We now consider the improper Lorentz transformation of reflection in space or the parity transformation:
We need to solve (5.122) for
We denote the Lorentz operator by for parity. Consider the following ansatz
where is an arbitrary phase. Using equation 5.122, we have
In analogy to the proper Lorentz transformation for which a rotation of reproduces the original spinors, we postulate that four space-inversions will reproduce the original spinors.
We see that
and is unitary.
The wave function thus transforms as
In the nonrelativisitc limit approaches an eigenstate of . The positive- and negative-energy states at rest have opposite eigenvalues, or intrinsic parities. The intrinsic parity of a Dirac particle and antiparticle are opposite. This is to be contrasted to the Klein-Gordon case wherein one finds identical parities for the particle and antiparticle solutions.