Lecture 25.

We considered the Dirac equation in the massless case, which imposes
on quantum states the linear relativistic dispersion relation p_0 {1}
= \alpha \dot \vec{p}, where the three matrices \alpha_i are
chosen so that (\alpha \dot \vec{p})^2 = |\vec{p}^2| 1. As seen
last time, this means that the matrices \alpha_i each square to
the identity and all anticommute with each other. There are two
independent possibilities for \alpha: plus or minus the Pauli
matrices \sigma. They determine the two possible linear
relativistic wave equations, in position space,  (1 \partial_t \mp
\sigma \dot \vec{p} ) \psi = 0 , called the Weyl equations.

A wave function that solves the Weyl equation must then be a
two-component column vector (since the operator equation involves 2 by
2 matrices). We attached meaning to these two components by
considering the wavefunction's behavior under Lorentz transformations.
We cited the result that \psi (solving either the plus or minus
version of Weyl's equation) transforms like a spinor under rotations,
and is matrix-multiplied by a nonunitary transformation under Lorentz
boosts, with minus solutions transformed by the inverse of the matrix
that transforms the plus solutions.

We thus saw that trying to extend a first order Schrodinger equation
to the relativistic regime automatically associated an intrinsic spin
of one-half with the quantum particle.

We reviewed why solutions to Weyl's equations must also solve the
Klein-Gordon equation, and used the Klein-Gordon equation to isolate
and solve for their space-time dependence, getting (for a particle
with momentum \vec{k}) that each spin component was some constant
times the wave equation solution for momentum \vec{k}.

We then returned to Weyl's equation to find the allowed constant spin
components. We found that solutions of the plus equation (which came
from the choice \alpha = + \sigma) must be in a spin up eigenstate
of the spin component along the direction of their motion. They have
maximum spin in the direction of their motion, and since the
projection of a particle's spin in its direction of motion is called
helicity, they have helicity +1. Physically, intrinsic spin along the
direction of motion corresponds to intrinsic rotation about the axis
of the motion in a righthanded sense, and so these particles are
called righthanded.

The alternate choice for linearizing the relativistic dispersion
relation --- choosing \alpha = - \sigma --- gives, by the same
analysis, solutions which must be in a spin down eigenstate of the
spin component along the direction of their motion. They have helicity
-1, corresponding to intrinsic rotation about the axis of the motion
in a lefthanded sense, and so are called lefthanded.

--- KB