22.7 The free Dirac action and Hamiltonian
The field equations (γμ∂μ−m)ψ= 0 may be derived from an action principle. The action involves
the fieldψ, as well as the fieldψ ̄, which is being considered as independent from the fieldψ, in the
same sense that the complex variableszand ̄zare considered as being independent variables. The
action is given by
S[ψ,ψ ̄] =∫
d^4 xψ ̄(γμ∂μ−m)ψ (22.64)Varyingψ ̄produces the Dirac equation, while varyingψproduces its conjugate equation. The
action is manifestly Lorentz invariant, since we have already shown the transformation laws of
(γμ∂μ−m)ψandψ ̄to be inverse of one another.
Note: for the time being, ψ ̄ should always be kept to the left of the fieldψ in the action.
Ultimately, we shall see thatψandψ ̄in the classical action are not ordinary complex-valued fields,
but rather Grassmann-valued.
The Hamiltonian formulation of the Dirac equation is ratherintricate. The momentum conju-
gate toψis given by
Πψ=∂L
∂∂ 0 ψ
=−ψγ ̄^0 =ψ† (22.65)but the momentum conjugate toψ ̄vanishes! This is a reflection of the fact that the Dirac system
is first-order in time-derivatives. The Hamiltonian becomes,
H =∫
d^3 x(Πψ∂ 0 ψ−L)=∫
d^3 xψ ̄(
−γi∂i+m)
ψ (22.66)22.8 Coupling to the electro-magnetic field
We now wish to include the interactions between the Dirac field and electro-magnetism, represented
by the electro-magnetic vector potentialAμ, and we want the final theory to be Lorentz invariant.
Recall from our discussion of Maxwell theory thatAμmust couple to a conserved electric current
density. We have already identified this current in Dirac theory : it must be proportional toψγ ̄ μψ,
and the factor of proportionalityqgives the strength of the coupling, which is nothing but the
electric charge of the unit element of the Dirac field. The corresponding action is
S[ψ,ψ,A ̄ ] =∫
d^4 x[
−1
4
FμνFμν+ψ ̄(γμ∂μ+iqγμAμ−m)ψ]
(22.67)This action is manifestly Lorentz-invariant. We shall now show that it is also gauge invariant. Under
a gauge transformation,ψis multiplied by anx-dependent phase factor, whileAμtransforms as,
Aμ(x) → A′μ(x) =Aμ(x) +∂μθ(x)
ψ(x) → ψ′(x) =e−iqθ(x)ψ(x) (22.68)