- #1
rntsai
- 80
- 1
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
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