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, Pµ, to be diagonal (they satisfy an Abelian subalgebra). The eigenstates of the momentum operators |p〉 are, of course, labelled by the momentum eigenvalues, pµ, satisfying
and in this basis, the eigenvalues of the operator P2 = PµPµ 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 Jk represents the total angular momentum of the particle, we obtain
The result in (4.102) can also be derived in an alternative manner which is simpler and quite instructive. Let us note that in the rest frame (4.99), the Pauli-Lubanski operator (4.82) has the form
where we have used (1.34) as well as (4.12). It follows now that
which is the result obtained in (4.102). Therefore, for a massive particle, we can think of W2 as being proportional to J2 and in the rest frame of the particle, this simply measures the spin of the particle. That is, for a massive particle at rest, we find
Thus, we see that the representations with p2 ≠ 0 can be labelled by the eigenvalues (m, s) of the two Casimir operators, namely the mass and the spin of a particle and the dimensionality of such a representation will be (2s + 1) (for both positive as well as negative energy states).
The dimensionality of the representation can also be understood in an alternative manner as follows. For a state at rest with momentum of the form pµ = (m, 0, 0, 0), we can ask what Lorentz transformations would leave such a vector invariant. Clearly, these would define an invariant subgroup of the Lorentz group and will lead to the degeneracy of states. It is not hard to see that all possible 3-dimensional rotations would leave such a vector invariant. Namely, rotations around the x or the y or the z axis will not change the time component of a four vector (recall that the time component is the spin 0 component of a four vector) and, therefore, would define the stability group of such vectors. Technically, one says that the 3-dimensional rotations define the “little” group of a time-like vector and this method of determining the representation is known as the method of “induced” representation. Therefore, all the degenerate states can be labelled not just by the eigenvalue of the momentum, but also by the eigenvalues of three dimensional rotations, namely, s = 0,
This can also be seen algebraically. Namely, a state at rest is an eigenstate of the P0 operator. From the Lorentz algebra, we note that (see (4.30))
Namely, the operators Mij, which generate 3-dimensional rotations and are related to the angular momentum operators, commute with P0. Consequently, the eigenstates of P0 are invariant under three dimensional rotations and are simultaneous eigenstates of the angular momentum operators as well and such spaces are (2s + 1) dimensional. In closing, let us note from (4.103) that, up to a normalization factor, the three nontrivial Pauli-Lubanski operators correspond to the generators of symmetry of the “little group” in the rest frame.
4.3.2 Massless representation. In contrast to the massive representations of the Poincaré group, the representations for a massless particle are slightly more involved. The basic reason behind this is that the “little” group of a light-like vector is not so obvious. In this case, we note that (we are assuming motion along the z axis and see (4.95))
Consequently, acting on states in such a vector space, we would have (see (4.83))
