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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Generators of the (1,2,2) of Pati-Salam

I am working on a project involving breaking SO(10) to its Pati-Salam sub group. In one of the path ways you can use, the broken generators fit in the (6,2,2) of Pati-Salam (recall the Pati-Salam ...
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Can we do better than “a spinor is something that transforms like a spinor”?

It's common for students to be introduced to tensors as "things that transform like tensors" - that is, their components must transform in a certain way when we change coordinates. However, we can do ...
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Fundamental representation in Group Theory

I am struggling to understand fundamental representations in Group Theory. I know that the fundamental representation of $SU(N)$, as assigned to a matrix $U=U^i_j \in SU(N)$ can be shown through the ...
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2answers
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How do we know there are only 16 Dirac bilinears?

We know there are 5 types of bilinears in 4 dimensions, all of them add up to contribute with 16 independent DoF (degrees of freedom). Namely, these bilinears are known as: scalar (1DoF), pseudoscalar(...
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Lorentz transformation of the spinor fields

I have been reading the Srednicki's QFT textbook (available online at https://web.physics.ucsb.edu/~mark/qft.html) and in Chapter 34 the left and right-handed spinors are discussed. There is a step in ...
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Complex conjugate Young Tableaux representation [duplicate]

I have been studying Young Tableaux representation from youtube to represent $2\times 2$ and other examples to in $SU(n)$ symmetry. But i am unable to understand nor able to find relevant answers of ...
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1answer
34 views

Is it possible to determine angular momentum state if expectation of projection onto direction is known for all directions?

Suppose we are in a total spin $J=j$ vector space and there is the angular momentum operator $\boldsymbol{J}$. The Hilbert space then has $2j+1$ states: $$ |m=-j\rangle, |m=-j+1\rangle, \ldots, |m=j-...
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An $SL(2,C)$ representation and Dirac Spinor

In PCT, spin and statistics, and all that book, the following example is given: Let $S(A)$ be a representation of $SL(2,C)$ given as : $$S(A)=\frac{1}{2}\left(a^{0} \mathbf{1}+\mathbf{a} \cdot \...
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PCT Theorem and PCT, spin and statistics, and all that book

I am reading through PCT, spin and statistics, and all that, and trying to understand the construction on page 15 specifically equation (1-26) and the calculations that follows, what I can't see is ...
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1answer
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How to prove $Q_u = -2Q_d$ from $SU(5)$?

I'm a beginner, and I'm trying to figure out how to prove that charge of Up quark is equal to 2 times the charge of down quark from the 10 representation of $SU(5)$. Please help.
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Lorentz Binary Group actions in Spin Statistic theorem: $D[-1] = (-1)^{2j}$ in Novozhilov

In Novozhilov's book "Introduction to Elementary Particle Field Theory" there is a reproduction of Weinberg's S-matrix covariant proof of the Spin Statistics Theorem. I've referenced this in other ...
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Normalization of Generators of $SU(N)$

I have given a finite-dimensional matrix-representation of $SU(N)$. In this representation, the generators are denoted by $G^{a}$ for $a=1,\dots N^{2}-1$. I have to show that I can choose the ...
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There are infinitely many equivalent irreducible representations of $SO(3)$ on $\mathbb R^3$ [migrated]

The irreducible representation of $SO(3)$ on $\mathbb R^3$ is the set of the matrices $M$ such that $MM^T=I$ and $\det(M)=1$. But this is not the only one, indeed if $A$ is an invertible matrix then ...
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Equivalence of the Dirac representation of the Lorentz algebra and its conjugate in even dimensions - Polchinski

In Polchinski's String Theory, Appendix B.1 we look at the smallest irreps of the Clifford algebra in even-dimensional spacetimes $d=2k+2$. In (B.1.16) he defines two operators from the gamma matrices ...
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Why Clebsch–Gordan coefficients does not have a recursion relation for $J$?

The Clebsch–Gordan coefficients had a recursion relation: Sakurai Eq 3.8.45 $$J_\pm|j_1j_2; jm\rangle =(j_{1\pm}+j_{2\pm}) \sum_{m_1} \sum_{m_2} |j_1j_2;m_1m_2\rangle\langle j_1j_2;m_1m_2| j_1j_2;jm\...
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Representation of the Poincaré group by means of exponential

Let $(\Lambda,a)$ be a Poincaré coordinate transformation. Let $U$ be an unitary representation of the Poincaré group on some vector space. Is it always possible to express $U(\Lambda,a)$ in the ...
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A particular representation of $SU(2)$ on $\mathbb R^3$

Studying physics I encounter group theory and it has told me that: the matrices that rotates $\mathbb R^3$ vectors in the Euclidean space are the representation of $SU(2)$. Namely, $SO(3)$ matrices ...
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What are the physical manifestations of the finite-dimensional irreducible non-unitary representations of the inhomogeneous Lorentz group?

