Quantum theory as a statistical theory under symmetry and complementarity

TL;DR

This paper derives core quantum mechanics principles from an extended statistical framework emphasizing symmetry, complementarity, and limited experimental bases, connecting quantum concepts with statistical theory.

Contribution

It introduces a novel statistical approach to quantum mechanics, deriving key elements like the Hilbert space, Born formula, and Schrödinger equation from symmetry and sufficiency concepts.

Findings

Derivation of the Hilbert space structure from statistical principles.

Motivation of the Born formula through recent analyses.

Discussion of Bell's inequality within the statistical framework.

Abstract

The aim of the paper is to derive essential elements of quantum mechanics from a parametric structure extending that of traditional mathematical statistics. The main extensions, which also can be motivated from an applied statistics point of view, relate to symmetry, the choice between complementary experiments and hence complementary parametric models, and use of the fact that there for simple systems always is a limited experimental basis that is common to all potential experiments. Concepts related to transformation groups together with the statistical concept of sufficiency are used in the construction of the quantummechanical Hilbert space. The Born formula is motivated through recent analysis by Deutsch and Gill, and is shown to imply the formulae of elementary quantum probability/ quantum inference theory in the simple case. Planck's constant, and the Schroedinger equation are also derived from this conceptual framework. The theory is illustrated by one and by two spin 1/2 particles; in particular, a statistical discussion of Bell's inequality is given. Several paradoxes and related themes of conventional quantum mechanics are briefly discussed in the setting introduced here.