What is the relativistic spin operator?

TL;DR

This paper evaluates various proposals for a relativistic electron spin operator, identifying the Foldy-Wouthuysen and Pryce operators as the only proper choices, and suggests experimental methods to distinguish them.

Contribution

It critically assesses existing relativistic spin operators and proposes experimental tests to validate the correct operator in quantum mechanics.

Findings

Most proposed operators fail to meet angular momentum criteria.

Only Foldy-Wouthuysen and Pryce operators are suitable as relativistic spin operators.

Experimental measurements on highly charged ions can identify the correct operator.

Abstract

Although the spin is regarded as a fundamental property of the electron, there is no universally accepted spin operator within the framework of relativistic quantum mechanics. We investigate the properties of different proposals for a relativistic spin perator. It is shown that most candidates are lacking essential features of proper angular momentum operators, leading to spurious Zitterbewegung (quivering motion) or violating the angular momentum algebra. Only the Foldy-Wouthuysen operator and the Pryce operator qualify as proper relativistic spin operators. We demonstrate that ground states of highly charged hydrogen-like ions can be utilized to identify a legitimate relativistic spin operator experimentally.