Historically, the main reason for Dirac to construct his equation was not the aim to include
particles with spin, but rather to present a solution for the problem concerning the solutions
with negative energies. He identified the quadratic form of the Klein-Gordon equation as the
source of the problem and, instead, pursued a linear form of the equations of motion and, thus
of the Lagrangian. This linearisation is the subject of this section.
PRACTISING: TWO-COMPONENT SPINORS
In a previous section, the spin expectation values have been constructed from the spin states,
Introducing a helping matrix
it is possible to construct a four-vector of Pauli matrices, ready to be contracted with
four-vectors such as momenta or derivatives. Then it is possible to construct some simple
kinetic term for these two-component spinors, namely
where the minus sign reflects the fact that spin-½ particles are fermions. When using a
plus sign instead, the result is nothing but a total derivative and thus vanishes. It is also easy
to check, that by construction this Lagrangian is real. In fact, it is also the simplest
possible kinetic term for spinors, using it implies that the canonical mass dimension of spinors
THE REAL THING: BI-SPINORS
Similar reasoning can be applied to construct a linear mass term for the Dirac (bi-) spinors.
There, however, instead of the Pauli matrices, suitable for two-component spinors, new matrices
suitable for four-component Dirac spinors need to be introduced. These are the Dirac- or
γ matrices. Without specifying them here, a kinetic term for Dirac spinors ψ thus reads
As will be seen in more detail later, the "barred" spinor is connected to the "daggered" one through
The reason behind this change from daggered to barred spinors is the ordering of the spinors,
the γ matrix and the partial derivative.