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The Foundations of Quantum Mechanics: Historical Analysis by Sandro Sozzo, Claudio Garola, Arcangelo Rossi

By Sandro Sozzo, Claudio Garola, Arcangelo Rossi

This quantity presents a different evaluation of contemporary Italian experiences at the foundations of quantum mechanics and similar historic, philosophical and epistemological themes. a meeting of students from diversified cultural backgrounds, the convention supplied a discussion board for a desirable alternate of rules and views on quite a number open questions in quantum mechanics. the various nature of the papers during this quantity attests to the success of that objective with many contributions delivering unique suggestions to validated difficulties by way of considering ideas from varied disciplines.

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Indeed every observable A : S -> M ^ S ) defines the unique EV measure EA • B(E) ->• £(S) by (EAX)(a) := (Aa){X) for every a € 5, X G B(E), and, conversely, every EV measure E : B(E) ->• £(S) defines the unique affine map AE : S -> M*(E) by (AEOL)(X) :— (EX)(a). The general frame summarized above characterizes what we call a convexity model, a name justified by the basic role played by the convexity of the set of states. As better specified in the next Section, it encompasses the standard classical frame which adds for S the requirement of being a simplex, and for the observables the deterministic requirement of having no dispersion on pure states.

Consider the Cartesian product A = A1 xA2 xA3 ={(A\,A 2 2 ,A\)} of the spectra, and the subset A ( A ) ={(A 1 ,,A 2 2 ,A 3 J ):A',+A 2 2 +A 3 3 =A}cA Lv \A) m m m ' m m J m satisfying condition (A). The discovery of an eigenvalue A^ (the value s = l,2 or 3 of the superscript is chosen by the experimenter, that of the subscript by nature) will determine a subset A , (A;m =n) ={(A1,,A\,A\):Al,+A22+A3]=A;m' = n}cAa), Lx m m m ' m 7 m m J {*) which would be a singleton if there were only two subsystems.

This notion mirrors the deterministic requirement that a random variable must take definite values on pure states, namely that it has no dispersion on pure states. Notice that a measurable function F : Q, -* S extends in a natural way to the afnne map Ap : M+(fi) -> Mf(E), denned by (AFfi)(X) := niF-1 (X)), X € B(E), (1) which will be called the distribution functional of F and corresponds to the notion of observable introduced in the previous Section. The pair (F, X) can be viewed as a two-valued experiment, which simply states whether the outcome of F does or does not fall in X.

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