The Gleason theorem for the field of rational numbers and residue fields

Abstract

Charges # taking values in a field F and defined on orthomodular partially ordered sets (logics) of all projectors in some finite-dimensional linear space over F are considered. In the cases where F is the field of rational numbers or a residue field, the Gleason representation p(P) ---tr(T~P), where T~ is a linear operator, is proved.KEY WORDS: Gleason representation, linear operator, trace-class operator, probability measure, charge, orthonormal partially ordered set (logic), logic of linear projectors.We consider charges p taking values in a field F and defined on orthomodulax partially ordered sets (logics) of all projectors in some finite-dimensional linear space over F. For certain fields F, we establish the Gleason representation [1] p(P) = tr(T#P),where T~ is a linear operator. According to the classical Gleason theorem, (1) applies to states (i.e., probability measures) on the logic II(H) of all Hermitian projectors in a real Hilbert space H of dimension at least 3. If the space is finitedimensional, then representation (1) is only valid when the boundedness condition [2] holds; otherwise, a charge, even if it satisfies (1), can be corrupted by a superimposed additive but irthomogeneous function on the real number line. In the infinite-dimensional case, Dorofeev and Sherstnev [3] removed the boundedness condition. The measures and charges can be defined on other logics; in particular, this author [4] considered the logic ~(H) of all continuous (not necessarily Hermitian) projectors in H. Taking into account the results of [3], we can state the theorem proved in [4] as follows.

Fermat’s Last Theorem Proved in Hilbert Arithmetic. Iii. the Quantum-Information Unification of Fermat’s Last Theorem and Gleason’s Theorem

“…-as well as in other, mainly mathematical sciences (e.g. Moretti, Oppio 2018;Marlow 2006;Mushtari 1998;1998a;Matvejchuk 1997;Dvurečenskij 1996;Dvurečenskij, Mišík 1988;Dvurečenskij 1987).…”