Lecture 34.
We discussed canonical quantization of Fermi fields (that is, fields obeying a Dirac equation). To obtain positive definite energies after quantization, we imposed canonical anticommutation relations, which also led to the Pauli exclusion principle.
We first displayed the complete set of plane wave solutions to the
Dirac equation, which involved the fixed spinor column vectors
, 
, 
,
and 
. We used these to write the most general
classical solution for the bispinor field 
as a Fourier integral
over all plane wave solutions. As for the complex Klein-Gordon field,
we allow to be complex, which means that complex conjugate
plane wave solutions need not have exactly conjugate coefficients.
There are thus 2 coefficients 
-- 1 for each positive
frequency solution 
-- and 2 independent
coefficients 
-- 1 for each negative frequency
solution 
. (Here the * is chosen by convention,
whenever the solution has negative frequency).
Upon quantization, the coefficients all become creation and annihilation operators. The extra set of allowed coefficients become, on quantization, creation operators for a whole new set of particles. As in the complex Klein-Gordon case, these new particles can be shown to have the same mass, but opposite Noether charge (associated with the field's phase symmetry), to the original particles. Thus we obtain, upon quantization, creation and annihilation operators for particles and antiparticles in the spin states (r).
We calculated the Hamiltonian, finding that -- unlike the
Klein-Gordon case -- adopting the field-dependent Hamiltonian does
not automatically eliminate the problem of states with negative
contributions to the energy. Instead, we saw that negative frequency
solutions make negative contributions to the classically-computed
energy. Upon quantization, this problem persisted -- while the
particles associated with the b creation and annihilation
operators each contribute the positive energy 
, those associated with the c creation and
annihilation operators naively seem to each contribute the negative
energy 
.
To get rid of this undesirable sign, we postulated that fermions obey canonical anticommutation relations. This made the number operators for the c-type particles (that is, the antiparticles) appear in the Hamiltonian with positive sign. Again, to get rid of infinite zero point energies, we have to normal order -- now defined as reordering to get all annihilation operators on the right, then discarding all the anticommutators that were generated in the reordering.
We showed that, even though they no longer have harmonic oscillator
commutation relations with each other, the creation and annihilation
operators still have harmonic oscillator commutation relations with
the Hamiltonian -- which means that they still raise and lower the
energy by the quantum . So the Fock space --
again spanned by energy eigenstates -- again contains a vacuum with
no particles, which is annihilated by all the annihilation operators,
as well as states with particles and antiparticles in them, obtained
mathematically by operating on the vacuum with various creation
operators.
However, a major difference from the Fock space of bosons arises when
we try to apply the creation operator for a single state twice, to
doubly occupy a single particle or antiparticle state. Since the
creation operator anticommutes with itself, its square must be zero:
that is, applying the creation operator twice to the vacuum always
annihilates it. Thus each state 
leads to a Hilbert
space that, unlike the harmonic oscillator, only has two states: an
empty state, which can be raised once to give a 1-particle state, but
which is then the highest state, annihilated when we try to raise it
again.
This fact, that only 1 particle can be put in a single state, is just the Pauli exclusion principle. The Dirac equation, along with canonical anticommutation relations, has automatically led to a theory with multiple particles and antiparticles, which are all fermions. The anticommutation relations also lead to the required antisymmetry of fermion wavefunctions.
Finally, we discussed the classical limit of quantum fields. We noted that Bose fields have both a particle limit, with finite number of arbitrarily high frequency particles; and a field limit, where large numbers of finite frequency particles have wave functions that add coherently. Fermions only have the particle limit, since the occupancy of any state is bounded by 1. Fermion fields also anticommute, even in the classical limit.
-- KB