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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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1answer
94 views

$\mathrm{SU}(2)$ as a representation of the rotation group

I have read in a book that the group $\mathrm{SU}(2)$ is one of the irreducible representations of the rotation group. The book begin saying that the rotation group has 3 generators $J_{1}, J_{2}$ and ...
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1answer
43 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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0answers
21 views

Normal mode decomposition of a triangular hexagonal lattice

I was trying to understand and redo the methods used in a previous question: Vibrational anharmonic coupling and noise-induced spontaneous symmetry breaking in a hexagonal finite mechanical lattice ...
3
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1answer
164 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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0answers
14 views

Dielcetric Tensor Transforms under Product Representation

How do I show that the Dielectric tensor in 2D transforms as a product of representations? I am told that the electric displacement and electric field transforms with the representation $D^v(g)$, and ...
2
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2answers
129 views

What does matrices act on different spaces mean in QFT?

I have a Dirac kinetic term in a Lagrangian. $$ i\bar{\psi}\gamma^\mu D_\mu\psi = i\bar{\psi}\gamma^\mu\partial_\mu\psi + g\bar{\psi}\gamma^\mu\psi A^a_\mu T^a,$$ However, I usually heard that ...
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1answer
54 views

Product of generators in fundamental representation of $SU(N)$

I'm trying to prove equation 25.20 in Schwartz: $$T^a T^b=\frac{1}{2N}\delta ^{ab}+\frac{1}{2}d^{abc}T^c + \frac{1}{2}if^{abc}T^c,\tag{25.20}$$ where $T^a$ are the fundamental representation ...
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4answers
249 views

Why in QFT what really matters is $\exp(\mathfrak{so}(1,3))$ instead of $O(1,3)$?

In QFT fields are classified according to representations of the Lorentz group $O(1,3)$. Now, most books when getting into this say that in order to understand the representations of $O(1,3)$ we need ...
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2answers
86 views

Orbital angular momentum quantum numbers - subtracted?

Reading Griffiths' Quantum Mechanics. We have the electronic confirmation of Carbon as $$(1s)^2 (2s)^2 (2p)^2$$ in the ground state. He says There are two electrons with orbital angular ...
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2answers
47 views

Confusion about Young tableaux composition rule

I'm following the rules in this document to combine irreps of $SU(N)$ using Young tableaux. If I'm not mistaken the product of two irreps should be symmetrical, that is $A \otimes B = B \otimes A$. I'...
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1answer
58 views

How are Dunkl operators used in Hamiltonian mechanics?

I am currently doing a math research project on the representation theory of Cherednik (double affine Hecke) algebras, specifically the algebra $\mathcal{H}_{t,c}(\mathfrak{S}_n,\mathfrak{h})$, which ...
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45 views

Are the ideals in two GNS constructions linked to the equivalence (or not) of the CCR representations?

Starting from the abstract C* algebra $A$ of canonical commutative relations, a state $\rho$ over this algebra enables to construct a Hilbert space $A/I$ where $I$ is the ideal of the elements $a$ ...
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62 views

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 ...
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2answers
146 views

Lorentz transformation of a Weyl Spinor?

A left handed Weyl Spinor belongs to the $(\frac{1}{2},0)$ representation of the Lorentz group. So given the Spinor, the unitary representation of the Lorentz transformation should look like $\exp{iA\...
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39 views

How does hexagonal boundary arise in $SU(3)$ representation?

