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### 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 ...
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### 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 ...
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### 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 $SU(3) \otimes SU(2) \otimes U(1)$...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...