Eeeeeeewww!! Equations!

[note: updated, fixed a few errors in the writing of this post]

I’ve had some wonderful insights this week about why quantum theory has not come up with the twist solutions that seem to be such good solutions to things like photons and electrons. There’s always the distinct possibility that the twist ideas are just dumb crackpot ideas, and I’d kind of like to know if that’s what a physicist would think if he were to sit and really look at them. To do that, I need to try and think like a physicist, and that means having a better understanding of the equations that drive quantum theory. This will tax my ability to write a coherent post, so I will try especially hard to make this readable. I’m going to experiment with changing the blog format to include a glossary on the side, but for now I’ll just create this post and see what happens. This is going to be simplified to the point that my feeble mind can understand it–let’s review the ground-breaking equations that form the basis for quantum theory.

OK, let’s start with the simplest form that Einstein discovered from the photomultiplier experiment–the equation E=hv. This says that for any frequency v (that should be the greek letter Nu) there is a fixed energy E related as Planck’s constant h. In other words, only one possible energy is possible for a given frequency and vice-versa; energy of a particle is quantized. You cannot change the energy of a particle without somehow changing its frequency.

Next comes the Heisenberg uncertainty relation x * p >= h. This means that at the quantum scale we have to work with probability distributions, not the actual x or p of a particle. I’ve more or less been ignoring this in my thinking but its always in the back of my mind that I cannot come up with equations that explicitly specify field vectors that are observable.

Now, look at the Schroedinger (non-relativistic) wave equation. This really is just a potential energy central force solution combined with E=hv. Potential energy central force solutions are only relevant in two (or more) body problems such as electrons about a massive center (proton-neutron nucleus of atoms, for example). In 3D, you get an interesting array of solutions that will give the orbital electron energies that can be thought of as waves that circle around the center–DeBroglie thought that quantization resulted from the fact that waves have to “fit” (phase of multiple go-arounds have to match). I always doubted that–it might be true if the uncertainty relation didn’t exist, but with a probability distribution, or electron cloud, it’s not clear to me why anything would have to phase match. In any event, it doesn’t explain the quantization of particles that don’t go around in an orbit, such as a free electron or a photon. There is a free particle version of the Schroedinger that sets the potential to zero, this yields wave solutions F = A e^i*Pi(kx -vt). But this doesn’t quantize the particle or where the particle is! All of these equations only describe probability distributions of where the particle is or what state it is in (actually, the related solution of probability density). Boundary conditions (information about the system outside of the particle) are necessary to set the constants here.

Now, suppose we want to add special relativity to the mix. What does that mean? Special relativity means that spatial and time dimensions are linearly altered according to the velocity and direction of the observer’s frame of reference. This means the solutions will have to obey the Lorentz transformations as you change the frame of reference. I thought this would be very hard to do mathematically, but in reading about how this was done, I realized how clever these guys were–instead of somehow trying to incorporate the Sqrt[1-beta^2] factors into the Schroedinger wave equation, they said–lets make the Schroedinger equation solutions invariant to frame of reference changes. This is what led to the Klein-Gordon equation–they did this by forcing the solution to be equivalent if there is a spatial or time-like displacement or rotation, using a matrix that is unitary in 3D + T. Expressing the wave solution in a form where it is valid whether or not there is a displacement or rotation in 3D + T constrains the solution set to work relativistically–meaning, if the particles involved are moving close to the speed of light.

It turns out that this approach does not take into account the electron spin property, so in some cases, the wrong energies are computed for electron orbital energies, so Dirac came up with an operator-based methodology that solves this problem. Schroedinger eventually fixed this with his wave equation but Klein-Gordon got there first and got their name on the solution. I’m not going to go there today.

The point of this exercise in reviewing these equations is to understand how something as simple as field twists could have been overlooked. Note that this history shows how these discoveries were atom-based. You really cannot detect an electron until it hits an atom, and observe the energies of, say, emitted photons that result. We see quantization based on these observations, and then it becomes logical why we have Schroedinger, Klein-Gordon, and Dirac equations. But, they don’t really work for free electrons and photons! The free electron solution in an unbounded box is just a wave solution–it could be valid, it says it will be analytic, but has no explanation for why its energy is quantized. It needs boundary conditions to provide definition for its solutions for location or state (electron in a box quantum mechanics).

DeBroglie was on a path to explain the quantization with his fitting waves around an orbit, but that begs the question of why phase has to match on each orbit. Since each orbit spatial starting point is time-wise separated, I see no reason why matching phases at this point would yield a lower energy state than if they weren’t matching.

However, twists in a background field have the appealing geometric property of a lowest energy state when the twists are quantized. Twists have lots of other appealing properties, such as deriving the Lorentz special relativity transforms and deriving the correct energy of an electron based on the energy connection between electrostatic and magnetic properties, explaining the polarization of photons, and lots of other things I’ve documented on this blog ad-nauseum. That’s what keeps me going on this–the current science hasn’t developed in a way where experiments and the resulting math could observe or deduce the twists.



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