We’re rewarding the question askers & reputations are being recalculated! Read more.

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.

Filter by
Sorted by
Tagged with
2
votes
2answers
124 views

Role of the special linear Lie algebra in general relativity (GR)

The Lie derivative measures the difference between two paths in the timespace manifold, and hence the commutator bracket occurs naturally, as explained in the presentation What is a Tensor? Lesson 21: ...
1
vote
2answers
129 views

Isometry group on a coset manifold

In ''Einstein Gravity in a Nutshell'' Zee says ''On a coset manifold $G/H$, the isometry group is evidently just $G$'' when discussing the relation between the Killing vector fields and Lie ...
0
votes
1answer
98 views

Generators in Field Theory and Derivatives

Let's consider a representation of the multiplicative group $(0,\infty)$ on Minkowski space $\mathbb{R}^4$ by dilations. \begin{align} \rho:(0,\infty)&\rightarrow\text{GL}(\mathbb{R}^4)&\\ a ...
1
vote
2answers
94 views

Commutator generating transformations

Lately I am encountering the commutator of variations of the variables and I'm not quite sure about its physical meaning. Some examples. 1) "The composition of two supersymmetries generates a time ...
0
votes
2answers
35 views

Clarification on statement in “Unitary Symmetry and Elementary Particles” by Lichtenberg

He says that: The set of values of the parameter or parameters which characterize a group element can be considered to be points in some kind of space. The number of parameters characterizes the ...
0
votes
0answers
13 views

Calculating adjoint representation of Lie group/algebra [duplicate]

How do I calculate adjoint representation of Lie group and Lie algebra? I would be thankful if anyone can give good example or general formula on calculating adjoint of any Lie group
1
vote
0answers
59 views

Are particles always represented by 4-volume preserving transformations?

I've been reading about affine gauge gravity, which uses the affine group A(4,R) (for example here ). If I'm getting it right there seems to be an “affine higgs” mechanism that breaks symmetry down ...
25
votes
4answers
1k views

What precisely is a *classical* spin-1/2 particle?

I was recently having a Twitter conversation with a UC Riverside Prof. John Carlos Baez about Geometric Quantization, and he said (about his work) that "Right. For example, you can get the ...
0
votes
0answers
64 views

Decomposition of the complex conjugate of the fundamental representation of $SU(5)$ in $SU(3)\times SU(2)\times U(1)$

I know I can decompose the fundamental representation (denoted as $5$) of $SU(5)$ as: $$ (3,1)_{-2c/3} \oplus (1,2)_{c} $$ But how do I get the decomposition of the complex conjugate of this ...
0
votes
0answers
89 views

Why are systems joined via a tensor product? [duplicate]

This question comes from seeing that the triangle addition rule for quantum mechanics comes out of groups/representation theory; I thought this was odd as we haven't used any group ideas in QM up to ...
2
votes
2answers
272 views

$SU(2)$ Invariant Lagrangian

Consider two arbitrary scalar multiplets $\Phi$ and $\Psi$ invariant under $SU(2)\times U(1)$. When writing the potential for this model, in addition to usual terms like $\Phi^\dagger \Phi + (\Phi^\...
0
votes
0answers
62 views

What should I read to understand this question?

I understand the strong force as a Yang-Mills theory with $SU(3)$ color invariance. I understand that the quarks live in the fundamental representation of $SU(3)$ and that gluons live in the adjoint. ...
3
votes
1answer
89 views

Wigner proof of the non-existence of finite unitary representation of the Lorentz group

I am reading Wigner's paper ”On unitary representations of the inhomogenous Lorentz group” (Annals of Mathematics, Vol. 40, No.1, p. 149) found here: https://www.maths.ed.ac.uk/~jmf/Teaching/Projects/...
3
votes
0answers
49 views

Particle statistics in fractal dimensions? [closed]

We know that fermions and bosons are the only two (indistinguishable) particle statistics for $d\geq 3$, and that anyons are for $d=2.$ What if the space were a fractal? Like the Sierpinski gasket, ...
2
votes
1answer
73 views

Supercharge transformation rules

Consider ${\cal N}=2$ supersymmetry with $SU(2)$ global symmetry group. Then both supercharges $Q_{ai},\bar{Q}_{\dot{a}\dot{j}}$ transform by 2 dimensional representation of $SU(2)$. Denote $SU(2)_I$ ...
0
votes
0answers
60 views

How to write the Poincare transformation for an arbitary path in Minkowski space?

