SIAI SM.84 by Cesare Gori

By Cesare Gori

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We divide the description in three steps: (1) composition and unitary evolution: after unitary evolution of the system composed by S and A , the two systems are left in some final (partial) states Ws and WA. (2) reading: a first cybernetic involvement, transition from physical to cybernetic reality as in Paradigm 1 applied to A, occurs when a family 77 = (77^) of attributes of A is "read". In fact, when 29 applying Paradigm 1 to A, the index set A which is supposed to label the composite samples, hence the samples of S as well as the samples of A, is partitioned in a family (At) of subsets according to the result of the measurement of (IIA) on A.

Let H be a Hilbert space, K a Hilbert subspace. Let CK, CJJ be the spaces of the linear operators in the respective Hilbert spaces K C H. Let W G CK- Then we consider as an identification the injective mapping L : CK —> £H '• W —• iW = W © 0\(H-K) (whose inverse is t'1 : W —• W\K). We need the following definition. Definition 4. Let (H,W) be a Hilbert space with states (Def. 1) and let n = (/7j)ier be an orthofamily of projectors. At least when H expresses a semiboolean structure for (H, W) (Prop.

The aforementioned trijective correspondence between the bases (again r = 1,2,3). The discovery of an eigenvalue therefore selects one in both of the other two factor spaces. ^)); for then the entire system possesses an amount A of the physical quantity 21 represented by A, whose exact distribution over all three subsystems would be determined by a measurement on any one of them. We expect this with two subsystems, maybe not with three. 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).

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