Quantum Theory from Symmetries in a General Statistical Parameter Space

Abstract

The aim of this paper is to show a connection between an extended theory of statistical experiments on the one hand and the foundation of quantum theory on the other hand. The main aspects of this extension are: One assumes a hyperparameter space $\Phi$ common to several potential experiments, and a basic symmetry group G associated with this space. The parameter \theta_{a} of a single experiment, looked upon as a parametric function $\theta_{a}(\cdot)$ on $\Phi$, is said to be permissible if G induces in a natural way a new group on the image space of the function. If this is not the case, it is arranged for by changing from G to a subgroup $G_{a}$. The Haar measure of this subgroup (confined to the spectrum; see below) is the prefered prior when the parameter is unknown. It is assumed that the hyperparameter itself can never be estimated, only a set of parametric functions. Model reduction is made by restricting the space of complex `wave' functions, also regarded as functions on $\Phi$, to an irreducible invariant subspace $\mathcal{M}$ under G. The spectrum of a parametric function is defined from natural group-theoretical and statistical considerations. We establish that a unique operator can be associated with every parametric functions $\theta_{a}(\cdot)$, and essentially all of the ordinary quantum theory formalism can be retrieved from these and a few related assumptions. In particular the concept of spectrum is consistent. The scope of the theory is illustrated on the one hand by the example of a spin 1/2 particle and a related EPR discussion, on the other hand by a simple macroscopic example.