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I'm reading Zee's book 'Quantum Field Theory in a Nutshell', the chapter about Lorentz group representations at the moment. In the end of the chapter there is suggested an exercise - "Show by explicit computation that (1/2,1/2) is indeed the Lorentz vector". And I just can't figure it out.

Here' s a little background. Lorentz group has 6 generators - 3 rotations [itex]J_i[/itex] and 3 boosts [itex]K_i[/itex] with commutation relations

[itex][J_i,J_j]=i\epsilon_{ijk}J_k, [J_i,K_j]=i\epsilon_{ijk}K_k, [K_i,K_j]=-i\epsilon_{ijk}J_k,[/itex]. We define [itex]J_{+}=1/2(J_i+i K_i), J_{-}=1/2(J_i-i K_i)[/itex], which now satisfy the SU2 groups commutation relations [itex][J_{+i},J_{+j}]=i\epsilon_{ijk}J_{+k},[J_{-i},J_{-j}]=i\epsilon_{ijk}J_{-k},[J_{+i},J_{-j}]=0[/itex], which means that Lorentz group is isomorphic to SU2[itex]\otimes[/itex]SU2.

Now coming back to (1/2,1/2) representation, it means that [itex]J_+[/itex] as well as [itex]J_-[/itex] are represented by 1/2 representation of SU2, i.e. by [itex]1/2\sigma_i[/itex], the Pauli matrices. Here comes the first misunderstanding, Pauli matrices are 2 by 2 matrices, Lorentz generators (therefore [itex]J_{+}, J_{-}[/itex] too, as they are just the sum of the Lorentz matrices) are 4 by 4.

As I couldn't solve it I decided to go backwards. J's and K's can be represented by

[tex]

J_1 =

\begin{bmatrix}

0 & 0 & 0 & 0\\

0 & 0 & 0 & 0\\

0 & 0 & 0 & -i\\

0 & 0 & i & 0\\

\end{bmatrix}, etc,

K_1 =

\begin{bmatrix}

0 & 1 & 0 & 0\\

1 & 0 & 0 & 0\\

0 & 0 & 0 & 0\\

0 & 0 & 0 & 0\\

\end{bmatrix}, etc

\\

[/tex]

therefore, for example [itex]J_{+1}, J_{-1}[/itex] have the following forms

[tex]

J_{+1} =1/2(J_1+i K_1)=1/2

\begin{bmatrix}

0 & i & 0 & 0\\

i & 0 & 0 & 0\\

0 & 0 & 0 & -i\\

0 & 0 & i & 0\\

\end{bmatrix},

J_{-1} =1/2

\begin{bmatrix}

0 & -i & 0 & 0\\

-i & 0 & 0 & 0\\

0 & 0 & 0 & -i\\

0 & 0 & i & 0\\

\end{bmatrix}.

\\

[/tex]

I just can't see how these matrices can be related 1/2 representation of SU2 group, i.e. to Pauli sigma matrices. While the words 'Show by explicit computation' in the exercise tell me, that they somehow should. Any help, even some general sketch of that computation would be much appreciated.

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# Why is (1/2,1/2) the vector representation?

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