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Lectures on Quantum Field Theory. Ashok Das
Читать онлайн.Название Lectures on Quantum Field Theory
Год выпуска 0
isbn 9789811220883
Автор произведения Ashok Das
Жанр Биология
Издательство Ingram
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 under a Lorentz transformation, Pµ behaves like a covariant four vector. (This is seen by recalling that
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,