# Formal motivation of using the Dirac equation

As is well-known, in 1928 Dirac has derived the same name equation by using the requirement of constructing a relativistic covariant equation describing the function $\psi$ with corresponding positive valued conserved density, which is interpreted as the probability density to find the one particle in the given point of space. But, as we clearly understand now, the localization of the particle with the wave function obeying any relativistic wave equation leads to creation of particle-antiparticle pairs, and therefore the one-particle wave function interpretation of $\psi$ fails.

Another reason to assume the Dirac equation is that, allegedly, only it describes the spin one half particle. But this is not correct, since actually the Dirac spinor $\psi$ is the direct sum of the two truly irreducible representations of the (double covering of the) Lorentz (Poincare) group - the two-dimensional spinors obeying the Klein-Gordon equation, each of which corresponds to the spin one half. Therefore, this formal argument also doesn't work.

As for me, I can only find the following formal argument to be true. The two-dimensional representations mentioned above are not invariant under $P, T, C$ transformations: these transformations convert one representation into another one. In order to work with P-, T-, C-invariant theory of free spin one half particles, one need to take the direct sum of these representations.

Does anyone know any other formal argument why do we need to use the Dirac equation? I don't ask about experimental motivation (the Dirac equation gives correct hydrogen atom spectrum corrections, while the KG one doesn't, and so on).

• @bolbteppa : by the way, it's interesting that elementary irreducible massive representation with the spin 1/2 is realized by the Klein-Gordon equation, while the elementary irreducible massless representation is realized by the Weyl equation. The formed isn't reduced to the latter in massless limit. But the Dirac equation is surprisingly reduced... – Name YYY Sep 9 '17 at 16:31