Mapping the transverse spin sum rule in position space

TL;DR

This paper explores the relativistic spatial distributions of transverse angular momentum, including orbital and intrinsic spin, in a quantum phase-space framework, verifying the transverse spin sum rule for spin-0 and spin-1/2 targets.

Contribution

It introduces the first derivation of relativistic transverse angular momentum distributions in position space and examines their dependence on target momentum.

Findings

Verified the transverse spin sum rule for spin-0 and spin-1/2 systems.

Derived relativistic distributions of transverse angular momentum in the transverse plane.

Found non-trivial transverse total angular momentum distributions even for spin-0 targets.

Abstract

We discuss in detail the relativistic spatial distribution of transverse angular momentum, including both orbital and intrinsic spin contributions. Using the quantum phase-space formalism, we begin with the definition of the three-dimensional spatial distributions of transverse orbital angular momentum and intrinsic spin in a generic Lorentz frame. By integrating these three-dimensional spatial distributions over the longitudinal axis, we derive for the first time the relativistic spatial distributions of transverse orbital angular momentum, intrinsic spin, and total angular momentum for spin-0 and spin-1/2 targets in the transverse plane. We verify the transverse spin sum rule about the relativistic center of spin for spin-0 and spin-1/2 systems, and find that the transverse total angular momentum distribution is non-trivial, even for spin-0 targets. We also show how the distributions of transverse orbital angular momentum, intrinsic spin, and total angular momentum change with the target momentum.