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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127 views

Why Lorentz algebra is not represented by the basis of antisymmetric $4\times 4$ tensors? Confusion building Lorentz Lie algebra

I am very confused when building the Lie algebra of the Lorentz Group. In every books, they expand $\Lambda^{\mu}_{\nu} = \delta^{\mu}_{\nu} + \omega^{\mu}_{\nu}$ at the origin and you end up with the ...
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Space of Operators on a vector space [migrated]

I am taking a course on Linear Algebra and Group Theory. So, we have a definition of an linear operator on a vector space, which is simply a map $O:\mathbb{V}\rightarrow\mathbb{V}$. with properties, 1....
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Anthony Zee's Proof of Schur's Lemma

In Anthony Zee's proof of Schur's lemma (in his book Group Theory in a Nutshell for Physicists, page 102), he used the following fact (summarized by myself) without proof: Proposition: Let $G$ be a ...
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1answer
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How do central charges affect $R$-symmetry group in extended SUSY?

When examining a ${\cal N}=1$ SUSY one finds that the corresponding $R$-symmetry group is simply $U(1)$. On the other hand, when considering extended SUSY (i.e. ${\cal N}>1$) the largest possible ...
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Is there a room for "another SUSY" to reduce extra dimensions?

I've heard that supersymmetry already dropped dimensions count in String theories from 26 to 10 (11 - after Witten). So, is there any space left in the rest of the math to introduce some "another ...
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A confusion about the notation in Ernst S. Aber Quantum Mechanics

I recently read the topic in chap 4 and chap 5 of Ernest S. Abers' book, Quantum Mechanics. In the section 4.2.5, he wrote: From Section 3.3.3 you know how the $D^{(j)}(J_i)$ acts on the $2j+1$ states ...
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Representations of $O(2)$ and related problems

I'm currently studying Group Theory in order make a further application to physics and understand the math of some physical theories. I know that $SO(2)$ literally is a special case for $O(2)$ and ...
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47 views

In an $SO(10)$ GUT, how many gauge fields acquire masses that are above the electro-weak scale?

The $SO(10)$ group is spontaneously broken down into the Standard Model gauge group at around $10^{16}$ GeV and the electroweak scale is ~246 GeV. I think there should be 45 gauge fields predicted by ...
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1answer
98 views

How does the wavefunction inherit the symmetries from the Hamiltonian?

The question is best explained with an example, Consider a particle in a 2D Square Box. The Hamiltonian has symmetry group $ C_{V4}$. Consider the ground state eigenfunction, $$\psi_0 = sin(x)sin(y)$$ ...
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1answer
67 views

How projective representations can lead to 't Hooft anomalies in quantum mechanics?

In Shao's talk https://youtu.be/2vTvHYYl1Qk?t=1554, he argues that in quantum mechanics "if a symmetry acts projectively on states, then we have a t' Hooft anomaly". But I'm having trouble ...
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1answer
128 views

Why must $U(1)$ matrices commute with $SU(2) \times SU(3)$ matrices in embedding within $SU(5)$?

I'm a physicist taking a groups course. I can believe that the direct sum of the fundmental representations for the $SU(2)$ and $SU(3)$ matrices will work as an embedding of the $SU(2) \times SU(3)$ ...
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2answers
73 views

Why are most QM Operator defined as Identity minus Generator? [duplicate]

I am currently rehearsing my lectures in quantum mechanics for the exam. I recognized that there is a pattern for different types of operators such as: Rotation operator, Time evolution operator and ...
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What related massive/massless degrees of freedom and the generators of a field?

A lecturer has stated that it is possible to determine the number of massive and massless degrees of freedom from the known generators of a $SO(3)$ field and a given v.e.v. In the given example, the ...
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Why do Higgs phase particles transform nontrivially under the gauge group?

In the electroweak theory, $SU(2)\times U(1)$ is Higgsed down to one of its $U(1)$ subgroups. The resulting theory consists of particles which have $U(1)$ charges. My understanding of gauge symmetry ...
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1answer
63 views

Does the generator of the group need to be physical observables?

In the Class, We have been told that the compact groups like $SO(3)$ have hermitian generators. Now in $SO(3)$, these generators turn out to be angular momentum components apart from some dimensions. ...
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Does the representation of the group also follow the properties of the group? [migrated]

Consider the finite group $\mathbb{Z}_2$ with two elements $\{e,g_1\}$ with $g_1^2=e$. We have been told in the class that there is a trivial representation of this group in which we have $$e\...
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1answer
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$SU(2)$ in two complex dimensions

I am just a beginner of group theory. I saw an $SU(2)$ example (example 4.16) in the book by Nadir Jeevanjee, An introduction to Tensors and group theory for Physicists. For $SU(2)$ elements, they ...
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1answer
102 views

Possible charge for Abelian and non-Abelian theory

I am reading Tong's lecture note gauge theory. On page 6 in chapter 1 he writes Instead, the key distinction is the choice of Abelian gauge group. A $U(1)$ gauge group has only integer electric ...
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67 views

Rotational generators in an arbitrary direction

In the Ernest Abers Quantum mechanic's book (p.107-p.108) he proved: $D(\vec{J}\cdot\hat{n})=\sum_{i}n_iD(J_i)=\hat{n}\cdot D(\vec{J})$ by using: $$ \hat{R}=e^{-i\theta\vec{J}\cdot\hat{n}}=\exp{(-i\...
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2answers
63 views

Does the number of broken generators in SSB depend on the choice of VEV?

