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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
154 views

2D anomaly-free condition for a gauge theory

Take a $SU(2)$ gauge theory in 2d spacetime, say there are $n_1$ left-handed Weyl fermion in spin-1 written as $$ 1_L, $$ and $n_0$ left-handed Weyl fermion in spin-0 written as $$ 0_L . $$ and $n_{1/...
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1answer
111 views

Lorentz Rotation Matrix as a Matrix Exponential

I am attempting two exercises from pages 20 - 21 of Srednicki's QFT book and for some reason cannot reproduce the required results (this is not homework and I am just working through the problems to ...
2
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1answer
29 views

For what angles (and why) does the equation for finite rotation fail to work?

I am learning rotations and group theory/representations and my lecturer's note mentioned that: "The group is considered connected, but not simply connected [...] As a result, the formula for a ...
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1answer
85 views

Quick Question About Conformal Blocks

When a conformal block has dimensions and spin that violates its unitary bounds, does that make the block equal to zero. I'm asking because I'm trying to calculate 3D conformal blocks via a recursion ...
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0answers
53 views

Is there a name for symmetry in which fermions and bosons are in identical adjoint representations?

In a Yang-Mills theory, with gauge group $G$, if the Fermions are in an adjoint representation then for every Fermion with "charge" $Q$ there is a boson with charge $Q$. i.e. there is no difference ...
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2answers
115 views

Are rotation matrices faithful representations of the rotation group?

I would like to use rotation matrices as representations of the rotation group. I would like to know if these representations are faithful, i.e. isomorphic to the rotational group elements. I read ...
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2answers
210 views

Prove that the singlet state of 3 quark flavors is fully antisymmetric

I have three quark flavors, according to group theory: $$3\otimes3\otimes3=10\oplus8\oplus8\oplus1$$ I have to show that the 1 is antisymmentric. My idea would be to try to construct the states ...
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1answer
83 views

Irreducibility of $SU(N)$ rank-2 tensors [closed]

Given a rank-2 $\mathrm{SU}(N)$ tensor $X^{ab}$, it transforms as $X'^{ab} = U^a{}_c U^b{}_d X^{cd}$, where $U \in \mathrm{SU}(N)$. We can decompose it into a symmetric and an anti-symmetric part $$ X^...
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3answers
183 views

In general, how are representations used in physics?

I want is a basic overview, if there is one, of the meaning (and purpose) of the word representation in general terms. I have looked up sources such as Particle Physics and Representation Theory, but ...
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0answers
42 views

Does an irreducible representation acting on operators imply that the states also transform in an irreducible representation?

I'm not exactly sure how to phrase my question, but I'm trying to ask the following: if I have an operator transforming in an irreducible transformation of some group, I get a corresponding symmetry ...
2
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1answer
164 views

Why not embed a spin-2 particle in an antisymmetric tensor field?

In writing down 4D relativistic field theories, we need to choose fields that have enough degrees of freedom to accommodate the degrees of freedom of the particles we want in our theory. So, if we ...
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0answers
44 views

Existence of an electric/magnetic dipole of a molecule

I have a molecule with a symmetry group $S_3=D_3$. I have to determine if it has a non-zero electric and/or magnetic dipole moment using the representation theory. I'm using the book of Jones "Groups, ...
2
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1answer
206 views

Weinberg's classification of one-particle states and representations of the Poincare group

A representation of a group $G$ is a pair $(\rho, V)$ where $V$ is a vector space and $\rho : G\to GL(V)$ is a homomorphism. If $V$ is actually a Hilbert space and $\rho : G\to \mathcal{U}(V)$ maps ...
2
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1answer
120 views

How to understand spinors in 1+1 spacetime?

I am struggling to understand spinors in 1+1 spacetime. I know in this case the Clifford algebra is realized by two by two matrices so the spinors have two components. Then what do we mean by spin or ...
3
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1answer
469 views

Spinor dotted and undotted indices

I have had an introduction to QFT following the book of Mandl and Shaw. However, I have been asked to write a report on the CPT theorem. For this, the main reference I'm using is PCT, spin and ...
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1answer
240 views

Spacetime dimension and the dimension of Clifford algebra

The dimension of the Clifford algebra $C_p$ generated by a vector space $V^p$ is given by $2^p$, where $p$ is the dimension of the vector space (T. Frankel, the geometry of physics). Based on the top-...
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1answer
27 views

Questions regarding the elements of vector space spin representations act on

Elements of vector space spin-$1/2$ representations act on are spinors. What about half-integers in general? And what about integer spins? Do spin-$0$,$1$ reps always act on vectors?
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1answer
321 views

Symmetry of Clebsch-Gordan coefficients

The symmetry of clebsch-gordan coefficients $\left< j_1j_2;m_1m_2 \middle| j_1j_2;JM \right>$ under exchange of $j_1,m_1$ and $j_2,m_2$ is \begin{equation} \left< j_1j_2;m_1m_2 \middle| ...
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1answer
206 views

Why are projective representations allowed in classical field theory?

