needed to prove a considered formula. Thus we are able to consider expressions which can become theorems by looking for a special set of assumptions (axioms) and then adding it to the standard set of axioms. Using this system the correctness and equivalence of programs can be determined. In the end we present some experimental results.

The first three of these axioms describe quantum theory and classical mechanics as statistical theories from the very beginning. With these, it can be shown in which sense a more general than the conventional measure theoretic probability theory is used in quantum theory. One gets this generalization defining transition probabilities on pairs of events (not sets of pairs) as a fundamental, not derived, concept. A comparison with standard theories of stochastic processes gives a very general formulation of the non existence of quantum theories with hidden variables. The Cartesian product of probability spaces can be given a natural algebraic structure, the structure of an orthocomplemented, orthomodular, quasi-modular, not modular, not distributive lattice, which can be compared with the quantum logic (lattice of all closed subspaces of an infinite dimensional Hubert space). It is shown how our given system of axioms suggests generalized quantum theories, especially Schrodinger equations, for phase space amplitudes.

The lattice Boltzmann concepts and applications described so far refer to classical (non-quantum) physics. Whether this limitation is strictly scientific or purely contingental in character, it is difficult to say. However, it appears plausible to speculate that at least part of the LBE assets should be exportable to the quantum world. In principle, this does not even require much of a conceptual jump, since it has been known for a long time that quantum mechanics can be formulated in a fluid dynamic language. This analogy is precisely the stepping stone for this final chapter: stay with us, LBE goes quantum mechanical! Intriguing analogies between quantum mechanics and fluid mechanics have been pointed out since the earliest days of quantum theory [212]. The orthodox tenet is that these analogies are purely formal in character and do not bear upon the basic physics of quantum phenomena. A less orthodox, albeit not minor, stream of thought insists instead that quantum mechanics and notably its philosophical hard-core, Heisenberg’s uncertainty principle, are nothing but a mirror of our ignorance of the underlying (hidden) microscopic physical level. This leads to the puzzling theory of ‘hidden variables’ which can be traced back to Einstein and subsequently to D. Bohm and others [213]. It is not our intention here to enter this fascinating and still open subject [214]. We turn to a practical question instead: what can the analogy do for us in terms of numerical modeling of evolutionary quantum mechanical phenomena? The question is legitimate because, regardless of its philosophical implications, the fluid analogy certainly provides an intuitive and sound basis to develop numerical methods for time-dependent quantum mechanics. In particular, it appears reasonable to ask whether the advantages brought about by LBE in fluid dynamics can (by means of the fluid analogy) be exported to the context of quantum mechanics. Before we put forward our discrete kinetic theory version of the analogy, it is useful to provide a cursory survey of the main ideas behind the analogy itself. To this end, a short recap of basic notions of quantum mechanics is in order.