The transverse angular momentum sum rule

TL;DR

This paper clarifies the controversy over the transverse angular momentum sum rule by correcting previous misconceptions and introducing a new classification of sum rules based on measurability.

Contribution

It provides a new, straightforward method for deriving correct angular momentum expectation values and establishes a clear distinction between primary and secondary sum rules.

Findings

Corrected the expression for angular momentum expectation values.

Established a valid transverse angular momentum sum rule.

Introduced a classification of sum rules into primary and secondary types.

Abstract

We explain the origin of the controversy about the existence of a transverse angular momentum sum rule, and show that it stems from utilizing an incorrect result in the literature, concerning the expression for the expectation values of the angular momentum operators. We demonstrate a new, short and direct way of obtaining correct expressions for these expectation values, from which a perfectly good transverse angular momentum sum rule can be deduced. We also introduce a new classification of sum rules into primary and secondary types. In the former all terms occurring in the sum rule can be measured experimentally; in the latter some terms cannot be measured experimentally.