Cellular Automata Lecture

CINVESTAV

Professor: Harold V. McIntosh.

It is interesting that the hardest part of the differential equation theory
that we are trying to study is just the part about plane waves. Especially
Dirac plane waves.

There are still several references about Schroedinger plane wave theory
which we will want to examine, but that theory isn't particularly hard, and
fits nicely into the part about uncertainty relations and coherent packets.
In contrast, the part about Dirac plane waves seems to suffer problems of
interpretation from the very outset.

Going on from there, most attention has naturally been given to the hydrogen
atom, in part because it is an important physical problem, and in part because
it is one of the few applications of the Dirac equation which can actually be
solved. For example, the Dirac equation separates in spherical polar
coordinates, which is good for a central force, but parabolic coordinates for
a uniform electric field or ellipsoidal coordinates for a two center problem
don't work for the Dirac equation, although they do for the Schrowdinger
equatilon. I'm not sure of the extent to which it even separates in cartesian
coordinates, given the angular-momentum-like behavior of the spin, and the
convenience of seeing it in polar coordinates.

And of course, that is one of the major obstacles --- separating spin from
those negative energies that are the objective of the Foldy-Wouthuysen paper.
Actually, it is not so hard to rewrite Foldy-Wouthuysen in one dimension, it's
just that nobody ever seems to have been interested in doing it. And even so,
the question remains as to what the transformation really means.

There is more to be read about things like Zitterbewegung, Foldy-Wouithuysen,
and Newton-Wigner in various textbooks which touch on relativistic quantum
mechanics, but for the most part they just repeat what was presented in the
original papers. So I'm not going to make any attempt to copy extracts from
them, but anyone who has the curiosity or interest to do so may find things
of interest by browsing the physics library.

So much for the literature. I looked again at Thaller's web site, and his
applet on the zitterbewegung claims to start out from gaussian functions in

Here is a good place to mention one of the connections between cellular
automata theory and differential equations. For both, the boundary conditions
are essential. The difference between the local and the global mapping for
cellular automata embodies the difference through the Welch Indices, among
other things. That is, for some automata, the effect of the boundary diminishes
and disappears as a finite automaton becomes longer and longer, whereas for
others (as with the XOR rule) the effect of the boundary pervades the whole
automaton from beginning to end.

In the realm of differential equations, it is the Sturm-Liouville theory which
is pertinent, by relating initial conditions to boundary conditions. Again this
dichotomy exists --- for some equations, the boundary has little effect on the
local properties of the solution, whereas for others the influence is never
lost. The first case is typical of bound states, the second of contunuum
states. For those, a typical boundary condition is that the current at infinity
on one side is unidirectional; on the other side there may be transmissions and
reflections. Problems arise, as they do for the plane waves, when there is no
effective way to impose a boundary condition because neither side is preferred.
Singular potentials create similar problems of another sort.

One of the articles which I have prepared for the next meeting is my
contribution to the Lowdin Festschrift: ``Quantization and a Green's function
for systems of linear differential equations.'' It is part of the CIEA web
page, and those who want to get a head start on the next meeting can read it
there, and even make themselves a hard copy. It can be read in conjunction
with ``Quantization as an Eigenvalue problem,'' available at the same location.

The basic mathematical problem (as distinguished from philosophical problems,
or with obtaining results to compare with experiment) with quantum mechanics
is representing elements of Hilbert space - the probabilities - by solutions of
differential equations - notably the Schroedinger equation or the Dirac
equation. Thus it is a branch of functional analysis, except that people
usually study Hilbert spaces as closed systems without taking into acount
that they may bave external bases. Notoriously it is those continuum waves,
plane waves included, which cause the problems.

The essential linkage was established in Hermann Weyl's dissertation of around
1910 written under David Hilbert's direction. He uses Stieltjes integrals, and
the theory in general was what enabled Schroedinger to derive his equation from
de Broglie's precepts in 1926. Much later, E. C. Titchmarsh took up the theory
in a two-volume treatise, which is the origin of the term Weyl-Titchmarsh
m-function. There are several other authors who have followed up on the theory,
mentioned in the bibliography of the article mentioned.

