1 Dirac Equation
The Dirac equation, introduced by Paul Dirac in 1928, describes relativistic spin-
particles. In
natural units (ℏ = c = 1), it is
where the Einstein summation convention is understood.
2 Derivation
The relativistic energy–momentum relation is
Dirac sought an equation that was first order in both time and spatial derivatives. He therefore
assumed a hamiltonian linear in momentum,
so that
with the momentum operator
For this first-order equation to reproduce the relativistic energy–momentum relation when squared,
the matrices must satisfy
and
together with
These relations cannot be satisfied by ordinary scalar coefficients. A representation therefore
requires matrices acting on a four-component spinor ψ.
In the Dirac representation,
where the Pauli matrices are
The gamma matrices are defined by
and satisfy the Clifford-algebra relation
Using the metric signature
the Dirac equation in conventional units can be written
when x0 = ct. In natural units, this becomes
3 Feynman Slash Notation
A convenient notation for contraction with the gamma matrices is
Similarly,
The Dirac equation can therefore be written compactly as
