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this question is 2 years old, but I thought it's never too late. I'm not sure about the definite answer, but here are my thoughts. Take the SO(6) algebra viewpoint. The $\mathbf{6}$ is the fundamental (vector) representation, and the $\mathbf{4}$ is the spinor representation. So we are looking for symbols $\Sigma_{AB}^I$ that combine two spinors into a ...

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Lucubration needs not light with insight. I fear you are expecting to make lemonade with apples. Here is why. The basic relation is the multiplication law of two Pauli vectors predicated on the abstract properties of the Pauli matrices, not their particular realization, $$(\vec{a} \cdot \vec{\sigma})(\vec{b} \cdot \vec{\sigma}) = (\vec{a} \cdot \vec{b}) ... 0 SO(3) symmetry means that the amplitudes |\langle \psi|\phi\rangle|^2 is invariant under rotations in the rays. Remember that a ray is specified by a family of vectors e^{i\phi}|\psi\rangle. This means that the linear or anti linear operators that describes how the vectors are changed by symmetric transformations obeys an projective representation. ... 2 Apologies for evincing magisterial cluelessness about what your diagrams represent and what you'd want to achieve, but I'd array the standard facts on tetraquarks avoiding Young diagrams, although they are self evident in the Dynkin labelling, which I also give, next to the tensor labelling. They may be useful to what you appear to be after--but I can't ... 3 Your group theory text probably betrayed you if it did not spend much time contrasting the two cases. A possibly related question is 254461. People use loose language and symbols that aggravate the confusion. Talking abstractly without explicit hands-on formulas clinches it (the confusion)! Let me stick to your 4-dimensional matrices and vectors, all tensor ... 1 Your equation (2) is right, in principle: it is the standard coproduct of Lie algebras, but it is irrelevant, and should have never been used for anything here. The language confused you. It should read$$ \boldsymbol{J^a} = \boldsymbol{j^a} \otimes 1\!\!1 +1\!\!1\otimes \boldsymbol{j^a} .$$If you wished to apply it to two doublet reps, you should have ... 1 @SAS answered most of the questions, however I believe there's a crucial point which still needs to be addressed: the chirality. Indeed, it is not obvious a priori why$$\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 ...

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The left hand: rotate the state $|JM\rangle$ by applying a rotation $R$ on it. Right hand side: insert completeness condition $\sum_{M'} |JM'\rangle\langle JM'|$ $D$ is the matrix representation of rotation matrix $R$ in basis ${|JM\rangle}$. The rotated state is expanded in terms of basis ${|JM\rangle}$ with coefficient $D$ in terms of rotation matrix.

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You can think, on the left is a short hand notation for a (2J+1) x (2J+1) matrix R applied to a (2J+1) component vector |J,M> with the components labelled by M. On the right, the matrix elements are explicitly shown, and the sum over M' is the matrix multiplication. Actually on the left is an abstract rotation operator R that will rotate any J. When ...

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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 ...

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The multiplicity $2S+1$ actually tells you how many degenerate spin states there are, each labelled with the total spin projection quantum number $M_S$ (this is from the total spin projection operator $\hat{S_z}$(conventionally taken to be in the z-direction) whose eigenvalues are $\hbar M_S$). The possible values of $M_s$ are $-S\le M_S\le S$ in integer ...

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Your feeling looks very misguided. Whatever you do, stay away from SU(3) for rotations. The rotation group and its Lie algebra are always linked to SO(3) ~ SU(2), to avoid formal forays into double covers and half angles. Read up on the spin matrices for any representation of the very same group (any spin). There are, in fact, simple systematic ...

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