Lectures on Quantum Field Theory. Ashok Das

in three dimensions of the form (repeated indices are summed)

      where α^k 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. ϵ_ijk denotes the three dimensional Levi-Civita tensor with ϵ₁₂₃ = 1. ϵⁱ_jk = η^iℓϵ_ℓjk, etc.) If we now identify (in the last chapter we had denoted the infinitesimal transformation matrices by ϵⁱ_j, ϵ^µ_ν, but here we denote them by ωⁱ_j, ω^µ_ν 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 M_ij = x_i∂_j − x_j∂_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, in (4.5) generates infinitesimal rotations and the operators, M_ij, are known as the generators of infinitesimal rotations.

      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 in (4.5), namely,

      we can obtain the algebra satisfied by the generators of infinitesimal rotations, M_ij, 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, M_ij, 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 J_i = −(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 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