According to https://en.wikipedia.org/wiki/Representation_theory_of_the_Lorentz_group: "The finite-dimensional irreducible non-unitary representations together with the irreducible infinite-...
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Wigner's classification in curved space

Wigner classfied elementary particle as unitary irreducible representations (UIR) of the Poincaré group. Suppose the spacetime is curved with symmetries $G$. Should the elementary particles in this ...
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1answer
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The word “representation” in the context of Lie groups

I'm still very new to learning about Lie groups, something I find particularly confusing is the use of the word representation in the context of Lie groups. Sources I've checked online go quite far ...
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2answers
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Physical interpretation of Lorentz group non-compactness in the case of Weyl spinors

If we consider the generators of Lorentz group $J$ and $K$, it is possible to indroduce the operators $J^{\pm}=\frac{J\pm iK}{2}$ which shows the $SU(2)\times SU(2)$ structure of the Lorentz group. ...
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Why is representation theory important in physics?

Given a certain group we can find many representations of it. And If I'm not wrong a representation is a group itself. For example, given the group of the unitary 2x2 matrices with determinant 1 $SU(2)...
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Some clarifications about the ideas of representation of a group

I started to study group theory but I have many doubts about the topic, so I'd like to share my current understanding together with some questions, my aim is to understand the general ideas and ...
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Terminology about the tensorial represention of a group

This could be extremely trivial but i need to be sure I am not wrong. I am encountering many times statements where the author says "Tensors are examples of representations for the Lorentz group". ...
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2answers
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Lorentz representation of totally symmetric tensors?

Pretty much the title. A rank 1 tensor is $(1/2,1/2)$, a symmetric rank 2 tensor is $(0,0)+(1,1)$. I'm curious how this generalizes to rank $n$ totally symmetric tensors.
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On the picture of Schur limit of the superconformal index

My goal is to understand qualitatively (hopefully quantitatively in future) the existence of the relationship of the Schur limit of the superconformal index given by \begin{align} \mathcal{I}(q)=\...
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1answer
49 views

About obtaining eigenvalues for angular momentum with ladder operators

$$ \newcommand\<\langle \newcommand\>\rangle $$ I'm following Griffiths' intro QM text, 2nd edition. We have defined the angular momentum operator and obtained the commutation relations $[L_i, ...
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Best way to find $SU(2)$ transformation representation in if I know the desired output? [closed]

I have two doublets in $SU(2)$: $\begin{pmatrix} 1 \\ 1\\ \end{pmatrix}$ and $\begin{pmatrix} 1 \\ i\\ \end{pmatrix}.$ What is the easiest way to figure out the vector $\vec{\theta} $ in the $SU(2)$ ...
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1answer
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Regarding the signs in the Clebsch-Gordan coefficients

Let's take, for example, the $\frac{1}{2}$ $\frac{1}{2}$ spin case. We have, for $J = 1, M = 0$ $$|1,0\rangle=\frac{1}{\sqrt{2}}(|-1 / 2,1 / 2\rangle+|1 / 2,-1 / 2\rangle),$$ and, if we follow the ...
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Decoupled spin vectors A and B

Let we have $J_i∈{J_1,J_2,J_3}$ ,and $K_i∈{K_1,K_2,K_3}$, rotation and boost generators respectable . $$A_i=\frac{1}{2}(J_i+iK_i)$$, and $$[A_i,A_j]=iϵ_{ijk}A_k$$ $$[K_i,K_j]=−iϵ{ijk}J_k$$ $$[J_i,...
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Are all familiar symmetry transformations, when they act on fields, linear? [duplicate]

Consider symmetry transformation acting on a field or a set of fields. For example, a gauge transformation of the form $$\phi^\prime_a(x)=U_{ab}(x)\phi_b(x)$$ where $U(x)$ is a matrix with elements ...
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1answer
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What's up with $\mathrm{U}(1)$ regarding the spin homomorphism?

