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Questions tagged [group-representations]

Use as a synonym to the representation-theory tag

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How do we know the group property of a new particle?

Suppose I have a particle $W'$ which can decay into $\mu$ and $\nu_{\mu}$ and $e$ and $\nu_{e}$. Suppose we know such new gauge bosons come from some additional gauge group added to the Standard ...
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What is the relation between degrees of freedom and the number of independent operator?

I studied $SU(n)$. I have two questions with that. Why are there $n$-1 diagonal operators? There are $n^{2} - 1$ degrees of freedom. Considering its operators (Hermitian and traceless matrices), ...
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How to construct invariant forms under the effect of an arbitrary group?

First I would like to mention that I do not know that should I post this question here or in the math community, but since my background is in physics and this kind of question is usually asked by ...
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1answer
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Combining SU(N) multiplets using Young diagrams

I am trying to follow the Particle Data Group's instruction (PDF link) to combine SU(N) multiplets. On page 3, they show an example calculation of SU(3)'s $\textbf 8\otimes \textbf 8$. I understand ...
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1answer
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Unitarity representations of CFT in arbitrary dimensions

There is a well defined notion of unitarity of representations in Euclidean Conformal field theories that follows from the requiring unitarity in the Lorentzian space. Under this notion, all states ...
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Wigner-Eckart theorem and vectors

Let's consider a system in state $^3$D$_1$: $$\vec{L}^2=L(L+1)=6 $$ $$\vec{S}^2=S(S+1)=2$$ $$\vec{J}^2=J(J+1)=2$$ According to Wigner-Eckart theorem, if this is an irreducible representation, all ...
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How to map the symmetry property of the lattice unit cell to the symmetry properties of eigen modes

For example, in $\mathbf{r}$ space, a honeycomb lattice (like graphene) has C6v symmetry about the center of the unit cell. The ground state (singlet) eigen mode has C6v symmetry and the 1st order ...
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Representation of $SO(3)$ acting on 3-index tensors in Zee's Group Theory Book

In page 193, the book Group Theory in a Nutshell for Physicists started to explain inductively how we need to consider only the traceless symmetric (in all indices) tensors to construct $3^j$-...
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What is the physical interpretation of the occurrence of a particular irreducible representation more than once?

I am working on a problem about the crystal field splitting of the five-fold degenerate $d$ states. First, in a crystal field of $O$ point group symmetry, the five-fold degenerate $d$ states are split ...
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Operators of the special orthogonal group $\mathrm{SO}(3)$ in 3 dimensions

My professor taught us that when we want to rotate a 3D vector we need a $3\times 3$ matrix $R$ that is a rotation matrix. The set of all these matrices is the special orthogonal group in three ...
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$SU(5)$ group theory: Contracting three adjoints to make a singlet

There's probably a trivial answer to my question, but I'm having trouble finding it. In an $SU(5)$ GUT, we have a gauge field, $A_{\mu} = A^{a}_{\mu}T^{a}$ which lives in the $24 (\text{Adjoint})$ ...
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1answer
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A question about Lorentz transformations in spinor representation

For $$\Lambda^{\mu}{}_{\nu}= \frac{1}{2} Tr(\bar{\sigma}^{\mu} S \sigma_{\nu}S^{\dagger}) $$ We need to prove that $$\Lambda (S)= \Lambda (-S)$$ Am I naive to say that by adding $-S$, $S^{\dagger}...
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Lie Algebra in Particle Physics

In his book " Lie Algebra in Particle Physics" Georgie directly put the relation $$(1-P)D(g)(1-P)=D(g)(1-P)...(1)$$ This came from the two previous relations: $$PD(g)P=D(g)P$$ $$PD(g)P=PD(g).$$ where ...
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1answer
62 views

Parameter space of $SO(3)$ and $SU(2)$

Is it parameter space of $SO(3)$ and $SU(2)$ are same? can we use quaternions to represent both groups? what about their connectedness?
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2answers
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Complex conjugated representation and its Young tableaux

This post is an exact copy of one that I posted in Math's site. I do this copy because people there suggested me to do it since, apparentely, in Mathematics and Physics we use different conventions ...
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1answer
59 views

Spinor representation

I am trying to study Special unitary group of order 2 and some textbooks mention objects transform under special unitary group are called Spinors. then How can we represent a spinor using matrix?
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Irrep decompositions for $SO(N)$ tensors for $N>3$

