Lecture 26.

We made some final comments on the problems with our solution of the
massless Dirac equation, then proceeded to solve the massive Dirac
equation.

We noted that our treatment of the massless Dirac equation only solved
1 of our three quantum-mechanics-associated problems: it gave us a
first order relativistic wave equation. We still have negative energy
solutions, since the Hamiltonian is linear in time derivatives, giving
energy plusorminus p_0 for solutions with spacetime dependences 
\exp{minusorplus ip_\mu x^\mu}, respectively. We have not yet addressed
whether the density j^0 will have a sensible interpretation as a
probability density; next time we will find that it does.

A new set of problems arises due to the fixed helicity of the two
solutions to the massless Dirac equation. We showed why fixed helicity
states are not invariant under parity, so that a theory with only
left-handed (or only right-handed) spinors violates parity. We then
showed that introduction of a finite mass makes a state's helicity
non-Lorentz invariant: that we could change a massive particle's
helicity by well-chosen Lorentz transformations. These 2 problems cure
each other when we return to the full Dirac equation, building a
massive, parity-invariant theory.

To solve the full Dirac equation, we need 4 matrices \alpha
and \beta that all anticommute, and each square to the identity.
Our desire for parity-invariance makes us consider a theory with 
*both* left-handed and right-handed spinors, since the two transform
into each other under parity. We thus consider 4-dimensional matrices,
acting on a right-handed spinor in the top two components, and a
left-handed spinor in the bottom two. This tells us that
\alpha must be block-diagonal, with +\sigma in the top
block acting on right-handed spinors, and -\sigma in the bottom
block acting on left-handed spinors. We claimed that this choice of
\alpha determines (up to similarity transformation) the fourth
anticommuting matrix \beta, given by
\beta = /    1  \
	|     1 |
	| 1     |    .
	\  1    /


This choice for \alpha and \beta is called the Weyl
basis. We worked out Dirac's equation component-by-component, finding
that our mass term led to coupled differential equations for the two
spinor wavefunctions \psi_L and \psi_R. From their Lorentz
transformation properties, we wrote out the Lorentz tranformation
properties of the full Dirac bispinor \psi = (\psi_R, \psi_L).

We then considered other 4-dimensional choices for \alpha and
\beta, which by Pauli's theorem are all related to our Weyl basis by
similarity transformation. We considered, as a particularly useful
basis, the Dirac basis, which diagonalizes the matrix \beta. In this
basis we were able to find a basis of energy eigenstate solutions to
the Dirac equation. We first found these eigenstate solutions in the
particle's rest frame, and solved for their spacetime dependence (by
requiring our solutions to be eigenstates of fixed energy, we
restricted them to either positive-frequency or negative frequency
Klein-Gordon solutions *only*). Solutions for arbitrary momentum
can be obtained by Lorentz boost, and we displayed their general form.

We noted that, as in the massless case, we have both positive and
negative energy solutions --- a conundrum that we will address next
time.

--- KB