Relation With The Standard Form
The spinor can be written in a null basis as
such that the representation of the spinor in terms of the Pauli matrices is
The standard form of the Dirac equation can be recovered by decomposing the spinor in its right and left-handed spinor components, which are extracted with the help of the projector
such that
with the following matrix representation
The Dirac equation can be also written as
Without electromagnetic interaction, the following equation is obtained from the two equivalent forms of the Dirac equation
so that
or in matrix representation
where the second column of the right and left spinors can be dropped by defining the single column chiral spinors as
The standard relativistic covariant form of the Dirac equation in the Weyl representation can be easily identified such that
Given two spinors and in APS and their respective spinors in the standard form as and, one can verify the following identity
,
such that
Read more about this topic: Dirac Equation In The Algebra Of Physical Space
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