130_notes.dvi

Writing theHamiltonian in terms of these fields, the formula can be simplified as follows

H =

∫

ψ†

(

̄hcγ 4 γk

∂

∂xk

+mc^2 γ 4

)

ψd^3 x

H =

∫∑

~p

∑^4

r=1

∑

p~′

∑^4

r′=1

√

mc^2

|E′|V

c∗p~′,r′u(r

′)†

~p′ e

−i(~p·~x−Et)/ ̄h

(

̄hcγ 4 γk

∂

∂xk

+mc^2 γ 4

)√

mc^2

|E|V

c~p,ru(~pr)ei(~p·~x−Et)/ ̄hd^3 x

H =

∫∑

~p

∑^4

r=1

∑

p~′

∑^4

r′=1

√

mc^2

|E′|V

c∗p~′,r′u(r

′)†

~p′ e

−i(~p·~x−Et)/ ̄h

(

̄hcγ 4 γk

ipk

̄h

+mc^2 γ 4

)√

mc^2

|E|V

c~p,ru(~pr)ei(~p·~x−Et)/ ̄hd^3 x

H =

∫∑

~p

∑^4

r=1

∑

p~′

∑^4

r′=1

√

mc^2

|E′|V

c∗p~′,r′u(r

′)†

~p′ e

−i(~p·~x−Et)/ ̄h(icγ

4 γkpk+mc

(^2) γ
4

)

√

mc^2

|E|V

c~p,ru(~pr)ei(~p·~x−Et)/ ̄hd^3 x

(

icγ 4 γjpj+mc^2 γ 4

)

ψ=Eψ

H =

∫∑

~p

∑^4

r=1

∑

p~′

∑^4

r′=1

√

mc^2

|E′|V

c∗p~′,r′u(r

′)†

~p′ e

−i(~p·~x−Et)/ ̄h(E)

√

mc^2

|E|V

c~p,ru(~pr)ei(~p·~x−Et)/ ̄hd^3 x

H =

∑

~p

∑^4

r=1

∑

p~′

∑^4

r′=1

√

mc^2

|E′|

c∗p~′,r′u

(r′)†

p~′ (E)

√

mc^2

|E|

c~p,ru

(r)

~p δ~pp~′

H =

∑

~p

∑^4

r=1

∑^4

r′=1

mc^2

|E|

c∗~p,r′c~p,ru(r

′)†

~p (E)u

(r)

~p

u(~pr)†u(r

′)

~p =

|E|

mc^2

δrr′

H =

∑

~p

∑^4

r=1

∑^4

r′=1

mc^2 E

|E|

c∗~p,r′c~p,r

|E|

mc^2

δrr′

H =

∑

~p

∑^4

r=1

E c∗~p,rc~p,r

where previous results from the Hamiltonian form of the Dirac equation and the normalization of
the Dirac spinors have been used to simplify the formula greatly.

Compare this Hamiltonianto the one used to quantize the Electromagnetic field

H=

∑

k,α

(ω

c

) (^2) [
ck,αc∗k,α+c∗k,αck,α

]

for which the Fourier coefficients were replaced by operators as follows.

ck,α =

√

̄hc^2

2 ω

ak,α

c∗k,α =

√

̄hc^2

2 ω

a†k,α