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Good evening to everyone,

At present I am studying QFT and foundations of General Relativity (GR) and Einstein-Cartan (EC) theory. Namely I have just studied the definition of the Belinfante-Rosenfeld stress-energy tensor and its particular equivalence with the ordinary symmetric stress-energy tensor of GR. Then I understand that this definition of the stress-energy tensor might be the most appropiate since it satisfies the conservation laws of QFT/GR and because of the empirical evidences of QFT/GR respectively.

However I don't understand the concept of "spin" when GR or EC are considered. I have seen from S. Weinberg - Gravitation and Cosmology the following definition for a spin vector:

$S_{\mu}=\epsilon_{\mu \nu \lambda \rho}\,J^{\nu \lambda}\,u^{\rho}$,

where $J^{\nu \lambda}=\int{(x^{\nu}\,T^{\lambda 0}-x^{\lambda}\,T^{\nu 0})\,dx^{3}}$, $T^{\lambda \rho}$ is the stress-energy tensor (I guess it's the canonical stress-energy tensor, although it is symmetric in the particular case considered by Weinberg) and $u^{\rho}$ is the four-velocity.

I suppose that this definition should reproduce the correct expressions for the spin of scalar, spinor and vector fields by taking into account the expression of the stress-energy tensor of such fields. Is this correct?

On the other hand, I don't understand the spin tensor quantity $\frac{\delta S}{\delta \Gamma^{\lambda}\,_{\rho \nu}}$ present in EC theory, where $\Gamma^{\lambda}\,_{\rho \nu}$ are the components of the affine connection with torsion. What's the difference between this spin tensor and the Weinberg's spin vector defined above? For example, according to EC theory the spin tensor of matter is identically zero if the torsion tensor vanishes and the GR is recovered, however there exist other quantities (as can be seen above) that represent the spin of matter in such a case of GR. Then, by regarding the empirical evidence too, what is the appropiate expression for the spin of matter?

To my mind, a possible answer is that the spin tensor present in EC theory is the unique quantity for representing the spin of matter in that theory and therefore it has the singular property of vanishing when torsion is zero, so that this framework demands the existence of torsion in order to introduce spinning sources in the universe (as GR demands curvature when a stress-energy tensor is present). In addition, the standard approach of GR given by Weinberg et al. doesn't need the presence of torsion and it achieves to describe the spin of matter without the presence of torsion (i.e. in a curved space-time alone), so that both approaches are compatible with the current empirical evidence but from a theoretical point of view they have fundamental differences. Is this correct?

In addition I know that the stress-energy tensor of EC is generally asymmetric and I don't have any problem with this result, only with the possible relations and differences between the spin quantities of the mentioned theories.

Best regards.

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  • $\begingroup$ I think the spin in QFT is really the unitary irreducible representation of the $\mathfrak{su}(2)$, which is not related with GR. $\endgroup$ – Xiaoyi Jing Feb 5 '17 at 0:17
  • $\begingroup$ I'm a bit unsure what the question is. The two "spin" quantities you are talking about appear to be (related to) the classical intrinsic angular momentum of the fields. The QFT notion of spin is wholly different; it is solely concerned with the representation of the Poincaré group a field/a particle on flat space transforms in, while this is also "intrinsic angular momentum", it is of a different kind than the "classical spin" of a field. So if you're asking solely about the differences of the classical spins of GR and EC theory, then QFT and quantum-spin are wholly irrelevant. $\endgroup$ – ACuriousMind Feb 5 '17 at 0:28
  • $\begingroup$ @PhilS I know is an OT, but where are you study EC theory? (I mean what book you use, or something like that) $\endgroup$ – Annibale Oct 3 '17 at 6:26
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Thanks for your answers.

Roughly speaking, according to my limited understanding it is possible to introduce in QFT the concept of the spin of a field by defining its associated Lagrangian (e.g. the Dirac Lagrangian for a Dirac spinor field) and analysing the invariance under space-time rotations. Therefore we have to deal with a set of $\{ S^{a b} \}$ generators associated with the Lorentz group and also with an irreducible representation of the mentioned group, which is different depending on the spin of the field considered. Is this correct?

In addition if we define the quantity $S_{a}=\epsilon_{a b c d}\,S^{b c}\,u^{d}$ it turns out that $S^{2} \equiv S_{a}\,S^{a}$ is a Casimir operator of the Lorentz group that commutes with all elements of the Lorentz group and the Schur's Lemma involves that all vectors of the irreducible representation are eigenvectors of $S^{2}$ with the same eigenvalue, so that I think this justifies that $S_{a}$ represents a fundamental quantity associated with the spin of the field and its transformation laws under the Lorentz group.

If this is correct, the next step would be to extend these notions to GR and maybe this is what appears in the analyses of Weinberg et al. Then my problem is the possible relation between $S_{\mu}$ (or an alternative expression associated with the spin tensor of matter) and the $\frac{\delta S}{\delta \Gamma^{\lambda}\,_{\rho \nu}}$ quantity defined in EC theory. For example, I read that "Rosenfeld demonstrated by Noether theorems that the Belinfante-Rosenfeld stress-energy tensor derived in QFT coincides with the ordinary symmetric stress-energy tensor of GR in presence of a curved space-time", so that both tensors are related and I consider the ordinary symmetric stress-energy tensor of GR as the natural stress-energy tensor of matter in curved space-time. As I think that today GR is the most accuracy theory of gravity (omitting other equivalent approaches from the current phenomenological point of view, as for example teleparallelism) then I consider the mentioned symmetric stress-energy tensor as the most accuracy and complete quantity for characterizing the stress-energy tensor of matter. Likewise I was wondering if such a relation might exist between a spin tensor in QFT (or in a extensión of QFT in curved space-time) and EC theory (i.e. in presence of torsion if it exists)? For example, it would be a relation between $S_{\mu}$ and $\frac{\delta S}{\delta \Gamma^{\lambda}\,_{\rho \nu}}$, in analogy to the relation between the Belinfante-Rosenfeld stress-energy tensor and the Einstein-Hilbert stress-energy tensor, but I don't know if this is possible or well-known, that's why I wanted to ask my questions here.

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