4Spinors
III Quantum Field Theory
4.4 The Dirac equation
Armed with these objects, we can now construct a Lorentz-invariant action. We
will, as before, not provide justification for why we choose this action, but as we
progress we will see some nice properties of it:
Definition (Dirac Lagrangian). The Dirac Lagrangian is given by
L =
¯
ψ(iγ
µ
∂
µ
− m)ψ.
From this we can get the Dirac equation, which is the equation of motion
obtained by varying
ψ
and
¯
ψ
independently. Varying
¯
ψ
, we obtain the equation
Definition (Dirac equation). The Dirac equation is
(iγ
µ
∂
µ
− m)ψ = 0.
Note that this is first order in derivatives! This is different from the Klein–
Gordon equation. This is only made possible by the existence of the
γ
µ
matrices.
If we wanted to write down a first-order equation for a scalar field, there is
nothing to contract ∂
µ
with.
We are often going to meet vectors contracted with
γ
µ
. So we invent a
notation for it:
Notation (Slash notation). We write
A
µ
γ
µ
≡
/
A.
Then the Dirac equation says
(i
/
∂ − m)ψ = 0.
Note that the
m
here means
m1
, for
1
the identity matrix. Whenever we have a
matrix equation and a number appears, that is what we mean.
Note that the
γ
µ
matrices are not diagonal. So they mix up different
components of the Dirac spinor. However, magically, it turns out that each
individual component satisfies the Klein–Gordon equation! We know
(iγ
µ
∂
µ
− m)ψ = 0.
We now act on the left by another matrix to obtain
(iγ
ν
∂
ν
+ m)(iγ
µ
∂
µ
− m)ψ = −(γ
ν
γ
µ
∂
ν
∂
µ
+ m
2
)ψ = 0.
But using the fact that ∂
µ
∂
ν
commutes, we know that (after some relabelling)
γ
ν
γ
µ
∂
µ
∂
ν
=
1
2
{γ
µ
, γ
ν
}∂
µ
∂
ν
= ∂
µ
∂
µ
.
So this tells us
−(∂
µ
∂
µ
+ m
2
)ψ = 0.
Now nothing mixes up different indices, and we know that each component of
ψ
satisfies the Klein–Gordon equation.
In some sense, the Dirac equation is the “square root” of the Klein–Gordon
equation.