How do I take a tensor products of $SO(N)$ irreps and decompose it in terms of irreps for $N>3$? (I understand the special case of $SO(3)$ we can use the nice $SU(N)$ technology of Young Tableauxs ...
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Clarification about confinement of colour charged objects

In lecture today we were reviewing the QCD lagrangian, and discussing hadronic wavefunctions. My lecturer said that QCD alone allows for states of colored hadrons, however because we do not see ...
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2answers
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Quantum mechanics and Group theory

Vectors are representations transform under $SO(3)$ Group, 4-vectors are representations transform under $SO(1,3)$ Group, Like wave function in discrete but infinite basis (hilbert space) are some ...
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How to write $3\otimes3\otimes 3=10\oplus8\oplus8\oplus1$ in a tensorial way? [duplicate]

I wasn't sure how to write the title because I don't really understand this topic. Here's my question: When we are constructing hadrons we put quarks together to form higher representations of the ...
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1answer
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How groups act on fields in QFT?

I read a lot a posts on how to verify what are the symmetries of a given Lagrangian but I really can't find what I need and can't even get it by myself, this because I don't actually understand how ...
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1answer
108 views

Non-abelian anyons, relation between representation of braid group and fusion rules

As far as I understand, anyons correspond to fields that live in the representation space of some (unitary?) representation of the braid group. One-dimensional representations commute and give rise to ...
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1answer
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Are the so-called representations of the Lorentz group actually all representations of it?

Fermionic fields change sign under a rotation by $2\pi$. However, in $SO^+\left(1,3\right)$ a rotation by $2\pi$ is the identity. For any representation $R$ of $SO\left(1,3\right)$ then we have $$R\...
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Rigorous Treatment of Quantum Tensor Operators

Recently, my classes have introduced me to the idea of spherical tensors and the Wigner-Eckart (WE) Theorem, but my previous classes on tensors had emphasis on things like covariance, contravariance, ...
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2answers
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Spin of 3 particles

I am trying to decompose the isospins of a three particle state using Clebsch-Gordan coefficients such as: $|1,1\rangle \otimes |1/2,-1/2\rangle \otimes |1,0\rangle$ Decomposing the first two ...
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1answer
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Doubts on representations of poincare group and QFT

I am studying Poincare group and encountered the term massless representations of the Poincare group. I know Poincare group is studied by the studying the little group of various momenta, massless and ...
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What is the Lorentz group composition of two electrons?

We know that the wavefunction of an electron transforms as Dirac spinor $(1/2,0)⊕(0,1/2)$ under the Lorentz group $SO(3,1) \sim SU(2)\times SU(2).$ Which representations can we form with two ...
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1answer
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Poincare group and free theories

How exactly is the Poincare group related to the free relativistic theories in quantum field theory? I know Poincare group is the Lorentz group along with translations but don't see any connected why ...
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Calculating adjoint representation of Lie group/algebra [duplicate]

How do I calculate adjoint representation of Lie group and Lie algebra? I would be thankful if anyone can give good example or general formula on calculating adjoint of any Lie group
2
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2answers
119 views

Direct Product vs Tensor Product

I am confused in the notation on page 67 and page 70 a text (http://www-pnp.physics.ox.ac.uk/~tseng/teaching/b2/b2-lectures-2018.pdf), whether it's talking about a direct product or an outer product: ...
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Is there a simple way to explain a fundamental representation of $O(N)$?

Is there a simple way to explain fundamental representation in Physics? For example, a fundamental representation of $O(N)$?
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Can fermions and bosons exist in the same representation?

Some people have made theories where they claim fermions and bosons exist in the same representation for example $E_8$. I can't see how this is possible. But say for example it is. This would imply ...
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1answer
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Can $E_8 \times E_8$ contain the standard model?

I know $E_8$ by itself can't be gauge group because it has no complex representation and so would not be chiral. But assuming the existence of mirror matter which also would have $E_8$ gauge group ...
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0answers
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$SU\left(N\right)$ Dynkin labels, how to compute

Let $V$ be somecomplex irreducible representation of $SU\left(N\right)$. I read that to compute the Dynkin labels of the weights, one can take the highest weight and then subtract from it the rows of ...
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1answer
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Weights of $SU\left(5\right)$ representation

Consider the representation $\Lambda^2V$ of $su\left(5\right)$ where $V$ is the fundamental representation. How can I work out the Dynkin labels of its weights? Are these the correct Dynkin labels ...
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1answer
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Branching of $SU\left(5\right)$

In the context of branching rules, what is a projection matrix for a subgroup. For instance, the projection matrix for the subgroup $SU\left(2\right)\times SU\left(3\right)$ of $SU\left(5\right)$ is ...
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1answer
54 views

What is the weight system for these SU(5) representations?

