# 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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### 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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### 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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### 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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### Rotation and translation of a function of a 3D vector

I want to change the frame by doing translation and rotation. $$f(\vec{v})=\sum_{n,l,m}R_{nl}(v)Y_{lm}(\hat{v})f_{nlm}^v.$$ Let, $\mathcal{R}$ be the rotation matrix and $\mathcal{T}$ be the ...
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### Isomorphism of Virasoro Algebra with Different Highest Weights

Recently I was reading the big yellow book on "Conformal Field Theory" by P. Francesco et.al, and in appendix 8.A.1, it defined a covariant linear map for fusion process among irreducible ...
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### Rotation of spherical harmonics

I have a question about the rotation of spherical harmonics. In Wikipedia it is mentioned that if we make a rotation in 3D space: $R\vec{r}=\vec{r}'$,then the Spherical Harmonics can be written as a ...
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### Generation of a non-accidental degenerate eigenspace carrying out the symmetry operations on one eigenstate

Let us consider a system described by an Hamiltonian $H$ over an Hilbert space $M$, and the finite group $G$ of symmetry operations w.r.t $H$, i.e. R_g : M \rightarrow M \qquad g\in G ...
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### Can the composition law of a group be defined only when considering a representation or realisation of the Group?

When we talk about, lets say, the Lorentz group, we define the action of the Lorentz transformation $\varLambda$ on \begin{alignat}{1} x^{\mu} & \in\mathbb{R}^{1,3},\\ x^{\mu} & \rightarrow x'^...
### Confusion about tensors in $SU(3)$
I have some confusion regarding the notion of tensors in $SU(3)$ (or some other matrix Lie group, but let's keep the discussion to $SU(3)$). For concreteness, I will refer to Peskin and Schroeder's ...