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Questions tagged [group-theory]

Group theory is a branch of abstract algebra. A group is a set of objects, together with a binary operation, that satisfies four axioms. The set must be closed under the operation and contain an identity object. Every object in the set must have an inverse, and the operation must be associative. Groups are used in physics to describe symmetry operations of physical systems.

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"Linear independency" of Lie Brackets [migrated]

I was watching this eigenchris video. At 21:49, he says: $$[g_i, g_j]=\Sigma_k {f_{ij}}^{k}g_k$$ for $\mathfrak{so}(3)$. Does this mean $[g_i, g_j]$ and $g_i, g_j$ can be linear independent? What ...
Cro's user avatar
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A reference for the fact that the second cohomology of the full Poincare algebra is zero

S. Weinberg in his book "The quantum theory of fields" vol. I says in page 86 that the full Poincare algebra is not semi-simple but its central charges can be eliminated (as he showed in the ...
Mahtab's user avatar
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Given a representation $(n, m)$ of the Lorentz group, is the little group representation just the tensor product $n \otimes m$?

I've been reading Weinberg's QFT Vol 1. and more specifically section 5.6. I would like to know if my understanding is correct or if I missed something. He starts with the full Lorentz group $\mathrm{...
Wihtedeka's user avatar
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How to find Casimir operator eigenvalues of $SU(N)$? [closed]

The $[f1, f2, f3…fn]$ in the image represent the irreducible representations of $SU[n]$. How to find the irreducible representations of $SU[n]$ that conform to the form $[f1, f2...fn]$. Can you give ...
snow snow's user avatar
3 votes
2 answers
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Does all symmetry breaking have corresponding unitary group?

In high energy physics. Symmetry breaking like electroweak's has corresponding $SU(2)\times U(1)$ unitary gauge group broken down to $U(1)$. Does it mean all kinds of symmetry breaking (even low ...
Jtl's user avatar
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What is the importance of $SU(2)$ being the double cover of $SO(3)$?

To my understanding, it is important that $SU(2)$ is (isomorphic to) the universal cover of $SO(3)$. This is important because $SU(2)$ is then simply-connected and has a Lie algebra isomorphic to $\...
Silly Goose's user avatar
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$Ad\circ\exp=\exp\circ ad$ and $e^{i(\theta/2)\hat{n}\cdot\sigma}\sigma e^{-i(\theta/2)\hat{n}\cdot\sigma}=e^{\theta\hat{n}\cdot J}\sigma$

This question is inspired by my recent question How to prove $e^{+i(\theta/2)(\hat{n}\cdot \sigma)}\sigma e^{-i(\theta/2)(\hat{n}\cdot \sigma)} = e^{\theta \hat{n}\cdot J}\sigma$? with answer https://...
Jagerber48's user avatar
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How to prove $e^{+i(\theta/2)(\hat{n}\cdot \sigma)}\sigma e^{-i(\theta/2)(\hat{n}\cdot \sigma)} = e^{\theta \hat{n}\cdot J}\sigma$?

Disclaimer: I'm sure this has been asked 100 times before, but I can't find the question asked or answered quite like this. If there are specific duplicates that could give me a simple satisfactory ...
Jagerber48's user avatar
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Help with Wigner-Eckert Theorem problem

Currently trying to solve the following problem: Consider an operator $O_x$ for $x = 1$ to $2$, transforming according to the spin $1/2$ representation as follows: $$ [J_a, O_x] = O_y[\sigma_a]_{yx} / ...
DingleGlop's user avatar
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Can you ever obtain a pure rotation from composing Lorentz transformations?

An exercise asks one to show that given $v, u$ speeds much smaller than $c$ and oriented orthagonally, the composition of the lorentz boosts $B(\mathbf{v})B(\mathbf{u})B(\mathbf{-v})B(\mathbf{-u})$ is ...
Y G's user avatar
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Rotation and translation of a function of a 3D vector

I want to change the frame by doing translation and rotation. $$f(\vec{v})=\sum_{n,l,m}R_{nl}(v)Y_{lm}(\hat{v})f_{nlm}^v.$$ Let, $\mathcal{R}$ be the rotation matrix and $\mathcal{T}$ be the ...
QED's user avatar
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From any element of $\mathrm{SO}(8)$, can we always find one corresponding $\mathrm{SU}(3)$ element?

