Lectures on Quantum Field Theory. Ashok Das

      Rather, it is what is known as a semi-direct sum of the two algebras. (The general convention is to denote groups by capital letters while the algebras are represented by lower case letters.)

4.2Representations of the Lorentz group

      Let us next come back to the homogeneous Lorentz group and note that the Lie algebra in this case is given by (4.30)

      We recall from (4.12), (4.26) and (4.27) that we can identify the angular momentum and the boost operators as

      Written out in terms of these generators, the Lorentz algebra takes the form

      where we have used (4.13) in the last relation.

      This is a set of coupled commutation relations. Let us define a set of new generators as linear superpositions of J_i and K_i as (this is also known as changing the basis of the algebra)

      which also leads to the inverse relations

      Parenthetically, let us note from the form of the algebra in (4.42) that we can assign the following hermiticity properties to the generators, namely,

      This unconventional hermiticity for J_i arises because, in choosing the coordinate representation for the generators, we have not been particularly careful about choosing Hermitian operators. As a consequence of (4.45), we have

      amely, the generators in the new basis are all anti-Hermitian. The opposite hermiticity property of the generators of boosts, K_i, (compared to J_i) is connected with the fact that such transformations are non-compact and, consequently, the finite dimensional representations of boosts are non-unitary (hence the opposite Hermiticity of K_i). However, infinite dimensional representations are unitary, as can be seen from the hermiticity of the generators in the coordinate basis, namely, if we define the generators with a factor of “i”,

      In the new basis (4.43), the Lorentz algebra (4.42) takes the form

      In other words, in this new basis, the algebra separates into two angular momentum algebras which are decoupled. Mathematically, one says that the Lorentz algebra is isomorphic to the direct sum of two angular momentum algebras,

      Incidentally, as we have already seen in the last chapter, the Lorentz group is double valued (doubly connected). Therefore, it is more meaningful to consider the simply connected universal covering group of SO(3, 1) which is to describe the Lorentz transformations, much the same way we consider the universal covering group SU(2) of SO(3) to describe rotations.

      The finite dimensional unitary representations of each of the angular momentum algebras are well known. Denoting by j_A and j_B the eigenvalues of the Casimir operators A² and B² respectively for the two algebras, we have

      An irreducible nonunitary representation of the homogeneous Lorentz group, therefore, can be specified uniquely once we know the values of j_A and j_B and is labelled as D^(jA,jB) (just as the representation of the rotation group is denoted by D^(j)). Namely, this represents the operator implementing finite transformations on the Hilbert space of states or wave functions as

      where Λ represents the finite Lorentz transformation parameter. (Note that we can write D^(jA,jB) = D^(jA)D^(jB), which is obvious in the first line of the following equation (4.52), since the operators A_i commute with B_i.) Explicitly, we can write (this is the generalization of the S(Λ) matrix that we studied in (3.37) in connection with the covariance of the Dirac equation)

      where the finite parameters of rotation and boost can be identified with

      Such a representation labelled by (j_A, j_B) will have the dimensionality (since it is a product representation)

      and its spin content follows from the fact that (see (4.44))

      Consequently, from our knowledge of the addition of angular momenta, we conclude that the values of the spin in a given representation characterized by (j_A, j_B) can lie between

      The first few low lying representations of the Lorentz group are as follows. For j_A = j_B = 0, we see from (4.54) and (4.56) that