The Dirac equation(γ 0∂
∂x 0+γ·∇−m)ψ(x) = 0can be written in the form of a Schr ̈odinger equation as
i∂
∂tψ(t,x) =HDψ(t,x) (47.4)with Hamiltonian
HD=iγ 0 (γ·∇−m) (47.5)
Fourier transforming, in momentum space the energy eigenvalue equation is
−γ 0 (γ·p+im)ψ ̃(p) =Eψ ̃(p)The square of the left-hand side of this equation is
(−γ 0 (γ·p+im))^2 =γ 0 (γ·p+im)γ 0 (γ·p+im)
=(γ·p−im)(γ·p+im)
=(γ·p)^2 +m^2 =|p|^2 +m^2This shows that solutions to the Dirac equation have the expected relativistic
energy-momentum relation
E=±ωp=±√
|p|^2 +m^2For eachp, there will be a two dimensional space of solutionsψ ̃+(p) to−γ 0 (γ·p+im)ψ ̃+(p) =ωpψ ̃+(p) (47.6)(the positive energy solutions), and a two dimensional space of solutionsψ ̃−(p)
to
−γ 0 (γ·p+im)ψ ̃−(p) =−ωpψ ̃−(p) (47.7)
(the negative energy solutions). Solutions to the Dirac equation can be identified
with either
- Four-component functionsψ(x), initial value data at a timet= 0.
- Four-component functionsψ ̃(p), Fourier transforms of the initial value
data. These can be decomposed as
ψ ̃(p) =ψ ̃+(p) +ψ ̃−(p) (47.8)into positive (solutions of 47.6) and negative (solutions of 47.7) energy
components.