So lets Say for arguments sake we have some vector $V^{a}$ and we drag it along some path $\gamma_{1}$ in Minkowski space $R^{3,1}$. For a straight path (represented by a vector $\Delta\overrightarrow{...
1
vote
1answer
92 views

Poincaré and Galilei group - notation

On this slide it just says that $\mathcal{P}$ and $\mathcal{G}$ are the Poincoré and Galilei groups, but I do not understand what they are made of. What does $\mathbb{R}^{1,3}$ mean? Why does $\...
2
votes
2answers
148 views

Direct Product vs Tensor Product

I am confused in the notation on page 67 and page 70 a text (http://www-pnp.physics.ox.ac.uk/~tseng/teaching/b2/b2-lectures-2018.pdf), whether it's talking about a direct product or an outer product: ...
5
votes
2answers
153 views

Spin statistics from the fundamental group of $SO(D)$

I read the answer to this question and am very intrigued by its simple and elegant explanation of the emergence of anyon, boson & fermion statistics. @Trimok basically says: In a space-time ...
2
votes
1answer
115 views

Doubt in Weinberg's book on Quantum Field Theory

In page number 59 of his book on QFT, Weinberg mentions that for the operator $U$, defined for infinitesimal parameters $\omega$ and $\epsilon$ as: \begin{equation} U(1+\omega,\epsilon)=1+\dfrac{1}{2}...
-2
votes
1answer
267 views

Is spin 1 described by $SO(3)$ or $SU(2)$ [duplicate]

What spin is described by which rotation group? I always only find information about spin-1/2
0
votes
1answer
139 views

Why do the $\gamma$ matrices behave like vectors (tensors)?

In the study of Quantum Field Theory and Group Theory for the spinor representation of $SO$ groups, we know the following correspondence: $\chi C\psi$ scalar $\chi C\gamma^\mu\psi$ vector $\chi C\...
1
vote
1answer
80 views

$(1,1)$ representation of $SL(2,\mathbb{C})$

How do you prove that the $(1,1)$ representation of the $SL(2,\mathbb{C})$ group acts on symmetric, traceless tensors of rank 2?
2
votes
2answers
132 views

Unlike rotation, why a $3\times 3$ translation matrix cannot be written in 3D? or can it be?

The effect of rotation in 3d on a vector, $\vec{r}=x\hat{x}=y\hat{y}+z\hat{z}$ is given in the form a matrix product:$$\vec{r}\to O\vec{r}$$ where $O$ is a $3\times3$ proper orthogonal matrix. Can we ...
0
votes
1answer
46 views

Relation of the Lorentz group to $O(1,3)$

Let $\Lambda$ be an element of the Lorentz group. It satisfies the identity:$${\Lambda}^T\eta\Lambda=\eta$$ where $\eta$ is the Minkowskii metric. Hence by the usual definition of orthogonality, $\...
2
votes
1answer
83 views

Is there a simple way to explain a fundamental representation of $O(N)$?

Is there a simple way to explain fundamental representation in Physics? For example, a fundamental representation of $O(N)$?
3
votes
1answer
72 views

Scattering matrix symmetries and standard model

I am not able to get around the following question (if it make sense): Suppose I can derive the scattering matrix S for any particle scattering process. Suppose that the standard model is actually ...
5
votes
2answers
115 views

Operational definition of rotation of particle

The question in brief: what does it mean, operationally, to rotate an electron? Elaboration/background: I am trying to understand how representation theory applies to quantum mechanics. A stumbling ...
0
votes
1answer
96 views

Can $E_8 \times E_8$ contain the standard model?

I know $E_8$ by itself can't be gauge group because it has no complex representation and so would not be chiral. But assuming the existence of mirror matter which also would have $E_8$ gauge group ...
1
vote
0answers
39 views

$SU\left(N\right)$ Dynkin labels, how to compute

Let $V$ be somecomplex irreducible representation of $SU\left(N\right)$. I read that to compute the Dynkin labels of the weights, one can take the highest weight and then subtract from it the rows of ...
0
votes
2answers
116 views

Why does the pion live in a representation of isospin SU(2) and is the mediator of the strong force generated by color SU(3)?

Why does the pion live in a representation of isospin $\rm SU(2)$ and is the mediator of the strong force generated by color $\rm SU(3)$? I somehow find strange that this is the case. Given that $\rm ...
0
votes
1answer
42 views

Weights of $SU\left(5\right)$ representation

Consider the representation $\Lambda^2V$ of $su\left(5\right)$ where $V$ is the fundamental representation. How can I work out the Dynkin labels of its weights? Are these the correct Dynkin labels ...
1
vote
1answer
63 views

Branching of $SU\left(5\right)$

In the context of branching rules, what is a projection matrix for a subgroup. For instance, the projection matrix for the subgroup $SU\left(2\right)\times SU\left(3\right)$ of $SU\left(5\right)$ is ...
1
vote
1answer
82 views

What is the weight system for these SU(5) representations?

I need to work out the weight systems for the fundamental representation $\mathbf{5}$ and the conjugate representation $\overline{\mathbf{5}}$. I'm not clear what this means. The $\mathbf{5}$ ...
2
votes
1answer
188 views

Why do we use infinitesimal forms of operators?

