130_notes.dvi

The Hamiltonian written in terms of the creation and annihilation operators is.

H =

1

2

∑

k,α

̄hω

[

ak,αa†k,α+a†k,αak,α

]

By analogy, we can skip the steps of making coordinates and momenta for the individual oscillators,
and justreplace 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

(Since the Fermi-Dirac operators will anti-commute, the analogy is imperfect.)

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

(r)†

~p ,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.

Astate of the electrons in a systemcan be described by the occupation numbers (0 or 1 for each
plane wave). The state can be generated by operation on the vacuum state with the appropriate set
of creation operators.

36.20The Quantized Dirac Field with Positron Spinors

The basis states in our quantized Dirac field can be changed eliminate the “negative energy” states

and replace them withpositron states. Recall that we can replace−u
(4)
−~pwith the positron spinor
v(1)~p andu(3)−~pwithv(2)~p such that the new spinors are charge conjugates of the electronspinors.

SCu(~ps)∗=v(~ps) s= 1, 2

The positron spinor is actually just the same as the negative energyspinor when the momentum is
reversed.