I am having trouble trying to understand the hexagonal boundary in $t_3$ and $y$ representation of $SU (3)$. I tried to work it out like we did in $SU \left( 2 \right)$ but got not luck. Could someone ...
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45 views

Unitary representations of a Kac-Moody algebra

I am reading one of the original papers about the GKO construction. They say that a representation of the (untwisted) Kac-moody algebra $\hat{g}$ is one where the generators are such that $T^{a\dagger}...
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39 views

Representations of the SUSY algebra

I am currently reading Wess & Bagger, and have trouble with a statement they make about representations. If we wish to study massive, one-particle representations of the SUSY algebra, we boost to ...
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0answers
46 views

Fields transforming under an exceptional Lie group

We may think of tensors as sections of an associated vector bundle to a principal $\mathrm{GL}(n,\mathbb R)$ bundle, with a fibre chosen to be $\mathbb R^m \times (\mathbb R^*)^n$ - these play a role ...
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65 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 ...
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1answer
58 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
2k views

In the quantum hamiltonian, why does kinetic energy turn into an operator while potential doesn't?

When we go from the classical many-body hamiltonian $$H = \sum_i \frac{\vec{p}_i^2}{2m_e} - \sum_{i,I} \frac{Z_I e^2 }{|\vec{r}_i - \vec{R}_I|} + \frac{1}{2}\sum_{i,j} \frac{ e^2 }{|\vec{r}_i - \vec{...
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2answers
79 views

Relation between giving the form of an operator in a given representation, and bra ket notation [closed]

So I understand that kets are abstract objects that are the elemnets of a Hilberts space. Say $|\psi \rangle$. We can write this ket in a position representation $\langle r|\psi \rangle = \psi(r)$, ...
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0answers
16 views

Representations of scalar fields from the expressions of fields

Consider scalar field $\phi(x)$, when we quantize this scalar field we get an expression in terms of creation and annihilation operators as $\phi(x) = \sum c(p)({ae^{ipx} + a^\dagger e^{-ipx}} )$. If ...
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1answer
211 views

Two spin-1 system and the projector onto total spin 2 subspace [closed]

I am having trouble grasping the projection operators in the context of composite spins system, e.g. with two spin-1. First off, a projector $P$ is said to be an operator that squares to itself, $P^2=...
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0answers
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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0answers
65 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 ...
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2answers
49 views

$j=\frac{1}{2}$ addition of angular momentum

For $j=\frac{1}{2}, j'=\frac{1}{2}$ we have $$|11\rangle=|\frac{1}{2}\frac{1}{2}\rangle$$ $$|10\rangle=\frac{1}{\sqrt{2}}(|-\frac{1}{2}\frac{1}{2}\rangle+ |\frac{1}{2}-\frac{1}{2}\rangle)$$ $$|10\...
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0answers
43 views

Physically, why don't we care about representations that differ only by a similarity transformation?

I was looking at how to derive the (1/2, 0) representation of the Lorentz group when acting on fields. Specifically, I'm interested in understanding the logic behind replacing the "symbols" $A,B$ with ...
6
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4answers
430 views

Weyl spinor representations and the Lorentz group

I'm currently trying to read up on the Lorentz-group and it's representations. I've found a couple of posts here on stack-exchange that I find helpful and confusing at the same time, so I would be ...
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0answers
66 views

Vector-like Representation of fermions

In the literature, they often extend the Standard Model by adding a so-called vector-like fermion which is a multiplet invariant under $SU(2)_L\times U(1)$. The left- and right-handed components of ...
3
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1answer
188 views

Nature of Spin in QFT

If the orbital angular momentum of an electron in an atomic orbital is associated with (generated by) an asymmetry in the orbital wave function, is it also the case that the intrinsic spin of a free ...
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1answer
51 views

Generator of 3D rotations in $\mathbb{C}^2 \otimes \mathbb{C}^2$

Let us consider a system of two spinors. The 3D rotation operator around the $\vec{n}$ axis in $\mathbb{C}^2$ is clearly $R(\theta) = \exp(i \frac{\theta}{2}\vec{n}\cdot\vec{\sigma})$. If I wish to ...
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1answer
80 views

Can we use the Pascal triangle as an aid to construct superpositions of wavefunctions corresponding to $n$ electron spins?

