(4.73) and (4.75) which look rather different are, in fact, related by a similarity transformation and, therefore, are equivalent.
4.3Unitary representations of the Poincaré group
Since we are interested in physical theories which are invariant under translations as well as homogeneous Lorentz transformations, it is useful to study the representations of the Poincaré group. This would help us in understanding the kinds of theories we can consider and the nature of the states they can have. Since Poincaŕe group is non-compact (like the Lorentz group), it is known that it has only infinite dimensional unitary representations except for the trivial representation that is one dimensional. Therefore, we seek to find unitary representations in some infinite dimensional Hilbert space where the generators Pµ, Mµν act as Hermitian operators.
In order to determine the unitary representations, let us note that the operator
defines a quadratic Casimir operator of the Poincaré algebra (4.38) since it commutes with all the ten generators, namely,
The second relation in (4.81) can be intuitively understood as follows. The operators Mµν generate infinitesimal Lorentz transformations through commutation relations and the relation above, which is supposed to characterize the infinitesimal transformation of P2, simply implies that P2 does not change under a Lorentz transformation (it is a Lorentz scalar) which is to be expected since it does not have any free Lorentz index.
Let us define a new vector operator, known as the Pauli-Lubanski operator, from the generators of the Poincaré group as
The commutator between Pµ and Mνλ introduces metric tensors (see (4.38)) which vanish when contracted with the anti-symmetric Levi-Civita tensor. As a result, the order of Pµ and Mνλ 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 Wµ is orthogonal to Pµ. (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 Wµ and the ten generators of the Poincaré group. First, we have
which follows from the the fact that momenta commute. Consequently, any function of Wµ and, in particular WµWµ, 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
In other words, we see that the operator Wµ transforms precisely the same way as does the generator of translation or the Pµ 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 WµWµ 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 Wµ 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 Mµν). There are other Lorentz scalars that can be constructed from Pµ and Mµν such as
However, it is easy to check that these do not commute with the
