In quantum logic there is well-known arbitrariness in choosing a binary operation for conditional. Currently, we have at least three candidates, called the Sasaki conditional, the contrapositive Sasaki conditional, and the relevance conditional. A fundamental problem is to show how the form of the conditional follows from an analysis of operational concepts in quantum theory. Here, we attempt such an analysis through quantum set theory (QST). In this paper, we develop quantum set theory based on quantum logics with those three conditionals, each of which defines different quantum logical truth value assignment. We show that those three models satisfy the transfer principle of the same form to determine the quantum logical truth values of theorems of the ZFC set theory. We also show that the reals in the model and the truth values of their equality are the same for those models. Interestingly, however, the order relation between quantum reals significantly depends on the underlying conditionals. We characterize the operational meanings of those order relations in terms of joint probability obtained by the successive projective measurements of arbitrary two observables. Those characterizations clearly show their individual features and will play a fundamental role in future applications to quantum physics.
In quantum logic there is well-known arbitrariness in choosing a binary operation for conditional. Currently, we have at least three candidates, called the Sasaki conditional, the contrapositive Sasaki conditional, and the relevance conditional. A fundamental problem is to show how the form of the conditional follows from an analysis of operational concepts in quantum theory. Here, we attempt such an analysis through quantum set theory (QST). In this paper, we develop quantum set theory based on quantum logics with those three conditionals, each of which defines different quantum logical truth value assignment. We show that those three models satisfy the transfer principle of the same form to determine the quantum logical truth values of theorems of the ZFC set theory. We also show that the reals in the model and the truth values of their equality are the same for those models. Interestingly, however, the order relation between quantum reals significantly depends on the underlying conditionals. We characterize the operational meanings of those order relations in terms of joint probability obtained by the successive projective measurements of arbitrary two observables. Those characterizations clearly show their individual features and will play a fundamental role in future applications to quantum physics.
Abstract A difficulty in quantum logic is the well-known arbitrariness in choosing a binary operation for conditional among three principal candidates called the Sasaki, the contrapositive Sasaki, and the relevance conditional, mainly chosen from syntactical grounds. A fundamental problem remains to clarify their semantical differences manifest in operational concepts in quantum theory. Here, we attempt such an analysis through quantum set theory, developing models of quantum set theory built upon quantum logics with those three conditionals, each of which defines different quantum logical truth-value assignment for set theoretical statements. We show that each of them satisfies the transfer principle to determine the truth values of theorems of the ZFC set theory and defines the internal reals bijectively corresponding to the observables of the quantum system under consideration. Then, the truth values of their equality relations are identical irrespective of the chosen conditionals. Interestingly, however, their order relations exhibit a strong dependence on the specific conditional employed, while the order relation attains full truth value if and only if Olson’s spectral order relation holds. We further characterize the order relation in terms of experimentally accessible relations for outcomes of successive projective measurements of the corresponding observables, showing that each choice has its own operational meaning with symmetry between the Sasaki and the contrapositive Sasaki conditionals, in contrast to the majority view that favors the Sasaki conditional. Our findings reveal that quantum set theory yields empirically testable predictions concerning state-dependent binary relations between quantum observables, thereby extending Born’s probabilistic interpretation from propositions to relations.
The purpose of this dissertation is to construct a radically new type of mathematics whose underlying logic differs from the ordinary classical logic used in standard mathematics, and which we feel may be more natural for applications in quantum mechanics. Specifically, we begin by constructing a first order quantum logic, the development of which closely parallels that of ordinary (classical) first order logic --- the essential differences are in the nature of the logical axioms, which, in our construction, are motivated by quantum theory. After showing that the axiomatic first order logic we develop is sound and complete (with respect to a particular class of models), this logic is then used as a foundation on which to build (axiomatic) mathematical systems --- and we refer to the resulting new mathematics as "quantum mathematics." As noted above, the hope is that this form of mathematics is more natural than classical mathematics for the description of quantum systems, and will enable us to address some foundational aspects of quantum theory which are still troublesome --- e.g. the measurement problem --- as well as possibly even inform our thinking about quantum gravity.\n\nAfter constructing the underlying logic, we investigate properties of several mathematical systems --- e.g. axiom systems for abstract algebras, group theory, linear algebra, etc. --- in the presence of this quantum logic. In the process, we demonstrate that the resulting quantum mathematical systems have some strange, but very interesting features, which indicates a richness in the structure of mathematics that is classically inaccessible. Moreover, some of these features do indeed suggest possible applications to foundational questions in quantum theory. \n\nWe continue our investigation of quantum mathematics by constructing an axiomatic quantum set theory, which we show satisfies certain desirable criteria. Ultimately, we hope that such a set theory will lead to a foundation for quantum mathematics in a sense which parallels the foundational role of classical set theory in classical mathematics. One immediate application of the quantum set theory we develop is to provide a foundation on which to construct quantum natural numbers, which are the quantum analog of the classical counting numbers. It turns out that in a special class of models, there exists a 1-1 correspondence between the quantum natural numbers and bounded observables in quantum theory whose eigenvalues are (ordinary) natural numbers. This 1-1 correspondence is remarkably satisfying, and not only gives us great confidence in our quantum set theory, but indicates the naturalness of such models for quantum theory itself. We go on to develop a Peano-like arithmetic for these new "numbers," as well as consider some of its consequences. Finally, we conclude by summarizing our results, and discussing directions for future work.
