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
1
vote
0
answers
38
views
The little group of the connected Lorentz group [duplicate]
For $n\geq 2$, the $n$-dimensional connected Lorentz group is defined as $$SO(n-1,1)^{\uparrow}=\{f\in M_n (\mathbb{R}):f^T\eta f=\eta, \det (f)=1,f^0_0>0\}$$ where $$\eta=\begin{bmatrix}-1 & ...
- 374
1
vote
0
answers
31
views
Wigner-Eckart for Finite groups
We know the Wigner-Eckart theorem generalizes to say $\mathrm{SU}(3)$ (see for example this answer). In a different direction, I am curious if/how it generalizes to finite groups of $\mathrm{SU}(2)$.
...
- 163
1
vote
2
answers
140
views
Why do representations of $SU(2)$ correspond to angular momentum eigenstates?
I have been learning about symmetry in one of my physics classes and specifically about $SU(2)$ and its irreducible representations. We can label a basis element of the vector space corresponding to a ...
3
votes
1
answer
58
views
Measurable effects of the global structure of the SM
It is known that the Lie algebra of the SM is
$$
\mathfrak{su}(3)\oplus \mathfrak{su}(2)\oplus \mathbb{R}\,,
$$
so that the Lie group is
$$
G_{\text{SM}} = \dfrac{SU(3)\times SU(2) \times U(1)}{\Gamma}...
- 1,611
1
vote
2
answers
73
views
Wigner $ D $ matrix equivalent for cyclic symmetry
$\newcommand{\ket}[1]{\left|#1\right\rangle}$The action of $ g \in SU(2) $ on a spin $ j $ system (with a Hilbert space of size $ 2j+1 $) is by the Wigner $ D $ matrix $ D^j(g) $. There are formulas ...
1
vote
0
answers
55
views
What is the connection between Lorentz transforms on spinors and vectors?
When deriving the (1/2,0) and (0,1/2) representations of the Lorentz group one usually starts by describing how points in Minkowski space transform while preserving the speed of light (or the metric).
...
1
vote
0
answers
29
views
Why every projective irreducible representation of $(ISO(2,1)^{\uparrow}=SO(2,1)^{\uparrow}\ltimes \mathbb{R}^3)$ is equivalent?
Why every projective irreducible representation of the connected poincare group $(ISO(2,1)^{\uparrow}=SO(2,1)^{\uparrow}\ltimes \mathbb{R}^3)$ is equivalent to a projective irreducible representation ...
- 374
3
votes
0
answers
35
views
Why a scalar particle with momentum orbit $\mathcal{O}_p$ is irreducible?
Let $G$ be a Lie group and $A$ a finite dimensional vector space. A scalar particle with momentum orbit $\mathcal{O}_p$ is a represenation $T: G\ltimes A\to GL(L^2 (\mathcal{O}_p,\mu,\mathbb{C}))$ ...
- 374
2
votes
1
answer
56
views
Helicity operator in spinor-helicity variables
How do I prove that the helicity operator is
$$
H = \frac{1}{2} (\tilde{\lambda}_\dot{\alpha} \frac{\partial}{\partial \tilde{\lambda}_\dot{\alpha}} - \lambda_\alpha \frac{\partial}{\partial \lambda_\...
- 191
2
votes
0
answers
59
views
English translation of Weyl's article "Quantenmechanik und Gruppentheorie"
Is there an English translation of Weyl's 1927 article Quantenmechanik und Gruppentheorie. Note tht I do not mean the book of the same name.
1
vote
0
answers
29
views
2-dimensional connected Lorentz group [closed]
Consider the connected Lorentz group $SO(1,1)^{\uparrow}$. I was wondering if someone could help me about showing that $SO(1,1)^{\uparrow}\cong \mathbb{R}\times \mathbb{Z}_2$. I just need a hint.
- 374
3
votes
0
answers
40
views
Gauging a finite non-abelian global symmetry in 2D
Consider a 2D system with a non-anomalous finite non-abelian global symmetry $G$, for example $$G = S_3=\{e,a,a^2,b,ab,a^2b\}$$ with $a^3=b^2=1$. One expects the local operators charged under the ...
- 189
0
votes
0
answers
25
views
How to show that $G_p=SO(D-1)$?
Let $G=SO(D-1,1)^{\uparrow}$ be the connected Lorentz group. Let $p$ be a timelike momentum with $p_0>0$. I want to show that $G_p=SO(D-1)$, the little group of $p=(M,0,\ldots,0)$ where $M>0$....
- 374
2
votes
0
answers
59
views
How do I relate objects in $SU(4)$ and $SO(6)$? [closed]
We know that $SU(4)$ is homomorphic $SO(6)$. I would like to understand how do we transform objects that we have, for example, in $SU(4)$, to objects in $SO(6)$.
For example, in $SO(6)$ we have ...
- 980
5
votes
0
answers
60
views
Group theoretical approach to conservation laws in classical mechanics
I'm doing some major procrastination instead of studying for my exam, but I wanted to share my thought just to confirm if I'm right.
Suppose that the action, $S(\mathcal{L})$ forms the basis of a ...
- 372