I first recap the relation between $\mathrm{SU}(2)$ and $\mathrm{SO}(3)$ and then raise my question concerning $\mathrm{SU}(3)$ and $\mathrm{SO}(8)$. Given any traceless hermitian matrix $H$, we can ...
narip's user avatar
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Is intrinsic spin a quantum or/and a relativistic phenomenon?

Ok, this is my reasoning. I am probably making some wrong assumptions here, pls tell me where I am going wrong. Spin as a quantum phenomenon: Quantum phenomena disappear as the Planck constant goes to ...
Saeed'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 ...
Thanos Athanasopoulos's user avatar
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What does the $N$ in $SU(N)$ mean?

So I know this is a very basic question, but I can't really wrap my head around it. I was told $N$ is the number of dimensions in the rotations of the group theory that we are considering, so I ...
minime's user avatar
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Do GUT's really explain parity violation?

Every book on the Standard Model introduces early on the concept of left and right-handed quantum fields, defined as \begin{align} (\psi_L)_{\alpha} = \left(\frac{1-\gamma_5}{2}\right)_{\alpha \beta}\...
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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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Why are Lorentz transformations singular at $i^0$?

On pg. 16 of Strominger's lectures, it is said Lorentz transformations themselves are not smooth at spatial infinity, because the vector fields that generate them are singular at $i_0$. A boost ...
Sanjana's user avatar
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Is the factorization method of Hamiltonian related to the theory of Lie groups?

I was learning about algebraic methods to solve the H atom, when I came across the factorization method. It is mentioned in various textbooks, notes and papers, like the one from Infeld and Hull. I am ...
Po1ynomial'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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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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Why do we study Heisenberg Lie group or Heisenberg Lie algebra?

Consider $\mathbb{R}^2$ as an Abelian Lie algebra and let $c$ be a non-zero antisymmetric bilinear form on $\mathbb{R}^2$. We then define the three-dimensional Heisenberg Lie algebra $\mathbb{R}^3=\...
Mahtab's user avatar
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Is it enough just to check the effect of a boost in the z-direction to prove the covariance of an expression for the lorentz transformations?

Is it enough to check the effect of a boost in the z-direction to prove the Lorentz invariance of an expression for the $L^\uparrow _+$ transformations? My argument why this is true is that we can ...
Xhorxho'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 ...
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Rotation of Pauli Vectors with $SU(2)$ reproduces the $SO(3)$ matrix. but do all $SU(2)$ matrices reproduces $SO(3)$?

So we can write the $SU(2)$ matrices multiplication as this. $$\begin{bmatrix}\alpha&\beta\\-\beta^*&\alpha^*\end{bmatrix}\begin{bmatrix}z&x-iy\\x+iy&-z\end{bmatrix}\begin{bmatrix}\...
abx_pradB's user avatar
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What is the meaning of the tensor product ($\otimes$) of groups? Is is the same as the cartesian product ($\times$)? [duplicate]

In the high energy physics literature and in some discussions on group theory for physics, the tensor product $\otimes$ of groups is sometimes mentioned. For example, the gauge group of the standard ...
Aqualone's user avatar
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1 answer
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Abelian vs non-abelian discrete symmetries in neutrino physics

I was reading about the parametrization of the PMNS matrix and stumbled upon an article of Serguey Petcov$^1$ about discrete flavour symmetries. It endeavors to see if there is a pattern induced by a ...
AZ0409's user avatar
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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 ...
Maple's user avatar
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4 votes
1 answer
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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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0 answers
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The little group of the connected Lorentz group [duplicate]

For $n\geq 2$, the $n$-dimensional connected Lorentz group is defined as $$SO(n-1,1)^{\uparrow}=\{f\in M_n (\mathbb{R}):f^T\eta f=\eta, \det (f)=1,f^0_0>0\}$$ where $$\eta=\begin{bmatrix}-1 & ...
Mahtab's user avatar
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0 answers
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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 $\mathrm{SU}(2)$. ...
Eric Kubischta's user avatar
1 vote
2 answers
139 views

Why do representations of $SU(2)$ correspond to angular momentum eigenstates?