I take the Lagrangian, $$\mathcal{L}=\frac{1}{2}\partial_\mu \phi^T\,\partial^\mu\phi\,-\, \frac{1}{2}\mu^2\phi^T\phi-\frac{\lambda}{4}(\phi^T\phi)^2~,$$ where $\phi=(\phi_1,\,\phi_2,\,\phi_3)$ (real ...
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1answer
67 views

Relationship between multiplicity in the $k$-fold product of fundamentals and irrep dimension at large $N$

Equation 3.5 of this paper by Gross and Klebanov makes the following interesting claim. Take a group $U(N)$, with $N$ large, and consider the reducible representation $\mathcal{H}_{fund}^{\otimes k}$ ...
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1answer
162 views

Trace identity for $SU(N)$ matrix integral

I would like to know if there's a nice way to compute the following: $$ \int_{SU(N)} \underbrace{ dU}_{\text{Haar Measure}} \mathrm{tr} \left(U^n \right)~?$$ The following is necessary: $U \in SU(N)$ $...
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Question on connections in general relativity and particle physics

$1$ Introduction It seems that when you learn General Relativity all the technology of bundles are irrelevant (at least in elementary discussions as $[1]$, $[2]$, $[3]$ and others). But, even the ...
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1answer
50 views

Why does the time component of a pseudovector reverse under parity?

Under parity, a four-vector $V^{\mu}=(V^0,\boldsymbol{V})$ transforms as $$(V^0,\boldsymbol{V})\rightarrow(V^0,-\boldsymbol{V})$$ which makes sense as parity only reverses the spatial components. ...
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Doubt on the gauge group of Gravitation

(I wrote the introduction section for the sake of completeness, notation and study. The question per se, is written in the section "My Question") Introduction On the one hand of nature, we ...
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1answer
56 views

Is there superconformal group? (Exponentiating the superconformal algebra)

I (only superficially) know that not every Lie algebra can be exponentiated to give a Lie group. I also have only heard about the superconformal algebras, and not the superconformal groups. This is in ...
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1answer
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(Georgi chapter 14) Why does the $n=2$ states transform under the 5+1 of the angular momentum?

In Georgi's Lie algebra in particle physics, chapter 14, the 3D harmonic oscillator is studied. The systems exhibits $SU(3)$ symmetry in the energy levels, we can construct the 8 generators of the $SU(...
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2answers
113 views

Why is working infinitesimally allowed here?

In page 42 of Peskin and Schroeder, for $$\left( \mathcal{J}^{\mu\nu} \right)_{\alpha\beta} = i \left( {\delta^\mu}_{\alpha} {\delta^\nu}_{\beta} - {\delta^\mu}_{\beta} {\delta^\nu}_{\alpha} \right)$$ ...
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1answer
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Proving that the adjoint representation of a simple Lie Algebra satisfying $Tr(T_aT_b)=\lambda \delta_{ab}$ is irreducible

I am following the book "Lie Algebras in Particle Physics" by Howard Georgi and on page 51 he claims the statement above and goes on to prove it. I am new to this so my doubt might be ...
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More rigorous explanation as to why $G/H = G'/H'$ for vacuum manifolds?

When studying topological defects, the parameter space (or vacuum manifold in QFT) is denoted as the coset space $G/H$, where $G$ corresponds to the symmetry group of the Lagrangian and $H$ the ...
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Homotopy group of $[SU(2) \times U(1)] / U(1)$ dependent on embedding of $U(1)$?

I have read, that the topology and thus the homotopy groups of $[SU(2) \times U(1)] / U(1)$ depend upon the embedding of $U(1)$ into $SU(2) \times U(1)$. For the Electroweak theory inside the standard ...
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1answer
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Deduction of weak boson hypercharge in Georgi–Glashow $SU(5)$ grand unified theory (GUT)

In the Georgi–Glashow model, the extra 12 bosons $X_\mu^a$, $Y_\mu^a$ that appear have weak hypercharge $y = -\frac{5}{6}$. I want to know how this is deduced from the weak hypercharge generator $$Y = ...
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1answer
214 views

Symmetry Properties of Wigner's Matrices

I have an expression of the form $$S=\sum_{m,n=-j}^{j}(-1)^{m-n}D^{j}_{mn}(g)D^{j}_{mn}(g)$$ This is the end result of a long calculation, from which I am pretty confident that it is correct. For a ...
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44 views

Is there a non-trivial unitary representation of the general linear group?