A quantum spin $1/2$ particle does not return to itself upon a rotation by $360^\circ$, but rather itself up to a sign. This is acceptable, because this extra phase is unobservable. In general the ...
2
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0answers
32 views

Interpretation of operators applied to systems with higher symmetry

Let me make an example from solid state physics to show what I mean: We can show that for a 2D hexagonal lattice the resistivity is isotropic by noticing that the resistivity tensor is symmetric ...
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73 views

Non-unitarity of finite dimensional Lorentz group and its implications

In Peskin and Schroeder, section 3.2, it is stated that Lorentz group being non-compact it does not have any finite dimensional, faithful unitary representation. But it has also been said that one ...
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3answers
247 views

Help trying to understand why the matrix representation of $J^2$ is what it is

Say we a basis of kets such as$$ \beta ~ := ~\left\{\left|j=1,\, m=-1\right> , ~ \left|j=1, \, m=0\right>, ~ \left|j=1, \, m=1\right>\right\} \,.$$ Then it's plain to see why in matrix ...
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1answer
44 views

Decomposition of $E_6$ into $SO(p,q)$

I've seen the following decomposition of the fundamental representation 27 of $E_6$ into $$E_6 \rightarrow SU(2) \times SO(5,2) \times SO(1,1)$$ $$27 \rightarrow (1,1)(-4) + (1,7)(-2) + (2,8)(+1) + ...
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1answer
173 views

Wigner Classification via the orbit structure of the Lorentz group

I have been reading and writing a lot about the Wigner Classification of irreducible unitaries of (the universal cover of) the Poincaré group lately, both from a physicist's and a mathematician's ...
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2answers
426 views

Are spin-1/4 theories forbidden?

Ok, this question looks a bit ridiculous at the outset. However, I was thinking, and I couldn't actually come up with a reason why there shouldn't exist a representation of the Lorentz group that was, ...
5
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2answers
317 views

Relation of Hilbert space to space-time

In Wigner's classification, one observes that the full automorphism group of the Lorentzian manifold $\mathbb{R}^{1,3}$ is precisely the well-known Poincare group. Motivated by this and basic ...
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2answers
115 views

Intuition for Total Angular Momentum Quantum Number

I am trying to build intuition about angular momentum states in quantum mechanics. I'll use $\vec{\boldsymbol{V}}$ to represent a quantum angular momentum operator. This could be orbital or spin ...
4
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2answers
150 views

Construct an SO(3) rotation inside the two SU(2) fundamental rotations

We know that two SU(2) fundamentals have multiplication decompositions, such that $$ 2 \otimes 2= 1 \oplus 3.$$ In particular, the 3 has a vector representation of SO(3). While the 1 is the trivial ...
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1answer
428 views

Is there a generalised Wigner-Eckart theorem?

The Wigner-Eckart theorem gives you the matrix element of a tensor transforming according to a representation of $\mathfrak{su}(2)$, when sandwiched between vectors transforming according to another (...
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2answers
120 views

Very basic question on group representation theory

I'm sorry for this naive question. I feel that since I was first introduced to the idea of group representation, I did not correctly grasp the idea. Unfortunately, therefore, several confusions keep ...
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1answer
171 views

Subtlety in the proof of 2-to-1 homomorphism between $SU(2)$ and $SO(3)$

In physics, it's common to use the relations $$\textbf{r}^\prime=\mathscr{R}\textbf{r};~~\text{and}~~\textbf{r}^\prime\cdot\boldsymbol{\sigma} =\mathscr{U}(\textbf{r}\cdot\boldsymbol{\sigma}) \mathscr{...
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0answers
26 views

Finding the generators of the adjoint representation [duplicate]

Let $G$ be a lie group and $\mathfrak{g}$ it's associated lie algebra.I can quite easily show that if there is a Lie group representation $\rho_{G}$ from $G$ into $L(G)$: $$ \rho_{G}: G \rightarrow \...
7
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1answer
349 views

Why aren't infinite-dimensional representations of the Poincaré group classified by *two* half-integers?

