Quantum theory as a statistical theory under symmetry

Abstract

Both statistics and quantum theory deal with prediction using probability. We will show that there can be established a connection between these two areas. This will at the same time suggest a new, less formalistic way of looking upon basic quantum theory. A total parameter space $\Phi$, equipped with a group $G$ of transformations, gives the mental image of some quantum system, in such a way that only certain components, functions of the total parameter $\phi$ can be estimated. Choose an experiment/ question $a$, and get from this a parameter space $\Lambda^{a}$, perhaps after some model reduction compatible with the group structure. The essentially statistical construction of this paper leads under natural assumptions to the basic axioms of quantum mechanics, and thus implies a new statistical interpretation of this traditionally very formal theory. The probabilities are introduced via Born's formula, and this formula is proved from general, reasonable assumptions, essentially symmetry assumptions. The theory is illustrated by a simple macroscopic example, and by the example of a spin 1/2 particle. As a last example we show a connection to inference between related macroscropic experiments under symmetry.