Is there any interest in a quantum field theory using a unitary representation of the general linear group?
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Gauge Groups in Nature [duplicate]

It is well known that the relevant gauge groups appearing in particle physics are $U(1)$, $SU(2)$, and $SU(3)$. In many respects these are among the simplest Lie groups, such as in the dimension of ...
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1answer
64 views

Confusion about relationship between $\mathfrak{so}^+(1,3)$ and $\mathfrak{su}(2) \oplus \mathfrak{su}(2)$

Following from this question and the links within, I have a couple of questions about the use of $\mathfrak{su}(2) \oplus \mathfrak{su}(2)$ for the classification of finite real restricted Lorentz ...
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1answer
63 views

Final steps in obtaining the $(m,n)$-labelled finite-dimensional irreps of the restricted Lorentz group

I've been trying to understand how we get the $(m,n)$-labelled irreps of $SO^+(1,3)$ by reading posts such as this, this, this, this and links within, as well as the Wikipedia article on the matter, ...
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1answer
134 views

Chiral spinor representations

I just want some suggestions because I cannot find anything relevant on the internet. I am working with $\mathfrak{so}(16)$ and $\mathfrak{so}(12)$. I know that their chiral spinor representations are ...
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1answer
67 views

Is representation theory in QM a real physical thing or just a mathematical tool? [closed]

I was studying group theory and representation theory in Quantum Mechanics and I really don't understand yet if it is just a mathematical tool of seen the operators as a representation of a group ...
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1answer
35 views

Casimir conformal generator of $SO(d+1,1)$

The purpose of this post is to ask the help of derivation of equation 2.8 of https://arxiv.org/abs/2106.10822 Let $P_i$ be a point on the conformal boundary of $AdS_{d+1}$ and $Z_i$ be a polarization ...
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2answers
108 views

Why do we demand $SU(2)$ and $SU(3)$ gauge invariance when we construct the standard model?

If one tries to verify the construction of the standard model, one has to find a Lagrangian that is invariant under $U(1)\times SU(2) \times SU(3)$. While it seems kind of logic that the Lagrangian ...
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0answers
45 views

Obtaining the unitary representation of the Lorentz group from infinitesimal transformations

My knowledge of Lie Groups and Lie Algebras is very limited, even more so when it comes to their representation theories. There is a difference that I can't quite understand, between the ...
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0answers
46 views

Orthogonality of basis functions of irreducible representations

Lets say I have two irreducible representations of a finite group $G$. Let the group have $l$ elements. The elements of the representation shall be linear operators $L$ that act on some function ...
2
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1answer
68 views

Is it correct to say that in QM Operators are a way of representing elements of a group acting on a state, as linear maps on a Hilbert space?

Sorry if my question is nonsense, but I was reading about representation theory in Quantum mechanics and I find it very interesting. As I understood, an example would be the charge operator, which is ...
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1answer
58 views

What are the $U(1)$ representations and their generators?

I previously asked a question about how it is possible for different fields in a gauge theory to have different $U(1)$ charges. I think the issue is that I do not actually know what the $U(1)$ irreps ...
3
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1answer
80 views

Confusion about $U(1)$ representations and charge quantisation in the context of gauge theory

In a gauge theory, the fields transform under representations of the gauge group. When studying a special unitary group $SU(n)$, I've usually thought of the elements of a representation as being the ...
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1answer
94 views

Spin of Fundamental Particles

Is there any explanation/theorem which justifies that most fundamental particles have spin half or spin one? Apriori, studying representations of symmetry groups and their connection with spin of ...
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3answers
101 views

Confusion about tensor products of states in the fundamental representation of $\mathrm{SU}(2)$

This question is an extension of this question, I asked previously. Let us denote the unique irreducible unitary representations of $\mathrm{SU}(2)$ by $V_{j}$, where $\mathrm{dim}(V_{j})=2j+1$. It is ...
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1answer
52 views

Tensor product of representation of $\mathrm{SU}(2)$ Identity

I have troubles to understand an equation, which was stated in a lecture. Consider the spin-j representation $V_{j}$ of $\mathrm{SU}(2)$ with its standard basis $$\{\vert j,m\rangle\}_{-j\leq m\leq j}$...
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1answer
49 views

What is meant by part (e) of MTW Exercise 9.13 involving the tangent to a curve on the manifold $\mathcal{SO}\left(3\right)$?

I asked a specific question about this exercise a while back. I put this exercise aside until now. I had hoped I might encounter something that would shed light on the discussion, but that didn't ...

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