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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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Isomorphism of Virasoro Algebra with Different Highest Weights

Recently I was reading the big yellow book on "Conformal Field Theory" by P. Francesco et.al, and in appendix 8.A.1, it defined a covariant linear map for fusion process among irreducible ...
Mohammad. Reza. Moghtader's user avatar
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Rotation of spherical harmonics

I have a question about the rotation of spherical harmonics. In Wikipedia it is mentioned that if we make a rotation in 3D space: $R\vec{r}=\vec{r}'$,then the Spherical Harmonics can be written as a ...
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Free fields in Weinberg QFT vol.1

Background: In section 5.1 Weinberg discusses free fields. He had shown that for interaction of the form, $V(t) = \int{d^3x \mathscr{H}(\mathbf{x},t)}$ if $$U_0(\Lambda,a) \mathscr{H}(x) U_0^{-1}(\...
Damo's user avatar
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One-Loop beta function for gauge couplings

I am currently doing my homework on Standard Model one-loop correction. When I am reading Quantum Field Theory by Mark Srednicki and Journeys Beyond the Standard Model by Pierre Ramond, I notice some ...
quantumology's user avatar
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Lorentz transformations of the E.M field from representation theory

I'm wondering if one can get the Lorentz transformations for the eletromagnetic field by considering any 6 dimensional representation of the Lorentz group $SO(1,3)^{+}$. I already know that $F_{\mu \...
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Wigner-Eckart theorem in classical physics?

The Wigner-Eckart theorem is a useful result in quantum physics and its many applications. Most presentations of this material in books on QM and online lecture notes seem to be variations on the same ...
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Finiteness of the Kac table for minimal model

I am currently reading CFT from Di Francesco. I am stuck at not understanding why the kac table for minimal models $(p, q)$ where $p$ and $q$ are co-prime, only has fields with conformal dimension $h_{...
Suriyah R K's user avatar
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Does the Wigner little group classification of particles have consequences for classical field theory?

Does the Wigner little group classification of particles have consequences for classical field theory? In particular, I'm curious whether it can be used to predict the two propagating modes for ...
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Followup question to "Invariant symbol, group representation"

There is a 2 year old answer by Cosmas Zachos which is very helpful regarding invariant symbols here. Aside this context, I have never encountered these and thus I have 3 questions: Why is it ...
JohnA.'s user avatar
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Physical interpretation of reducible but indecomposable reps of the Poincaré group?

As I understand it, there are representations (reps) of the Poincaré group that are reducible but still indecomposable (i.e., cannot be expressed as a direct sum of two subreps). This would be ...
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How does the Hamiltonian act on the multiplicity space of irreps?

My question in the title stems from not completely understanding the last three lines of this answer. I list specific questions at the end of this post. Setup. Consider a quantum system described over ...
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Why do physicists refer to irreducible representations as "charges" or "charge sectors"?

My question is in the title: Why do physicists refer to irreducible representations (irreps) as "charges" or "charge sectors"? For concrete examples, irreps are referred to as &...
Maple's user avatar
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Why are spinors members of minimal ideals?

Why do we require that spinors live in minimal left ideals of Clifford algebras and not just left ideals? I assume that it has something to do with irreps but a Dirac spinor also lives in an minimal ...
Silas's user avatar
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Wigner-Eckart for Finite groups

We know the Wigner-Eckart theorem generalizes to say $\mathrm{SU}(3)$ (see for example this answer). In a different direction, I am curious if/how it generalizes to finite groups of $\mathrm{SU}(2)$. ...
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Meaning of “transforms like the adjoint” in the context of Yang Mills Theory and connection to Lie Algebra

In Srednicki Chapter 69, we say something transforms like the adjoint if its transformation under the $SU(N)$ group action is $$W\rightarrow UWU^\dagger$$ The Field strength and the covariant ...
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Wigner $ D $ matrix equivalent for cyclic symmetry

$\newcommand{\ket}[1]{\left|#1\right\rangle}$The action of $ g \in SU(2) $ on a spin $ j $ system (with a Hilbert space of size $ 2j+1 $) is by the Wigner $ D $ matrix $ D^j(g) $. There are formulas ...
Ian Gershon Teixeira's user avatar
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What is the connection between Lorentz transforms on spinors and vectors?

When deriving the (1/2,0) and (0,1/2) representations of the Lorentz group one usually starts by describing how points in Minkowski space transform while preserving the speed of light (or the metric). ...
Alexander Haas's user avatar
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Why every projective irreducible representation of $(ISO(2,1)^{\uparrow}=SO(2,1)^{\uparrow}\ltimes \mathbb{R}^3)$ is equivalent?

Why every projective irreducible representation of the connected poincare group $(ISO(2,1)^{\uparrow}=SO(2,1)^{\uparrow}\ltimes \mathbb{R}^3)$ is equivalent to a projective irreducible representation ...
Mahtab's user avatar
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Why a scalar particle with momentum orbit $\mathcal{O}_p$ is irreducible?

