Clarifying the relation between covariantly conserved currents and Noether's theorem (2506.14454v1)

Abstract: On one hand, proper conservation laws, that is, the existence of local functions of the fields whose divergence vanishes on-shell, are associated with global symmetries and are a consequence of Noether's first theorem. On the other hand, covariant conservation laws, i.e., the same type of laws but featuring a covariant divergence, are associated with local symmetries and are a consequence (under certain conditions) of Noether's second theorem. In this paper, we clarify under which conditions Noether's second theorem, which is mostly known for providing relations between the equations of motion of a theory that possesses some local symmetries, can be used to generate covariantly conservation laws. We also investigate the relation between the conserved current of a theory with global symmetry and the covariantly conserved current of the corresponding local symmetry. We treat these issues using only the tools of Calculus, in the fashion of Noether's original article, avoiding the somewhat complicated mathematical tools that plague much of the literature on this topic, thus making the article readable for the general physicist.

• The paper demonstrates that specific symmetry conditions in non-dynamical fields and incomplete systems lead to covariantly conserved currents.
• It analyzes examples from General Relativity and Yang-Mills theories to show how minimal coupling aligns canonical and covariant currents.
• The findings underscore significant implications for theoretical consistency and quantization in gauge theories and gravity.

Clarifying the Relation Between Covariantly Conserved Currents and Noether's Theorem

In the paper under discussion, the authors delve into a nuanced exploration of Noether's theorems and their implications for conserved currents within physical theories, particularly in the context of gravity and gauge theories. Noether's first theorem traditionally connects global symmetries with conserved quantities, such as energy-momentum or electric charge, while the second theorem is mostly associated with local symmetries, such as those found in gauge theories, producing identities relating the equations of motion.

Key Arguments and Methodology

The authors' primary objective is to clarify under what circumstances Noether's second theorem can indeed generate covariantly conserved currents. They argue that these conservation laws do not universally arise solely from the local symmetry properties of a theory, but under certain specific conditions:

1. Non-Dynamical Fields or Incomplete Dynamical Systems: The paper posits that if a theory possesses fields that transform under local symmetries but aren't dynamical, one can derive non-trivial identities that entail covariant conservation — provided certain criteria are met. Similarly, if an action is partially invariant under a symmetry due to a subset of variables, covariant conservation laws may result.
2. Specific Formulations in Gravity and Gauge Theories: Using the examples of General Relativity and Yang-Mills theories, the authors illustrate how these special on-shell identities manifest. In these theories, energy-momentum tensors and currents, influenced by their interactions with respective geometrical or gauge structures, adhere to covariant conservation laws.
3. Minimal Coupling and Canonical Current Equivalence: The authors elucidate the conditions under which the canonical currents of a globally invariant theory match up with covariantly conserved currents when the theory is locally extended (by introducing gauge fields, for example). The principle of minimal coupling ensures such alignment at the level of conserved currents.

Implications and Conclusions

The conclusions drawn address fundamental insights into the workings of gauge symmetries and gravitational dynamics, reinforcing the understanding of:

• Covariant Conservation in Non-Conventional Theories: Even when actions aren't fully symmetric under transformations, conditions are outlined whereby covariant conservation properties still hold, such as in certain modified gravity theories or when fields are considered with background metrics.
• The Significance of Gauging: The work emphasizes the canonical currents before and after gauging a symmetry, highlighting how the introduction of additional fields and transformations alters the conserving properties of original quantities, with implications for theoretical consistency and quantization.

This acute investigation into conservation principles encourages further theoretical investigations into formulations of physical laws where symmetries and conservation tangentially intertwine in non-intuitive manners. The implications for the understanding of fundamental forces and particles, as well as potential new forms of coupling in field theories, offer a significant point of departure for future research in theoretical and mathematical physics. The insights are especially pertinent for exploring theories beyond the standard model or in contexts where gravity is reconciled with quantum mechanics.