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 components of the wave equation, so I'm not so sure how helpful his book is actually going to be when it finally appears. What I mean by this is that part of the whole problem seems to lie in setting up suitable initial conditions, and that is something which it is now time to discuss. So it will be interesting to see how he does it, but there is no assurance that it will be done any better than what is already implicit in the articles we have seen.

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 solutions in a place where eigenfunction is to be expected.

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 to graph some solutions!

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