Another important contribution was Birkhoff's exposition of how to use the
solutions of a differential equation to form a basis for a function space. The
difference between the standard form and the canonical form of a differential
equation is important, the latter leading to self-adhount operators which are
highly esteemed in quantum mechanics.

In all these expositions there exists a conflict between the idea of an initial
value problem and that of a boundary value problem. Moreover, an additional
conflict arises according to whether the initial value is taken at a boundary,
or at an interior point. Singularities often occur at boundaries, such as when
the coulomb potential becomes infinite in a radial equation; when they do they
further complicate and obscure the analysis.

Coddington and Levinson's book goes into great detail for the Sturm-Liouville
problem on a two-sided infinite interval, as does a book of Atkinson when it
is a question of systems of differential equations. The Green's function paper
in the Festschrift was intended to show these results more explicitly than
they appear in the references, and to emphasize three levels of symmetry which
one encounters in the process.

The essence of this whole procedure is that it is possible to transfer a
geometry from the Hilbert space of functions to the symplectic space of
boundary values. The essential tool is Green's formula, which in turn is a
device for integrating by parts. If that is the tool, the finished product is
a series of orthogonality relations which allow certain functions to be taken
as a basis for the function space. There is an accompanying normalization
problem, which is resolved in different ways.

All of which is why those boundary values are so important, even though they
refer to exceedingly remote regions of space.

Linear differential equations are characterized by having exponential
solutions, although ``exponential'' is a generic term subject to individual
variation. Nevertheless the essential fearures are an essential singularity
at infinity without any other singularities; and the general properties of
either decreasing to zero or growing to infinity. Although orthogonality of
eigenfunction solutions is realatively easy to demonstrate, the touchy details
come from normalization, which also depend on how the boundary conditions
enter into the calculations.

The best situation is the one which prevails for bound states. For some
combinations of initial values and eigenvalues, there are solutions which
diminish exponentially in both directions and hence are normalizable without
any additional fuss. Sometimes the solutions are complex exponentials and
refuse to diminish and therefore are not normalizable. But the growth of a
normalizing integral is linear, not exponential; moreover by giving the
eigenvalue a small imaginary part, the exponential can be made to decrease
and the first set of conditions apply.

Next best may be where things work out well approaching infinity in one
direction, but not the other. Problems where a current has a definite sign at
infinity are an example, which includes reflections from a step, reflection
from barriers and wells, encounters with odd potentials, such as the
gravitational field at the surface of the earth (although too weak to be
quantum mechanical) or a constant electric field - the Stark effect or the
Auger effect. One can imagine ``stealth potentials'' which refuse to reflect
oncoming particles.

With unidirectional currents, it is only uecessary to satisfy joining
conditions to get the current on the other side of the space. That was
evident is the two sample articles where movement of wave packets was
displayed, and is responsible for introducing the two coefficients, of
reflection and transmission.

Then there is the worst case, where there is no guidance from either direction
and a single eigenvalue in the differential equation can result in two
equally acceptable solutions, not just one, or just one in exceptional
circumstances to produce a discrete spectrum. So, beyond the nice algebraic
trick involved in forming the wave packet to demonstrate the Heisenberg
uncertainty principle, one has to wonder what boundary condition it was that
alows exp(ipx) and exp(-ipx) both to be included in an integral over all
momenta.

Some time ago, at Salazar, I worked this out in terms of sines and cosines
vanishing at the edges of a long box, and it can be done. Bit I don't recall
much of the details any more, and I don't even have the programs, as they were
on a magnetic tape whose whereabouts is no longer known, even supposing that
it could be read on today's equipment. I don't seem to have paper copies, or
if so, they are buried in all the paper laying about.

After all, is it so hard to apply Weyl-Titchmarsh theory to plane waves?

The Dirac equation has been a different matter. It has been possible to get
nice representations of the wave functions and probabilities for several
situations, notably including the relativistic harmonic oscillator and the
relativistic mathiey equation; even the Kronig-Penny modle, taking due
account of the Klein paradox for supercritical potentials. On the other hand
trying to form wave packets in a way which I no longer remember how, the
packet split in two and the parts went zooming off in opposite directions.

