Linked Questions

49
votes
2answers
8k views

How do I construct the $SU(2)$ representation of the Lorentz Group using $SU(2)\times SU(2)\sim SO(3,1)$ ?

This question is based on problem II.3.1 in Anthony Zee's book Quantum Field Theory in a Nutshell Show, by explicit calculation, that $(1/2,1/2)$ is the Lorentz Vector. I see that the ...
28
votes
2answers
5k views

What's the relationship between $SL(2,\mathbb{C})$, $SU(2)\times SU(2)$ and $SO(1,3)$?

I'm a beginner of QFT. Ref. 1 states that [...] The Lorentz group $SO(1,3)$ is then essentially $SU(2)\times SU(2)$. But how is it possible, because $SU(2)\times SU(2)$ is a compact Lie group ...
6
votes
1answer
1k views

Difference between Cartesian product and tensor product on gauge groups

After a comment of John Baez to a question I asked on MathOverflow, I would like to ask what the difference between, for example, $SU(3)\times SU(2) \times U(1) $ and $SU(3) \otimes SU(2) \otimes U(1)$...
4
votes
1answer
1k views

What is the meaning of a (1/2, 1/2) representation?

A spin-1 representation is equivalently a (1/2, 1/2) representation of the Lorentz group. Does this mean we are summing together two irreducible representations labelled by the 'quantum number' a 1/2 ...
4
votes
2answers
296 views

A whole lot of doubts on Lorentz representation

Can someone tell me in layman's language how the $(1/2,1/2)$ represents a vector field and $(0,1/2)$ or $(1/2,0)$ represents spinors and $(0,0)$ represents scalar field. Please don't be pedantic on ...
2
votes
2answers
107 views

Does $\mathcal{M} = AdS_2 \otimes S_2$ makes any sense as a manifold?

I'm not a topologist or a group theorist and I need a clarification about some notations. Consider the Bertotti-Robinson metric in General Relativity (relativity students should study this metric, by ...
0
votes
1answer
364 views

Lorentz Group Representations

Consider for example the (trivial) spin-1/2 representation of the $SU(2)$ group. This representation has dimension two, which is clear from a quantum mechanical perspective since we need to specify ...
3
votes
1answer
254 views

How does the Lorentz group act on a 4-vector in the spinor-helicity formalism $p_{\alpha\dot{\alpha}}$?

Given a 4-vector $p^\mu$ the Lorentz group acts on it in the vector representation: $$ \tag{1} p^\mu \longrightarrow (J_V[\Lambda])^\mu_{\,\,\nu} p^\nu\equiv \Lambda^\mu_{\,\,\nu} p^\nu. $$ However, I ...
2
votes
2answers
158 views

Direct Product vs Tensor Product

I am confused in the notation on page 67 and page 70 a text (http://www-pnp.physics.ox.ac.uk/~tseng/teaching/b2/b2-lectures-2018.pdf), whether it's talking about a direct product or an outer product: ...
3
votes
1answer
175 views

Building $\mathfrak{so}(1,3)$ reps using $\mathfrak{so}(1,3)\cong \mathfrak{su}(2)\oplus \mathfrak{su}(2)$

I'm going through the representation theory of $\mathfrak{so}(1,3)$, building Dirac/Weyl spinors and vectors, and I'm a bit confused on the mathematical definitions involved. We have $\mathfrak{so}(1,...
3
votes
0answers
221 views

Show the Lie algebra is the same for $SU(2) \times SU(2)$ and Lorentz group [duplicate]

So I know: $$[\sigma_{I},\sigma{j}] = 2i \epsilon_{ijk} \sigma_{k}$$ So two products of this should give us the Lorentz group: $SO(4) = SU(2) \times SU(2)$ Where $SO(4)$ has 3 Lie algebra which can ...