On reconstructing parts of quantum theory from two relates maximal conceptual variables

TL;DR

This paper reconstructs the Hilbert space formulation of quantum theory using the concept of accessible and inaccessible variables influenced by group actions, broadening the foundational understanding of quantum states and observables.

Contribution

It generalizes previous reconstruction results by removing measurement assumptions and allowing arbitrary phase factors, thus extending the applicability to decision scenarios and more general states.

Findings

Hilbert space formalism derived from conceptual variables

No measurement assumption needed for the reconstruction

States can have arbitrary phase factors

Abstract

In the book [4] the general problem of reconstructing the Hilbert space formulation in quantum theory is discussed from the point of view of what I called conceptual variables, any variables defined by a person or by a group of persons. These variables may be inaccessible, i.e., impossible to assign numerical value to by experiments or by measurements, or accessible. One basic assumption in [4] and here is that group actions g 2 G are defined on a space where some maximally accessible variable varies, and then accessible functions of these maximal variables are introduced. By using group representation theory the basic Hilbert space formalism is restored under the assumption that the observator or the set of observators has two related maximally accessible variables in his (their) mind(s). The notion of relationship is precisely defined here. Symmetric (self-adjoint) operators are connected to each variable, and in the discrete case the possible values of the variables are given by the eigenvalues of the operators. In this paper the main results from [4] are made more precise and more general. It turns out that the conditions of the main theorem there can be weakened in two essential ways: 1) No measurements need to be assumed, so the result is also applicable to general decision situations; 2) States can have arbitrary phase factors. Some consequences of this approach towards quantum theory are also discussed here.