9
votes
2answers
131 views

Applications of Algebra in Physics

Often I have heard about the link between Algebra (in particular Representations of Groups and Algebras) and some "indefinite" field of Physics. I have a good preparation in Algebra and ...
4
votes
2answers
79 views

Identifying the algebra

In order to solve an obscure (physics) problem I have been considering whose details are not important, I am looking for elements (I am thinking in terms of matrices and their products but this may ...
0
votes
0answers
28 views

Formal proof of Clebsch Gordon sum

physicist here. When looking at the irreducible representations of $so(3)$, i.e. the set of all real valued anti-symmetric matrices, one can parametrize those irreps with an index $j$ which can be ...
1
vote
0answers
16 views

Reference request: what is the relation between classical r-matrices and quantum R-matrices?

I learned from a professor that $$ R=Id+(q-1)r+ o(q-1), $$ where $R$ is a quantum $R$-matrix and $r$ is the corresponding classical $r$-matrix. Here $o(q-1)$ denotes a term of the form $A(q-1)^2$, ...
2
votes
1answer
42 views

What is the explicit formula for classical r-matrices?

It is said that classical r-matrices are those satisfy the classical Yang-Baxter equation $[r_{12}, r_{13}] + [r_{12}, r_{23}] + [r_{13}, r_{23}] = 0$, where $r \in \mathfrak{g} \otimes \mathfrak{g}$. ...
3
votes
1answer
74 views

Representation theory and particle physics

Are there good books which explain clearly explain the connections between modern particle physics and representation theory of groups and lie algebras?
2
votes
1answer
45 views

How to show that Yang-Baxter equation is the same as braid equation?

The quantum Yang-Baxter equation is $R_{12}R_{13}R_{23} = R_{23}R_{13}R_{12}$. The braid equation is $R_{12}R_{23}R_{12}=R_{23}R_{12}R_{23}$. It is said that these two equations are equivalent. How to ...
0
votes
0answers
24 views

How to decompose a representation of $so(n)$ into representations of a subalgebra

In some cases, it is possible. For instance the representation $16$ of $so(9)$ decomposes as $8_c+8_s$ of $so(8)$. Now I would like to do the same with representations of $so(8)$ into a sum of ...
1
vote
0answers
38 views

What does “The Hilbert space carries a representation of […] group” means?

Often, in quantum mechanics I found the sentence "The Hilbert space carries a representation of $SU(2)$ group" (in particular when dealing with anglar momenta). Effectively, I know that this means ...
4
votes
1answer
73 views

Infinite dimensional reps of the rotation group

$\mathbf{Background:}$ The following is paraphrased from ``Representations of the rotation and Lorentz groups and their applications,'' by Gel'fand. Consider a finite-dimensional representation $T: ...
2
votes
2answers
60 views

Application of representation theory

I often read that one can use representation theory in the field of quantum physics or for the analysis of symmetries in physics or chemistry. Unfortunately I coundn't find a concrete example for ...
4
votes
1answer
64 views

Reference request for Lorentz group and unitary representations

More precisely, I often read or listen that Lorentz group has not (non trivial) unitary finite dimensional irreducible representations because it is not compact. Now, I know that the "converse" part ...
2
votes
1answer
228 views

Lie algebras and physics

I often hear physicists talk about Lie algebras and their representation theory, but most of the time hardly understand them because my knowledge of physics is very limited. Does anyone know of any ...
4
votes
0answers
147 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 ...
2
votes
2answers
117 views

Are spinors, at least mathematically, representations of the universal cover of a lie group, that do not descend to the group?

Following on this question about how to characterise Spinors mathematically: First, given a universal cover $\pi:G' \rightarrow G$ of a lie group $G$, is it correct to say we can always lift ...
6
votes
1answer
188 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
360 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?
8
votes
1answer
408 views

Applications of representation theory in physics

The notes of a lecture on basic group and representation theory I attended last semester begin with a bit of motivation for the argument. They give the following examples for applications in physics: ...
7
votes
1answer
343 views

Do these two sets of matrices form groups?

Stimulated by some Physics backgrounds, consider the following two sets of matrices. Notations and definitions:Let $A,B$ be two complex $n\times n$ matrices, then $\left [ A,B \right ...
0
votes
0answers
113 views

Choosing a spinor space

I still have a hard time on trying to fully understand spinors (on real spaces). Let $V$ be a $n$-dimensional real quadratic space $V$ with a non degenerated signature $(p, q)$. From what I had ...
2
votes
1answer
136 views

Conjugate Representations of Lie Algebra of Lorentz Group

I'm trying to understand the Lie algebra of the Lorentz group and am almost there, but am stuck at the final hurdle! It's easy to prove that $$\frak ...
1
vote
1answer
246 views

Lie Algebra of the Lorentz Group $SO(1,3)^{\uparrow}$

I'm trying to get my head around the Lie algebra of the Lorentz group once and for all, but have got tied up in knots. Where is my error in the following? The universal covering group of the Lorentz ...
5
votes
1answer
386 views

Mathematical significance of the “Dirac conjugate”

Let $\psi$ be a Dirac spinor. The so-called "Dirac conjugate" of $\psi$ is defined to be $\widetilde{\psi}:=\psi ^*\gamma ^0$, where $^*$ denotes the adjoint and the gamma matrices $\gamma ^\mu$ ...
2
votes
1answer
70 views

Spinor Mapping is Surjective

I'm (still) trying to prove that $SL(2,\mathbb{C})$ is the universal covering group the the proper orthochronous Lorentz group $L$. I have completed the following steps. (1) Prove that the vector ...
6
votes
2answers
262 views

Universal Covering Group of $SO(1,3)^{\uparrow}$

I'm trying to prove that $SL(2,\mathbb{C})$ is the universal covering group for the proper orthochronous Lorentz group $SO(1,3)^{\uparrow}$. The standard way goes as follows. (1) Exhibit a real ...
0
votes
0answers
122 views

Hermitian conjugation and representations of the Lorentzian Clifford algebras

The Clifford algebra $\mathcal{C}\ell _{1,2d-1}$ is central and simple (L), and hence has a unique faithful, irreducible representation (over $\mathbb{R}$) (A). Denote this representation by $\gamma ...
2
votes
1answer
313 views

Finding All Irreducible Representations of $SO(3)$

I've read that one may prove that all irreducible representations of $SO(3)$ are tensor product representations of the fundamental representation (or tensor product representations of the spin 1/2 ...
4
votes
2answers
405 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 ...
2
votes
1answer
279 views

Difference between the SU(2) and SO(3) lie groups and their lie algebras

In many places I have seen the SU(2) and SO(3) lie algebras used interchangeably. How are they exactly identical? Moreover, what about their lie groups? Are they identical as well. It would be great ...
5
votes
1answer
623 views

Some questions about representations of $SO(6)$

I would like to know the proof/explanation for the following three properties of the representation of $SO(6)$, What is the importance of symmetric traceless tensors of arbitrary rank w.r.t $SO(6)$ ...
4
votes
1answer
402 views

Fundamental and the anti-fundamental representation of $U(n)$

I guess that conventionally one thinks of the fundamental representation and the anti-fundamental representation of $U(n)$ as the complex $n-$dimensional representation and its complex conjugate. ...
2
votes
0answers
191 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 ...