Lectures on Quantum Field Theory. Ashok Das

      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 as the vector operator generating infinitesimal proper Lorentz transformations and the operators, M_µν = −M_νµ, as the generators of the infinitesimal transformations. We also note that we can identify the infinitesimal generators of spatial rotations with (see (4.12))

      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 σ_µν 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 ϵ^µ, ω^µ_ν 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 [M_µν, M_λρ] as well as [P_µ, P_ν] (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 under a Lorentz transformation, P_µ behaves like a covariant four vector. (This is seen by recalling that ω^µνM_µν generates infinitesimal Lorentz transformations. The commutator of a generator (multiplied by the appropriate transformation parameter) with any operator gives the infinitesimal change in that operator under the transformation generated by that particular generator. For change in the coordinate four vector under an infinitesimal Lorentz transformation, see, for example, (4.22) and (3.59).)

      Thus, combining with our earlier results on the algebra of the translation group, (4.21), as well as the homogeneous Lorentz group, (4.30), we conclude that the Lie algebra associated with the Poincaré transformations (inhomogeneous Lorentz group) is given by

      We note that the algebra of translations defines an Abelian sub-algebra of the Poincaré algebra (4.38). However, since the generators of translations do not commute with the generators of Lorentz transformations, Poincaré algebra cannot be written as a direct sum of those for translations and Lorentz transformations. Namely,