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 Wµ in such a space is proportional to that of the momentum operator, namely, acting on states in such a space, Wµ 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) that, in the massless case, W0, indeed represents the helicity operator up to a normalization as we had noted in (4.113). It follows now from (4.115) that (the contributions from W0 and W3 cancel out)
Let us now determine the dimensionality of the massless representations algebraically. Let us recall that we are considering a massless state with momentum of the form pµ = (p, 0, 0, −p) and we would like to determine the “little” group of symmetries associated with such a vector. We recognize that in this case, the set of Lorentz transformations which would leave this four vector invariant must include rotations around the z-axis. This can be seen intuitively from the fact that the motion of the particle is along the z axis, but also algebraically by recognizing that a light-like vector of the form being considered is an eigenstate of the operator P0 − P3, namely,
Furthermore, from the Poincaré algebra in (4.38), we see that
so that rotations around the z-axis define a symmetry of the light-like vector (state) that we are considering. To determine the other symmetries of a light-like vector, let us define two new operators as
It follows now that these operators commute with P0 − P3 in the space of light-like states, namely,
and, therefore, also define symmetries of light-like states. These represent all the symmetries of the light-like vector (state). We note that the algebra of the symmetry generators takes the form
Namely, it is isomorphic to the algebra of the Euclidean group in two dimensions, E2 (which consists of translations and rotation). Thus, we say that the stability group or the “little” group of a light-like vector is E2. Clearly, M12 is the generator of rotations around the z axis or in the two dimensional plane and Π1, Π2 have the same commutation relations as those of translations in this two dimensional space. Furthermore, comparing with Wi, i = 1, 2, 3 in (4.115), we see that up to a normalization, the three independent Pauli-Lubanski operators are, in fact, the generators of symmetry of the “little” group, as we had also seen in the massive case. This may seem puzzling, but can be easily understood as follows. We note from (4.90) that in the momentum basis states (where pµ is a number), the Pauli-Lubanski operators satisfy an algebra and, therefore, can be thought of as generators of some transformations. The meaning of the transformations, then, follows from (4.86) as the transformations that leave pµ invariant. Namely, they generate transformations which will leave the momentum basis states invariant. This is, of course, what we have been investigating within the context of “little” groups.
Let us note from (4.121) that Π1 ∓ iΠ2 correspond respectively to raising and lowering operators for M12, namely,
Let us also note for completeness that the Casimir of the E2 algebra is given by
