DIRAC'S FAMOUS ALGEBRA
CUT IN HALF
Dr. James Edmonds, Professor of Physics at McNeese State University in the southern USA, has discovered that Dirac's 1928 factorization of the so called Klein-Gordon (relativistic quantum) equation was not done most efficiently. In fact, only half of the Dirac algebra is really needed. (Details can be found on the web at mcneese.edu/colleges/science/physics, if interested.) This has been confirmed by Dr. Hans Ohanian, prolific text book writer and Professor of Physics at Rensselaer Polytech. Inst., USA. It is quite a shock, since the Dirac equation has been looked over by every theoretical physicist since about 1930. How could it take so long to notice this, and what does this cut mean for the left out parts?
Edmonds did not find this easily. It took 30 years of refusing to let go of this fundamental description, while others went on, through quantum electrodynamics (QED) and into quantum charmodynamics (QCD) — the theory of quarks in protons and neutrons. Richard Feynman had stressed in his two years of lectures (at the Hughes Research Lab in Malibu, which Edmonds attended) that something radical but simple was yet to be found. Without it, we cannot progress beyond the QED theory that describes electrons in atoms so well. Dirac shared these misgivings. Edmonds got started on the quest then in 1969. He left Hughes to teach and has worked hard on the search ever since, though some hard times in the 70's. The year 1999 has seen this work begin to really open new insights, but the basic outlook is already contained in Edmonds' 1997 book from World Scientific, entitled: Relativistic Reality: A Modern View. Sales have been very weak, but that may change soon as more physicists become convinced that he really is onto something that their professors should have discovered in the 1930's and 40's. These new ideas are simple but radical, just as Feynman predicted. He died a decade ago, so he missed this enjoyment. Professors Bethe and Dyson contributed to QED along with Feynman, and they are still alive, but neither has ever responded to Edmonds' pre-prints or letters over the past decades.
The Dirac algebra consists of a 4-by-4 array of numbers which are complex, so there are 2X16 total real numbers in any one of these matrices. Edmonds has shown that only 16 of these numbers need be non-zero to factor the Klein-Gordon equation. This opens the important question, "What are the other 16 parts really used for in nature, if anything?" We believe spacetime is four dimensional. However, there are two ways to cut Dirac in half. One suggests 4 and 8 dimensions, whereas, the other suggests 5 and 10 dimensions. The full Dirac algebra even suggests 15 dimensions. These were ridiculous ideas for physics before about 1980 when, in desperation, physicists started to explore the so called String Theory. It led to 10-spacetime in order to begin to develop quantum gravity without infinities popping up in an uncontrolled way. Edmonds' discovery may enrich the String Theory studies, now that the language which Edmonds has developed will be taken up by the younger mainstream theorists and their PhD students. His notation is really the key.
Matrices have been used since 1928 to do quantum, but Edmonds went off into hypercomplex numbers from the start in the early 70's. These generalize from complex numbers: a+ib, (a+ib)* = a-ib, (a+ib)(a+ib)* = ... = a2+b2+i0. The quaternions have four parts, a+Ib+cJ+dK, and like ii = -1, we define II = JJ = KK = -1, but IJ = K = -JI,..., and things get complicated. Lanczos, in 1929, tried to recreate Dirac's matrix theory using complex quaternions, but this algebra is too small, having only 8 parts. Edmonds has explored this number system approach to Dirac's relativistic quantum theory for over three decades, trying to write the old theory in as pretty a form as possible. He has Maxwell's equation of electricity in the form PF = µoc J and Dirac's equation for electrons is Py = ym.Mc, with the same P operator in both equations. It is possible to have everything here as "block diagonal" 4X4 matrices, with 16 parts rather than 32. Certainly this is the way to teach Dirac theory in the future (no gµ's at all here), even if no new physics comes out of this new formulation of our 1930's knowledge. Others have come and gone in similar studies, since the 50's, but Edmonds' approach is emerging as the best.
The Eur. J. Phys., finally in mid-1999, accepted Edmonds' paper which summarizes the new notation and outlines how to set up the Dirac Hydrogen Atom equations, to find the energy levels of hydrogen without using any matrices or vectors at all! Without having to learn matrices first, undergraduate students of physics will now be able to see the beauty of Maxwell and Dirac on an equal footing. The so called form covariance of these equations is also easy to show here, and the Lorentz group itself emerges here in a natural way from the math. The higher dimension extensions of these equations naturally lead to Lorentz symmetry being generalized as well. Einstein's old motivation for Lorentz symmetry will lose most of its support, should these higher dimensions prove themselves in the real world.
Without matrices, Edmonds had to invent rules for multiplying all pairs, of the 32 basic elements of Dirac theory, in his head. That is 1,024 combinations! This was very successful and greatly streamlines many of the old calculations which where very tedious. We can now begin to see the "forrest for the trees," Edmonds says.
Besides the natural extensions to specific higher dimensional systems, Edmonds' work has also lead to the discovery of three mass pieces in Dirac's equation. These replace the single mass term that everyone has assumed should be found inside of protons — since one mass number is easily seen outside of protons. So Edmonds' factorization goes beyond Dirac's in two ways: new dimensions and multi-mass. When "processing" the new equation back to Klein-Gordon, the three mass pieces amazingly fit together to give one mass squared term; the six cross-terms cancel out. The Klein-Gordon equation is next approximated to predict Schrödinger's non-relativistic quantum and Einstein's relativistic classical worlds. Thus, only this composite mass shows up in these two crude, approximate descriptions.
There is no proof yet of multi-mass existing for quarks, but certainly QCD will need to be expanded to include Edmonds conjugations in the basic algebra, and maybe the multi-mass concept as well. Quantum gravity would also be dramatically impacted if there are three kinds of mass inside protons.
The USA journals have continued to reject Edmonds papers but that may finally change, once the Eur. J. Phys. paper gets circulated widely, and his book is read by more young theorists. Radical change is hard to sell, as Feynman knew well. He said in his lectures, Edmonds remembers, that his early paper was rejected by a USA journal also, in the 1940's. QED was finished by the early 50's and we may have made little progress since, if higher dimensions are essential to further progress, not to mention multi-mass, perhaps. Since Edmonds also has found a third way to factor K-G using quaternions directly as coefficients, it may be that the future involves all three Dirac forms being merged, and Dirac's algebra actually being doubled, rather than cut in half. This leads rather directly to 36 dimensional spacetime, which is really a shock. Could nature actually be this complicated? The next generation will find out, we hope.