it is also clear that, for the case of general motion, where
the solutions take the forms (with p0 = E± = ±E =
which can be explicitly verified. (The change in the sign in the dependent components in (2.34) compared to (2.30) comes from raising the index of the momentum, namely, pi = −pi = −(p)i.)
2.2Normalization of the wave function
Let us note that if we define
then, we can write the solutions (2.34) for motion along a general direction as
Here α and β are normalization constants to be determined. The two component spinors
For different spin components, this product vanishes.
Given this, we can now calculate
where we have used the familiar identity satisfied by the Pauli matrices, namely
Similarly, for the negative energy solutions we have
It is worth remarking here that although we have seen in (2.37) that, for the same spin components,
In dealing with the Dirac equation, another wave function (known as the adjoint spinor) plays an important role and is defined to be
Thus, for example,
Thus, we see that the difference between the Hermitian conjugate u† and the adjoint
We can also calculate the product
Similarly, we can show that
Our naive instinct will be to normalize the wave function, as in the non-relativistic case, by requiring (for the same spin components)
However, as we will see shortly, this is not a relativistic normalization. In fact, u†u, as we will see, is related to the probability density which transforms like the time component of a four vector. Thus, a relativistically covariant normalization would be to require (for the same spin components)
(Remember that this will correspond to the probability density and, therefore, must be positive. By the way, the motivation for such a normalization condition comes from the fact that, in the rest frame of the particle, this will reduce to
With the requirement (2.46), we determine from (2.38) and (2.40) (for the same spin components when (2.37) holds)
Therefore, with this normalization, we can write the normalized positive and negative energy solutions of the Dirac equation to be
It is also clear that, with this normalization, we will obtain from (2.43) and (2.44) (for the same spin components)
This particular product, therefore, appears to be a Lorentz invariant (scalar) and we will see later that this is indeed true.
Let us also note here that by construction the positive and the negative energy solutions are orthogonal.
