0.1 Klein-Gordon Equation 0.2 Dirac Equation

expressed in terms of the fields. Although this looks like the expectation
value of the Dirac Hamiltonian in the single particle theory, it now signifies
the field Hamiltonian with ψ and ψ interpreted as fields to be quantized later.
0.7.1 Dirac Field in Fourier Space
To formulate the theory in Fourier space, we begin by expressing ψ as a sum
of free particle solutions of the free Dirac equation. Denote the two positive
energy solutions by
ψ = u s ( ⃗ k)e i⃗ k⃗x e −iωt
and the two negative ones by
ψ = ṽ s ( ⃗ k)e i⃗ k⃗x e iωt
with the usual definition of ω and s = 1, 2. The tilde is used because we will
later wish to redefine the negative energy states to make them more easily
identifiable as antiparticle states. Because u s and ṽ s are eigenstates of the
Hermitian H belonging to different eigenvalues, they satisfy u † s (⃗ k)ṽ s ′( ⃗ k) = 0.
They can also be chosen orthogonal among themselves by appropriate choices
of the spinor χ in Eq.(??). This can be done simply by choosing the χ s ’s
to be orthogonal. However, unless the χ’s are eigenstates of the helicity
h = σ · ˆp which is conserved by the free H, the resulting u’s or v’s will not
be eigenstates of any component of the spin even though χ is always such an
eigenstate. In fact, the components of the spin are not conserved by the free
Dirac Hamiltonian unless they are in the direction of the momentum. We
normalize the states to satisfy the orthonormality relations
u † s( ⃗ k)u s ′( ⃗ k) = 2ωδ ss ′,
ṽ † s (⃗ k)ṽ s ′( ⃗ k) = 2ωδ ss ′,
u † s (⃗ k)ṽ s ′( ⃗ k) = 0.
Beginning with the superposition
∫ (
ψ(x) = a s ( ⃗ k)u s ( ⃗ k)e i⃗k·⃗x + ˜b s ( ⃗ k)ṽ s ( ⃗ )
k)e i⃗ k·⃗x
˜dk ∑ s
where the quantities with tildes will be modified in anticipation of their role
as antiparticle amplitudes. Since an antiparticle associated with a negative
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