It is known that to specify a finite-dimensional irreducible representation of the Lorentz group, one needs to specify two half-integers, $(j_1,j_2)$. For instance, the left-handed and right-handed ...
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0answers
243 views

A few doubts with showing Lorentz invariance of Dirac equation and probability current

Trying to understand some about Lorentz invariance and representation theory, I thought that the best way is with an example of application: Show the Lorentz invariance of the Dirac Equation $$(i \...
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1answer
68 views

How to count states in SUSY multiplets?

There is an easy proof of the structure of multiplet that I don't reproduce here (it can be found in Bertolini, Lecture on Supersymmetry, pp.40-41 for the massless case and p.47 for the massive one)....
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4answers
541 views

Deriving the unitary operator $U(R)$ associated with a rotation $R$ using Wigner's theorem

A rotation $R(\hat{\textbf{n}},\phi)$ about an arbitrary axis $\hat{\textbf{n}}$ through an angle $\phi$ in the three-dimensional physical space is given by $$R(\hat{\textbf{n}},\phi)=e^{-i(\textbf{j}\...
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0answers
44 views

Representations of a symmetry group: what is actually being represented? [duplicate]

For definiteness, consider the group $SO(3)$. There is a Lie algebra $so(3)$ given by $$ [T_a, T_b] = if_{abc}T_c $$ The generators of this algebra can be exponentiated to form the elements of $SO(3)...
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0answers
82 views

What is the physical meaning of Lie congruence classes?

Any weight $\lambda$ characterising a representation of $\mathfrak{su}(N)$ is an element of one of the $N$ congruence classes defined by (ref.1) $$ \lambda_1+2\lambda_2+\cdots+(N-1)\lambda_{N-1}\quad\...
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0answers
157 views

What is the spin of an operator in QFT?

Operators in quantum field theory with $n$ Lorentz indices that are symmetrized and traceless are referred to as spin-$n$ operators. For example, a spin two operator would be \begin{equation} \bar{\...
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3answers
1k views

Confusion about rotations of quantum states: $SO(3)$ versus $SU(2)$

I'm trying to understand the relationship between rotations in "real space" and in quantum state space. Let me explain with this example: Suppose I have a spin-1/2 particle, lets say an electron, ...
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3answers
2k views

What is the difference between the dimension of a group and the dimension of its representation?

I'm reading the following post to learn about QCD interactions: Why are 3 colors used in QCD? However, I can't seem to grasp the conceptual difference between the dimension of a group and the ...
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2answers
261 views

Adjoint representation in Liouville-von Neumann equation

I am having trouble understanding the adjoint representation of a Lie algebra in the scope of a very specific example, so I thought physics.SE would be the best place to ask. Background: A $N \times ...
5
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2answers
488 views

Choice of Dirac gamma matrix representation and definition of adjoint spinor

Is the definition of the adjoint spinor $\bar{\psi}=\psi^\dagger \gamma^0$ forcing a particular choice of representation of the Dirac matrices (or a subset of the possible choices)? More precisely, I ...
5
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2answers
753 views

spinor vs vector indices of Dirac gamma matrices

I am struggling to understand the nature of the components of the Dirac matrices. If we view the four components of a Dirac spinor as $\psi^a$ with $a$ being a 'spinor' index, then if a gamma matrix ...
4
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0answers
167 views

Infinite-dimensionality of unitary representations of non-compact simple Lie Groups

I have a question about the argument given in On finite-dimensional unitary representations of non-compact Lie groups. I have been looking for a good proof for this claim for a little while now. I ...
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0answers
52 views

Representation theory of AdS and its classical field theory

I am looking for any resources which review the representation theory of the (Anti)-de Sitter group and its Lie algebra and its application to (classical) field theory. I am familiar with how this is ...
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0answers
140 views

Why is a spinor not a tensor?

The title says it. why is a spinor not a tensor? I know the transformation rules for a spinor but I cant see why it is not a tensor?
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2answers
2k views

How many states for two spin 1 particles?

A fairly simple question: If we have a composite system of two spin-1 particles, where $J_1=1$ and $J_2=1$, how many possible states $|Jm\rangle$ are there? I know |$J_2 - J_1$| < $J$ < |$J_2 +...
2
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1answer
115 views

axial anomaly for adjoint fermion v.s. fundamental fermion

It is known that the axial anomaly (chiral anomaly, the left L- right R) shows that $U(1)_A$-axial symmetry is not a global symmetry at quantum level. In particular, one can consider the (1) ...
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1answer
176 views

“Color charge” of the adjoint fermion?

What kind of "color charge" does the adjoint fermion carry? Let us consider the SU(N) gauge theory. The gauge field is in the adjoint representation (rep). Well-Konwn: If the fermion is in SU(N) ...