The angular momentum controversy: What's it all about and does it matter? (1309.4235v1)

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.

• The paper presents a master framework unifying various angular momentum decompositions, clarifying issues of gauge invariance and non-uniqueness in QED and QCD.
• It systematically compares methods like the canonical, Belinfante, Ji, Jaffe-Manohar, and Chen decompositions to highlight their experimental relevance and theoretical implications.
• The paper elucidates how ambiguities in gauge potential splits and Stueckelberg symmetry affect the measurability of spin contributions in hadronic physics.

Overview of the Angular Momentum Controversy in Gauge Theories

The disentanglement of angular momentum into spin and orbital contributions in gauge theories, such as QED and QCD, has been an enduring and intricate challenge, emphasized by Leader and Lorcé. Conventional QED textbooks persistently affirm the impossibility of a gauge-invariant decomposition of a photon's angular momentum, extending to gluons by analogy, despite experimental endeavors to measure gluon spin polarization $\Delta G(x)$. This controversy stems partly from the notable assertion in 2008 that a gauge-invariant decomposition is feasible. However, such decompositions lack uniqueness, prompting inquiry into the physical or natural choices. Although experimental measurability does not definitively resolve these ambiguities, it underscores the viability of different approaches.

Key Contributions of the Paper

The paper by Elliot Leader and Cedric Lorcé comprehensively articulates the nuanced debates surrounding angular momentum decomposition in gauge theories. It presents various decompositions, scrutinizing their implications concerning gauge invariance, frame dependence, uniqueness, and measurability.

1. Exploration of Sum Rules and Experimental Relevance:
   • The work outlines sum rules and examines relations that bear experimental significance, laying out a framework for empirical exploration.
2. Pedagogical Introduction:
   • The initial sections familiarize readers with concepts like energy-momentum and angular momentum densities, crucial in field theory, by deploying both QED and QCD to illustrate foundational principles while minimizing technical obfuscation.
3. Analytic Dissection of Angular Momentum Decomposition:
   • The manuscript delineates primary decompositions—canonical, Belinfante, Ji, Jaffe-Manohar, and the Chen et al. decomposition.
   • It expounds the virtues and shortcomings of each, with a robust focus on the necessity for gauge-invariant formulations, even as it acknowledges their non-uniqueness due to ambiguous Stueckelberg symmetry.
4. Stueckelberg Symmetry and Transformation Complexity:
   • The discussion extends into the complexities of Lorentz-covariant decomposition frameworks, emphasizing the need to navigatively align the fixed gauge with a specified gauge potential split.
5. Master Decomposition Model:
   • Leader and Lorcé propose a master framework unifying the disparate decomposition theories, accommodating the different interpretations of orbital angular momentum allocation between quarks and gluons.

Implied Theoretical and Practical Impacts

The theoretical implications are far-reaching, challenging the long-held views on gauge theories' angular momentum frameworks. Practically, the paper guides the development of sum rules essential for interpreting experimental data from hadronic physics experiments. Until a broader consensus is reached, different communities might lean towards decompositions aligning with their expertise or technological feasibilities.

Speculation on the Future Trajectory and Developments

Future research may refine the understanding of gauge non-invariant operators' measurability and delineate more profound applications of Chen et al.'s non-local decompositions or gauge-invariant extension mechanisms. This intellectual progress signals pathways to improved lattice QCD computations, advancing insights into gluon contributions to spin and establishing robust corroborations across measurement paradigms.

In summation, Leader and Lorcé's paper offers an in-depth dialogue on angular momentum decomposition's tenacious issues, stabilizing the discourse towards an experimentally testable and theoretically sound comprehension of spin dynamics in gauge theories. This endeavor enriches the broader physics community's repository of methodologies for understanding hadronic structures, potentially unveiling new vistas in both theoretical depth and empirical reach.