Lorentz Algebra Vector Rep: How to Derive 4x4 Matrices?

[Lapidus]

The vector representation of the Lorentz algebra in 4 dimensions can be very explicitly given by six 4x4 matrices. Peskin/Schroeder has it on page 39, formula 3.18, for example

But then a four-vector is also a tensor product of a left-handed and a right-handed Weyl spinor!

Knowing the Weyl spinor represenation of the Lorentz algebra, how do I arrive at these explicit matrices for the vector representation?

Strangely, no book explains that. Though, many make a lot of effort showing how vectors, Dirac spinors and tensors are direct sums and/ or products of Weyl spinors, they just give the explicit formula for the vector representation.

 

[fzero]

You can map a 4-vector to a bispinor via

$$V^\mu \rightarrow V^\mu (\sigma_\mu)_{\alpha \dot{\alpha}}.$$

More generally, you can use this to map any representation of $SO(3,1)$ to a representation of $SU(2)\times SU(2)$. In principle you can work out the representations from one side to the other via

$${\Lambda^\mu}_\nu V^\nu (\sigma_\mu)_{\alpha \dot{\alpha}} = {M_\alpha}^\beta V^\mu (\sigma_\mu)_{\beta \dot{\beta}} {M^{\dot{\beta}}}_{\dot{\alpha}}.$$

 

[Lapidus]

Thank fzero!

I saw that in Srednicki's book, too, but could not quite decipher what he means. But that's probably because I have not bothered yet to learn this funny dot notation...

Luckily I discovered that in this new book Symmetries and the Standard Model by Robinson, that there is a lovely section on Lorentz symmetry and all its representation. By the way the whole book is great!