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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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Terms in Lagrangian and it's connection with Poincare group

For electromagnetic theory, there are different lagrangians, in QFT. My question is can I study the properties of Poincare group by knowing all the terms in the Lagrangian. For example, the Chern-...
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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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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 ...
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1answer
128 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
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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
63 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 ...
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0answers
27 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
50 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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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
60 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 ...
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1answer
39 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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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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Poincare group during interactions

Weinberg in Vol 1 says for non-interacting particles, we can take the state of particles to direct product of one particle states. I want to know what happens during the representations of the ...
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1answer
64 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
43 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
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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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1answer
38 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
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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
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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
44 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
37 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 ...
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1answer
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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 ...
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1answer
60 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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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
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Are particles always represented by 4-volume preserving transformations?

I've been reading about affine gauge gravity, which uses the affine group A(4,R) (for example here ). If I'm getting it right there seems to be an “affine higgs” mechanism that breaks symmetry down ...
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1answer
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How to build an antisymmetric selfdual tensor out of two 4-vectors?

In problem C of section 1.4 of Ramon's Field Theory: A Modern Primer, we are asked to build a field bilinear in $\chi_L$ and $\psi_L$, two left-handed weyl spinors, which transforms as the (1,0) ...
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1answer
68 views

Explanation of the sign of Clebsch-Gordan coefficients

These are the Clebsch-Gordan coefficients when the orbital and spin-angular momenta of a single spin 1/2 particle are added. I'm not able to understand the explanation. What I can understand is that: ...
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0answers
41 views

Decomposition of the complex conjugate of the fundamental representation of $SU(5)$ in $SU(3)\times SU(2)\times U(1)$

I know I can decompose the fundamental representation (denoted as $5$) of $SU(5)$ as: $$ (3,1)_{-2c/3} \oplus (1,2)_{c} $$ But how do I get the decomposition of the complex conjugate of this ...
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2answers
94 views

$SU(2)$ Invariant Lagrangian

Consider two arbitrary scalar multiplets $\Phi$ and $\Psi$ invariant under $SU(2)\times U(1)$. When writing the potential for this model, in addition to usual terms like $\Phi^\dagger \Phi + (\Phi^\...
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0answers
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What should I read to understand this question?

I understand the strong force as a Yang-Mills theory with $SU(3)$ color invariance. I understand that the quarks live in the fundamental representation of $SU(3)$ and that gluons live in the adjoint. ...
3
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1answer
77 views

Wigner proof of the non-existence of finite unitary representation of the Lorentz group

I am reading Wigner's paper ”On unitary representations of the inhomogenous Lorentz group” (Annals of Mathematics, Vol. 40, No.1, p. 149) found here: https://www.maths.ed.ac.uk/~jmf/Teaching/Projects/...
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1answer
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Supercharge transformation rules

Consider ${\cal N}=2$ supersymmetry with $SU(2)$ global symmetry group. Then both supercharges $Q_{ai},\bar{Q}_{\dot{a}\dot{j}}$ transform by 2 dimensional representation of $SU(2)$. Denote $SU(2)_I$ ...
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Theory for free, non-interacting anyons?

This link suggests that one cannot make a free theory out of anyons, because of its Lorentz representation. How exactly does the $SO(2,1)$ representation enforce the $\pm1$ eigenvalues? How can one ...
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2answers
75 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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1answer
87 views

Doubt in Weinberg's book on Quantum Field Theory

In page number 59 of his book on QFT, Weinberg mentions that for the operator $U$, defined for infinitesimal parameters $\omega$ and $\epsilon$ as: \begin{equation} U(1+\omega,\epsilon)=1+\dfrac{1}{2}...
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1answer
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Is spin 1 described by $SO(3)$ or $SU(2)$ [duplicate]

What spin is described by which rotation group? I always only find information about spin-1/2
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1answer
75 views

Why do the $\gamma$ matrices behave like vectors (tensors)?

In the study of Quantum Field Theory and Group Theory for the spinor representation of $SO$ groups, we know the following correspondence: $\chi C\psi$ scalar $\chi C\gamma^\mu\psi$ vector $\chi C\...
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1answer
62 views

$(1,1)$ representation of $SL(2,\mathbb{C})$

How do you prove that the $(1,1)$ representation of the $SL(2,\mathbb{C})$ group acts on symmetric, traceless tensors of rank 2?
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1answer
59 views

Constraints on RH component of fermion triplet under $SU(2)_L$

Consider a fermion $\chi$ whose left-handed part is in a triplet representation of $SU(2)_L$: $$ \chi_{L} = (\chi^1,\chi^2,\chi^3)_L^{\ \ \text{T}}. $$ The charged current of $\chi_L$ (i.e. its ...
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0answers
59 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 $d$ ...
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2answers
87 views

Operational definition of rotation of particle

The question in brief: what does it mean, operationally, to rotate an electron? Elaboration/background: I am trying to understand how representation theory applies to quantum mechanics. A stumbling ...
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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
33 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 ...
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1answer
40 views

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
47 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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Why do we use the $\left(\tfrac{1}{2}, \tfrac{1}{2}\right)$ rep for spin $1$ particles and not $(0, 1)$? [duplicate]

The spin 1 $A^\mu$ field transforms under the $\left(\tfrac{1}{2}, \tfrac{1}{2}\right)$ representation of the Lorentz field. When restricted to the $SO(3)$ subgroup, it decomposes into the $0 \oplus 1$...
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1answer
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Does dimension of irreducible representations of the double cover $SU(2)$ of the 3D rotation group define spin of particle?

In quantum field theory, does dimension of irreducible representations of the double cover $SU(2)$ of the 3D rotation group conclusively define spin? In other words, Is spin 1 particle only thing ...
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2answers
77 views

What is the way to represent a Lorentz tensor field?

For a vector field one can represent this with an array of arrows. There is a standard sort of way to represent tensors in Euclidean space as small ellipses. Is there any standard way of ...
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75 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 ...
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1answer
36 views

Why does the upper component of a $SU(2)$ doublet has $T^3=1/2$ and lower component $T^3=-1/2$ and not the opposite?

For a $SU(2)$ doublet, why does the upper component have $T^3=1/2$ and lower component $T^3=-1/2$? I know that this can be answered in the Standard Model by using $Q=T^3+Y/2$. But that is because we ...