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The proof of the Wigner-Eckart theorem for irreducible tensor operators

Thank you for your clarification on $\alpha$, so I can provide the answer more accurately. The trick to deriving the last equation is to consider the product $\langle e^{l}_{\lambda}|O^{\mu}_i|e^{\nu}...
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About obtaining eigenvalues for angular momentum with ladder operators

I am not sure to understand what you mean by non-normalisable. However, the only non-normalisable vector in a vector space with scalar product is the zero vector. So what it is actually obtained is ...
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Sym$^2\mathbb{C}^2$ as the unique 3-dimensional irrep of $\operatorname{SU}(2)$

If $$V_L~\cong~\mathbb{C}^2\tag{1}$$ denotes the fundamental/defining/left Weyl/spin-$(\frac{1}{2},0)$ representation of the Lie group $\operatorname{SL}(2,\mathbb{C})$, then $$\begin{align} \{M\in{\...
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Addition of Two Elements of Group Representation (Quantum Mechanics angular momentum)

You appear to know a bit more about the rotation group than the authors assume: They start from basic infinitesimal rotations in three dimensions, standard orthogonal matrices, and steer you in ...
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Addition of Two Elements of Group Representation (Quantum Mechanics angular momentum)

A group representation is, by definition, a homomorphism from the group into the group of linear operators on some vector space $V$. In less technical terms, a representation is a way of writing the ...
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In the Poincaré group, what are explicit representations of translations, boosts, and rotations?

In general, there are two classic ways to represent the Poincaré group. The first one comes from its definition. They are the "rigid" motions for a 4D Minkowski spacetime. Identifying space-...
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Interpretation of Feynman Slash Notation

Yes, of course; people do this routinely. Read your texts on. The essence of the map is its invertibility through $$\operatorname{Tr} \left(\gamma^\mu\gamma^\nu\right) = 4\eta^{\mu\nu},$$ so that $$v_\...
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How (if) is the gauge group for gravity incorporated in the Calabi-Yau manifold of string theory?

The Yang-Mills type gauge groups of an effective theory in string theory do not arise solely from the compactification, but from "wrapping" D-branes (to which open strings can be attached) ...
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How to contract spinor indices?

Undotted upper and lower indices can be contracted. Undotted indices are raised and lowered with the Levi-Civita symbol/tensor $\epsilon_{ab}$. Dotted indices work similarly. Dotted indices are ...
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Are Pauli matrices invariant tensors in the representation of $\frac12 \otimes \frac12 \otimes 1$?

Yes, the Pauli matrices are invariant in the sense that $$\sum_{j=1}^3 U \sigma_j U^{-1}(R^{-1})^j{}_k~=~ \sigma_k ,\tag{A} $$ where $U\in SU(2)$ is a $2\times 2$-matrix in the spin-$1/2$ ...
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