Let $G$ be a Lie group and $A$ a finite dimensional vector space. A scalar particle with momentum orbit $\mathcal{O}_p$ is a represenation $T: G\ltimes A\to GL(L^2 (\mathcal{O}_p,\mu,\mathbb{C}))$ ...
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Lie algebra adjoint representation

I'm currently reading Maggiore's book on QFT. He's explains Lie groups and Lie algebra right now. There is a point he makes which I cannot really follow. Its about how the adjoint represantation ...
Gogoman96 X's user avatar
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What physical properties give rise to abelian anyons as opposed to non-abelian anyons?

As far as I understand, abelian anyons are those which have a particle exchange operator which is a one-dimensional representation of the braid group. Non-abelian anyons, then, have a higher-...
Lucas's user avatar
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Concept about the table of Clebsch–Gordan coefficients

Reference:https://en.wikipedia.org/wiki/Table_of_Clebsch%E2%80%93Gordan_coefficients The above picture is when $j_1 = 1$ and $j_2 = 1$, why do there exist three values for $j$? I know when $j = 2$,it ...
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Gauging a finite non-abelian global symmetry in 2D

Consider a 2D system with a non-anomalous finite non-abelian global symmetry $G$, for example $$G = S_3=\{e,a,a^2,b,ab,a^2b\}$$ with $a^3=b^2=1$. One expects the local operators charged under the ...
JQ Skywalker's user avatar
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Proof Majorana spinors exists if maximal commutant of Clifford algebra is $\mathbb{R}$

I am searching for a proof of the claim made in this post. It states that Majorana spinors (I refer to both complex pinor and spinor representations which are restricted to the real Spin group and ...
anonymous250's user avatar
5 votes
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58 views

Group theoretical approach to conservation laws in classical mechanics

I'm doing some major procrastination instead of studying for my exam, but I wanted to share my thought just to confirm if I'm right. Suppose that the action, $S(\mathcal{L})$ forms the basis of a ...
Ilya Iakoub's user avatar
2 votes
1 answer
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Are projective representiations of a Lie group a representation of the semi-direct product of the group with $U(1)$ if the norm is preserved?

Let's say we have a function $f(x_{\mu},t)$ that transforms under the action of an $N$-parameter group $G(a_{\nu})$. Then a projective representation of $G(a_\nu)$ in the $f(x_\mu,t)$ basis would ...
Ilya Iakoub's user avatar
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Proving $so(3,1)\simeq sl(2,\mathbb{C})$ by redefining generators

First of all, I am a pedestrian in group theory. I have a general question and two particular ones. General question: I am trying to show that $so(3,1)\simeq sl(2,\mathbb{C})$ by redefining its ...
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Using Galilean covariance to find conditions on physical observables

Let's suppose that coordinates have to transform accoring to the Inhomogenous Galilean Group. Then $$ x' = x + a + v(t+b) $$ $$ t' = t + b $$ Let's use a funtion $\psi(x,t)$ of $x$ and $t$ as the ...
Ilya Iakoub's user avatar
2 votes
1 answer
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Generation of a non-accidental degenerate eigenspace carrying out the symmetry operations on one eigenstate

Let us consider a system described by an Hamiltonian $H$ over an Hilbert space $M$, and the finite group $G$ of symmetry operations w.r.t $H$, i.e. \begin{equation} R_g : M \rightarrow M \qquad g\in G ...
Gippo's user avatar
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2 answers
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Can the composition law of a group be defined only when considering a representation or realisation of the Group?

When we talk about, lets say, the Lorentz group, we define the action of the Lorentz transformation $\varLambda$ on \begin{alignat}{1} x^{\mu} & \in\mathbb{R}^{1,3},\\ x^{\mu} & \rightarrow x'^...
HypnoticZebra's user avatar
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1 answer
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Confusion about tensors in $SU(3)$

I have some confusion regarding the notion of tensors in $SU(3)$ (or some other matrix Lie group, but let's keep the discussion to $SU(3)$). For concreteness, I will refer to Peskin and Schroeder's ...
Quercus Robur's user avatar
8 votes
1 answer
351 views

Can we make a Bloch sphere for Weyl spinors?

If spinors are the "square root" of 3-vectors [$\mathrm{SU}(2)$ double cover of $\mathrm{SO}(3)$], Weyl spinors can be thought of as the "square root" of 4-vectors [$\mathrm{SL}(2,\...
Mauricio's user avatar
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2 votes
2 answers
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The dimension of the Clifford algebra for the Dirac equation

The Dirac algebra contains sixteen linearly independent elements. In general, a Clifford algebra $\mathcal{C}\!\ell(V,Q)$ generated from a vector space $V$ equipped with a quadratic form $Q$ has ...
Nada Band's user avatar
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1 answer
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Is there a systematic way to construct the parity and charge conjugation operator for any Poincaré irreducible representation?

I am currently taking an undergraduate introductory QFT course. However, the proceeding will be about classical field theory, the results of which I assume will carry over mutatis mutandis into ...
Silly Goose's user avatar
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17 votes
4 answers
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How to rotate an electron mathematically?

