Lecture 24.
Handouts: 
17. Exam 1 Solutions (at 5pm)
18. Homework 6.


We discussed the challenges of making quantum mechanics relativistic,
derived the Dirac equation, and began a discussion of the Dirac
equation for the massless case.

We pointed out that the salient features of quantum mechanics,
summarized last time, all rely strongly on the nonrelativistic limit.
Schrodinger's equation explicitly enforces a nonrelativistic
dispersion relation on quantum states, E\  |\psi > = 
(|\vec{p}^2\, | /2m + V ) \ |\psi >. Moreover,

1) Positive definiteness of the Hamiltonian comes from the fact that,
in the nonrelativistic limit, the energy depends *quadratically*
on the momentum. This quadratic dependence doesn't occur for
relativistic particles.

2) The probability density \rho = \psi* \psi and a probability
current \vec{j} = -i\hbar/ 2m\ (\psi* \grad \psi - \psi
\grad \psi*) determined by Schrodinger's equation are
manifestly not of Lorentz-covariant form; they do not form a
Lorentz four-vector j^\mu = (c\rho, \vec{j}). Something about their
relationship will have to change for a relativistic theory.

3) Schrodinger's equation is first order in time because the
nonrelativistic energy-momentum dispersion relation is linear in
energy. This again will not be true in the relativistic case.

These points, particularly 2) which establishes the probabilistic
interpretation of the wave function, are fundamental to the way we
understand quantum mechanics. So, as we try a naive extension of
Schrodinger's equation to a relativistic dispersion relation, we must
be prepared to have the very foundations of quantum mechanics shaken.

With that caveat, we tried to write a relativistic Schrodinger
equation: that is, we imposed the relativistic dispersion relation
E^2 = |\vec{p}|^2 + m^2 on quantum states, using the canonical
commutation relations to write E and \vec{p} as i\hbar
\partial_t and -i\hbar \grad. This reproduced an equation
we've already seen: the Klein-Gordon equation. Here, however, we
obtain the Klein-Gordon equation for a wave function \psi. We showed
how each of the features mentioned above make no sense: 1) we have
energy eigenvalues E = - \sqrt{\|\vec{p}|^2 + m^2} with
arbitrarily low values; 2) we obtain a conserved current j^\mu =
-i\hbar/ 2m\ (\psi* \partial^\mu \psi - \psi \partial^\mu \psi*)
which --- while a relativistic generalization of the Schrodinger
probability current --- does not give a positive definite
``probability density'' j^0; and 3) we got an equation that was
second order in time.

We went on to address the third --- and most abstract --- problem. Our
theory of relativistic quantum mechanics must reproduce Schrodinger
quantum mechanics in the nonrelativistic (low-energy) limit.  But for
a second order wave equation, both \psi(\vec{x}) and
\dot{\psi}(\vec{x}) are independent degrees of freedom, and must be
specified as initial conditions to fully determine the future wave
function \psi. It is hard to see how an equation with independent
\psi and \dot{\psi} at high energies can suddenly involve fewer
degrees of freedom (only \psi which determines \dot{\psi}) at low
energies.

We suggested that the other two problems can be solved by involving
field theory in our interpretation, since the Klein-Gordon equation,
as a field equation, does determine a positive definite Hamiltonian as
well as a sensible charge current.

We then followed Dirac's course of attempting to impose the
relativistic energy-momentum dispersion relation in a way that is 1)
linear in p_0; and 2) Lorentz-covariant, so that p_0 and \vec{p} are
treated symmetrically. We found that we had to introduce matrices to
obtain a linear dispersion relation: particularly, we found the
dispersion relation p_0 1 = \alpha \dot \vec{p} + \beta m. Here 1,
\alpha_i, and \beta are matrices, and reproduce the quadratic
energy-momentum dispersion relation if and only if \alpha_i and \beta
1) each squares to the identity; and 2) all anticommute with each
other.

Given such matrices, we used the canonical commutation relations to
impose this linear relativistic dispersion relation on the wave
function in position space, obtaining the Dirac equation. We then
began investigating the meaning of this equation by considering the
massless case, where we need only 3 anticommuting matrices \alpha_i
that each square to one. This is a challenge we know how to face, and
we will develop the theory next time.

--- 
KB