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

 

[sir-pinski]

.


Yes, your calculations and approach seem correct. The Pauli-Lubanski pseudo vector is an important quantity in the representation theory of the Poincaré group, and it is used to classify different representations of the group.

In your first representation, the generators of the Lorentz part are given by the usual angular momentum and boost operators, and the momentum part is simply the derivative operator. In this case, the Pauli-Lubanski pseudo vector is indeed zero, as you have found.

In the spin representation, the generators of the Lorentz part are still given by the angular momentum and boost operators, but now they act on an N-dimensional spinor space. The momentum part is still the derivative operator, but now it acts on an N-dimensional vector space. In this case, the Pauli-Lubanski pseudo vector is a 4x4 matrix, with each component being an NxN matrix.

As for finding explicit examples, you can look at the representations of the Poincaré group in the context of quantum mechanics or quantum field theory. In quantum mechanics, the spinor space is usually taken to be 2-dimensional, so you can work with 2x2 matrices. In quantum field theory, the spinor space can be higher dimensional, so you can work with higher dimensional matrices.

You can also refer to textbooks or papers on representation theory of the Poincaré group for more detailed calculations and examples. Some good references are "Group Theory and Its Applications to the Quantum Mechanics of Atomic Spectra" by Wigner, and "Lie Groups, Lie Algebras, and Representations: An Elementary Introduction" by Hall.

I hope this helps. If you have any further questions, please feel free to ask.