Ai commute with Bi.) 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 (jA, jB) 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 (jA, jB) can lie between
The first few low lying representations of the Lorentz group are as follows. For jA = jB = 0, we see from (4.54) and (4.56) that
which corresponds to a scalar representation with zero spin (and acts on the wave function of a Klein-Gordon particle). Similarly, for jA =
corresponds to a two component spinor representation with spin
which also corresponds to a two component spinor representation with spin
is known as a four component vector representation and can be identified with a spin content of 0 and 1 for the components. (Note that a four vector such as xµ has a spin zero component, namely, t and a spin 1 component x (under rotations) and the same is true for any other four vector.) It may be puzzling as to where the four component Dirac spinor fits into this description. It actually corresponds to a reducible representation of the Lorentz group of the form
This discussion can similarly be carried over to higher dimensional representations.
4.2.1 Similarity transformations and representations. Let us now construct explicitly a few of the low order representations for the generators of the Lorentz group. To compare with the results that we had derived earlier, we now consider Hermitian generators by letting Mµν → iMµν as in (4.47). (Namely, we scale all the generators Ji, Ki, Ai, Bi by a factor of i.)
From (4.50), we note that for the first few low order representations, we have (we note here that the negative sign in the spin
Using (4.44), this leads to the first two nontrivial representations for the angular momentum and boost operators of the forms
and
Equations (4.63) and (4.64) give the two inequivalent representations of dimensionality 2 as we have noted earlier. Two representations are said to be equivalent, if there exists a similarity transformation relating the two. For example, if we can find a similarity transformation S leading to
then, we would say that the two representations
which is clearly impossible. Therefore, the two representations labelled by
From (4.63)