Im a mathematics student who just learned about the fact that if you rotate an electron by $2 \pi$ its spin state changes but if you turn it by $4 \pi$ it stays the same. I understand all the ...
Henry T.'s user avatar
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How can I construct a projective representation when the group is not simply connected?

S. Weinberg, in his book "The quantum theory of fields", states this theorem (page 83): The phase of any projective representation $U(T)$ of a given group can be chosen so that $\phi =0$ if ...
Mahtab's user avatar
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1 answer
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Can we prove in general that gauge fields associated with broken generators form representations of the unbroken group?

The title is a bit ambiguous. More specifically, I'm asking: Are all coupling between massive gauge fields (associated with broken generators) and massless gauge fields of the unbroken group are in ...
Bababeluma's user avatar
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2 answers
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Conserved current transforming under adjoint

If we have a Lagrangian with a global internal symmetry $G$. Why do the conserved currents transform under the adjoint representation of $G$? Is it a general statement (if this is the case, how can we ...
Nathex's user avatar
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1 vote
1 answer
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How can I calculate action of $\mathfrak{su}(3)$ or other simple algebra ladder operators on "states" from the algebra commutators?

I wanted a way to "derive" Gell-Mann matrices for $\mathfrak{su}(3)$ and generalise this to other semi-simple algebras $\mathfrak{g}$. The way I wanted to approach this is start from the ...
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Multiplying two $SO(3)$ representations

In Group Theory by Zee in Chapter IV.2, he discusses the multiplication of two $SO(3)$ representations on p. 207. Suppose you have a symmetric traceless tensor $S^{ij}$ which furnishes a $5$-...
mathemania's user avatar
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1 answer
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What is the difference between a vector and a representation of a vector in QM?

What does the phrase The wave function is a representation of the abstract quantum state. Or more generally, $A$ is a representation of a vector $\vec V $ mean? What is the difference between a ...
GedankenExperimentalist's user avatar
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Difference between Mutlipole moments calculated with normal integration and with Wigner-$D$ matrices

Im learning about Wigner-$D$ matrices and the applications to spherical harmonics. Now I wanted to test my knowledge, but i failed miserably :( (worked on this the whole day). So here is my problem: I ...
Stefan283's user avatar
2 votes
3 answers
156 views

Why do QM books point out that $S^2$ commutes with $S_x$, $S_y$, and $S_z$?

The spin angular momentum magnitude squared operator: $$S^2=S_x^2+S_y^2+S_z^2=\frac{3\hbar^2}{4} \begin{pmatrix}1&0\\0&1\end{pmatrix}$$ Obviously $S^2$ commutes with everything, so why do QM ...
hbar's user avatar
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On the $0$ representation of massive multiplets

So my doubt involves the massive multiplet of $\mathcal{N}=2$. I am not being able to deduce what particles does the states represents. For example, The $\mathcal{N}=2$ massive short hypermultiplet $s=...
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1 answer
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Simplification of $\sum_{a=1}^3\sigma^a_{\alpha\beta} S^a_{\gamma\delta}$ where $\sigma^a,S^a$ are representations of $SU(2)$?

As the question asks, I am dealing with a problem where I'd like to simplify $$\sum_{a=1}^3 \sigma^a_{\alpha\beta} S^a_{mn}$$ where $\sigma^a$ are the spin-1/2 Paulis and $S^a$ are some higher-spin ...
N Paul's user avatar
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3 votes
1 answer
106 views

Classifying projective representations

Blagoje Oblak in their thesis "BMS particles in three dimensions” says that "Given a group $G,$ suppose we wish to find all its projective unitary representations. The above considerations [...
Mahtab's user avatar
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2 answers
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Confusion with Weinberg's QFT book, Volume I, Equation 2.5.3 (one-particle states as irreps of Poincare group)

I am reading Sec. 2.5 of Weinberg's Quantum Theory of Fields, Volume I. There he talks about the classification of relativistic one-particle states according to their transformation under the Poincare ...
Solidification's user avatar
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Fermion boundary conditions vs projective representations

I have a basic confusion about the two different boundary conditions for a fermion around a compact direction, periodic versus anti-periodic. We learn in a QM class that a fermion wavefunction picks ...
mkn's user avatar
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0 votes
2 answers
61 views

Can the generators of a Lie group furnish its adjoint representation?

For generators of the Lie group under an arbitrary representation: $[T^a,T^b]=if^{abc}T^c$ $[T_A^c]^{ab}=-if^{cab}$ is the generator of the adjoint representation. Is $\ \ e^{i\theta^d T^d}T^ae^{-i\...
Bababeluma's user avatar
1 vote
1 answer
45 views

What is a non-linear space of connections

In the book "Loops Knots Gauge Theory and Quantum Gravity" when trying to define a loop representation, one needs to integrate over the space of connections (modulo Gauge transformations). ...
Confuse-ray30's user avatar

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