What is the Pauli-Lubanski Pseudo-Vector and How Can its Invariance be Proven?

[udaraabey]

Hi

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

 

[meopemuk]

Hi

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

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

Eugene.

 

[udaraabey]

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

W_a=(1/2)E_abcdM^bcP^d

And to end with

W²=-(1/2)M_abM^abP²+M_acM^bcP^aP_b

Could anybody tell me how to derive this

 

[samalkhaiat]

Could anybody tell me how to derive this

use the identity

<br /> \epsilon_{a}^{bcd}\epsilon^{a \bar{b} \bar{c} \bar{d}} = - \left| \begin{array}{ccc} \eta^{\bar{b}b} &amp; \eta^{\bar{c}b} &amp; \eta^{\bar{d}b} \\ \eta^{\bar{b}c} &amp; \eta^{\bar{c}c} &amp; \eta^{\bar{d}c} \\ \eta^{\bar{b}d} &amp; \eta^{\bar{c}d} &amp; \eta^{\bar{d}d} \end{array} \right|<br />

and

M_{ab} = - M_{ba}


regards


sam

 

[rntsai]

use the identity

<br /> \epsilon_{a}^{bcd}\epsilon^{a \bar{b} \bar{c} \bar{d}} = - \left| \begin{array}{ccc} \eta^{\bar{b}b} &amp; \eta^{\bar{c}b} &amp; \eta^{\bar{d}b} \\ \eta^{\bar{b}c} &amp; \eta^{\bar{c}c} &amp; \eta^{\bar{d}c} \\ \eta^{\bar{b}d} &amp; \eta^{\bar{c}d} &amp; \eta^{\bar{d}d} \end{array} \right|<br />

and

M_{ab} = - M_{ba}


regards


sam

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