Value Assignments to Observables Depending on a History of Context Change

The functional composition principle is generalized by taking into account of history of context change. Analysis of Peres’ example shows hysteresis of value assignments. It is shown that value assignments which depend on the history of context change are possible in the case that the Hilbert space of state vectors is finite dimensional. 1 A generalized functional composition principle depending on history of context change The Kochen–Specker theorem states that there is no value assignment for quantummechanical observables that satisfies the functional composition principle (FUNC) when the dimension of the Hilbert space of state vectors is greater than two [1, 2]. By analogy with Riemannian surfaces which settle contradictive concept of multivalued functions by analytic continuation, we investigate a value assignment depending on a history of context change. The eigenvalues of an observable represented by a self-adjoint operator are values which are obtainable when one makes an observation. If there was hidden variables that specify the observed value, we can assign the value to the pair of the observable and the hidden variables. If we denote the hidden variables, the observable, and the eignevalue by ω, Ô, and oi, respectively, then a value assigning map v is defined as a mapping of the set of all observables into a set of random variables such that for the possible ω, v(Ô)(ω) = oi. (1) email: [email protected] Address untill the end of Feb. 2007: Centro Vito Volterra, Università degli Studi di Roma “Tor Vergata”, Via Columbia, Rome 00133, Italy.