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Pauli lubanski pseudo vector in spin representation

  1. Oct 4, 2009 #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


    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
  2. jcsd
  3. Oct 6, 2009 #2
    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.

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