Suppose we have n electrons and want to construct the wavefunction corresponding to the spins of the electrons. Can we construct this wavefunction (in the $(s_1,s_2,s_3 ... s_n)$ representation, so ...
2
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1answer
51 views

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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2answers
166 views

Clebsch-Gordan coefficients for more than 2 particles

I need to couple arbitrary spins together and need Clebsch-Gordan coefficients for them. This should be just coupling the last two particles, then couple the next until the first particle is coupled. ...
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0answers
111 views

Books, papers, etc on Lorentz and Poincare groups/algebras/etc

I'm currently trying to learn more about the Lorentz- and Poincare Lie-algebras and the representation theory about them. But I'm really struggling with the material that we were given and I'm also ...
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2answers
103 views

Are representations of $\text{SL}(2,\mathbb{C})$ indexed by one half-integer or two?

I am very confused by this. In Hall's book on Lie theory, he states that the representations of $\text{sl}(2,\mathbb{C})$ are indexed by a half-integer. This is the usual result for $\text{su}(2)$ in ...
2
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1answer
55 views

Quantisation of $z$-angular momentum eigenvalues

Consider the eigenvalue equation for the $\hat{l}_z$ angular momentum operator: $$\hat{l}_zY_{lm_l}(\theta,\phi)=m\hbar Y_{lm_l}(\theta,\phi)$$ with separable solution $$Y_{lm_l}(\theta,\phi)=\Theta_{...
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0answers
67 views

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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1answer
99 views

$(1,0)$ representation of $\text{SL}(2,\mathbb{C})$ and selfdual antisymmetric tensors

The $(1,0)$ representation of $\text{SL}(2,\mathbb{C})$ is realized on two indexed symmetric spinors $\psi^{ab}$ transforming like $$D^{(1,0)}(A)\psi^{ab}=\sum_{c,d=1}^2A^a_cA^b_d\psi^{cd}$$ for all $...
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1answer
156 views

Gauge Field Transformation Properties

I'm a bit confused about the gauge transformation properties of non-abelian gauge fields, and I just wanted some clarification. I keep seeing the statement that "gauge fields transform in the adjoint ...
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0answers
71 views

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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1answer
46 views

Root weights and states in orbifold compactifications

I have the following question regarding orbifold compactifications of the heterotic string: What is the relation between a certain representation and the weights of the root lattice? I mean: take ...
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2answers
354 views

Do I need Gamma matrices in Majorana representation in the Lagrangian of a Majorana fermion?

I understand that the Majorana representation of the Gamma matrices are the real representations of the associated Clifford algebra. A Majorana fermion is defined as a fermion that equals to its ...
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2answers
148 views

Computing the spin degrees of freedom for a massless particle in $D$ dimensions

According to the paper A Lagrangian formulation of the classical and quantum dynamics of spinning particles, a relativistic spinless particle in $D$ spacetime dimensions can be described by the ...
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1answer
133 views

What is the meaning of “representation of the canonical commutation relation in the form of Heisenberg for symplectic locally convex space”?

What is the meaning of "representation of the canonical commutation relation in the form of Heisenberg for symplectic locally convex space"?
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1answer
39 views

About $(0,1/2)$ representations

While studying representations of Lorentz group, we get the generators to be $J_{i}$ - rotations and $K_{i}$ - boosts. We define $N_{i}^+$ and $N_{i}^-$ operators and these operators obey the same ...
2
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1answer
60 views

What is the dimensionality of each part of a covariant derivative?

In the standard model, we have the following covariant derivative: $$D_\mu = \partial_\mu - ig_sG_\mu^a\lambda_a-igW_\mu^a\frac{\sigma^a}{2}-ig'B_\mu\frac{Y}{2}$$ If we let this work in on e.g. the ...
3
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1answer
76 views

Intuition for the supertrace identity in supersymmetry

In pretty much every introductory book/lecture notes I've come across, one finds the expression for the mass matrices for scalars, fermions and vector bosons for a generic Lagrangian, and simply ...
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0answers
13 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