0
votes
0answers
52 views

SU(2) representations and differential equations in physics.

I studied that $SU(2)$ has a spin $j$ representation $U_j$on a homogeneous space of 2 variables with dimension $2j+1$. Now I am trying to understand the following sentences. Suppose $\phi: ...
1
vote
0answers
80 views

Sources for learning Lie groups and symplectic geometry for Quantum optics

I am asking this question on behalf of my junior who has recently joined in the graduate programme. As suggested by my boss, the student wants to work on quantum optics from a symplectic geometric ...
4
votes
0answers
149 views

How to write $SO(2n)$ characters in terms of rotation angles?

Say one is working in a representation of $SO(2n)$ such that it has the highest weights $(h_1,...,h_n)$. And let $\{H_i\}_{i=1}^{n}$ be a basis in the Cartan of $so(2n) = Lie(SO(2n))$. Now one says ...
7
votes
1answer
203 views

What are spinors mathematically?

In the wikipedia article on spinors a number of mathematical definitions are given of spinors which I find slightly confusing. There are essentially two frameworks for viewing the notion of a ...
5
votes
1answer
394 views

What are defining & fundamental representations?

In physics terminology, one hears of the fundamental & defining representations of lie algebras or groups - are these the same as irreducible representations?
2
votes
1answer
103 views

Show that the SU(2) group is a Lie group

How can I prove that the SU(2) is a Lie group with the Pauli matrices as generators?
1
vote
1answer
37 views

Is this a set of generators for the conformal group of Minkowski space?

My physics textbook asserts that the group of maps $f: M \rightarrow M $ ($M$ is the Minkowski space, i. e. $\Bbb R^4$ with the pseudonorm $||x||=x_0^2-x_1^2-x_2^2-x_3^2$ and scalar product $x\dot{} ...
4
votes
2answers
420 views

$SU(2)$ Representation of $SO(3)$

I've often seen it written that $SU(2)$ is a "two-valued representation" of $SO(3)$ (in theoretical physics books mainly). I have a major conceptual issue with this however. I know there is a Lie ...
4
votes
1answer
343 views

Galilean transformations

How do you prove that every galilean transformation of the space $\mathbb R \times \mathbb R^3$ can be written in a unique way as the composition of a rotation, a translation and uniform motion? ...
-1
votes
2answers
105 views

Questions about $su(2)$. [closed]

Edit: In physics, it seems that people usually study $su(2)$ but not only $sl(2)$? Why people study $su(2)$ but not only $sl(2)$?
2
votes
0answers
196 views

Decomposing products of spinor representations into anti-symmetric tensors

There is always a natural $2^{[\frac{d}{2}]}$ dimensional spinorial representation of $SO(d-1,1)$ (..induced from a representation of the related Clifford algebra..) and if $[m]$ denote the space of ...
4
votes
3answers
569 views

Physical interpretation of the Lie Bracket

I've come accross this physical interpretation for $ [X,Y] $ which I don't understand : Follow $X$ for some time $\epsilon$; Follow $Y$ for $\epsilon$; Follow -X for $\epsilon$; Follow -Y for ...
0
votes
0answers
89 views

Is there a rigorous exposition of 'tensor methods' for finding lie group representations?

I've seen tensor methods in physics for finding lie group representations, as in Wu-Ki Tungs Group Theory in Physics, which uses tensors physics style, ie with indices; and Cvitonovics Birdtracks, ...
2
votes
2answers
115 views

Is this a one dimensional Lorentz Boost? And can you have a 1-d Boost without group structure?

Someone has claimed that he has constructed a quaternion representation of the one dimensional (along the x axis) Lorentz Boost. His quaternion Lorentz Boost is $v'=hvh^*+ 1/2( ...
0
votes
2answers
270 views

Is this proof that SU(2) cannot be isomorphic to SO(1,3) valid?

It seems intuitively obvious to me that there cannot be an isomorphism between $\mathrm{SU}(2)$ and $\mathrm{SU}(2)\times\mathrm{SU}(2)$ where SU(2) is the Lie Group with the Pauli matrices as ...
2
votes
1answer
253 views

What physical meaning do the dimension of Wigner d-matrices have?

Wigner's D-matrices is defined as $D_{m'm}^j(\phi,\theta,\psi)=\langle jm'|R(\phi,\theta,\psi)|jm\rangle$; it produces a square matrix (indices $m$ and $m'$) of dimension $2j+1$. It is also ...
1
vote
0answers
81 views

Does triality survive in product Lie groups?

Look at the following diagrams of Lie groups ...
7
votes
1answer
666 views

The mathematics behind Clebsch-Gordan Coefficients

In quantum physics we have to work a lot with Clebsch-Gordan coefficients and generalizations like the Wigner 3j,6j, and 9j symbols. In our coursework we are taught that the coefficients are coupling ...
6
votes
1answer
428 views

Representations of a non-compact group are labeled by its maximal compact subgroup?

I don't have much of any awareness about the representation theory of non-compact Lie groups but I bumped into it for my work. Is there some idea that the representations of a non-compact group are ...