and (4.64), we can obtain the representation of the Lorentz generators for the reducible four component Dirac spinors as
However, we note that these do not resemble the generators of the Lorentz algebra defined in (3.71) and (2.99) (or (3.73) and (3.80)). This puzzle can be understood as follows. We note that in the Weyl representation for the gamma matrices defined in (2.120),
As a result, we note that the angular momentum and boost operators in (4.67) are obtained from
and, consequently, give a representation of the Lorentz generators in the Weyl representation. On the other hand, if we would like the generators in the standard Pauli-Dirac representation (which is what we had used in our earlier discussions), we can apply the inverse similarity transformation in (2.122) to obtain
Therefore, we note that the generators in (4.67) and in our earlier discussion in (3.71) and (2.99) (see also (3.73) and (3.80)) are equivalent since they are connected by a similarity transformation that relates the Weyl representation of the Dirac matrices in (2.120) to the standard Pauli-Dirac representation.
There is yet another interesting example which sheds light on similarity transformations between representation. For example, from the infinitesimal change in the coordinates under a Lorentz transformation (see, for example, (3.20)), we can determine a representation for the generators of the Lorentz transformations belonging to the representation for the four vectors. On the other hand, as we discussed earlier, from the Lie algebra point of view the four vector representation corresponds to jA = jB =
Let us consider a three dimensional infinitesimal rotation of coordinates around the z-axis as described in (3.5). (Here we will use 3-dimensional Euclidean notation without worrying about raising and lowering of the indices.) Representing the infinitesimal change in the coordinates as
we can immediately read out from (3.5) the matrix structure of the generator J3 to be
Similarly, considering infinitesimal rotations of the coordinates around the x-axis and the y-axis respectively, we can deduce the matrix form of the corresponding generators to be
It can be directly checked from the matrix structures in (4.72) and (4.73) that they satisfy
and, therefore provide a representation for the generators of rotations. This is, in fact, the representation in the space of three vectors which would correspond to j = 1.
On the other hand, it is well known from the study of the representations of the angular momentum algebra that the generators in the representation j = 1 have the forms1
which look really different from the generators in (4.72) and (4.73) in spite of the fact that they belong to the same representation for j = 1. (The superscript (LA) denotes the standard representation obtained from the study of the Lie algebra.) This puzzle can be resolved by noting that there is a similarity transformation that connects the two representations and, therefore, they are equivalent.
To construct the similarity transformation (which actually is a unitary transformation), let us note that the generators obtained from the Lie algebra are constructed by choosing the generator J3(LA) to be diagonal. Let us note from (4.72) that the three normalized eigenstates of J3 have the forms
Let us construct a unitary matrix from the three eigenstates in (4.76) which will diagonalize the matrix J3,
If we now define a similarity (unitary) transformation
then, it is straightforward to check
This shows explicitly that the two representations for J corresponding to j = 1 in (4.72),
