All Questions
Tagged with group-theory topology
80
questions
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
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
2
votes
0
answers
57
views
How can I construct a projective representation when the group is not simply connected?
S. Weinberg, in his book "The quantum theory of fields", states this theorem (page 83): The phase of any projective representation $U(T)$ of a given group can be chosen so that $\phi =0$ if ...
- 374
0
votes
0
answers
37
views
Conformal symmetry and group in arbitrary dimensions [duplicate]
As far as i understand, the full symmetry of relativity is conformal symmetry.
This is represented by the conformal group $ \operatorname{Conf}(1, 3) $
Of Minkowski spacetime which is $ \mathbb{R}^{1, ...
- 309
0
votes
0
answers
114
views
Relationship between holonomy and fundamental group
In my notes of topological QFT we demonstrated that the holonomy associated with a path in $\mathbb{R}^3$ is invariant under smooth deformation of the path if the connection is flat.
Then I wrote:
If ...
- 167
3
votes
1
answer
112
views
Quantization of charge from the path integral
Consider a complex scalar field, with the usual Lagrangian:
$$
\mathcal{L} = | \partial_{\mu} \phi|^2 - V(|\phi|^2).
$$
This theory has a $U(1)$ symmetry, $\phi \to e^{i \alpha} \phi$, and the ...
- 3,098
2
votes
0
answers
50
views
Domain walls in theories of axions
I'm stuck in figuring out why some theories of axions predict the existence of domain walls.
Axions are NG bosons associated with the chiral $U(1)_{PQ}$ symmetry, which was spontaneously broken at ...
- 117
2
votes
2
answers
228
views
Topological proof of spin-statistics theorem confusion
I am currently studying the spin-statistics theorem. I have found a section on John Baez's website which presents a "proof" of the spin-statistics theorem. He states the theorem as:
This is ...
- 1,261
0
votes
0
answers
48
views
General element of the Poincare group
Consider the generators of the Poincare group are $M^{\mu\nu}$ and $P^\mu$. The Lorentz group elements are $g(\omega)\in SO(1,3)=e^{-\frac{i}{2}\omega_{\mu\nu}M^{\mu\nu}}$ and the translation group ...
- 534
3
votes
2
answers
218
views
Why must boosts be non-compact?
It is a common argument in the theory of kinematic groups (the groups of motions for a spacetime) that the subgroups generated by boosts must be non-compact[1][2][3]. This is true of all commonly used ...
- 16.5k
2
votes
1
answer
70
views
Is the electroweak gauge group a semidirect product?
In the typical treatment of electroweak theory, the gauge group is $G = \mathrm{SU}(2)_I \times \mathrm{U}(1)_Y$. This group is broken by the Higgs mechanism, while the combination of generators $Q = ...
- 744
1
vote
0
answers
57
views
Analogue of Bargmann's theorem for Super Lie groups
Bargmann's theorem gives the criteria under which a projective representation of a Lie group $G$ can be lifted to a representation of its universal cover. More generally, if this criterion, namely $H^...
- 689
5
votes
3
answers
682
views
In what way are Lie groups generated by the basis of their Lie algebra?
In this question, the answer by twistor59 says
by using the exponential map on linear combinations of [the lie algebra basis vectors], you generate (at least locally) a copy of the Lie group.
I'm ...
- 201
0
votes
0
answers
85
views
What is the rotational symmetry of the theorized spin-2 graviton?
We know and have actually measured in the lab with self-interference neutron experiments the 4π-symmetry (720° rotation Dirac Belt characteristic) of all spin-1/2 particles (except the neutrinos) thus ...
- 4,170
4
votes
0
answers
73
views
Fundamental group of configuration space of gauge theories
If I consider the space of all the gauge fields $A_{\mu}$ (call this $A$) and a proper gauge group $\Omega_*$, I know that the fundamental group $\pi_1(A)=0$ and the for the gauge group, for example $...
- 159