23 Quantization of the Dirac Field
We proceed to quantizing first the free Dirac field, and shall then motivate the quantization of the
interacting field. Recall that we have,
(
γμ∂μ−m)
ψ= 0- H=
∫
d^3 xψ†(
−γ^0 γi∂i+mγ^0)
ψ- Q=
∫
d^3 xψ†ψ (23.1)Recall that the key new feature of the Dirac field is that it must describe fermions, in contrast to
the scalar or electro-magnetic field which describes bosons. The Fock space of states for the Dirac
field should automatically reflect the the corresponding Fermi-Dirac statistic, just as the Fock space
for bosons automatically reflected Bose-Einstein statistics. While bosonic fields are decomposed
into bosonic oscillators, we should expect the Dirac field todecompose into fermionic oscillators,
as had already been guessed in earlier sections.
23.1 The basic free field solution
Recall that the general solution to the free Dirac equation is given by by a linear superposition of
Fourier modes of the form,
ψ(x) = u(k)e−ik·x
(ikμγμ+m)u(k) = 0 (23.2)with the “on-shell” conditionkμkμ=k^2 =m^2. The equation foru(k) has solutions for bothk^0 > 0
andk^0 <0. It is conventional to reorganize the solutions with the help of the following notation,
ψ+(x) = u(k)e−ik·x k^0 > 0
ψ−(x) = v(k)e+ik·x k^0 < 0 (23.3)where now,
(ikμγμ+m)u(k) = 0
(ikμγμ−m)v(k) = 0 (23.4)Note that each equation actually has two linearly independent solutions, which we may label by an
extra index,s, standing for the spin of the particle. Thus, the 4 independent solutions to the Dirac
equation areus(k) andvs(k) fors= 1,2. The solutionsψ+admit the standard interpretation as
particle withpositive energy.
Historically, the interpretation of the solutionsψ−andv(k) was at first confusing, as they
seemed to correspond to particle with negative energy, which is absurd. The issue was first clarified
by Feynman in 1948. The solutionψ−cab be obtained formally fromψ+by “reversing the arrow