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Oct
8
comment Spacetime Torsion, the Spin tensor, and intrinsic spin in Einstein-Cartan theory
Sorry, the first sentence in my comments should be: The scalar and Dirac field examples were not given for the purpose of showing how to couple torsion to classical fields, they were given as examples of matter.
Oct
8
comment Spacetime Torsion, the Spin tensor, and intrinsic spin in Einstein-Cartan theory
The answer to your second comment is that nothing was added by hand, the coupling term is obtained according to the rules of the minimal coupling. The only manipulation done is to separate the symmetric part of the affine connection and include it in the spin connection, and writing the antisymmetrical components separately giving the interaction term. In summary, the coupling of the matter fields to Einstein-Cartan theory can be viewed as a stage in the determination of its spin tensor.
Oct
8
comment Spacetime Torsion, the Spin tensor, and intrinsic spin in Einstein-Cartan theory
This is given in equation (2.20) in Shapiro. The spin tensor is also analogous to the energy momentum tensor being a Noether current of the Lorentz symmetry as the energy momentum tensor is the Noether current of the translation symmetry. (Together they generate the Poincare symmetry).
Oct
8
comment Spacetime Torsion, the Spin tensor, and intrinsic spin in Einstein-Cartan theory
This is exactly analogous to the energy momentum tensor: In order to obtain the energy momentum tensor, first one minimally couples the matter theory to gravity, and then varies the Lagrangian with respect to the metric. The analogy for the spin tensor: first one couples the matter theory to gravity with torsion (i.e., with a nonsymmetrical affine connection), then varies the Lagrangian with respect to the contorsion.
Oct
8
comment Spacetime Torsion, the Spin tensor, and intrinsic spin in Einstein-Cartan theory
The scalar and Dirac field examples were given for the purpose of showing how to couple torsion to classical fields, they were given as examples of matter, please see that these examples are given in Shapiro's review in section 2.3 titled as "Interaction of torsion with matter". The interaction appears as a term of the form: Spin tensor $\times$ contorsion tensor as can be seen in the Dirac example.
Oct
2
revised Spacetime Torsion, the Spin tensor, and intrinsic spin in Einstein-Cartan theory
added 15 characters in body
Oct
2
answered Spacetime Torsion, the Spin tensor, and intrinsic spin in Einstein-Cartan theory
Aug
4
comment What does it mean to renormalize an effective field theory?
@ New_new_newbie It may happen that some process (like photon-photon scattering in QED) does not exist at the tree level approximation, then you can say that radiative corrections predict new physics. But apart from that, the control of the quantum corrections is mainly required to ensure that these corrections will not spoil the leading order.
Jul
31
revised Instantons in Witten's supersymmetry and Morse theory
added 1 character in body
Jul
31
answered Instantons in Witten's supersymmetry and Morse theory
Jul
17
awarded  Enlightened
Jul
17
awarded  Nice Answer
Jul
3
comment Expressing an adjoint representation Wilson line in terms of the fundamental representation
Sorry, $E^{ij}$ is the matrix with all elements zero except a unit at the place $i,j$
Jul
3
answered Expressing an adjoint representation Wilson line in terms of the fundamental representation
Jun
5
comment The $U(1)$ charge of a representation
@user41746, cont.Now, Slansky insists to identify the highest weight of the $SU(5) $ representation with the highest weight of the quark representation after the symmetry breaking. For that he needs to act on the weights with a “projection matrix” given in equation 6.7. After the action of the projection matrix the fourth component of the weights is lost, but nevertheless, the dual U(1) vector can be obtained by insisting that it has a constant scalar product with each subrepresentation weights. As you can see all the exercises lead to the same result for the hypercharges.
Jun
5
comment The $U(1)$ charge of a representation
@user41746, Sorry I noticed that I used $a_1, a_2$ for the $SU(3)$ labels and $a_4$ for the $SU(2)$ and in your example you used $a_3, a_4$ for $SU(3)$ labels and $a_1$ for SU(2). Thus you should use the second row in $G$ for your example. You will get the same result with the normalization factor $\frac{1}{3}$.
Jun
5
comment The $U(1)$ charge of a representation
@user41746 $SU(5)$ is $A_4$, so the metric tensor should be $4\times 4$ matrix, please see the first matrix in table 7 page 82. Its third row for n=4 is just the given vector. I can’t see this exercise on page 84 (This page belongs to the appendix and it contains tables 10 and 11a. However, on page 16 equation (3.3), Slansky obtained the same hypercharges as in the question.
Jun
5
revised The $U(1)$ charge of a representation
deleted 4 characters in body
Jun
5
answered The $U(1)$ charge of a representation
Jun
3
revised Coadjoint orbits in physics
added 678 characters in body