Lectures on Quantum Field Theory. Ashok Das

the order of P_µ and M_νλ are irrelevant in the definition of the Pauli-Lubanski operator. Furthermore, we note that

      which follows from the fact that the generators of translation commute. It follows from (4.83) that, in general, the vector W_µ is orthogonal to P_µ. (However, this is not true for massless theories as we will see shortly.) In general, therefore, (4.83) implies that the Pauli-Lubanski operator has only three independent components (both in the massive and massless cases). Let us define the dual of the generators of Lorentz transformation as

      With this, we can write (4.82) also as

      where the order of the operators is once again not important.

      Let us next calculate the commutators between W^µ and the ten generators of the Poincaré group. First, we have

      which follows from the the fact that momenta commute. Consequently, any function of W_µ and, in particular W_µW^µ, will also commute with the generators of translation. We also note that

      Here we have used the identity satisfied by the four dimensional Levi-Civita tensors,

      Equation (4.87) simply says that under a Lorentz transformation, the operator behaves exactly like the generators of Lorentz transformation (see (4.30)). Namely, it behaves like a second rank anti-symmetric tensor under a Lorentz transformation. Using this, then, we can now evaluate (see (4.37) and (4.87))

      In other words, we see that the operator W^µ transforms precisely the same way as does the generator of translation or the P^µ operator under a Lorentz transformation. Namely, it transforms like a vector which we should expect since it has a free Lorentz index. Let us note here, for completeness as well as for later use, that

      It follows now from (4.89) that

      which is to be expected since W_µW^µ is a Lorentz scalar. Therefore, we conclude that if we define an operator

      then, this would also represent a Casimir operator of the Poincaré algebra since W_µ commutes with the generators of translation (see (4.86)). It can be shown that P² and W² represent the only Casimir operators of the algebra and, consequently, the representations can be labelled by the eigenvalues of these operators. In fact, let us note from this analysis that a Casimir operator for the Poincaré algebra must necessarily be a Lorentz scalar (since it has to commute with M_µν). There are other Lorentz scalars that can be constructed from P_µ and M_µν such as

      However, it is easy to check that these do not commute with the generators of translation and, therefore, cannot represent Casimir operators of the algebra.

      The irreducible representations of the Poincaré group can be classified into two distinct categories, which we treat separately.

4.3.1 Massive representation. To find unitary irreducible representations of the Poincaré algebra, we choose the basis vectors of the representation to be eigenstates of the momentum operators. Namely, without loss of generality, we can choose the momentum operators, P_µ, to be diagonal (they satisfy an Abelian subalgebra). The eigenstates of the momentum operators |p〉 are, of course, labelled by the momentum eigenvalues, p_µ, satisfying

      and in this basis, the eigenvalues of the operator P² = P^µP_µ are obvious, namely,

      where

      Here m denotes the rest mass of the single particle state and we assume the rest mass to be non-zero. However, the eigenvalues of W² are not so obvious. Therefore, let us study this operator in some detail. We recall that

      Therefore, using (4.88), we have

      where we have simplified terms in the intermediate steps using the anti-symmetry of the Lorentz generators.

      To understand the meaning of this operator, let us go to the rest frame of the massive particle. In this frame,

      and the operator W² acting on such a state, takes the form

      Recalling that (see (4.26))

      where J_k represents the total angular momentum of the particle, we obtain

      The