Lecture 27.

Handouts:
19. Homework 6 Solutions.
20. A Summary of the Dirac Equation.
21. Homework Assignment 7.

We discussed Dirac's solution to the negative energy level problem of
the Dirac equation, and began putting the Dirac equation in a
manifestly Lorentz-covariant form.

Interpreting the Dirac equation as the equation for the quantum
mechanical wave function \psi of the electron, we find the electron
has an energy spectrum as follows: we have a continuum of energy
levels above E=m, and a continuum of energy levels below E=-m. In
between there is a gap with no energy eigenstates. The physical
problem in having a continuum of negative energy states is that an
electron could make a transition to a state of arbitrarily low energy,
emitting a photon with arbitrarily large energy in the process --- a
phenomenon which is neither stable nor observed.

Dirac solved this problem using the Pauli exclusion
principle. Electrons are fermions, so that if all the negative energy
states are occupied, the undesirable transitions are
forbidden. Dirac's claim was that in the ``vacuum'' (which means
ground state), all the negative energy levels are filled; the vacuum
contains a ``Dirac sea'' of an infinite number of negative energy
electrons. To be precise, in the vacuum *all* of the negative
energy levels and *none* of the positive energy levels are filled.

This brings up 3 issues:

1) When we calculate the raw charge or energy of the vacuum we get
infinite answers. While this is unsettling, Dirac pointed out that the
observable characteristics of physical states describe only how they
*differ* from other physical states (including the vacuum). Thus
to describe the energy or charge of a physical state, we must always
compare it with the vacuum, subtracting off the vacuum's
contribution. This is a first taste of ``renormalization'', where to
make finite physical predictions we will have to deftly cancel formal
infinities.

2) Say we have excited our ground state, so that there is a vacancy
(or ``hole'') in the negative energy sea. Then a positive energy
electron can make an allowed transition down to the negative energy
state, emitting a photon. This is a prediction: if (compared to the
ground state) we start with an electron and a ``hole'', those two
objects can annihilate each other and emit a photon. A quick
inspection of the hole's dispersion relation reveals that it has the
same mass as the electron; while charge conservation during the event
tells us it must have the opposite charge. This object --- with the
same mass and opposite charge as the original fermion we were studying
--- is called its ``antiparticle''; for the electron specifically,
it's called the ``positron''. We have made two kinds of predictions
here: 1) for every fermion, there must be an antiparticle. Specifically,
the positron was first observed --- in cosmic rays --- within a year
of Dirac's prediction due to this argument. 2) We predict emission
events, in which an electron and positron annihilate, leaving a
photon; and absorption events, where an initial photon becomes an
electron positron pair (called pair creation). These events are
observed as well.

3) This solution contains the seeds of its own destruction, as it
involves intrducing into the theory an essentially incompatible
element. Dirac developed the Dirac equation in a quest for a single
particle relativistic wave equation, whose conserved charge density
would be the probability density to find the single particle at a
particular position. To solve the negative energy level problem,
however, he had to introduce antiparticles, and to consider physical
states containing infinite numbers of antiparticles. This is a sign of
what we anticipated earlier: relativity is pushing us toward a
multiparticle, and not single particle, theory.

We then hoped to address more of the challenges mentioned earlier ---
probabilistic interpretation of j^0, parity invariance. To do this,
it is helpful to leave this form of the Dirac equation and go to a
form with manifest Lorentz covariance. We will take an even easier way
out: to show Lorentz covariance of the Dirac equation, we will show
that it comes from a Lorentz-invariant Lagrangian.

We built the Dirac Lagrangian --- that is, the Lagrangian for complex
\psi whose Euler-Lagrange equations give the Dirac equation. We then
wanted to see if its terms were all Lorentz invariant. We showed that
this determination could be made in the Weyl basis, since the result
is basis-independent. We then listed more carefully the Lorentz
transformation properties of the component fields \psi_R, \psi_L,
and their adjoints. We showed which products of adjoint fields with
fields --- such as occur in our Lagrangian --- are invariant under
rotations, and which under boosts. Next time we will start looking
specifically at the terms in our Lagrangian, to see how they manage to
come together in Lorentz-invariant combinations.

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KB