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Explanation of the meaning of those singlets, doublets

Abstractly, $\hat s_i$ are spin operators that act on general states $|s,s_z \rangle$ where the first and second labels refer to the total spin and spin-projection along an arbitrary axis respectively....
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Could the Universe have monster symmetry?

A cheap point is that there is no symmetry we can ever rule out. Just let all of the fields we've observed be singlets under that symmetry and posit that new fields transforming faithfully under it ...
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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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Symmetry and Symplectic Group of Hydrogenic Atom

Even though the intro to your question conjures up Pauli's legendary quantization of the Hydrogen atom using the rotational so(3) symmetry and the suitably normalized LRL vector which can be combined ...
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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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Do Legendre transformation form a group?

Think about a system with the so-called $C_2$ rotational axis. That is a system such that a rotation by an angle $\pi$ around that axis corresponds to a configuration indistinguishable from the ...
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How the factor of $1/2$ is purely conventional?

Lie Algebras are vector spaces, and the commutation relations in then are simply relations between elements of a basis of the vector space. Both in $SU(2)$, $SO(3)$, and in any other group, the ...
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Which is the correct way to write a Lorentz group component in exponential form?

You are writing the same thing in different ways. You can indeed write a general Lorentz transformation in the form of $$\Lambda = e^A e^B \cdots,$$ where each term corresponds to a "simple" ...
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Are the linear Lie groups matrices, tensors, or both?

On one hand, a multiplicative Lie group $G$ is not a vector space, and hence cannot be a tensor space. On the other hand, a Lie algebra $\mathfrak{g}$ can sometimes be a tensor space. Example: The EM ...
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Are the linear Lie groups matrices, tensors, or both?

Any type of mathematical object in the right representation can belong to a group as long as the necessary requirements are met. Tensors may in fact be brought in a form as to belong to some group. A ...
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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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