The angular momentum controversy: What's it all about and does it matter?

TL;DR

This review dissects the long-standing angular momentum decomposition problem in gauge theories, clarifying why splitting photon/gluon total angular momentum into spin and orbital parts is gauge- and frame-sensitive. It juxtaposes canonical (JM) and Belinfante formalisms, and surveys gauge-invariant covariant decompositions (Ji, Chen et al., Wakamatsu, and gik/gic variants) that trade locality for gauge invariance. The authors unify these approaches under a master decomposition, discuss Stueckelberg symmetry that underlies non-uniqueness, and derive angular-momentum sum rules tied to GPDs and twist-3 distributions, with careful treatment of Lorentz transformation issues and non-locality. The discussion culminates in applications to the proton spin problem, including lattice and model approaches to orbital angular momentum and the interpretation of gluon helicity in the parton framework. Overall, the work clarifies measurability versus locality, guides experimental extraction of spin and orbital contributions, and frames the proton spin puzzle in terms of covariant, gauge-consistent decompositions.

Abstract

The general question, crucial to an understanding of the internal structure of the nucleon, of how to split the total angular momentum of a photon or gluon into spin and orbital contributions is one of the most important and interesting challenges faced by gauge theories like Quantum Electrodynamics and Quantum Chromodynamics. This is particularly challenging since all QED textbooks state that such an splitting cannot be done for a photon (and a fortiori for a gluon) in a gauge-invariant way, yet experimentalists around the world are engaged in measuring what they believe is the gluon spin! This question has been a subject of intense debate and controversy, ever since, in 2008, it was claimed that such a gauge-invariant split was, in fact, possible. We explain in what sense this claim is true and how it turns out that one of the main problems is that such a decomposition is not unique and therefore raises the question of what is the most natural or physical choice. The essential requirement of measurability does not solve the ambiguities and leads us to the conclusion that the choice of a particular decomposition is essentially a matter of taste and convenience. In this review, we provide a pedagogical introduction to the question of angular momentum decomposition in a gauge theory, present the main relevant decompositions and discuss in detail several aspects of the controversies regarding the question of gauge invariance, frame dependence, uniqueness and measurability. We stress the physical implications of the recent developments and collect into a separate section all the sum rules and relations which we think experimentally relevant. We hope that such a review will make the matter amenable to a broader community and will help to clarify the present situation.

Figures (11)

• Figure 1: The longitudinal $L_z$ (left) and transverse $L_x$ (right) components of the orbital angular momentum.
• Figure 2: The Wakamatsu classification of proton spin decompositions into two families. See text for more details.
• Figure 3: A schematic view of the Belinfante, Ji, gauge-invariant kinetic and gauge-invariant canonical decompositions. In white are depicted the pieces that are Stueckelberg invariant, and in gray are depicted the pieces that are Stueckelberg non-invariant. The potential OAM corresponds to the piece delimited by a dashed line common to both $\boldsymbol{L}^e_\text{gic}$ and $\boldsymbol{L}^\gamma_\text{gik}$.
• Figure 4: Refinement of the Wakamatsu classification depicted in Fig. \ref{['classification_short']}. See text for more details.
• Figure 5: Relations between the Chen et al., Hatta, Bashinsky-Jaffe and Jaffe-Manohar decompositions. The Chen et al. decomposition reduces to the Jaffe-Manohar decomposition in the Coulomb gauge. The Hatta decomposition reduces to the Bashinsky-Jaffe decomposition in the light-front gauge and reduces further to the Jaffe-Manohar decomposition once appropriate boundary conditions are imposed on the gauge potential.