On The Peierls Interpretation of Quantum Mechanics

TL;DR

This paper explores Peierls's interpretation of quantum mechanics, emphasizing the subjective nature of the wavefunction and density matrix, and discusses how observers can reconcile different knowledge states without violating quantum principles.

Contribution

It extends Peierls's ideas by demonstrating the subjective interpretation of density matrices and the importance of their commutation for consistent knowledge sharing among observers.

Findings

Density matrices can be expressed in terms of observable outcome probabilities.

Observers' density matrices must commute to combine information without violating uncertainty.

The commutation criterion extends to mixed states, aligning with standard quantum mechanics.

Abstract

The brief works of Peierls on the role of the observer in quantum mechanics are examined, interpreted and expanded to widen accessibility and understanding of these works. The approach followed here is very much in the spirit adopted by Peierls who eschewed a `rigorous axiomatic' and aimed at using `logic at the level of the working physicist'. The fundamental tenet of his work is that the wavefunction or density matrix represents the knowledge of an observer and that two observers of the same system may well have different knowledge and will use different density matrices to describe it. Essential to the understanding of Peierls's approach is the demonstration given here that the density matrix generally can be expressed entirely in terms of probabilities of observable outcomes and that such probabilities are subjective in the first instance.The process by which the knowledge of two observers may be combined is discussed in detail for some simple cases and, by using Bayes's theorem, the progress to a common understanding of the correct density matrix for a given situation demonstrated. Peierls gave a criterion to ensure that the amalgamation of data from two observers can never lead to violation of the uncertainty principle : that the density matrices of two observers must commute. It is pointed out that it is essential that the density matrices of two observers commute under all circumstances no matter how inaccurate or even fanciful the data or beliefs used to construct them. Once this is appreciated it can be shown that Peierls's density matrix commutation criterion is fully equivalent to the standard result in quantum mechanics that two observables must commute if they are to be known precisely and simultaneously,but now extended to imperfectly known mixed states rather than a single eigenstate.