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Your group theory text probably betrayed you if it did not spend much time contrasting the two cases. A possibly related question is 254461. People use loose language and symbols that aggravate the confusion. Talking abstractly without explicit hands-on formulas clinches it (the confusion)! Let me stick to your 4-dimensional matrices and vectors, all tensor ...

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I gather your unfortunate edit, which further confused things, to the point of intractability, came from verbatim reproduction of eqn (3.23) of the Duff et al paper on Stochastic Local Operations and Classical Communication. One of about a dozen problems with your malformed question is that you used the wrong coset space to be described by the Cartan pair ...

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1) Universal covering groups are groups with the property of being simply connected. Each algebra has a unique covering group. The other groups, $\{G\}$, associated to the same algebra can be obtained from the covering group in the following way $$G=\frac{\tilde G}{Ker(\rho)},$$ where $Ker(\rho)$ is the kernel of the group homomorphism $\rho:\tilde ... 2 Your$SU(3)\otimes SU(3)={\bf 1}\oplus {\bf 8}$above is a chimaeric typo from hell. OK, I'll just give you the self-evident answers, but they would be meaningless junk numbers if you failed to reproduce them directly on the basis of your SU(3) text or the WP article which explains the rules and the Dynkin representation notation, D(p,q), which connects to ... 2 1) Note that the non-relativistic gluon model of glueballs has even less justification than the non-relativistic quark model of baryons and mesons. This is because the model messes up the spin assignments: Massless gluons have only two spin states, but non-relativistic gluons have three. 2) The color quantum numbers are trivial: The product $$[8]\times ... 1 These branes are being smashed together so that once they are closer than the string length they become indistinguishable from a single brane. These branes have U(N_c) gauge group, and in this space there is a vector. In a generic sense all Lie algebras are like the harmonic oscillator with a, a^\dagger and a^\dagger a in the structure of roots and ... 1 Let \theta>0 denote the angle between the \hat{\mathbf x} and \hat{\mathbf n}. Notice that$$ \hat{\mathbf u} = \frac{\hat{\mathbf x}\times\hat{\mathbf n}}{\sin\theta} $$Is a unit vector perpendicular to both \hat{\mathbf x} and \hat{\mathbf n}. The desired rotation is a right-handed rotation around \hat{\mathbf u} by the angle \theta. ... 1 For any crystal, the First Brillouin Zone is found using the Wigner-Seitz construction for the reciprocal lattice. The high-symmetry points are labeled by certain letters mainly as a convention--like you said Gamma for (0,0,0) etc. The important thing to realize as far as the group theory, is that the group of the wavevector at the Gamma point has the full ... 1 @SAS answered most of the questions, however I believe there's a crucial point which still needs to be addressed: the chirality. Indeed, it is not obvious a priori why$$\Psi^T C \Psi\,\Phi\,,$$(where \Phi is some Higgs representation) leads to a Dirac-type masses instead of Majorana masses. Why not the common \bar\Psi \Psi? It turns out to be the ... 1 Your equation (2) is right, in principle: it is the standard coproduct of Lie algebras, but it is irrelevant, and should have never been used for anything here. The language confused you. It should read$$ \boldsymbol{J^a} = \boldsymbol{j^a} \otimes 1\!\!1 +1\!\!1\otimes \boldsymbol{j^a} .$\$ If you wished to apply it to two doublet reps, you should have ...

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