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.$$
Let, $\mathcal{R}$ be the rotation matrix and $\mathcal{T}$ be the ...
- 11
-1
votes
1
answer
74
views
From any element of $\mathrm{SO}(8)$, can we always find one corresponding $\mathrm{SU}(3)$ element?
I first recap the relation between $\mathrm{SU}(2)$ and $\mathrm{SO}(3)$ and then raise my question concerning $\mathrm{SU}(3)$ and $\mathrm{SO}(8)$. Given any traceless hermitian matrix $H$, we can ...
- 307
19
votes
5
answers
2k
views
Is intrinsic spin a quantum or/and a relativistic phenomenon?
Ok, this is my reasoning. I am probably making some wrong assumptions here, pls tell me where I am going wrong.
Spin as a quantum phenomenon:
Quantum phenomena disappear as the Planck constant goes to ...
- 309
3
votes
1
answer
132
views
Rotation of spherical harmonics
I have a question about the rotation of spherical harmonics. In Wikipedia it is mentioned that if we make a rotation in 3D space: $R\vec{r}=\vec{r}'$,then the Spherical Harmonics can be written as a ...
3
votes
5
answers
219
views
What does the $N$ in $SU(N)$ mean?
So I know this is a very basic question, but I can't really wrap my head around it.
I was told $N$ is the number of dimensions in the rotations of the group theory that we are considering, so I ...
- 99
1
vote
0
answers
54
views
Do GUT's really explain parity violation?
Every book on the Standard Model introduces early on the concept of left and right-handed quantum fields, defined as
\begin{align}
(\psi_L)_{\alpha} = \left(\frac{1-\gamma_5}{2}\right)_{\alpha \beta}\...
- 131
0
votes
0
answers
38
views
Free fields in Weinberg QFT vol.1
Background:
In section 5.1 Weinberg discusses free fields. He had shown that for interaction of the form, $V(t) = \int{d^3x \mathscr{H}(\mathbf{x},t)}$ if $$U_0(\Lambda,a) \mathscr{H}(x) U_0^{-1}(\...
- 56
1
vote
1
answer
76
views
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 ...
- 785
4
votes
1
answer
128
views
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 ...
- 105
1
vote
0
answers
44
views
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 ...
- 81
0
votes
0
answers
59
views
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 ...
- 257
3
votes
0
answers
36
views
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=\...
- 374
2
votes
0
answers
21
views
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 ...
- 189
0
votes
1
answer
57
views
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 ...
- 2,292
1
vote
0
answers
40
views
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 ...
- 1,713
0
votes
1
answer
94
views
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}\...
- 15
2
votes
0
answers
72
views
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 ...
- 376
2
votes
1
answer
42
views
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 ...
- 21
3
votes
0
answers
48
views
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 ...
- 171
4
votes
1
answer
377
views
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 &...
- 171