# Pauli-lubanski pseudo-vecor

1. Jul 15, 2008

### udaraabey

Hi

I’m wondering if anybody could help me to prove the result of W2 (square of the pauli-lubanski pseudo-vecor).

2. Jul 15, 2008

### meopemuk

Re: Pauli-Lubanski

See first equation on page 117 of http://www.arxiv.org/abs/physics/0504062

Eugene.

3. Jul 15, 2008

### udaraabey

Re: Pauli-Lubanski

Thank you for your reply but in my case I need to start with

Wa=(1/2)EabcdMbcPd

And to end with

W2=-(1/2)MabMabP2+MacMbcPaPb

Could any body tell me how to derive this

4. Jul 15, 2008

### samalkhaiat

Re: Pauli-Lubanski

use the identity

$$\epsilon_{a}^{bcd}\epsilon^{a \bar{b} \bar{c} \bar{d}} = - \left| \begin{array}{ccc} \eta^{\bar{b}b} & \eta^{\bar{c}b} & \eta^{\bar{d}b} \\ \eta^{\bar{b}c} & \eta^{\bar{c}c} & \eta^{\bar{d}c} \\ \eta^{\bar{b}d} & \eta^{\bar{c}d} & \eta^{\bar{d}d} \end{array} \right|$$

and

$$M_{ab} = - M_{ba}$$

regards

sam

5. Sep 8, 2008

### rntsai

Re: Pauli-Lubanski

i tried to get an explicit form of this (Pauli-Lubanski pseudo vector) and i keep getting
zero. i.e W=(0,0,0,0)...which would still make it an invariant; albeit a boring one.
Looking at the form too, with M_ab=-M_ba it does seem that it should be zero.
Can someone tell me what i might be doing wrong. Thanks