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Diffraction and "the Theory of Everything" By Andy Everett   Recent Thoughts   1. Something Feynman Wrote. 2. Elastodynamics Compared With Electrodynamics. 3. Tinker Toy spin. 4. Hyperspherical Harmonics and QM in S3. 5. Fun With Capacitors.and Inductors, Massive and Massless Fields. 6. Electrons, Photons, and the Pendulum. 7. A weak argument for neutrino mass. 8. Interpreting Curved 3-Manifolds With Color. 9. Further Questions. 10. A Particle in S3, A post From UseNet.   1. Something Feynman Wrote.   After summing up the rudiments of quantum mechanics Feynman tries to answer a question that readers might have at this point of his book1. He writes, (likeness of Feynman by my daughter)   "One might still like to ask: "How does it work? What is the machinery behind the law?" No one has found any machinery behind the law. No one can "explain" any more than we have just "explained". No one will give you any deeper representation of the situation. We have no ideas about a more basic mechanism from which these results can be deduced. ... And no one has figured a way out of this puzzle. So at the present time we must limit ourselves to computing probabilities. We say "at the present time," but we suspect very strongly that it is something that will be with us forever--that it is impossible to beat that puzzle--that this is the way nature really is." Like someone who believes in a God never seen I also have faith that there exists an explanation which answers the question, "how does it work". Further, if someone of good science background could read all possible one page compositions (about 10^800), composed using only words from a medium sized dictionary, then one would know "how does it work". Sadly this report does not contain that composition.   2. Elastodynamics Compared With Electrodynamics.   The equations of linear elastodynamics (in this case for an isotropic, homogeneous elastic body) are close to those of classical electrodynamics. Compare2 3, In (3) m and l are constants, r is the mass density, u is the displacement field, and f is the body force density. The symmetry is also spoiled somewhat because in solids the transverse wave speed and the longitudinal wave speed are different. One can compare many phenomenon in electrodynamics with similar ones in elastodynamics, for example, a transverse electromagnetic wave with a transverse wave in an elastic solid. Let the vector potential A be compared to the displacement field u then the partial time derivative of u is likened to the electric field E and the curl of u is likened to the magnetic field B. If one could grab and shake a single point of an infinite elastic solid one would get a displacement field u nearly identical to the vector potential A of the analogous situation, a short oscillating dipole (at least for the far field). Unfortunately the comparison of electrodynamics and elastodynamics starts to breakdown when we try to consider relativistic or gauge transformations of Am.as compared with u.   3. Tinker Toy spin.   I call the function TTS = exp[-iwt] exp[-iNf/2] (here f is the cylindrical coordinate) my Tinker Toy Spin. If one sets TTS equal to some complex constant C of unit magnitude one can get the pretty graph of a helicoid (plot those points x, y, and t (suppress z) that satisfy TTS = C). With a rudimentary knowledge of relativistic transformations one can figure out what this function looks like in different relativistic frames. There are some pretty interesting transformations one can apply to TTS. Freeze time and take a region of the space part of this function which includes a piece of the z axis and bend that piece of the z axis into a loop or any other closed curve with the possible addition of a twist. Now continue this new function into all space in some minimal way.   4. Hyperspherical Harmonics and QM in S3.   In the rudiments of quantum mechanics one deals with the wave functions and energy levels of a particle in a box. In the basics of general relativity one deals with the concept that our Universe may in fact be closed and we are introduced to the 2-sphere and 3-sphere to get a feel for this. Now let us combine these two ideas and consider the wave functions for a particle in a 3-sphere (one of constant radius). I think this leads us to consider hyperspherical harmonics, the higher dimensional analogs of the familiar spherical harmonics. Below is a short table of the harmonics appropriate for a 3-sphere with the normalization constants left out for simplicity4,   Y0,0,0 ~ constant   Y1,0,0 ~ -u_4 Y1,1,1 ~ -i(u_1 + iu_2) Y1,1,0 ~ iu_3 Y1,1,-1 ~ i(u_1 - iu_2)   Y2,0,0 ~ 4(u_1)^2 - 1 Y2,1,1 ~ iu_4(u_1 + iu_2) Y2,1,0 ~ -iu_3u_4 Y2,1,-1 ~ -iu_4(u_1 - iu_2)   Y2,2,-2 ~ -(u_1 - iu_2)^2 Y2,2,-1 ~ -u_3(u_1 - iu_2) Y2,2,0 ~ -2(u_3)^2 - (u_1)^2 - (u_2)^2 Y2,2,1 ~ u_3(u_1 + iu_2) Y2,2,2 ~ -(u_1 + iu_2)^2   Where u1, u2, u3, and u4 are the coordinates of a point on the unit 3-sphere,   u_1^2 + u_2^2 + u_3^2 + u_4^2 = 1 also u1 = sincsinqcosf u2 = sincsinqsin f u3 = sinccos q u4 = cos c   At this point one might also like to consider what are the fundamental modes for the electromagnetic field in the 3-sphere. I suspect that the above hyperspherical harmonics will again be useful. After we have this figured out of coarse we might want to consider interactions between a charged spinless particle and the electromagnetic field in S3, this we save for another sleepless night.   go to Next Section of this paper or go back to Index Of Thoughts