Currents and the Energy-Momentum Tensor in Classical - arXiv
|
Currents and the Energy-Momentum Tensor in Classical Field
We give a comprehensive review of various methods to define currents and the energy-momentum tensor in classical field theory with emphasis on a geometric point of view The necessity of “improving” the expressions provided by the canonical Noether procedure is addressed and given an adequate geometric frame-work |
|
Noether’s first theorem and the energy-momentum tensor
Dedicated to the late Bessel-Hagen who when alive had his habilitation thesis thrown into the sea and even now must feel as if his work was lying somewhere on the seabed |
|
Classical Electrodynamics ofExtended Bodies
of the electromagnetic field tensor provides sufficient stress-energy terms to allow for conservation of energy-momentum There are three such independent terms: a direct current-current interac-tion and two curvature-mediated (non-minimally coupled) short-range interactions one of which changes sign under a parity transformation |
|
New Improved Energy-Momentum Tensor and Its
energy-momentum tensor The new tensor differs from the traditional canonical and symmetric ones and can be derived as N ̈ other current from a Lagrangian with second derivative We also attempt to construct a gravitational coupling in such a way that the new energy-momentum tensor becomes the source of the gravitational field |
Does Noether's theorem yield the correct energy-momentum tensor?
Particularly striking is the existence of inequivalent definitions of the energy-momentum tensor, a central physical quantity in any classical field theory. Typical textbook presentations leave the impression that Noether’s theorem fails to yield the correct energy-momentum tensor.
Is the gravitational source a covariant extension of the energy-momentum tensor?
In this section, we present explicitly the new gravitational coupling for the scalar, spinor, and vector fields, respectively, and verify that the gravitational source is indeed the covariant extension of the energy-momentum tensor and spin tensor currents as required to properly describe the fluxes of conserved quantities in quantum measurement.
What is a symmetric energy-momentum tensor?
One is the canonical energy-momentum tensor, derived from N ̈ other’s theorem: where Lst(φ, ∂φ) is the standard expression of matter Lagrangian in terms of the field φ and its first derivative, and the Minkowski metric tensor ημν has signature (−+++). The other is the symmetric energy-momentum tensor, known as the Belinfante tensor:
What is the canonical energy-momentum tensor for classical electrodynamics?
the canonical Noether energy-momentum tensor for classical electrodynamics. By contrast, the accepted energy-momentum tensor Tμν for the theory is given by eqn (1) above. This specific case illustrates two distinct problems for the canonical energy-momentum tensor that hold more broadly.
July 23, 2021
Dedicated to the late Bessel-Hagen, who when alive had his habilitation thesis thrown into the sea, and even now must feel as if his work was lying somewhere on the seabed. arxiv.org
Abstract
Noether’s theorems are widely praised as some of the most beautiful and use-ful results in physics. However, if one reads the majority of standard texts and literature on the application of Noether’s first theorem to field theory, one imme-diately finds that the “canonical Noether energy-momentum tensor” derived from the 4-parameter translation of
3.1.2 Contragredient transformations
The non-bar transformation of fields δAν (second term in (3)) is referred to by Bessel-Hagen as being associated to the “contragredient” transformations of the fields, which in current treatments follow simply from the definition of a con-travariant tensor (in this case vector): ∂x′ν A′ν(x′) = arxiv.org
3.2 Gauge (field) symmetries of the action
There is also the possibility of gauge (field) symmetries of the action, often over-looked from the perspective of Noether’s first theorem because they are thought to be relevant only to Noether’s second theorem. The Bessel-Hagen et al. approach uses these symmetries as well to derive the known conservation laws of electrody-namics directly from No
B AνEμ. (27)
The Belinfante prescription alone does not yield the correct expression without adding this additional term (−AνEμ) proportional to the equations of motion; equivalence to Tμν alone can only be established after imposing the on-shell con- B dition Eμ = 0. Note that requiring such an on-shell condition for just formulat-ing the energy-momentum tenso
4.3 Converse of Noether’s first theorem as a test for Noethe-rian currents
We now use the converse of Noether’s first theorem relative to the Lagrangian density of electrodynamics and the accepted energy-momentum tensor (1) in order to arrive at the relevant variational symmetry linked to this Tμν. As we will see, the converse can generally be used to decide whether an energy-momentum tensor can be directly derived from N
7 Conclusions
Noether’s first theorem is one of the most celebrated results in physics. Yet, standard textbook and literature presentation gives the picture that this method fails to derive standard physical conservation laws: the canonical Noether energy-momentum tensor, which is derived using a restricted condition placed on Noether’s first theorem, does not g
8 Acknowledgments
We are grateful to James Read, Bryan Roberts and Nicholas Teh for the invitation to contribute to this volume. We thank Harvey Brown for encouraging comments and helpful suggestions on the paper. We also thank an anonymous reviewer for their helpful input. arxiv.org
|
Noethers first theorem and the energy-momentum tensor ambiguity
21?/07?/2021 2 Energy-Momentum Tensors. 6. 3 Noether's first theorem for classical electrodynamics. 8. 3.1 Transformations associated to the Lie ... |
|
Towards an effective action for relativistic dissipative hydrodynamics
15?/05?/2014 the energy-momentum tensor and global symmetry currents [1]. ... account by supplementing the classical hydrodynamic equations with ... |
|
Improvement of a conserved current density versus adding a total
04?/05?/2022 matical physics and in particular to classical and quantum field theory. ... Jackiw improvement of the canonical energy-momentum tensor for ... |
|
Relativistic magnetohydrodynamics with spin
22?/02?/2022 We extend the classical phase-space distribution function to include the spin ... baryon charge current energy-momentum tensor |
|
On the number and spin of photons in classical electromagnetic field
25?/10?/2018 free classical electromagnetic field through the currents that created the field |
|
The Energy-Momentum Tensor(s) in Classical Gauge Theories
29?/06?/2016 The relationship with the Einstein-Hilbert tensor following from the coupling to a gravitational field is also discussed. arXiv:1605.01121v2 [ ... |
|
Noether symmetries energy-momentum tensors and conformal
27?/02?/2009 arXiv:0902.4871v1 [hep-th] 27 Feb 2009. UB-ECM-PF-09/05. Noether symmetries energy-momentum tensors and conformal invariance in classical ... |
|
A New Improved Energy-Momentum Tensor and Its Possible Role in
03?/03?/2019 gravitational theory the symmetric energy-momentum tensor leads to ... similar correlation of currents as for classical particles: If the. |
|
New conserved tensors and Brans-Dicke type field equation using
20?/01?/2022 The study of the scalar field is as old as the classical field theory. ... equations apparently admit energy-momentum tensor to be the only ... |
|
ArXiv:2102.11098v2 [hep-th] 9 Mar 2022
09?/03?/2022 possible that a current is conserved at the classical level ... ical energy-momentum tensor to the quantum effective. |
![Quantum Physics authors/titles \](https://i1.rgstatic.net/publication/1964646_Energy-Momentum_Tensor_of_Field_Fluctuations_in_Massive_Chaotic_Inflation/links/00463518bd6cc37ceb000000/largepreview.png "Quantum Physics authors/titles \")