Arturo Cisneros worked out a wave packet for the Dirac harmonic oscillator,
and although he told me that I had used the wrong initial condition, I do not
have the one he used. All that was part of a paper that was underway when he
went off to the United States. But anyway, the difficulties can hardly have
been unrelated to the zitterbewegung and the Foldy-Wouthuysen, and the real
question is, ``In just what do these two phenomona consist?''

The articles tell you, and everybody who has copied them repeats, that the
negative energy states interfere with the positive energy states; and
furthermore they are responsible for the zitterbewegung. That is only
poetically true, but what we need to do is look at this in detail. Especially
because it is possible to write down an explicit solution as a matrix
exponential. So it is pure laziness not to form some wave packets and graph
them and their time evolution.

I've had various thoughts about explaining the Foldy-Wouthuysen transformation.
The tempting thing to suppose, is that it is a transformation to one or the
other of the wave-number eigenstates, that is, to a wave travelling in one
determinate direction. But it is not, amongst other things, there is the
abrupt change in components on exchanging the waves, so that if one writes
a uniform integral over all momenta in trying to form a gaussian wave packet,
this yields a genuine discontinuity which is not found in the Schroedinger
equation. But form the foregoing discussion, just what is it that allows one
to use

Well, if it isn't a transformation to a travelling wave state, it can hardly
be a WKB approximation to the same thing. Besides, these things would multiply
a solution matrix on the right, equivalent to choosing a selected initial
equation. But the paper of Foldy-Wouthuysen puts the projection operator on
the left, which has no interpretation in terms of initial conditions. Rather,
it is a reassignment of the components in the solution matrix; in particular
it is intended to suppress one of them at the expense of the other.

As a parenthetical remark: now that the distinction between phase velocity
and group velocity has been extablished, is it clear that for the Dirac plane
wave they are reciprocals? Thus phase velocity starts out infinite, as is
worthy of having to include the rest mass in the energy, while the momentum is
zero. But the group velocity is zero for zero momentum, as one would
naturally expect, rising to light velocity only at infinite momentum.

After considering these various possibilities, it seems that what is being
done can be seen between equation 4) and equation 5) of the festschrift
article. Namely, in defining the inner product for two-component spinors in
function space, it should not be (x^T y), but (x^T P y) where P is the
Foldy-Wouthuysen projection operator, (1 - beta) and so on. In other words,
they are going to just pluck out one of the components, in an invariant
fashion as expressed by the projector P, and work with it.

Well, the details of this need to be worked out. Can anyone do it? Or is
anyone sufficiently a master of Mathematica to let Wolfram's little gremlins
do it? What has to be done is to put the P in the equation and then follow
through with the rest of the derivation. What should result is the same
dispersion relation, the two waves probably won't be separated, but there
should be a smoother transition between them when trying to construct a
Gaussian wave packet from all possible momenta.

Those who are working on their English comprehension can take the Newton-Wigner
and Foldy-Wouthuysen articles as prime exhibits. Aside from the fact that they
deal with an advanced subject in a technical area such as physics, there is a
lot of poetry floating around.

That is: It is true that for every positive energy solution there is a negative
energy solution, because the quaternion j (F-W beta, I guess) will leave the
mass alone and negate the energy, if applied to the Dirac equation for a free
particle. This is one of the noteworthy properties of the Dirac equation. In
fact, j just exchanges the two components, which establishes the relation
between the solution for E and the solution for -E. It is also true that of
the two components, with the actual electron mass, one is much larger than
the other.

But, it is an extrapolation to say that one component "belongs" to one energy
and the other to the negative energy. We get used to saying this, but it is
still blatant poetic license. We say something which sounds nice and which is
easy to remember --- but, is it really an explanation?

Then there is the question of whether a wave packet jiggles back and forth,
just gets fatter, or goes zooming off in two directions. We've really

Let me put up a reminder: if anyone can get us a copy of the Huang article,

Kerson Huang, ``On the Zitterbewegung of the Dirac Electron,'' {\em American Journal of Physics} {\bf 20} 479-484 (1952).

we can make more copies. However it is doubtful that the CIEA has it, so the only likely place would be the Institute of Physics at the UNAM, and even if they have it, it may not be so easy to go and pick up a copy, Maybe they could mail it.

Also, I asked about Mielnik's article --- yes, that is one of some that were brought a couple or three meetings ago, so we have it now.

- hvm