Expectation Pauli-Lubanski vector and intrinsic angular momentum of relativistic wavepackets

TL;DR

This paper introduces a unified formalism for describing the intrinsic angular momentum of relativistic wavepackets, incorporating both spin and orbital contributions, and resolving issues related to zero-mass singularities.

Contribution

It develops the expectation Pauli-Lubanski vector formalism that generalizes the traditional approach, allowing intrinsic AM to have arbitrary orientation even for massless particles.

Findings

The formalism avoids zero-mass singularity issues.

Intrinsic AM can be arbitrarily oriented relative to momentum.

Examples demonstrate the theory with relativistic wave beams and packets.

Abstract

In non-relativistic mechanics, the total (orbital) angular momentum (AM) of a spatially-distributed system can be decomposed into intrinsic and extrinsic contributions. In relativistic quantum mechanics, intrinsic AM is typically associated with spin, which can be described using the Pauli-Lubanski four-vector. Here, we develop a unified formalism that combines the main features of both approaches and describes the intrinsic AM of a relativistic wavepacket, including both spin and orbital contributions. Our approach is based on the "expectation Pauli-Lubanski vector" constructed from the expectation values of the wavepacket's momentum and AM. Equivalently, it defines the intrinsic AM relative to the wavepacket's energy centroid. In contrast to the conventional Pauli-Lubanski formalism, the zero-mass singularity does not occur for the expectation Pauli-Lubanski vector. Consequently, the intrinsic AM of a wavepacket may have an arbitrary orientation with respect to its momentum, even for massless particles. We illustrate the general theory with a number of examples of relativistic wave beams and packets carrying spin and orbital AM.