Harmonic solutions as Riemannian oscillators ?

[glengarry]

Harmonic solutions as "Riemannian oscillators"?

Has anyone heard of this idea before? Basically, you just solve the Schrodinger equation using n-dimensional spherical boundary conditions. Given n=2, you get what look like atomic orbitals. But rather than going the Born probalisitic route by squaring the result, you take the Einstein GR route by viewing it as oscillations of a Riemannian manifold. (Honestly, I've never liked the Standard Model as a conceptual framework anyway. Call me old-fashioned, but I much prefer the tangibility of differential geometry.)

Using wavefunctions of amplitude→∞ and [itex]\nu[/itex]→0, we can model gravity. But if amplitude→0 and [itex]\nu[/itex]→∞, we can model the tiny oscillators of Planck's blackbody work. I think I might have heard a disillusioned string theorist kicking around something like this a long time ago.

 

[glengarry]

Anyhoo, I was thinking that since we are dealing with geometric objects (even if they *are* undergoing periodic transformations wrt time), we can deform them into the 4th dimension in order to create hyperspherical oscillators. In other words, the bounding surface is now connected into a point. Given that the boundary has a constant amplitude=0, this models a gravity field very well (as long as we keep the frequency→0).

Then, you can add as many of these "Riemannized wavefunctions" together as you wish, in the manner of simple wave addition. You can do the same thing with oscillators of very high frequency in order to model bits of matter. To make it "realistic", you don't want any of the boundary points to initially coincide. Now everything occupies the same space. Mathematically, all you should be trying to do is to minimize the maximum amplitude of the composite waveform. This allows us to obey what I see as the most important physical principle... the 2nd law of thermodynamics.

I've been thinking about this quite alot, and I fully realize it has nothing to do with the canonical/statistical understanding of QM. (I considered putting this into the GR forum, but I figured that the QM stuff was more conceptually crucial than the GR stuff.) I just think that there has been too much theoretical focus on conflating "quantization" with "smallness". It is *always* said that QM works beautifully with the world of the very small and that GR works beautifully with the world of the very large. But it seems more theoretically pleasing to just write scalar differences off as being relative purely to the natural resolving limits of our "real world" measuring devices.

From this picture, the concept of atomic matter seems to be able to emerge without too much trouble. After all, we are still using the very same "electron orbital" picture as ever. The only question to ask ourselves is whether the minimization process that I mentioned above is at all compatible with what we observe in nature. That is, whether atoms will combine into interesting molecular complexes, whether these complexes will be able to effectively signal one another over arbitrary distances, and whether everything above will occur within the large scale structure of space that we see as planets, stars, galaxies, superclusters, etc.

 

[tom.stoer]

I think the principle idea is to generalize

[tex]H = p^2 \to -\Delta[/tex]

on arbitrary Riemann manifolds M; using the metric g on M one can construct the Laplace-Beltrami operator and therefore the Hamiltonian

[tex]H \to -\Delta_g[/tex]

 

[glengarry]

It's really tough for me to see how the ordinary language of QM can be applied here, since this entire picture just consists of universally defined Riemannian oscillators. Mass is not involved, and neither, therefore is energy. It also seems that algebraic operators only make sense when dealing with precisely defined locations. All of the objects here are defined within the same scale. We can think of this scale as a being the entirety of a theoretical "universe", and we can just arbitrarily set the size to unity.

A conceptual problem that one may have with this picture is how to recover particle-like entities from it. The clue lies in what happens with the mapping of the "flat" harmonic solutions into a spherical form. Thinking one-dimensionally, we have what look like the oscillating loops of string theory. But you have to go up the 2nd dimension to see the metric deformations that occur when bending a flat disc into a sphere. In this case, the closer you get to the "boundary point" while traveling along a longitude, the more the latitudinal metric shrinks until it finally disappears. We can think of the north pole as being the original center of the disc, and the south pole as being the original circular boundary.

Using 3 dimensions, there are *two* shrinking latitudinal dimensions. This fact allows us to recover the particulate nature of the quantum world. That is, even though an electron or atom is mathematically defined within the same boundary conditions as a galaxy's gravity field, the fact that the metric of a given wavefunction shrinks faster than its amplitude as we approach its boundary point causes it to approach infinite "amplitudinal density", for want of a better term. Particles will emerge.

To derive a universe like ours, it is important to use a general rule that amplitude and oscillator frequency are inversely related. Quantum particles (eg. electrons, quarks) will have effective [itex]amplitude \to 0[/itex] and [itex]\nu \to \infty[/itex]. Gravity fields will have effective [itex]amplitude \to \infty[/itex] and [itex]\nu \to 0[/itex]. Atoms will be somewhere in between.

Mathematically, everything fully occupies the same universal space. But we just need to develop an effective theory of signal propagation within this space in order to recover the EM phenomena. We can say that everything is always entangled within everything else; I think many QM theorists would be happy to entertain this concept.

Gravity (and everything related to 2nd law/thermodynamics) is included within the picture when we always keep the maximum amplitude of the composite/universal waveform to a minimum.