# Pauli matrices with two spacetime indices

1. Nov 3, 2010

### John Corn

"Pauli matrices with two spacetime indices"

Hi all. This is my first post so forgive me if my latex doesn't show up correctly. I am familiar with defining a zeroth Pauli matrix as the 2x2 identity matrix to construct a four-vector of 2x2 matrices, $\sigma^\mu$. I'm trying to read a paper which uses the notation $\sigma^{\mu \nu}$. This is between a 4-spinor and a gamma matrix. Can someone please enlighten me about what this notation means? Thanks so much.

2. Nov 3, 2010

### DrDu

Re: "Pauli matrices with two spacetime indices"

I vaguely remember it to be the (anti-?) commutator of two gamma matrices.

3. Nov 3, 2010

### John Corn

Re: "Pauli matrices with two spacetime indices"

Thanks for the quick response Dr. Du. The anticommutator of gamma matrices is just $2 \eta^{\mu \nu} I_{4 \times 4}$, which hardly calls for new notation. One usually doesn't discuss commutators in relation to Clifford algebra, but I can't rule that out.

4. Nov 3, 2010

### dextercioby

Re: "Pauli matrices with two spacetime indices"

As far as I remember

$$\Sigma^{\mu\nu} := \frac{i}{2}\left[\gamma^{\mu},\gamma^{\nu}\right]_{-}$$

It has to do with the spin operator for the quantized massive Dirac field.

5. Nov 3, 2010

### tom.stoer

Re: "Pauli matrices with two spacetime indices"

The Sigma matrices are usually used during the derivation of the Lorentz covariance and transformation properties of the Dirac equation. Later it is usually shown how to represent the Sigma matrices using thre gamma matrices.

So strictly speaking you don't need them (or you only need them in an intermediate step)