Pauli lubanski pseudo vector in spin representation

[rntsai]

I'm trying to calculate the pauli-lubanski pseudo vector for different representations
of the poincare group. The first rep is the infinite dimensional "angular momentum"
rep where the generators of the lorentz part take the form :

M_ab = x_a*d_b - x_b*d_a (for 3 rotations)
M_ab = x_a*d_b + x_b*d_a (for 3 boosts)

(here d_a is partial differentiation with respect to x_a, the indices...should be obvious).

the momentum part of the generators are :

P_a = d_a (4 translations)

The pauli-lubanski pseudo vector is defined :

W_a = e_abcd * M_bc * P_d

(e_abcd is antisymmetric levi-civita symbol)

A bit of a surprise (to me) is that W_a = 0 for this rep! (check it if you like).
I moved to calculating W_a in a "spin" rep of say dimension N; so now :


M_ab -> M_ab*I_N + S_ab

P_a -> P_a * I_N

where

S_ab = NxN matrices (6 constant matrices satissfying the lorentz algebra multiplication).
I_N is NxN identity matrix

(S_ab and P_c commute : S_ab * P_c = 0) and the pauli-lubanski pseudo vector becomes :

W_a = e_abcd * S_bc * P_d

So it seems like each of the four components is an NxN matrix. Even the invariant
W^a*W_a is an NxN matrix...I assume with eigenvalues equal to some
multiple of spin(spin+1)...although looking at this matrix that doesn't look obvious.

Anyway, my question is this : does the above look right? where can I find an
explicit example where the above calculations are carried out in detail. Also please
let me know if there's a better place to post this if this is outside the forum's area

 

[Wriju]

Couple of things I'd like to point out (unless you've figured it out by yourself)
1) The P-L (pseudo) tensor is constructed in a manner so that it receives NO contribution from orbital ang. mom. since that can take any arbitrarily large/small integer multiple of h-bar while spin ang. mom. is a CHARACTERISTIC of the particle (i.e represenatation) like mass and hence provides the only non-vanishing contribution. No wonder you found vanishing answer with M_ab.

2)S_ab-matrix for a spin-j particle (i.e. Lorentz group representation) is (2j+1) dimensional. S_0i=0, S_ij=e_ijk*J^K where J^K's are the usual spin-j matrices e.g. J^3=diag(-j,-j+,...,j-1,j).

Hope that helps.

Wriju