Abstract
The notion of analytical apparatus was introduced in the paper by Hilbert, Nordheim and von Neumann (1928) ≪ Über die Grundlagen der Quantenmechanik ≫ : The analytical apparatus <der analytische Apparat> is simply the mathematical formalism or the set of logico-mathematical structures of a physical theory. Von Neumann used the notion extensively in his 1932 seminal work on the mathematical foundations of Quantum Mechanics and he stressed particularly the auxiliary notion of conditions of reality <Realitätsbedingungen>. I want to show that this last notion corresponds grosso modo to our notion of model in contemporary philosophy of physics. Hermann Weyl, who has initiated group theory in Quantum Mechanics (1928), also exploited the idea in his conception of the parallelism between a mathematical formalism and its physical models.
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Notes
- 1.
I am drawing here parts from various recent papers of mine, especially from three different sources “Hilbert’s idea of physical axiomatics: the analytical apparatus of quantum mechanics” in Journal of Physical Mathematics, Vol. 2 (2010), 1–14, “Hermann Minkowski:From Geometry of Numbers to Physical Geometry”, Chapter 10 in Minkowski Spacetime: A Hundred Years Later, ed. by Vesselin Petkov, Springer, 2010, 247–257 and “A General No-Cloning Theorem for an Infinite Multiverse”, in Reports on Mathematical Physics, Vol. 2 (2013), no. 2, 191–199.
- 2.
Von Neumann’s dogma has been challenged in 1952 by Wick, Wightman and Wigner who introduced superselection rules showing that there exist Hermitian operators that do not correspond to observables; on the other side, Park and Margenau argue that there are observables, for example, the non-commuting x and z—components of spin which are not represented by Hermitian operators.
- 3.
Orthocomplementation requires that \(\left (a^{-}\right )^{-} = a,a^{-}\cap a =\emptyset\) and \(a \leq b \leftrightarrow b^{-}\leq a^{-}\).
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Gauthier, Y. (2015). Arithmetical Foundations for Physical Theories. In: Towards an Arithmetical Logic. Studies in Universal Logic. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-22087-1_5
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