In many undergraduate texts on quantum mechanics (I'm using Modern Quantum Mechanics 2nd Edition by Sakurai as reference here), the treatment of angular momentum goes something along the lines of: ...
4
votes
0answers
88 views

Why do we use the $\left(\tfrac{1}{2}, \tfrac{1}{2}\right)$ rep for spin $1$ particles and not $(0, 1)$? [duplicate]

The spin 1 $A^\mu$ field transforms under the $\left(\tfrac{1}{2}, \tfrac{1}{2}\right)$ representation of the Lorentz field. When restricted to the $SO(3)$ subgroup, it decomposes into the $0 \oplus 1$...
1
vote
1answer
89 views

Does dimension of irreducible representations of the double cover $SU(2)$ of the 3D rotation group define spin of particle?

In quantum field theory, does dimension of irreducible representations of the double cover $SU(2)$ of the 3D rotation group conclusively define spin? In other words, Is spin 1 particle only thing ...
1
vote
4answers
227 views

What causes $A^{\mu\nu}_{\pm}=F^{\mu\nu}\pm i \tilde{F}^{\mu\nu}$ to have three independent components rather than six?

Both the elctromagnetic field strength tensor $F^{\mu\nu}$ and its dual $\tilde{F}^{\mu\nu}$ defined as $\tilde{F}^{\mu\nu}=\frac{1}{2}\epsilon^{\mu\nu\lambda\rho}F_{\lambda\rho}$ are examples of ...
5
votes
2answers
212 views

How do the properties of a Lie group (represented as a manifold) manifest in the metric tensor of that manifold?

I know this is a math question; however, physicists are more likely to be familiar with what I'm asking (also, I'm directly trying to utilize it in the context of general relativity). I may have ...
1
vote
2answers
94 views

Clebsch-Gordan coefficient for 1x0

I'm trying to work out the combination of $|1\ 0 \rangle|0\ 0 \rangle$ (in this case they represent isospin, $|I\ I_3 \rangle$) using Clebsch-Gordan coefficients, but the table for $j_1\times j_2=1\...
0
votes
1answer
95 views

What is a good basis for this Hamiltonian with reduced symmetry?

What would be a good basis for a modified Hamiltonian that reads: $$ H_1 = \frac{1}{2}(L_+S_- + L_-S_+) + c_1 L_x + c_2 S_x,$$ from a symmtry point of view? The Hamiltonian itself is not difficult ...
0
votes
1answer
42 views

Why does the upper component of a $SU(2)$ doublet has $T^3=1/2$ and lower component $T^3=-1/2$ and not the opposite?

For a $SU(2)$ doublet, why does the upper component have $T^3=1/2$ and lower component $T^3=-1/2$? I know that this can be answered in the Standard Model by using $Q=T^3+Y/2$. But that is because we ...
0
votes
1answer
56 views

$C_4$ symmetry of Chern insulator

I was reading a paper that claimed the following Hamiltonian had $C_4$ rotational symmetry, $$ \hat{H} = \sum_{k} c^\dagger_k h_s(\bf{k}) \sigma_s c_k$$ where the Bloch hamiltonian is given by \...
3
votes
0answers
63 views

Can you do gauge theories over topological groups?

Quantum gauge theories involve (functional) integration over a Lie group. Is there any meaningful generalisation to (non-manifold) topological groups? Consider for example the Whitehead tower $$ \...
1
vote
1answer
297 views

Relation between Spin 1 representation and angular momentum and $SO(3)$

This is a naive question. It occurred to me while studying in detail the the Spin 1 angular momentum matrices. The generators of $SO(3)$ are $J_x= \begin{pmatrix} 0&0&0 \\ 0&0&-1 \\ ...
0
votes
1answer
106 views

Character expansion and Casimir

Is there a simple way to extract the quadratic Casimir of a representation from the character? I keep hearing things such as "Chern characters have an expansion that goes like" $$\chi(r) = dim(r) ...
2
votes
0answers
35 views

Why does the hybridization occur between the orbitals that are even under a certain symmetry operations?

For example, in monolayers MoS$_2$, considering the prismatic coordination of the metal atom, the $d$-orbitals split into three categories: {$d_{z^2}$} {$d_{xy}, d_{x^2-y^2}$} {$d_{xz},d_{yz}$}. ...
1
vote
1answer
152 views

Matrix of SU(2) representation

I'm reading D. Gross' Lecture Notes on QFT. In finding the representations of the Poincare Group he finds the represenations of SU(2), which we know are classified by $j=0, \frac12,1,...$ and are $2j+...
4
votes
2answers
400 views

Trace of generators of Lie group

In most textbooks (Georgi, for example) a scalar product on the generators of a Lie Algebra is introduced (the Cartan-Killing form) as $$tr[T^{a}T^{b}]$$ which is promptly diagonalised (for compact ...
4
votes
2answers
223 views

What is the relationship between the Lorentz group and the $CL(1,3)$ algebra?

In my classes the dirac equation is always presented as the "square root" of the Klein Gordon equation, then from this you can demand certain properties from the Matrices (anticommutation relations, ...