0.1 Klein-Gordon Equation 0.2 Dirac Equation

i.e., α and β i are a set of 4 anticommuting matrices whose square is unity.
In terms of derivatives the Dirac equation is:
i∂ 0 ψ = −iβ i ∂ i ψ + αmψ
To give it a covariant appearence (deceptive because the γ’s are numerical
matrices that do not change under LT, but see below), introduce γ i = αβ i
and γ 0 = α, multiply the equation by α and obtain
The γ µ satisfy
−iγ µ ∂ µ ψ + mψ = 0 (2)
{γ µ , γ ν } = −2g µν (3)
where {} means anticommutator. A possible explicit form is
( )
( )
1 0
0 σ
γ 0 =
γ i i
=
0 −1
−σ i 0
Each entry is a 2 by 2 matrix, making γ 4 by 4. An equivalent explicit form
is
( )
( )
0 1
0 σ
γ 0 =
γ i i
=
1 0
−σ i (5)
0
The first form is the standard form, good for taking the NR limit. The
second form is the chiral form, good for neutrinos and things. Changing
involves reshuffling components of ψ. Both forms can be represented as
direct products of two sets of Pauli matrices τ i for the skeleton and σ i for
the entries: the first set is (τ 3 , iτ 2 σ i ) and the second is (τ 1 , iτ 2 σ i ). Both
forms (and others you can easily invent) satisfy Eq.(3). In both forms, the
hermitian conjugate of γ 0 is γ 0 and of γ i is −γ i . A very useful relation valid
for both forms is
γ µ† = γ 0 γ µ γ 0 .
0.3 Free Particle Solutions
Look for solution in the form of an eigenstate of the energy-momentum vector:
ψ = ψ 0 e ipx ,
(4)
2