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.
206
questions
0
votes
0
answers
16
views
"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 ...
- 137
2
votes
0
answers
35
views
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 ...
- 374
2
votes
2
answers
125
views
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{...
- 2,068
1
vote
0
answers
35
views
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 ...
- 19
3
votes
2
answers
111
views
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 ...
- 425
3
votes
1
answer
85
views
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 $\...
- 2,676
2
votes
1
answer
98
views
$Ad\circ\exp=\exp\circ ad$ 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://...
- 14.4k
2
votes
4
answers
408
views
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 ...
- 14.4k
1
vote
0
answers
35
views
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:
$$ [J_a, O_x] = O_y[\sigma_a]_{yx} / ...
- 111
0
votes
1
answer
168
views
Can you ever obtain a pure rotation from composing Lorentz transformations?
An exercise asks one to show that given $v, u$ speeds much smaller than $c$ and oriented orthagonally, the composition of the lorentz boosts $B(\mathbf{v})B(\mathbf{u})B(\mathbf{-v})B(\mathbf{-u})$ is ...
- 75
0
votes
0
answers
52
views
Rotation and translation of a function of a 3D vector
I want to change the frame by doing translation and rotation.
$$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 ...
- 11
-1
votes
1
answer
74
views
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 ...
- 307
19
votes
5
answers
2k
views
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 ...
- 309
3
votes
1
answer
132
views
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 ...
3
votes
5
answers
219
views
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 ...
- 99