130_notes.dvi

The Hamiltonian density may be derived from the Lagrangian in the standard way and the total
Hamiltonian computed by integrating over space. Note that the Hamiltonian density is the same as
the Hamiltonian derived from the Dirac equation directly.

H=

∫

ψ†

(

̄hcγ 4 γk

∂

∂xk

+mc^2 γ 4

)

ψd^3 x

We may expandψin plane waves to understand the Hamiltonian as a sum of oscillators.

ψ(~x,t) =

∑

~p

∑^4

r=1

√

mc^2

|E|V

c~p,ru(~pr)ei(~p·~x−Et)/ ̄h

ψ†(~x,t) =

∑

~p

∑^4

r=1

√

mc^2

|E|V

c∗~p,ru

(r)†

~p e

−i(~p·~x−Et)/ ̄h

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

H =

∑

~p

∑^4

r=1

E c∗~p,rc~p,r.

By analogy with electromagnetism, we can replace the Fourier coefficients for the Dirac plane waves
by operators.

H =

∑

~p

∑^4

r=1

E b

(r)†

~p b

(r)

~p

ψ(~x,t) =

∑

~p

∑^4

r=1

√

mc^2

|E|V

b(~pr)u(~pr)ei(~p·~x−Et)/ ̄h

ψ†(~x,t) =

∑

~p

∑^4

r=1

√

mc^2

|E|V

b(r)

†

~p u

(r)†

~p e

−i(~p·~x−Et)/ ̄h

Thecreation an annihilation operatorsb(r)

†

~p andb

(r)

~p satisfyanticommutation relations.

{b(~pr),b(r

′)†

p~′ } = δrr′δ~pp~′

{b(~pr),b(~pr)} = 0

{b(~pr)†,b(r)

†

~p } = 0

N~p(r) = b(r)

†

~p b

(r)

~p

N~p(r)is the occupation number operator. The anti-commutation relations constrain theoccupation
number to be 1 or 0.

The Dirac field and Hamiltonian can now berewrittenin terms of electron and positron fields for
which the energy is always positive by replacing the operator to annihilate a “negative energy state”