that
and we can write
The Lie algebra associated with translations is then obtained from
In other words, two translations commute and the corresponding relation for the generators is
Namely, translations form an Abelian (commuting) group while the three dimensional rotations form a non-Abelian group.
4.1.3 Lorentz transformation. As we have seen in the last chapter, a proper Lorentz transformation can be thought of as a rotation in the four dimensional Minkowski space-time and has the infinitesimal form
where, as we have seen in (3.20), the infinitesimal, constant parameters of transformation satisfy
As in the case of rotations, let us note that if we define an infinitesimal vector operator (see (4.10))
then, we obtain
Therefore, we can think of
and the generators of infinitesimal boosts with
As before, we can determine the group properties of the Lorentz transformations from the algebra of the vector operators generating the transformations. Thus,
where, as in the case of rotations (see (4.8) and (4.9)), we have
This shows that the algebra of the vector operators is closed and that Lorentz transformations define a non-Abelian group.
The algebra of the generators can also be calculated directly and has the form
This, therefore, gives the Lie algebra associated with Lorentz transformations. As we have seen these transformations correspond to rotations, in this case, in four dimensions and, therefore, the Lie algebra of the generators is isomorphic to that of the group SO(4). In fact, we note that the number of generators for SO(4) which is (for SO(n), it is
coincides exactly with the six generators we have (namely, three rotations and three boosts). However, since the rotations are in Minkowski space-time whose metric is not Euclidean it is more appropriate to identify the Lie algebra as that of the group SO(3, 1). (Namely, Lorentz transformations (boosts) are non-compact unlike rotations in Euclidean space.)
We end this section by pointing out that the algebra in (2.110) coincides with (4.30) (up to a scaling). This implies that, up to a scaling, the matrices σµν provide a representation for the generators of the Lorentz group. This is what we had seen explicitly in (3.71) in connection with the discussion of covariance of the Dirac equation.
4.1.4 Poincaré transformation. If, in addition to infinitesimal Lorentz transformations, we also consider infinitesimal translations, the general transformation of the coordinates takes the form
where ϵµ, ωµν denote respectively the parameters of infinitesimal translation and Lorentz transformation. The transformations in (4.32) are known as the (infinitesimal) Poincaré transformations or the inhomogeneous Lorentz transformations. Clearly, in this case, if we define an infinitesimal vector operator as
then, acting on the coordinates, it generates infinitesimal Poincaré transformations. Namely,
The algebra of the vector operators for the Poincaré transformations can also be easily calculated as
where we have identified
We can also calculate the algebra of the generators of Poincaŕe group. We already know the commutation relations [Mµν, Mλρ] as well as [Pµ, Pν] (see (4.30) and (4.21)). Therefore, the only relation that needs to be calculated is the commutator between the generators of translation and Lorentz transformations. Note that
which simply shows that