I have been learning about symmetry in one of my physics classes and specifically about $SU(2)$ and its irreducible representations. We can label a basis element of the vector space corresponding to a ...
Physics_Boss_India's user avatar
3 votes
1 answer
58 views

Measurable effects of the global structure of the SM

It is known that the Lie algebra of the SM is $$ \mathfrak{su}(3)\oplus \mathfrak{su}(2)\oplus \mathbb{R}\,, $$ so that the Lie group is $$ G_{\text{SM}} = \dfrac{SU(3)\times SU(2) \times U(1)}{\Gamma}...
Gabriel Ybarra Marcaida's user avatar
1 vote
2 answers
73 views

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
1 vote
0 answers
55 views

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
1 vote
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29 views

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 $T: G\ltimes A\to GL(L^2 (\mathcal{O}_p,\mu,\mathbb{C}))$ ...
Mahtab's user avatar
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2 votes
1 answer
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Helicity operator in spinor-helicity variables

How do I prove that the helicity operator is $$ H = \frac{1}{2} (\tilde{\lambda}_\dot{\alpha} \frac{\partial}{\partial \tilde{\lambda}_\dot{\alpha}} - \lambda_\alpha \frac{\partial}{\partial \lambda_\...
michael pasqui's user avatar
2 votes
0 answers
59 views

English translation of Weyl's article "Quantenmechanik und Gruppentheorie"

Is there an English translation of Weyl's 1927 article Quantenmechanik und Gruppentheorie. Note tht I do not mean the book of the same name.
1 vote
0 answers
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2-dimensional connected Lorentz group [closed]

Consider the connected Lorentz group $SO(1,1)^{\uparrow}$. I was wondering if someone could help me about showing that $SO(1,1)^{\uparrow}\cong \mathbb{R}\times \mathbb{Z}_2$. I just need a hint.
Mahtab's user avatar
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3 votes
0 answers
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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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25 views

How to show that $G_p=SO(D-1)$?

Let $G=SO(D-1,1)^{‎\uparrow‎}$ be the connected Lorentz group. Let $p$ be a timelike momentum with $p_0>0$. I want to show that $G_p=SO(D-1)$, the little group of $p=(M,0,\ldots,0)$ where $M>0$....
Mahtab's user avatar
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2 votes
0 answers
59 views

How do I relate objects in $SU(4)$ and $SO(6)$? [closed]

We know that $SU(4)$ is homomorphic $SO(6)$. I would like to understand how do we transform objects that we have, for example, in $SU(4)$, to objects in $SO(6)$. For example, in $SO(6)$ we have ...
LSS's user avatar
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5 votes
0 answers
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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
46 views

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
3 votes
0 answers
59 views

Global properties of the gauge group

In this very good P.E. answer, it is explained precisely what it means for a quantum system/theory to have a symmetry group $G$ (where $G$ is a Lie group): going back to first principles, it means ...
SolubleFish'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 ...
hyriusen's user avatar
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4 votes
0 answers
68 views

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
3 votes
1 answer
58 views

Why semi-simple and compact Gauge Group in YM Theory? [duplicate]

I'm studying the Yang-Mills theory, with the Action: $$ S=-\frac{1}{2}\int\mathrm{tr}_{\rho}(\mathcal{F}\wedge\star\mathcal{F}) $$ where $\mathcal{F}:=\mathrm{d} \mathcal{A}+\frac{1}{2}[\mathcal{A},\...
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