Abstract
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 nite 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.
Highlights
First we have a mathematical probability theory which serves as the basic analytical apparatus for the physical theory; follows a physical interpretation of the analytical structure and if the basis is fully determined, the analytical structure should be canonical
The usual presentation of QM requires the analytical apparatus of Hilbert space as a linear vector space with complex coefficients; among all linear manifolds that constitute a Hilbert space, the closed ones or the subspaces are of special interest for physics (i.e., QM here), since notions like orthogonal vectors, orthogonal complements, projections, etc. can be defined on them
Hilbert introduced the notion of analytical apparatus drawn from the general structure of an axiomatic system in physics and he made no mystery of his intention to provide physics with the same kind of axiomatic foundations as geometry
Summary
The usual presentation of QM requires the analytical apparatus of Hilbert space as a linear vector space with complex coefficients; among all linear manifolds that constitute a Hilbert space, the closed ones or the subspaces are of special interest for physics (i.e., QM here), since notions like orthogonal vectors, orthogonal complements, projections, etc. can be defined on them. Orthocomplementation induces an involutive antiautomorphism (a ) on the field vector space It is such an antiautomorphism which yields Gleason’s important theorem [6] stipulating that any probability measure μ(A) on the subspaces of H has the following form: μ(A) = T r W PA , where T r means T rX = ∑r(φr, Xφr) for any complete system of normalized orthogonal vectors, PA denotes the orthogonal projection of A, and W is a Hermitian operator which satisfies. Vectors σnαn have a well-defined value since projections are in bijection with the subspaces of the Hilbert space, but the system is no more in a pure state, but in a mixture. The lesson to be drawn here is perhaps that a paraconsistent logic that accommodates contradictions as well as tautologies can take care of a “quasi-consistency” for the “quasi-classicality” in a mixture of coherent histories in quantum systems and decoherent histories in classical (macroscopic) systems, as quantum decoherence theory seems to indicate, but the term “consistency histories” would sound like a misnomer for a theory which makes room for too many divergent histories, as the universal ramification of the wave function would have it in Everett’s multiverse interpretation
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