Abstract
In this chapter we will discuss the general mathematical structure of Quantum Mechanics. The procedure to achieve this is due to von Neumann, essentially, and will be presented here in its modern account via Gleason’s theorem: that is, an extension of classical (Hamiltonian) mechanics that keeps in account the experimental evidence about the nature of quantum systems, seen in the previous chapter.
Some historians claim that it is very difficult, nowadays, to Find the line separating - and at the same time joining - the Experimental level from the so-called theoretical one. But their View already includes several arbitrary elements, the so-called Approximations.
Paul K. Feyerabend
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Notes
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2 ℋ n + 1 is the total space of a fibre bundle with base ℝ (the time axis) and fibres ℱt given by 2n-dimensional symplectic manifolds. There is an atlas on ℋ n + 1 whose local charts have coordinates t, q 1,..., q n, p 1,..., p n , where t is the natural parameter on the base ℝ while the remaining 2n coordinates define a local simplistic frame on each ℱ t .
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3 ℱ t is a smooth manifold hence a locally compact Hausdorff space (since locally homeomorphic to ℝn). As μ t is defined on ℬ(ℱ t ) and ρ t is continuous, ν t is well defined on ℬ(ℱ t ).
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4 Sikorski S.: On the representation of Boolean algebras as field of sets. Fund. Math. 35, 247–256 (1948).
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5 Stone M.H.: The Theory of Representations of Boolean Algebras. Trans. AMS 40, 37–111
(1936).
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6 Solèr M.P.: Characterization of Hilbert spaces by orthomoduler spaces. Communications in Algebra 23, 219–243 (1995).
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7 Holland S.S.: Orthomodularity in infinite dimensios; a theorem of M. Solèr. Bulletin of the American Mathematical Society 32, 205–234 (1995).
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8 Aerts D., van Steirteghem B.: Quantum Axiomatics and a theorem of M.P. Solèr. International Journal of Theoretical Physics 39, 497–502 (2000).
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9 Although we will not do so, one could also use two-parameter groupoids of unitary transformations between different Hilbert spaces.
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10 Particles with spin 1/2 admit a Hilbert space – in which the observable spin is defined – of dimension 2. The same occurs to the Hilbert space in which the polarisation of light is described (cf. helicity of photons). When these systems are described in full, however, for instance including freedom degrees relative to position or momentum, they are representable on a separable Hilbert space of infinite dimension.
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Kochen S., Specker E.P.:The problem of Hidden Variablesin Quantum Mechanics. J. Math. Mech. 17(1), 59–87 (1967).
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We cannot but notice how this interpretation muddles the semantic and syntactic levels. Although this could be problematic in a formulation within formal logic, the use physicists make of the interpretation eschews the issue.
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If the charge is taken to be continuous and Hq is the subspace where it equals q ∈ R, i.e. the eigenspace relative to eigenvalue q for a self-adjoint operator Q, the Hilbert space (nonseparable) is still a direct sum⊕q∈ℝHq. ℝ coincides then with the point spectrum of Q. Some authors, instead, prefer to think the Hilbert space as a direct integral, thereby preserving its separability, and in this case bR =σc(Q).
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Moretti, V. (2013). The first 4 axioms of QM: propositions, quantum states and observables. In: Spectral Theory and Quantum Mechanics. UNITEXT(). Springer, Milano. https://doi.org/10.1007/978-88-470-2835-7_7
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