REVIEW
Symmetric energy-momentum tensor in Maxwell, Yang-Mills, and Proca theories obtained using only Noether's theorem
High Energy Physics - Theory · Submitted 20 Feb 2006 · 12 references
Pith's reading: A careful application of Noether's theorem, including Bianchi identities, reproduces the known symmetric energy-momentum tensors for Maxwell, Yang-Mills, and Proca fields without Belinfante's symmetrization.
The paper
The paper claims
A careful application of Noether's theorem, including Bianchi identities, reproduces the known symmetric energy-momentum tensors for Maxwell, Yang-Mills, and Proca fields without Belinfante's symmetrization.
The reading
This paper shows that the extra term is not needed if the calculation is done more carefully. When the coordinates are shifted, the authors keep all boundary and off-shell terms. They use both the equations of motion and the Bianchi identity, an identity that the field strength obeys, before concluding that the current is conserved. With that, the symmetric tensor appears directly. They repeat the calculation for three theories: Maxwell, Yang-Mills, and massive Proca. In each case they recover the standard energy-momentum tensor used in textbooks and in coupling matter to gravity.
The physics does not change: the final tensors are the same ones everyone uses. The contribution is at the level of method and teaching. It clarifies that Noether's theorem is not incomplete, but that the usual shortcut omits half of the field equations. The paper does not present new experimental predictions or new field theories.
Extended reading notes
Core claim
The systematic implementation of Noether's theorem to the source-free Maxwell and Yang-Mills theories leads to symmetric and gauge-invariant expressions for their energy-momentum tensors, while for the Proca field the procedure yields the symmetric energy-momentum tensor (from the abstract and Eqs. (24)-(29), (48), (65)). The paper further claims that the action 'has all the information required to uniquely determine the right energy-momentum tensor' (Concluding Remarks, first paragraph).
Load-bearing premise
The paper's stronger conclusion depends on the premise that the energy-momentum tensor extracted from the Noether current is the unique symmetric conserved tensor determined by the action. This premise is not proven and is not generally true, since any conserved antisymmetric superpotential can be added to a symmetric conserved tensor without altering total momentum or angular momentum. Sections I and V invoke uniqueness to argue that Belinfante's method is unnecessary; if uniqueness fails, the claim that the action uniquely fixes the tensor is unsupported, even though the specific tensor derived remains a valid symmetric energy-momentum tensor.
Original abstract
The symmetric and gauge-invariant energy-momentum tensors for source-free Maxwell and Yang-Mills theories are obtained by means of translations in spacetime via a systematic implementation of Noether's theorem. For the source-free neutral Proca field, the same procedure yields also the symmetric energy-momentum tensor. In all cases, the key point to get the right expressions for the energy-momentum tensors is the appropriate handling of their equations of motion and the Bianchi identities. It must be stressed that these results are obtained without using Belinfante's symmetrization techniques which are usually employed to this end.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
The symmetric and gauge-invariant energy-momentum tensors for source-free Maxwell and Yang-Mills theories are obtained by means of translations in spacetime via a systematic implementation of Noether's theorem. For the source-free neutral Proca field, the same procedure yields also the symmetric energy-momentum tensor. In all cases, the key point to get the right expressions for the energy-momentum tensors is the appropriate handling of their equations of motion and the Bianchi identities. It must be stressed that these results are obtained without using Belinfante's symmetrization techniques which are usually employed to this end.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The symmetric and gauge-invariant energy-momentum tensors for source-free Maxwell and Yang-Mills theories are obtained by means of translations in spacetime via a systematic implementation of Noether's theorem. | contribution 0.66 | 0.92 | 0.67 | asserted 1.00 |
| s1 | For the source-free neutral Proca field, the same procedure yields also the symmetric energy-momentum tensor. | result 0.95 | 0.92 | 0.16 | asserted 1.00 |
| s2 | In all cases, the key point to get the right expressions for the energy-momentum tensors is the appropriate handling of their equations of motion and the Bianchi identities. | method 0.56 | 0.75 | 0.25 | asserted 1.00 |
| s3 | It must be stressed that these results are obtained without using Belinfante's symmetrization techniques which are usually employed to this end. | contribution 0.77 | 0.91 | 0.21 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v1
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
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Human review
References
How the literature uses it
3 arXiv papers cite this in 4 sentences
Papers citing it · 2 sentences without a confident label
Compares with it 2 papers
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From [ref], we may directly express the stress-energy-momentum tensor $free-ym$ as [equation] with on- and off-diagonal components respective given by [ref] and [ref], [equation] in agreement with [ref] and [ref] .
An exterior-algebraic derivation of the symmetric stress-energy-momentum tensor in flat space-time · Yang-Mills fields · 0.86
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Our result for the symmetrized energy-momentum tensor is in perfect agreement with the standard result [ref] but we arrived at it by purely geometric considerations and a much less painful, almost trivial, computation.
Belinfante-Rosenfeld Symmetrization from Metric-Affine Conservation Laws: Hypermomentum as the Improvement Term -- The Cases of QED and QCD · QED Energy-momentum tensor · 1.00
Discussion
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