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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Highest weight theorem of $SU(2)$

In Woit's: Quantum Theory, groups and representations, p109, 110, is proved the "Highest Weight Theorem": Finite dimensional irreducible representations of $SU(2)$ have weights of the form $...
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What is the status of orbifolded 10D $E_8$ theory?

A recent trio of papers presents an $E_8$ GUT in 10 dimensions, where compactification on a $\mathbb T^6/(\mathbb Z_3\times\mathbb Z_3)$ orbifold avoids the usual problem of no complex representations ...
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Can we write the mass $M$, a Casimir invariant of the Galilean group, as a function of its generators?

According to Wikipedia, the mass $M$ is one of the Casimir invariants of the Galilean group. Casimir invariants of a group are made out of the generators, and they commute with all the generators of ...
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Why is the gauge group of pseudo-fermion mapping referred to as $\mathrm{SU}(2)$ and not $\mathrm U(2)$?

The representation of spin $\frac{1}{2}$ operators $\hat{S}^{a}$ by pseudo-fermions (also called Abrikosov fermions) is defined by the mapping $$ \hat{S}^{a} = \frac{1}{2} \text{Tr}\big[ \hat{\psi}^{\...
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Can the gauge boson respective of a spontaneously broken generator remain massless in the context of the Higgs Mechanism?

I'm studying a 3-3-1 model which is an extension of the standard model. The breaking $$SU(3)\times U(1) \to SU(2)\times U(1)'$$ occurs in a single step and through a single scalar VEV. The problem is ...
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Why are there only two 496-dim. gauge groups $E_8\times E_8$ and $SO(32)$ allowed in string theory? Why not $E_8\times U(1)^{248}$ or $U(1)^{496}$?

While constructing anomaly-free string theories with $\mathcal N=1$ supersymmetry (16 supercharges constituting a Majorana-Weyl spinor), we learn that the gauge group must be 496-dimensional in order ...
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How to calculate the Dynkin index and the Casimir operator for arbitrary representation knowing the fundamental ones

Suppose I want to find the Dynkin index of the sextet representation $S_2(6)_{SU(3)}$ of $SU(3)$ given that the fundamental is normalized in such a way that $S_2(3)_{SU(3)}=1/2$. What is the best way ...
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Topological Band Representations

I have a question regarding the little group representations on the topologicalquantumchemistry webpage. There is a link for the compound RhSi below. The site lists representations like 6+7 (4), which ...
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Proof of form of 4D rotation matrices

I am considering rotations in 4D space. We use $x, y, z, w$ as coordinates in a Cartesian basis. I have found sources that give a parameterization of the rotation matrices as \begin{align} &R_{...
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How to generate a complete set of doped (substituted) structures?

I'm trying to analyse doping in a certain material in low concentrations. I'm approaching the problem by taking crystallographic unit cell of that material, then expanding that to a 2x2x2 supercell ...
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Definition of pull-back in gauge transformation [closed]

In an online lecture given by a famous physicist Youshi Wu, he says: Consider a map $g:\quad M\rightarrow G:x\rightarrow g(x)\in G$ ($G$ is a Lie-Group manifold, $M$ can be physical space or ...
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Can I say two physical systems are isomorphic?

I wonder if it makes sense to say two physical systems are isomorphic to each other. Say if I have a system of electron spins in a magnetic field, and another system with an ammonia molecule. Since ...
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Application of Lie groups in particle physics [duplicate]

Is there a simple answer to the question, how lie groups are used in particle physics?
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Basis functions in group theory and wave functions

I just started to study group theory by reading the book Group theory: Applications to the physics of condensed matter by M. S. Dresselhaus. In chapter 4 it was mentioned: Suppose that we have a ...
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Writing generic Lorentz transformation as product of boosts and rotations

Consider the scattering of two particles with momenta $p_1$ and $p_2$ ($p_1^2, p_2^2\ge 0$). One can boost in the rest frame of the collision, such that $q_+=p_1+p_2=(\sqrt{q_+^2},0,0,0)$. Then, one ...
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Why “identity” is selected as “generator” in the Int. Tables of Crystallography?

In chapter 8.3.5 of the Int. Tables for Cryst. Vol A it is stated that “in group theory, a set of generators of a group is a set of group elements such that each group element may be obtained as an ...
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Using $SU(3)$ Clebsch-Gordan coefficients to find the singlet state

I tried to use this online CG calculator to find the singlet state in the $SU(3)$ relation: $$ \textbf{3}~\otimes~\bar{\textbf{3}} =\textbf{1}~\oplus~\textbf{8} $$ Taking the basis of $\bf 3$ and $\...
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Higher order expansion term of Baker–Campbell–Hausdorff (BCH) formula [migrated]

I want to calculate unto 20th order, the expansion of Baker–Campbell–Hausdorff formula. In Wikipedia up to 4th order expansion is available. Is it possible to get the expansion term up to 20th order ...
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Linear combination of group generators

In Matthew Robinson's book Symmetry and the Standard Model he explains that we have generators for rotations $J$ and for boosts $K$. To analyse the group structure, we will look at $N^\pm = J \pm i K$ ...
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How to choose Clebsch-Gordan coefficients?

