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Given the matrix γ5, we can define four new matrices as
Since we know the explicit forms of the matrices
Finally, we can also define six anti-symmetric matrices, σµν, as (µ, ν = 0, 1, 2, 3)
whose explicit forms in our representation can be worked out to be (i, j, k = 1, 2, 3)
Here we have used the three dimensional notation ϵijk = ϵijk. We have already seen in (2.71) that the matrices
can be identified with the spin operators for the Dirac particle. (This relation can be obtained from (2.99) using the identity for products of Levi-Civita tensors, namely, ϵijkϵℓjk = 2 δiℓ.)
We have thus constructed a set of sixteen Dirac matrices, namely,
where the numbers on the right denote the number of matrices in each category and these, in fact, provide a basis for all the 4 × 4 matrices. Here, the notation is suggestive and stands for the fact that
Let us note here that each of the matrices, even within a given class, has its own Hermiticity property. However, it can be checked that except for γ5, which is defined to be Hermitian, all other matrices satisfy
In fact, it follows easily that
where we have used the fact that γ5 is Hermitian and it anti-commutes with γµ. Finally, from
it follows that
The Dirac matrices satisfy nontrivial (anti) commutation relations. We already know that
We can also calculate various other commutation relations in a straightforward and representation independent manner. For example,
In this derivation, we have used the fact that
We note here parenthetically that the commutator in (2.108) can also be expressed in terms of commutators (instead of anti-commutators) as
However, since γµ matrices satisfy simple anti-commutation relations, the form in (2.108) is more useful for our purpose.
Similarly, for the commutator of two σµν matrices, we obtain
Thus, we see that the σµν matrices satisfy an algebra in the sense that the commutator of any two of them gives back a σµν matrix. We will see in the next chapter that they provide a representation for the Lorentz algebra.
The various commutation and anti-commutation relations also lead to many algebraic simplifications in dealing with such matrices. This becomes particularly useful in calculating various amplitudes involving Dirac particles. Thus, for example, (these relations are true only in 4-dimensions)
where we have used (γµ = ηµνγν)
and it follows now that,
Similarly,
and so on.
The commutation and anti-commutation relations also come in handy when we are evaluating traces of products of such matrices. For example, we know from the cyclicity of traces that
Therefore, it follows (in 4-dimensions) that
Here
