Lecture 29.
Today we tied up some loose ends about the Dirac equation, and started to discuss canonical quantization of classical field theory.
First, we noted that the Dirac equation meets the goal of having a
conserved current whose density can be interpreted as the probability
density for finding a single particle. We calculated this probability
current, associated with a phase symmetry of the Dirac Lagrangian,
showing that 
, with density 
as in quantum
mechanics.
Second, we mentioned some handy identities obeyed by the 
matrices.
Third, we discussed parity transformation properties of fermion
bilinears (objects that look like products 
, for some inserted 4-dimensional spin matrix x). We
showed that the bilinears that appear in the Dirac Lagrangian,
and 
,
transform as scalars and vectors respectively under parity. Thus all
terms in the Dirac Lagrangian are parity-invariant, as are electron
interactions in the real world. We built other bilinears,
and 
, which transform as pseudoscalars and
pseudovectors under parity. We will keep them in our toolkit for when
we build Lagrangians for fermions with non-parity-invariant
interactions.
We have exhausted our initial goals in relativizing quantum mechanics,
solving, with the Dirac equation, all but the negative energy level
problem to a purist's satisfaction. Recall that the Klein-Gordon
equation, as a relativized equation for a quantum wavefunction, had
additional problems with 1)being second order; and 2) having 
interpretable as a single particle probability density.
Now we attempt to get a relativistic quantum theory from the other
direction: instead of trying to relativize quantum mechanics, we will
try to quantize relativistic field theory. We have studied three
important relativistic field equations this semester: for bosons, 1)
the Klein-Gordon equation for real and complex scalar fields; and 2)
Maxwell's equation with no sources for the vector field 
; and
for fermions, 3) the Dirac equation for the bispinor field 
. The
last one came from thinking about quantum mechanics, but we can treat
it on the same footing as the others as simply a relativistic field
equation.
We reviewed quickly how we quantize classical mechanics. Classically, a Lagrangian determines all the conjugate momenta as well as the Hamiltonian, which gives Hamilton's equations (which are equivalent to the Euler-Lagrange equations). We quantize the system by 1) making all observables operators; 2) imposing canonical commutation relations between operators for the particles' coordinates and their conjugate momenta; and 3) imposing the Heisenberg equations of motion on all operators. Due to the canonical commutation relations, Heisenberg equations of motion for the positions and momenta reproduce Hamilton's equations, but as operator equations relating position and momentum operators.
We discussed how the above description ( 
) can be grafted onto
classical field theory, with Lagrangians and Hamiltonians becoming
densities, and the role of the conjugate momentum being played -- for
a field 
-- by its conjugate 
. The only
unpleasantness in this parallel structure is that we move from a
Lorentz-invariant object -- the Lagrangian density -- to a
non-Lorentz-invariant one -- the Hamiltonian density. But we do get
Hamilton's field equations being equivalent to the Euler-Lagrange equations.
Given this parallel structure, we tried to quantize the theory in an
exactly parallel way, called canonical quantization. We first imposed
the canonical commutation relations. We saw how to do this by
discretizing our field theory, giving a classical particle theory with
many coordinates -- one for the field variable at each grid
point. Here the coordinates and their conjugate momenta -- and hence
the commutation relations that must be imposed -- are clear. We took
a continuum limit to return from particle mechanics to field theory,
obtaining the right canonical commutation relations for field theory.
Again 
played the role of conjugate momentum to the field
, with -- at equal times -- noncommutation between the
fields and only at the same point in space.
-- KB