Let $\mathcal H(2)$ be the space of hermitian matrices of size $2\times 2$, and let $\sigma:\mathbb R^{4}\rightarrow\mathcal H(2)$, $$ \sigma(x)=x^\mu\sigma_\mu=\left(\begin{matrix} x^0+x^3 & x^1-...
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1answer
48 views

Product of representations of Lorentz group

How represent angular momentum in representation $(a,b)\otimes (c,d)=(a\otimes c,b\otimes d)$ ? I also post Tensor product of representations of Lorentz group Agreements: $I_a$ is $(2a+1)$-...
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1answer
119 views

Tensor product of representations of Lorentz group

Where is the rules for tensor product of representations of Lorentz group $(a,b)\otimes (c,d)$ without decomposition of one of these in orthogonal sum?
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1answer
42 views

Gauge covariant derivative and Leibniz rule

Let's say I've got 2 different fields $a, b$ and I want to compute its covariant derivative $D_\mu = \partial_\mu + iA_\mu^a T^a$ where $\{A_\mu^a\}$ is the set of gauge fields and $\{T^a\}$ the ...
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Casimir operators of de Sitter space

De-Sitter space can be thought of as a 4 dimensional hyperboloid embedded in 5D Minkowski space. Hence, the symmetry group of dS is $SO(1,4)$ whose generators are, $J_{AB}=i\left(X_A\frac{\partial}{\...
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Irrep of stress energy tensor

We have 4-tensor of second rank. For example energy-momentum tensor $T_{\mu\nu}$, which is symmetric and traceless. Then $T_{\mu\nu}=x_{\mu}x_{\nu}+x_{\nu}x_{\mu}$ where $x_{\mu}$ is 4-vector. Every ...
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Little group property as effect of rotation the spectator (Dual vectors in Hilbert space)

Plz read my note. The main insight is to consider little group property as effect of rotation the spectator. But since a twistor transforms under SU(2), and we have SO(3,R)=SU(2)/Z_2, so there ...
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Spinful time reversal and SOC

I'm having some trouble with understanding the distinction between spinful and spinless time-reversal in a real materials context. 1) Does spinful time-reversal (e.g. $\mathcal{T}^2 = -1$) imply spin-...
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Lie group and corresponding smooth manifold, and also why $SO(3)$ have a 3-dimensional manifold embedded in 4-dimensional Real space?

I think I have some loop holes on a connecting smooth manifold to a lie group. I state what my concepts are, Lie groups are expressed as manifold because the parameters in corresponding metric form a ...
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How to find representation of symmetry operation in magnon system?

Suppose an anti-ferromagnetic system with $n$ inequivalent atoms which is described by a Heisenberg Hamiltonian. The symmetry group of a given wave vector $G_k$ and how those inequivalent atoms ...
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Irreducible representation and character number

I am doing a course on "Symmetry of molecules and crystals". We talked about character tables and irreducible representation, as well as the notation for the representation. I got the following ...
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1answer
62 views

Addition of angular momentum and (anti)symmetry under exchange of constituent angular momenta

I have two potentially distinguishable particles, each with spin $s_1=s_2=s$. I'll only be looking at spin degrees of freedom. I'll write the total spin eigenvectors $|s_{tot} m\rangle$ in terms of ...
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1answer
110 views

Standard model: Why linear representation?

I'm trying to understand better the idea of the standard model, where particle states are described within vector spaces corresponding to irreducible representations of the group of symmetry of ...
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1answer
66 views

Deriving an identity of Lorentz group representation

I have a representation of Lorentz group on Hilbert space by following rule: $$|\alpha\rangle_{F'}=U(\Lambda)|\alpha\rangle_{F}$$ where $\Lambda $ is Lorentz transformation satisfying $$x^{\mu'}=\...
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1answer
47 views

How to prove that different squeezed vacua are the ground states of inequivalent CCR representations?

one can find on wikipedia articles on squeeze operators and squeeze coherent states these squeezed coherent states depend on a squeezed parameter r. the usual coherent states have r = 0 i have to show ...
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1answer
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Deriving masses after spontaneous symmetry breaking with a field in a peculiar representation

I am attempting to break the Pati-Salam group $SU(4)_{c'} \times SU(2)_L\times SU(2)_R$ with a field $\psi$ that fits in the following representation: $(4,\bar 2,1)$. My objective is to derive the ...
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1answer
74 views

Finite conformal transformations of fields

I want to work out the finite change of a Field in Conformal Field Theory. In Di Francesco's Conformal Field Theory he states "In principle we can derive [it] from the [local generators at x=0]" but ...
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1answer
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Representations of Conformal Group

I want to work out the Representations of the Conformal Group. I work with Francesco's Conformal Field Theory. He stats in equation 4.30 that $$e^{i x^\rho P_\rho}K_\mu e^{-i x^\rho P_\rho}= K_\mu +...
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2answers
182 views

Why ${\rm SL(2,C)}$ if everything can be derived with ${\rm SO(3,1)}$ and ${\rm SU(2)}$?

By showing that the complexified Lie algebra of the proper Lorentz group ${\rm SO(3,1)}$ is equivalent to two the direct sum of two complexified ${\rm SU(2)}$ Lie algebras $$\mathfrak{so}(3,1)_\mathbb{...

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