theories. In this chapter, therefore, we will study the symmetry algebras of the Lorentz and the Poincaré groups as well as their representations which are essential in constructing physical theories. But, let us start with rotations which we have already discussed briefly in the last chapter. In studying the symmetry algebras of continuous symmetry transformations, it is sufficient to study the behavior of infinitesimal transformations since any finite transformation can be built out of infinitesimal transformations. Furthermore, the symmetry algebra associated with a continuous symmetry group is given by the algebra of the generators of infinitesimal transformations. It is worth noting here that, for space-time symmetries, the symmetry algebras can be easily obtained from the coordinate representation of the symmetry generators and that is the approach we will follow in our discussions.
4.1.1 Rotation. Let us consider an arbitrary, infinitesimal rotation in three dimensions of the form (repeated indices are summed)
where αk represents the infinitesimal constant parameter of rotation around the k-th axis (there are three of them). (Let us recall our notation from (1.34) and (1.35) here for clarity. ϵijk denotes the three dimensional Levi-Civita tensor with ϵ123 = 1. ϵijk = ηiℓϵℓjk, etc.) If we now identify (in the last chapter we had denoted the infinitesimal transformation matrices by ϵij, ϵµν, but here we denote them by ωij, ωµν in order to avoid confusion with the Levi-Civita tensors)
then, we note that
and that the infinitesimal rotation around the k-th axis (alternatively in the i-j plane) can also be represented in the form
This is, of course, the form of the rotation that we had discussed in the last chapter.
Let us next define an infinitesimal vector operator (also known as the tangent vector field operator) for rotations (an operator in the coordinate basis) of the form
where we have identified Mij = xi∂j − xj∂i. It follows now that
In other words, we see from (4.4) that we can write the infinitesimal rotations in the i-j plane also as
Namely, the vector operator,
The Lie algebra of the group of rotations can be obtained from the algebra of the vector operators themselves. Thus, we note that
where we have identified
Namely, two rotations do not commute, rather, they give back a rotation. Such an algebra is called a non-Abelian (non-commutative) algebra. Using the form of
we can obtain the algebra satisfied by the generators of infinitesimal rotations, Mij, from the algebra of the vector operators in (4.8). Alternatively, we can calculate them directly as
This is the Lie algebra for the group of rotations. If we would like the generators to be Hermitian quantum mechanical operators corresponding to a unitary representation, then we may define the operators, Mij, with a factor of “i”. But up to a rescaling, (4.11) represents the Lie algebra of the group SO(3) or equivalently SU(2). To obtain the familiar algebra of the angular momentum operators, we note that we can define (recall that in the four vector notation Ji = −(J)i)
which gives the familiar orbital angular momentum operators. Using this, then, we obtain (p, q, r, s = 1, 2, 3)
where in the last step we have used the Jacobi identity for the structure constants of SO(3) or SU(2) (or the identity satisfied by the Levi-Civita tensors), namely,
where we have used the anti-symmetry of the Jacobi identity in the i, j indices. This, in turn, leads to (see (4.12))
The algebra of the generators in (4.11) or (4.13) is, of course, the Lie algebra of SO(3) or SU(2) (or the familiar algebra of angular momentum operators) up to a rescaling.
4.1.2 Translation. In the same spirit, let us note that a constant infinitesimal space-time translation of the form
can be generated by the infinitesimal vector operator (repeated indices are summed)
so
