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 j^\mu = \hbar \bar{\psi} \gamma^\mu \psi, with density
j^0 = \psi^\adjoint \psi as in quantum mechanics.

Second, we mentioned some handy identities obeyed by the \gamma^\mu
matrices.

Third, we discussed parity transformation properties of fermion
bilinears (objects that look like products \bar{\psi} x \psi, for some
inserted 4-dimensional spin matrix x). We showed that the bilinears
that appear in the Dirac Lagrangian, \bar{\psi} \psi and \bar{\psi}
\gamma^\mu \psi, 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, \bar{\psi} \gamma^5 \psi and \bar{\psi} \gamma^\mu \gamma^5
\psi, 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 j^0
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 A^\mu; and
for fermions, 3) the Dirac equation for the bispinor field \psi. 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 positions and momenta reproduce Hamilton's
equations, but as *operator* equations relating position and momentum
*operators*.

We discussed how the above description (L --> p --> H --> Hamilton =
E-L) 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 \phi_a --- by its conjugate
\pi_a^0. 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 \pi_a played the role of conjugate momentum to the field
\phi_a, with --- at equal times --- noncommutation between the
fields \phi_a and \pi_a only *at the same point in space*.

--- 
KB