Questions tagged [group-representations]

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Why do all fields in a QFT transform like *irreducible* representations of some group?

Emphasis is on the irreducible. I get what's special about them. But is there some principle that I'm missing, that says it can only be irreducible representations? Or is it just 'more beautiful' and ...
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1answer
144 views
+100

$z$ component of angular momentum under Lorentz transformation for massless particle

This question is related to this Helicity states. Suppose we have $k=[\omega,0,0,\omega]$. In Weinberg's book The Quantum Theory of Fields: Volume I he defines the state $|k,\sigma\rangle$ as an ...
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1answer
66 views

Helicity states

On page 71 of Weinberg's book The Quantum Theory of Fields: Volume I, he defines the operators $$A=J_2+K_1$$and $$B=-J_1+K_2$$ where ${\mathbf{J }}=(J_1,J_2,J_3)$ are the rotation generators and ${\...
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1answer
60 views

Helicity under rotation

Suppose that the state $|p,\sigma\rangle$ (for a massless particle) has 3 momentum ${\bf p}=p_3$ (that is the momentum is in the $z$ direction) and that $J_3|p,\sigma\rangle=\sigma|p,\sigma\rangle$ ...
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0answers
28 views

There is a classification of matrix representations of Euclidean group $\mathrm{E}(n)$ (also called $\mathrm{ISO}(n)$)?

I'm wondering if someone has already done a classification of matrix representations of the Euclidean group $\mathrm{E}(n)$, more specifically I'm trying to classify $\mathrm{E}(2)$. I watched a ...
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2answers
326 views

Why are physicists so interested in irreps if in their non-block-diagonal form they mix all components of a vector?

Consider a group $\{G,\circ\}$, with elements $e,g_1,g_2,...$, represented by the matrices $\{D(e), D(g_1), D(g_2)...\}$. If all the matrices can be brought to block diagonal forms by a similarity ...
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1answer
34 views

How to make a triplet out of 2 doublets in the $SU(2)$ representation?

In Y.Grossman and Y.Nir "The Standard Model" book in chapter 4 (non abelian symmetrys) they present the law of whom we can have a triplet and singlet out of 2 doublets name them $\phi_a$ and $\phi_b$, ...
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28 views

Generators of 2D global conformal group in terms of differential operators?

I'm looking for a reference that lists generators of two dimensional global conformal group on a complex sphere in terms of differential operators that may act on quasi primary fields $\phi(z,\bar z)$....
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33 views

What does it mean to take the tensor product of two reps of the Lorentz group? [duplicate]

If I reduce the Lorentz group to the representation $\mathfrak{su}(2)\oplus \mathfrak{su}(2)$, I can write left and right-handed Weyl spinors respectively as $\left( \frac{1}{2},0 \right)$ and $\left(...
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0answers
44 views

Spinor transformations as representations of $\mathrm{SL}(2, \mathbb{C})$

Background In the Weyl representation of the Dirac $\gamma$-matrices, the spinor transformations $S=e^{\frac{1}{2} \omega_{\alpha \beta} \Sigma^{\alpha \beta}} \in \,\mathrm{G}_{\mathrm{L}} \leq \...
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2answers
66 views

$3+3$ representation of $SO(4)$

In Zee's Group Theory in a Nutshell book, he says that the antisymmetric tensor $A^{ij}$ furnishes a 6 dimensional representation of $SO(4)$. He further argues that this 6 dimensional representation ...
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0answers
57 views

Representation of Poincaré group

Let's consider the most general Lorentz transformation: $x'^{\mu} = \Lambda^{\mu}_{\ \ \nu} x^{\nu} + a^{\mu}$. These transformations form the Poincaré group. The generators of translations of this ...
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0answers
37 views

Connection between $2n$ real fermions and $SO(2n)$

In section 11.4 of "Basic Concepts of String Theory" by Blumenhagen et al, they say: Consider a system of $2n$ two-dimensional real fermion (...) transforming as a vector of $SO(2n)$. I guess they ...
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39 views

Representation of the Lorentz group using matrices of $SL(2,\mathbb{C})$

There is a correspondence between the Lorentz group and the group $SL(2,\mathbb{C})$. To each Lorentz transformation $\Lambda$ we can associate two matrices $\pm A(\Lambda) \in SL(2,\mathbb{C})$ such ...
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0answers
29 views

Representation of the Lorentz group and correspondence with the $SL(2,\mathbb{C})$ group

We can find a correspondence between the restricted Lorentz group and the group $SL(2,\mathbb{C})$ if to each coordenate $x^{\mu}$ we associate a $2\times 2$ hermitian matrix $X$ given by $$X = x^{\mu}...
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2answers
85 views

What is a rotation group and how do we get its unitary representation?

The rotation group is ${\rm SO(3)}$. It is the group of $3\times 3$ orthogonal matrices $\{g(\theta)\}$ with unit determinant. So these are already defined in terms of $3\times 3$ matrices. But we use ...
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1answer
44 views

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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1answer
36 views

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
35 views

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
128 views

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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2answers
110 views

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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0answers
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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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1answer
73 views

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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1answer
25 views

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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1answer
57 views

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 ...
2
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1answer
46 views

$SU(5)$ 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
55 views

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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0answers
34 views

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
87 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?
3
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2answers
152 views

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
80 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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0answers
56 views

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 ...
0
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1answer
55 views

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
155 views

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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0answers
63 views

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 ...
4
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1answer
81 views

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 ...
1
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1answer
118 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 ...
2
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1answer
38 views

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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0answers
34 views

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, ...
2
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2answers
182 views

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 ...
3
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1answer
93 views

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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0answers
119 views

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
69 views

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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0answers
11 views

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
132 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: ...
2
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1answer
66 views

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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1answer
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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
88 views

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
36 views

$SU\left(N\right)$ 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
39 views

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 ...