I need to work out the weight systems for the fundamental representation $\mathbf{5}$ and the conjugate representation $\overline{\mathbf{5}}$. I'm not clear what this means. The $\mathbf{5}$ ...
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2answers
559 views

Is there a simple way to calculate Clebsch-Gordan coefficients?

I was reading angular momenta coupling when I came across these CG coefficients, there is a table in Griffith's but doesn't help much.
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Plausibility of the Weyl-Relations of position and Momentum - Physical meaning of the Heisenberg group

In this question I asked about the uniqueness of the momentum operator $\hat{p}$ for a given position operator $\hat{x}$, and wether the uniqueness was fixed by the commutation relations that position ...
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1answer
112 views

How to determine isospin $T$ (not just $T_z$) of a nuclear ground state

I’m trying to work out the total isospin and the $z$-component of the isospin for specific elements like $^{20}O$ and $^{20}F$ in the ground state. I’ve worked out the $z$-component, as I concluded ...
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1answer
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Why is $4=3\oplus 1$? What are propagating modes? Etc

In Schwartz's QFT book, he said that the vector representation of the Lorentz group, $V_\mu$ that is four-dimensional, is the direct sum of two irreducible representations of $SO(3)$: a spin-0 ...
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1answer
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Explicit construction of $(\frac{1}{2}, \frac{1}{2})$ representation of Lorentz group

For the vector representation of the Lorentz group (actually the algebra), the $J^1$ generator is $$J_1 = i \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 &0 \\ 0 & 0 & 0 ...
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1answer
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Wigner $D$ matrix

How to derive symmetry relation of Wigner $D$ matrix? I mean this relation $$ D_{m',m}^j (\alpha,\beta,\gamma) = (-1)^{m'-m} D_{-m',-m}^j (\alpha,\beta,\gamma)^*. $$ I want to derive this, but I ...
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1answer
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Relation between Spin 1 representation and angular momentum and $SO(3)$

This is a naive question. It occurred to me while studying in detail the the Spin 1 angular momentum matrices. The generators of $SO(3)$ are $J_x= \begin{pmatrix} 0&0&0 \\ 0&0&-1 \\ ...
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Trace of generators of Lie group

In most textbooks (Georgi, for example) a scalar product on the generators of a Lie Algebra is introduced (the Cartan-Killing form) as $$tr[T^{a}T^{b}]$$ which is promptly diagonalised (for compact ...
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3answers
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Given the transformation of $SU(2)$ triplet $\vec{\phi}$ how to find the transformation of ${\Phi}\equiv\vec{\phi}\cdot\vec{\tau}$?

Given the transformation of a $SU(2)$ triplet $\vec\phi$ $$\phi\to \exp{(-i\vec{T}\cdot\vec{\theta})}~\vec\phi\tag{1}$$ (in the question here by @physicslover) how does obtain the transformation of $\...
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1answer
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How is the complexification of the Lorentz Lie algebra related to the need for Dirac's 4-component spinor in QFT?

There have been several questions with good answers in physics.stackexchange about the motivation of the complexification of the Lorentz Lie algebra, basically as a mathematically nice way to deal ...
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2answers
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Notation for the Representations of $SO(3,1)$

I was reading Zee's book "Quantum field theory in a nutshell" and came to the chapter regarding representations and algebra. He writes that the representations of $SO(3,1)$ are $(0,0),(0, 0),( \frac{...
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1answer
159 views

General Irreducible Representation of Lorentz Group

Background (just for context, you can skip it if you're familiar with Lorentz representations) A Lorentz transformation can be represented by the matrix $M(\Lambda)=exp(\frac{i}{2}\omega_{\mu\nu}J^{\...
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Arbitrary functions can be expanded by basis functions of invariant subspaces of a group

On page 96 of M. S. Dresselhaus's Applications of Group Theory to the Physics of Solids, it is said that Any arbitrary function $F$ can be written as a linear combination of a complete set of basis ...