Quantum mechanics is basically a mathematical recipe on how to construct physical models. Historically its origin and main domain of application has been in the microscopic regime, although it strictly seen constitutes a general mathematical framework not limited to this regime. Since it is a statistical theory, the meaning and role of probabilities in it need to be defined and understood in order to gain an understanding of the predictions and validity of quantum mechanics. The interpretational problems of quantum mechanics are also connected with the interpretation of the concept of probability. In this thesis the use of probability theory as extended logic, in particular in the way it was presented by E. T. Jaynes, will be central. With this interpretation of probabilities they become a subjective notion, always dependent on one's state of knowledge or the context in which they are assigned, which has consequences on how things are to be viewed, understood and tackled in quantum mechanics. For instance, the statistical operator or density operator, is usually defined in terms of probabilities and therefore also needs to be updated when the probabilities are updated by acquisition of additional data. Furthermore, it is a context dependent notion, meaning, e.g., that two observers will in general assign different statistical operators to the same phenomenon, which is demonstrated in the papers of the thesis. It is also presented an alternative and conceptually clear approach to the problematic notion of "probabilities of probabilities", which is related to such things as probability distributions on statistical operators. In connection to this, we consider concrete numerical applications of Bayesian quantum state assignment methods to a three-level quantum system, where prior knowledge and various kinds of measurement data are encoded into a statistical operator, which can then be used for deriving probabilities of other measurements. The thesis also offers examples of an alternative quantum state assignment technique, using maximum entropy methods, which in some cases are compared with the Bayesian quantum state assignment methods. Finally, the interesting and important problem whether the statistical operator, or more generally quantum mechanics, gives a complete description of "objective physical reality" is considered. A related concern is here the possibility of finding a "local hidden-variable theory" underlying the quantum mechanical description. There have been attempts to prove that such a theory cannot be constructed, where the most well-known impossibility proof claiming to show this was given by J. S. Bell. In connection to this, the thesis presents an idea for an interpretation or alternative approach to quantum mechanics based on the concept of space-time.
The notion of equality between two observables will play many important roles in foundations of quantum theory. However, the standard probabilistic interpretation based on the conventional Born formula does not give the probability of equality between two arbitrary observables, since the Born formula gives the probability distribution only for a commuting family of observables. In this paper, quantum set theory developed by Takeuti and the present author is used to systematically extend the standard probabilistic interpretation of quantum theory to define the probability of equality between two arbitrary observables in an arbitrary state. We apply this new interpretation to quantum measurement theory, and establish a logical basis for the difference between simultaneous measurability and simultaneous determinateness.
Quantum logic has been introduced by Birkhoff and von Neumann as an attempt to base the logical primitives, the propositions and the relations and operations among them, on quantum theoretical entities, and thus on the related empirical evidence of the quantum world. We give a brief outline of quantum logic, and some of its algebraic properties, such as nondistributivity, whereby emphasis is given to concrete experimental setups related to quantum logical entities. A probability theory based on quantum logic is fundamentally and sometimes even spectacularly different from probabilities based on classical Boolean logic. We give a brief outline of its nonclassical aspects; in particular violations of Boole-Bell type consistency constraints on joint probabilities, as well as the Kochen-Specker theorem, demonstrating in a constructive, finite way the scarcity and even nonexistence of two-valued states interpretable as classical truth assignments. A more complete introduction of the author can be found in the book "Quantum Logic" (Springer, 1998)
In the absence of a satisfactory interpretation of quantum theory, physical law lacks physical basis. This paper reviews the orthodox, or Dirac–von Neumann interpretation, and makes explicit that Hilbert space describes propositions about measurement results. Kets are defined as conditional clauses referring to measurements in a formal language. It is seen that these clauses are elements of a Hilbert space, such that addition is logical disjunction, the dual space consists of consequent clauses, and the inner product is a set of statements in the subjunctive mood. The probability interpretation gives truth values for corresponding future tense statements when the initial state is actually prepared and the final state is to be measured. The mathematical structure of quantum mechanics is formulated in terms of discrete measurement results at finite level of accuracy and does not depend on an assumption of a substantive, or background, space–time continuum. A continuum of kets, ∣x〉 for x ∈ ℝ3, is constructed from linear combinations of kets in a finite basis. The inner product can be expressed either as a finite sum or as an integral. Discrete position functions are uniquely embedded into smooth wave functions in such a way that differential operators are defined. It is shown that the choice of basis has no effect on underlying physics (quantum covariance). The Dirac delta has a representation as a smooth function. Operators do not in general have an integral form. The Schrödinger equation is shown from the requirements of the probability interpretation. It is remarked that a formal construction of quantum electrodynamics (qed) avoiding divergence problems has been completed using finite dimensional Hilbert space. I conclude that quantum mechanics makes statements about the world with clear physical meaning, such that space is emergent from particle interactions and has no fundamental role.