A quantum theory for a total system including a reference system

TL;DR

This paper extends quantum mechanics to include a reference system within the total system, defining a framework where measurement and reference properties are incorporated into the formalism, leading to a probabilistic relationship akin to Born's rule.

Contribution

It introduces a new quantum formalism that explicitly includes a reference system and its properties, providing a consistent way to describe measurements and states within the total system.

Findings

Defines a frame of reference for the total system.

Establishes postulates linking reference properties to stable states.

Derives a probabilistic relationship similar to Born's rule.

Abstract

The standard formalism of quantum mechanics is extended to describe a total system including the reference system (RS), with respect to which the total system is described. The RS is assumed to be able to act as a measuring apparatus, with measurement records given by the values of some reference properties of the RS. In order to describe the total system, we define a frame of reference (FR) as a set of states that can be used to express all other states of the total system. The theory is based on four basic postulates, which have, loosely speaking, the following contents. (i) A reference property of a RS has a definite value and is sufficiently stable in the FR directly related to the reference property. (ii) States of the total system are associated with vectors in the Hilbert space. (iii) Schr\"odinger equation is the dynamical law in each valid FR. (iv) Under certain condition a property of a system can be regarded as a reference property; vector descriptions of the total system given in different FRs of the same RS may have a probabilistic relationship like in Born's rule.