All Questions
Tagged with group-theory quantum-mechanics
327
questions
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
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
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
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
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
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
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
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 ...
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 ...
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.
2
votes
1
answer
46
views
Are projective representiations of a Lie group a representation of the semi-direct product of the group with $U(1)$ if the norm is preserved?
Let's say we have a function $f(x_{\mu},t)$ that transforms under the action of an $N$-parameter group $G(a_{\nu})$. Then a projective representation of $G(a_\nu)$ in the $f(x_\mu,t)$ basis would ...
- 372
4
votes
0
answers
68
views
Using Galilean covariance to find conditions on physical observables
Let's suppose that coordinates have to transform accoring to the Inhomogenous Galilean Group. Then
$$ x' = x + a + v(t+b) $$
$$ t' = t + b $$
Let's use a funtion $\psi(x,t)$ of $x$ and $t$ as the ...
- 372
19
votes
4
answers
3k
views
How to rotate an electron mathematically?
Im a mathematics student who just learned about the fact that if you rotate an electron by $2 \pi$ its spin state changes but if you turn it by $4 \pi$ it stays the same.
I understand all the ...
- 512