I just started learning Clebsch-Gordan coefficients recently. I want to use the expression on Wikipedia (relation to Wigner $3j$ symbols): $$ \langle j_{1},m_{1},j_{2},m_{2}|J,M\rangle=(-1)^{-j_{1}+j_{...
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Relevance of Unitary Inequivalence of Representations

In my QFT book the author (Schwartz) mentioned, that we found two unitarily inequivalent representations of the Lorentz group. However, he never really introduced the idea of unitarily equivalent ...
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Irreducible Representations of a Lie Algebra and the Exponential Map [migrated]

Having found an irreducible representation of a Lie Algebra, we get the representation of the Lie group using the exponential map. Is the representation of the group then irreducible too? If so, how ...
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What is the general definition of symmetry in quantum mechanics

Consider a quantum system with Hilbert space $\mathcal{H}$ and Hamiltonian $H$. Let $G$ be a Lie group and $U$ a unitary representation of $G$ on $H$. What are the most general conditions that $H$, $G$...
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Traceless symmetric tensor's transformation under a Lie group

Usually, one decomposes a tensor product whose elements are transformed under a Lie group into its trace part, traceless symmetric part and antisymmetric part to obtain an irreducible representation ...
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Can anyone help me derive and/or define the infinitesimal generators of a Lie group? [migrated]

I'm doing a project involving Lie groups in Physics, and a part of the project involves generators. I initially used Robert Gilmore's "Lie Groups, Lie Algebras, and some of their Applications&...
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Antiunitary operators and compatibility with group structure (Wigner's theorem)

From Wigner's theorem, we get that a physical symmetry can be described either by a unitary or antiunitary operator, eventually with a phase factor, as in here. However we have to respect the group ...
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Group theory and electronic energy bands in solids books

I am trying to study about group theory and energy bands to try and learn and understand this scary graph: So far I'm having serious difficulties understanding the subject of Symmetrization of ...
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Fixing non-calculus proof that Lorentz transformations are linear

Define the Lorentz group to be $$O(1,3)=\{\Lambda:\mathbb{R}^4\rightarrow\mathbb{R}^4|\eta(\Lambda u,\Lambda v)=\eta(u,v)\},$$ where $\eta$ is the Minkowski inner product. One could try to mimic the ...
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Derivation of rotation matrices in A. Zee's Einstein gravity in a nutshell

In page-40 of A. Zee's Einstein Gravity book, rotations are generated by considering the powers of the infinitesimal rotation given as: $$ R( \theta) \approx I+A$$ Now, he to establish the form of $A$,...
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Lorentz scalar of chiral fermions

In lectures on Gauge Theory by David Tong (sec. 5.6.1), the author suggests a Lorentz scalar $\epsilon^{\alpha\beta} \psi_{+\alpha}\psi_{+\beta}$ where $\psi_+$ is a right-handed fermion and $\alpha,\...
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Poincaré bundle, a way of understanding it

So I will call the Poincaré bundle $(FM,\pi, M)$ the principal fiber bundle that has the Poincaré group as a structure group, the space of linear frames as total space and $M$ as the Riemann-Cartan ...
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How different solutions for gauge transformations $G_U$ of a PSG are in one-to-one correspondence with the elements in IGG?

Between eqs. (47) and (48) of Wen's article "Quantum Orders and Symmetric Spin Liquids" (https://arxiv.org/abs/cond-mat/0107071) is the statement "Thus for each symmetry transformation $...
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Derivative of rotation matrix in a form skew-symmetric matrix

I am working on an application of CV, in which a way to calculate the derivative of rotation matrix is involved. $$R$$ is the rotation matrix and $$R \in SO(3)$$ Also, $R$ is changing with $t$ giving $...
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Representation of $SO^{+}(3,1)$ for scalar fields

As far as i know, the generators of the representation of the group of the orthochronous Lorentz transformations $SO^{+}(3,1)$ can bewritten in the following form: $$J^{\mu \nu} = i(x^{\mu}\partial^{\...
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The Cartan sub algebra and Killing form of the Poincaré algebra

Doing some studies on Group theory, I worked Frederic Schuller's lectures on youtube where he classifies all semisimple Lie algebras by the Dynkin's diagrams; I should say it was interesting. Trying ...
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Lorentz boost transformations form a group?

