Lectures on Quantum Field Theory. Ashok Das

      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 p_µ = (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, 1, · · · and m_s = −s, −s + 1, · · · , s − 1, s. This defines the 2s + 1 dimensional representation for a massive particle of spin s.

      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 M_ij, 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))

      However, from (4.83) we see that our states in the representation should also satisfy

      There now appear two distinct possibilities for the action of the Casimir W² on the states of the representation, namely,

      In the first case, namely, for a massless particle if W² ≠ 0, then it can be shown (we will see this at the end of this section) that the representations are infinite dimensional with an infinity of spin values. Such representations do not correspond to physical particles and, consequently, we will not consider such representations.

      On the other hand, in the second case where W² = 0 acting on the states of the representation, we can easily show that the action of W^µ in such a space is proportional to that of the momentum operator, namely, acting on states in such a space, W^µ has the form

      where h represents a proportionality factor (operator). To determine h, let us recall that

      from which it follows that acting on a general momentum basis state |p〉 (not necessarily restricting to massless states), it would lead to (see (4.88))

      Comparing with (4.111) we conclude that in this space

      This is nothing other than the helicity operator (since L · p = 0) and, therefore, the simultaneous eigenstates of P² and W² would correspond to the eigenstates of momentum and helicity. For completeness, let us note here that in the light-like frame (4.107), the Pauli-Lubanski operator (4.82) takes the form

      We see from both (4.103) and (4.115) that the Pauli-Lubanski operator indeed has only three independent components because of the transversality condition (4.83), as we had pointed out earlier. We also note from (4.115)

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