130_notes.dvi

(

γμ

∂

∂xμ

+

mc

̄h

)

ψ= 0

Thisgives us the Dirac equationindicating that this Lagrangian is the right one. The Euler-
Lagrange equation derived using the fieldsψis theDirac adjoint equation,

∂

∂xμ

(

∂L

∂(∂ψ/∂xμ)

)

−

∂L

∂ψ

= 0

∂

∂xμ

(

−c ̄hψγ ̄μ

)

+mc^2 ψ ̄ = 0

−

∂

∂xμ

ψγ ̄μ+mc

̄h

ψ ̄ = 0

again indicating that this is thecorrect Lagrangian if the Dirac equation is assumed to be
correct.

To compute theHamiltonian density, we start by finding the momenta conjugate to the fieldsψ.

Π =

∂L

∂

(

∂ψ

∂t

)=−c ̄hψγ ̄ 4

1

ic

=i ̄hψ†γ 4 γ 4 =i ̄hψ†

There is no time derivative ofψ ̄so those momenta are zero. The Hamiltonian can then be computed.

H =

∂ψ

∂t

Π−L

= i ̄hψ†

∂ψ

∂t

+c ̄hψγ ̄μ

∂

∂xμ

ψ+mc^2 ψψ ̄

= −c ̄hψ†

∂ψ

∂x 4

+c ̄hψ†γ 4 γ 4

∂ψ

∂x 4

+c ̄hψγ ̄k

∂

∂xk

ψ+mc^2 ψψ ̄

= ̄hcψ†γ 4 γk

∂

∂xk

ψ+mc^2 ψ†γ 4 ψ

= ψ†

(

̄hcγ 4 γk

∂

∂xk

ψ+mc^2 γ 4

)

ψ

H =

∫

ψ†

(

̄hcγ 4 γk

∂

∂xk

+mc^2 γ 4

)

ψd^3 x

We may expand the fieldψin thecomplete set of plane waveseither using the four spinorsu
(r)
~p
forr= 1, 2 , 3 ,4 or using the electron and positron spinorsu
(r)
~p andv

(r)
~p forr= 1,2. For economy of
notation, we choose the former with a plan to change to the later once the quantization is completed.

ψ(~x,t) =

∑

~p

∑^4

r=1

√

mc^2

|E|V

c~p,ru(~pr)ei(~p·~x−Et)/ ̄h

The conjugate can also be written out.

ψ†(~x,t) =

∑

~p

∑^4

r=1

√

mc^2

|E|V

c∗~p,ru(~pr)†e−i(~p·~x−Et)/ ̄h