In the QFT book of Ryder, he states that Lorentz boost transformations do NOT form a group. This is due to the boost generators $\textbf{K}$, i.e. they do not form a closed algebra under commutation. ...
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Are Lie algebras of groups unique?

Take for example $GL(2,\mathbb R)$, the group of $2\times2$ invertible matrices with real entries. By considering small variations from the identity, it is clear that one needs four parameters to ...
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Infinite dimensional gauge theories [duplicate]

Gauge theories depend on finite-dimensional symmetry groups like $SU(2)$. Is it possible to construct sensible gauge theories (or at least something similar in spirit) in QFT based on infinite ...
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What is the motivation for the $\mathrm{Spin}(n)$ group to be the double cover of $SO(n)$? [duplicate]

The $\mathrm{Spin}(n)$ group is defined to be the double cover of $\mathrm{SO}(n)$. In the case of $n > 2$, this agrees with the universal cover. However, for $n=2$, the physically relevant group ...
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Meaning of the symbol: top clockwise arrow $\curvearrowright$ (html code ↷)

In Tachikawa's book on ${\cal N}=2$ Supersymmetric dynamics for pedestrians (pg 72 https://arxiv.org/abs/1312.2684), he uses a symbol ↷. $F$ is the flavor symmetry group and $G$ is the gauge group. $F,...
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What is the inverse map of $R_{jk}(U)=\frac{1}{2}{\rm Tr}(\sigma_j U \sigma_kU^\dagger)$?

Given a $2\times 2$ unitary, unimodular matrix $U\in {\rm SU}(2)$, the (elements of the) corresponding $3\times 3$ rotation matrix $R\in {\rm SO}(3)$ can be obtained from the map $$R_{jk}(U)=\frac{1}{...
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The $3×3$ representation of weak $SU(2)$

I'm reading chapter 11.2 of the Cheng and Li textbook 'Gauge theory of elementary particle physics'. It says that $T_+$, $T_-$ and $Q$ do not form a closed algebra. In order to fix this problem the ...
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How is $j=1/2$ representation, $U(R(\theta,\hat{\bf n}))=e^{i{\sigma}\cdot{\hat {\bf n}}\theta/2}$, is a projective representation of ${\rm SO}(3)$?

A projective unitary representation of ${\rm SO(3)}$ satisfies $$U(R_1)U(R_2)=e^{i\phi(R_1,R_2)}U(R_1R_2)\tag{1}$$ where $R_1,R_2\in {\rm SO(3)}$. How to show that the $j=1/2$ representation, $U(R(\...
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Book recommendation on group theory [duplicate]

Currently I am learning Quantum field theory, however I notice the lack of group theory knowledge especially Lie Algebra is quite fatal to my understanding. I hope to find some materials that can self ...
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Is Galilean Conformal Algebra (GCA) isomorphic to any other algebra in $d$ dimension?

I was recently studying stuffs related to Conformal Field theory and its Galilean version. It's known that CFT algebra in $d$ dimension is isomorphic to $SO(d+1,1)$ algebra. We also know that Galilean ...
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Respresentation Dirac group and Lie algebra

I am reading Peskin and Schroeder's book on QFT and have some difficulties with representation groups. Let's start with the Lorentz group since it is easier. let $\Lambda$ be a Lorentz transformation, ...
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1answer
32 views

How to compute the tensor product $[0,1] \times [2,k]$ in $SP(4)$?

In this paper the authors give in eq.(A.4) the tensor product $[0,a] \otimes [0,b]$, with $[a,b]$ the Dynkin labels for irreps of $SP(4)$. How can one compute the tensor product $[0,1] \otimes [2,k]$, ...
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What is the beta function of $\mathrm{SO}(N)$ Yang-Mills?

What is the beta function of $\mathrm{SO}(N)$-Yang-Mills? I know that $\mathrm{SO}(4)\cong\mathrm{SU}(2)\times \mathrm{SU}(2)$ and that the beta function of $\mathrm{SU}(2)$ is well known.
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Little Group of massive particles in moving frame

My understanding of the little group till now was, that we take some standard-momentum and define the little Group as the subgroup of the Poincaré group that leaves this standard-momentum invariant. E....
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Feriz rearrangement

For a tetraquark system $QQ\bar{Q}\bar{Q}$, with diquark-antiquark configuration, the color configuration can be $|6_{QQ}\otimes \bar{6}_{\bar{Q}\bar{Q}}\rangle_{1}$ or $|\bar{3}_{QQ}\otimes 3_{\bar{Q}...

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