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Journal of Physical Mathematics

Hilbert's idea of a physical axiomatics: the analytical apparatus of quantum mechanics

Yvon Gauthier

Published Jan 1, 2010• 2 citations
Publisher

Abstract

We discuss the Hilbert program for the axiomatization of physics in the contextof\nwhat Hilbert and von Neumann came to call the analytical apparatus and\nitsconditions of reality. We suggest that the idea of a physical logic is the\nbasisfor a physical mathematics and we use quantum mechanics as a paradigm case\nforaxiomatics in the sense of Hilbert. Finite probability theory requires\nfinitederivations in the measurement theory of QM and we give a polynomial\nformulationof local complementation for the metric induced on the topology of\nthe Hilbertspace. The conclusion hints at a constructivist physics.

DOI
10.4303/jpm/p100601

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We discuss the Hilbert program for the axiomatization of physics in the contextof\nwhat Hilbert and von Neumann came to call the analytical apparatus and\nitsconditions of reality. We suggest that the idea of a physical logic is the\nbasisfor a physical mathematics and we use quantum mechanics as a paradigm case\nforaxiomatics in the sense of Hilbert. Finite probability theory requires\nfinitederivations in the measurement theory of QM and we give a polynomial\nformulationof local complementation for the metric induced on the topology of\nthe Hilbertspace. The conclusion hints at a constructivist physics.
We discuss the Hilbert program for the axiomatization of physics in the context of what Hilbert and von Neumann came to call the analytical apparatus and its conditions of reality.We suggest that the idea of a physical logic is the basis for a physical mathematics and we use quantum mechanics as a paradigm case for axiomatics in the sense of Hilbert.Finite probability theory requires finite derivations in the measurement theory of QM and we give a polynomial formulation of local complementation for the metric induced on the topology of the Hilbert space.The conclusion hints at a constructivist physics.
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.
Analysing Quantum Measurement requires analysing the physics of amplification since amplification of phenomena from one scale to another scale is essential to measurement. There still remains the task of working this into an axiomatic logical structure, what should be the foundational status of the concepts of measurement and probability. We argue that the concept of physical probability is a multi-scale phenomenon and as such, can be explicitly defined in terms of more fundamental physical concepts. Thus Quantum Mechanics can be given a logically unexceptionable axiomatisation. We introduce a new definition of macroscopic observable which implements Bohr's insight that the observables of a measurement apparatus are classical in nature. In particular, we obtain the usual non-abelian observables as limits of abelian, classical, observables. This is the essential step in Hilbert's Sixth Problem.
The emergence of an objective reality in line with the laws of the microscopic world has been the focus of longstanding debates. Recent approaches seem to have reached a consensus at least with respect to one aspect, namely, that the encoding of information about a given observable in a physical degree freedom is a necessary condition for such an observable to become an element of the physical reality. Taking this as a fundamental premise and inspired by quantum information theory, here we build an axiomatization for quantum realism---a notion of realism compatible with quantum theory. Our strategy consists of listing some physically motivated principles able to characterize quantum realism in a ``metric'' independent manner. We introduce some criteria defining monotones and measures of realism and then search for potential candidates within some celebrated information theories---those induced by the von Neumann, R\'enyi, and Tsallis entropies. We explicitly construct some classes of entropic quantifiers, among which some are shown to satisfy all the proposed axioms and hence can be taken as faithful estimates for the degree of reality (or definiteness) of a given physical observable. Hopefully, our framework may offer a formal ground for further discussions on foundational aspects of quantum mechanics.