The natural numbers play a key role in axiomatic set theory and hence in the foundations of mathematics. The natural numbers also have a major role in quantum mechanics. The idea is explored that von Neumann's construction (vNC) of the natural numbers, within the framework of Zermelo-Fraenkel (ZF) axiomatic set theory, can serve as a blueprint for deriving some key basic equations of quantum mechanics. This approach obviates any need for quantizing classical mechanics and provides further support for the view that quantum mechanics is perfectly applicable to fields beyond those in physics for which it was originally intended.This article is part of the theme issue 'Quantum theory and topology in models of decision making (Part 1)'.
The natural numbers play a key role in axiomatic set theory and hence in the foundations of mathematics. The natural numbers also have a major role in quantum mechanics. The idea is explored that von Neumann's construction (vNC) of the natural numbers, within the framework of Zermelo-Fraenkel (ZF) axiomatic set theory, can serve as a blueprint for deriving some key basic equations of quantum mechanics. This approach obviates any need for quantizing classical mechanics and provides further support for the view that quantum mechanics is perfectly applicable to fields beyond those in physics for which it was originally intended.This article is part of the theme issue 'Quantum theory and topology in models of decision making (Part 1)'.
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 logic as it is presented in Birkhoff-von Neumann’s 1936 paper arose within the framework of the mathematical elaboration of the foundations of quantum mechanics. Birkhoff-von Neumann’s axiomatic was an alternative of the traditional formulation of foundations of quantum mechanics proceeding from the technique of the Hilbert space. It provided more economic formulation of the foundations of the theory. In Mackay’s writings it has been transformed into the axiomatic of quantum theory of probability. However, Mackay came back to the concept of Hilbert space as the fundamental for quantum mechanics. He also adopted the rules of super-selections in his presentation of the foundations of quantum mechanics. The contemporary approach to Birkhoff-von Neumann’s axiomatic is twofold: the criticism of their too rigid construction from the point of view of physics and the development in connection with mathematical intuitionism.
The set-theory twist of quantum mechanics uncovers forcing in axiomatic Zermelo-Fraenkel set theory as a viable tool to understand the singularities in a physical spacetime and serves as a link between the quantum and classical worlds. The random forcing explains the emergence of time in quantum mechanics on infinite-dimensional Hilbert spaces. A natural flow of randomness from fully random to deterministic and classical, as in the measurement procedure and the decoherence process, becomes responsible for the cosmological arrow of time and also for the local-coordinate time in spacetime.
This paper deals with the foundations of quantum mechanics. We start by outlining the characterisation, due to Birkhoff and Von Neumann, of the logical structures of the theories of classical physics and quantum mechanics, as boolean and modular lattices respectively. We then derive these descriptions from what we claim are basic properties of any physical theory - i.e. the notion that a quantity in such a theory may be analysed into parts and that the results of this analysis may be treated in languages with an underlying boolean structure. We shall see that in the course of constructing a model of a theory with these properties different indistinguishable possibilities will arise for how the elements of the model may be named, that is to say different possibilities arise for how they can be associated with points from Set. Taking a particular collection of possibilities gives the usual boolean lattice of the propositions of classical physics. Taking all possibilities - in a sense, the set of all things that may be described by physical theories - gives the lattice of quantum mechanical propositions. This gives an interpretation of quantum mechanics as the complete set of such possible descriptions, the complete physical description of the world.
The Orthodox interpretation of quantum mechanics, which followed the Copenhagen Interpretation but was enhanced by primarily Werner Heisenberg and John von Neumann into a fully developed theory, brought in, among others, the role of measurement, available choices and response of the quantum system.It is, more consistent and clear than other interpretations of quantum mechanics as it provides account of the interactions of observers with the external world.As such, the Orthodox interpretation does a lot more than just account for physical interactions in the atomic world, which was the original goal of quantum mechanics in the early part of the twentieth century.In this article we present several issues that may have been answered or need further development, such as measurement and observation, what is Nature and who the observer is.Extending Orthodox quantum mechanics, leading to universal non-dual Awareness may provide a consistent and integrated view of reality and is consistent with advances in mathematical theory.An issue of paramount importance is what are the philosophical underpinnings or ontological view of the quantum nature of the universe and the role of human minds, observations and choices.
Our purpose in this chapter is to present the key concepts of quantum mechanics in the language of Hilbert spaces. The reader who has not previously met the physical ideas motivating quantum mechanics, and some of the more elementary applications of Schrödinger's equation, is encouraged to read any of a number of excellent texts on the subject such as [1–4]. Otherwise, the statements given here must to a large extent be taken on trust – not an altogether easy thing to do, since the basic assertions of quantum theory are frequently counterintuitive to anyone steeped in the classical view of physics. Quantum mechanics is frequently presented in the form of several postulates, as though it were an axiomatic system such as Euclidean geometry. As often presented, these postulates may not meet the standards of mathematical rigour required for a strictly logical set of axioms, so that little is gained by such an approach. We will do things a little more informally here. For those only interested in the mathematical aspects of quantum mechanics and the role of Hilbert space see [5–8]. Many of the standard applications, such as the hydrogen atom, will be omitted here as they can be found in all standard textbooks, and we leave aside the enormous topic of measurement theory and interpretations of quantum mechanics. This is not to say that we need be totally comfortable with quantum theory as it stands. Undoubtedly, there are some philosophically disquieting features in the theory, often expressed in the form of socalled paradoxes.