Journal of Physical Mathematics

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

Yvon Gaulthier

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

Article information

J. Phys. Math., Volume 2 (2010), Article ID P100601, 14 pages.

First available in Project Euclid: 22 September 2011

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Digital Object Identifier

Primary: 46K15: Hilbert algebras 60B11: Probability theory on linear topological spaces [See also 28C20] 81P10: Logical foundations of quantum mechanics; quantum logic [See also 03G12, 06C15]

Topological algebras Hilbert algebras Probability theory Topological structures Linear topological spaces Quantum theory Quantum logic


Gaulthier, Yvon. Hilbert's idea of a physical axiomatics: the analytical apparatus of quantum mechanics. J. Phys. Math. 2 (2010), Article ID P100601, 14 pages. doi:10.4303/jpm/P100601.

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