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# 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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144 views

### Representation Theory of $SL(2,\mathbb R)$

The representation theory regarding the finite-dimensional representations of $SL(2,\mathbb C)$ is well-understood; namely, they all decompose into irreducibles $V_n$, $\dim(V) = n > 0$. ...
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### explicit matrix elements for a representation decomposed into subgroup by branching rules

I'm looking for a way to construct a representation for a simple Lie group such that one particular subgroup is manifest. I learned the branching rules from Cahn, Georgi and Slansky, but I'm still not ...
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### What type of fields are continuous spin representations?

Continuous spin representations (infinite dimensional representations of the Lorentz group) are pretty rarely discussed, and usually not in that much mathematical details. And usually it is done in a ...
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### What does $\Lambda^{-1}_{\frac{1}{2}}\gamma^\mu\Lambda_{\frac{1}{2}}=\Lambda^\mu_{\phantom{\mu}\nu}\gamma^\nu$ mean?

\begin{equation} \Lambda^{-1}_{\frac{1}{2}}\gamma^\mu\Lambda_{\frac{1}{2}}=\Lambda^\mu_{\phantom{\mu}\nu}\gamma^\nu \end{equation} In P&S, p. 42: Equation (3.29) says that the $\gamma$ ...
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### Infinite-dimensionality of unitary representations of non-compact simple Lie Groups

I have a question about the argument given in On finite-dimensional unitary representations of non-compact Lie groups. I have been looking for a good proof for this claim for a little while now. I ...
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### Is there an easy way to compute $\exp(-i\pi J_2) |jm\rangle = (-1)^{j-m} |j,-m\rangle$

Is there an algebraic way to compute $\exp(-i\pi J_2) |jm\rangle = (-1)^{j-m} |j,-m\rangle$. I know this is basically the Wigner $d$-matrix (which I can just look up), but how is it derived in this ...
43 views

### Weight $h$ as a function of the central charge $c$ in a Unitary Highest Weight Irrep of the Virasoro algebra

A highest weight irreducible representation of the Virasoro algebra can be labelled uniquely by $(c,h)$, with $c$ the central charge and $h$ the "Witt weight" (the weight of the Witt algebra part of ...
121 views

### Dimension of gamma matrices in dimensional regularization

When performing loop integrals in theories containing Dirac fermions, one almost always confronts terms of the form $$\text{Tr}\left[\gamma^{\mu_1}\cdots\gamma^{\mu_n}\right].$$ For instance, in $d$ ...
95 views

### Geometry of Affine Kac-Moody Algebras

One can reconstruct the unitary irreducible representations of compact Lie groups very beautifully in geometric quantization, using the Kähler structure of various $G/H$ spaces. Can one perform a ...
170 views

### General Commutator for Spherical Tensors (reformulated)

Edit: I think what I was looking for wasn't very well understood, so I'm reformulating my question to make it clearer. Hope this helps. In the book of "Angular Momentum: An Illustrated Guide to ...
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### Wicks contractions of stress-energy tensor and plane partitions

I am working out the number of wick contraction of a number $n$ of stress-energy tensor in 4D CFT. The strategy is as follows: For 1 stress energy tensor $T_{\alpha\beta}$, you have only one ...
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### Understanding the idea behind the super-Poincaré algebra

On the Super-poincaré algebra wiki page (https://en.m.wikipedia.org/wiki/Super-Poincaré_algebra), it says: "If Minkowski space-time belongs to the adjoint representation, then can Poincaré symmetry ...
101 views

### Transformations of contravariant and covariant tensor operators

I've been able to convince myself that a set of contravariant tensor operators $\hat{O}^{x}$ for $x=1,2,...,n$ respond to a small transformation $\hat{A}$ as, \begin{equation} [\hat{A},\hat{O}^{x}]=-(\...
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### Classical mechanics in coadjoint orbits

We know that coadjoint orbits are symplectic manifolds, and they can be used to find unitary representations of Lie groups and stuff, and it's also related to quantization. However, is it true that ...
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### Young tableaus for $SO(n)$

I know how to use young tableaus to find irreducible representations and their dimensions of $SU(n)$. Are there similar rules for $SO(n)$?
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### Are there supersymmetry algebras with higher spinor representations?

The super-Poincare algebra contains supersymmetry generators $Q^I$ which satisfy fermionic anticommutation relations. By the higher-dimensional analogue of the spin-statistics theorem, they must ...
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### For what angles (and why) does the equation for finite rotation fail to work?

I am learning rotations and group theory/representations and my lecturer's note mentioned that: "The group is considered connected, but not simply connected [...] As a result, the formula for a ...
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### Existence of an electric/magnetic dipole of a molecule

I have a molecule with a symmetry group $S_3=D_3$. I have to determine if it has a non-zero electric and/or magnetic dipole moment using the representation theory. I'm using the book of Jones "Groups, ...
32 views

### Interpretation of operators applied to systems with higher symmetry

Let me make an example from solid state physics to show what I mean: We can show that for a 2D hexagonal lattice the resistivity is isotropic by noticing that the resistivity tensor is symmetric ...
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### What is the spin of an operator in QFT?

Operators in quantum field theory with $n$ Lorentz indices that are symmetrized and traceless are referred to as spin-$n$ operators. For example, a spin two operator would be \begin{equation} \bar{\...
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### Rotation representations for spin higher than 1/2

I was reading through Feynman's Lectures Vol. III when I stumbled upon Eqs. 5.38 which describe the transformation amplitudes for a Stern-Gerlach experiment rotated about the y-Axis for an arbitrary ...
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### Implications of Poincare symmetry: spin and mass

Is it correct for me to say that the symmetry of a quantum system with respect to the Poincare group leads to the concept of mass and spin? The postulates of the special theory of relativity demand ...
I am trying to pin down what spaces a spinor and gluon gauge field exactly occupy. I know that the spinor is a quantity $\psi_{i\alpha f}(\vec x, t)$ where $i$ is a color index in the fundamental ...