Questions tagged [representation-theory]
The systematic study of group representations, which describe abstract groups in terms of linear transformations of vector spaces, such that group elements or their generators are represented as matrices, reducing group-theoretic problems to linear-algebraic ones.
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A puzzle about relativistic spin
I'm suffering from a confusion about relativistic spin. I don't believe my question has been asked before, and I'm sure I've made some silly mistake somewhere, but I can't spot it. So I'm appealing to ...
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"Linear independency" of Lie Brackets [migrated]
I was watching this eigenchris video. At 21:49, he says:
$$[g_i, g_j]=\Sigma_k {f_{ij}}^{k}g_k$$
for $\mathfrak{so}(3)$.
Does this mean $[g_i, g_j]$ and $g_i, g_j$ can be linear independent? What ...
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The meaning of a representation in one-dimensional quantum mechanics
In many places, one reads about chosing a representation for studying a particular one-dimensional quantum system. Usual representations are the position representation, the momentum representation or ...
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Orthogonal singlet states?
I recently encountered a problem where the inner product of the two product states
$|0,0\rangle\otimes|0,0\rangle$ and $|\frac{1}{2},\frac{1}{2}\rangle\otimes |\frac{1}{2},\frac{-1}{2}\rangle$
...
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Given a representation $(n, m)$ of the Lorentz group, is the little group representation just the tensor product $n \otimes m$?
I've been reading Weinberg's QFT Vol 1. and more specifically section 5.6. I would like to know if my understanding is correct or if I missed something. He starts with the full Lorentz group $\mathrm{...
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How to find Casimir operator eigenvalues of $SU(N)$? [closed]
The $[f1, f2, f3…fn]$ in the image represent the irreducible representations of $SU[n]$. How to find the irreducible representations of $SU[n]$ that conform to the form $[f1, f2...fn]$. Can you give ...
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Why is half-integer spin not observed classically? [duplicate]
It is usually stated that half-integer phenomena is purely quantum. The way in which "half-integerness" manifests itself seems very counterintuitive to me, or I simply do not understand it. ...
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What is the importance of $SU(2)$ being the double cover of $SO(3)$?
To my understanding, it is important that $SU(2)$ is (isomorphic to) the universal cover of $SO(3)$. This is important because $SU(2)$ is then simply-connected and has a Lie algebra isomorphic to $\...
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Generators of rotations: $[J_i, J_j] = \epsilon_{ijk} J_k$ and $(J_i)_{jk} = -\epsilon_{ijk}$. Is this a coincidence?
Thinking about $SO(3)$. Any rotation matrix $R$ can be written
$$
R = e^{\theta \hat{n}\cdot J}
$$
where $J$ is a vector the three skew-symmetric generators of rotation $J_x$, $J_y$, and $J_z$. In ...
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Why order matters when combining angular momentum
This seems like the answer should be trivial but when decomposing the direct product of 4 spin-$\frac{1}{2}$ states into a direct sum, one gets two singlets, namely
$$\frac{1}{\sqrt{2}} \left(\mid{\...
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$Ad\circ\exp=\exp\circ ad$ and $e^{i(\theta/2)\hat{n}\cdot\sigma}\sigma e^{-i(\theta/2)\hat{n}\cdot\sigma}=e^{\theta\hat{n}\cdot J}\sigma$
This question is inspired by my recent question How to prove $e^{+i(\theta/2)(\hat{n}\cdot \sigma)}\sigma e^{-i(\theta/2)(\hat{n}\cdot \sigma)} = e^{\theta \hat{n}\cdot J}\sigma$? with answer https://...
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Why Is There No Oscillator Representation for Operators in Planar ${\cal N}=4$ SYM Theory?
I'm studying the planar ${\cal N}=4$ Super Yang-Mills (SYM) theory and I'm curious about the representations of its operators, specifically the Hamiltonian and the dilatation operator. In many quantum ...
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How to prove $e^{+i(\theta/2)(\hat{n}\cdot \sigma)}\sigma e^{-i(\theta/2)(\hat{n}\cdot \sigma)} = e^{\theta \hat{n}\cdot J}\sigma$?
Disclaimer: I'm sure this has been asked 100 times before, but I can't find the question asked or answered quite like this. If there are specific duplicates that could give me a simple satisfactory ...
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Help with Wigner-Eckert Theorem problem
Currently trying to solve the following problem:
Consider an operator $O_x$ for $x = 1$ to $2$, transforming according to the spin $1/2$ representation as follows:
$$ [J_a, O_x] = O_y[\sigma_a]_{yx} / ...
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Is the Dirac adjoint in the representation dual to Dirac spinor?
As seen in this Wikipedia page, the Lorentz group is not compact and the Dirac spinor (spin $\frac{1}{2}$) representation is NOT unitary.
Therefore, the complex conjugate representation does NOT ...
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