Dr. Larson's Blog

Relativistic Quantum Mechanics

 

Preonic Force 1. One of the reasons my ABC Preon Model hasn't been considered seriously within the physics community concerns the lack of highly complex math. Highly complex math tends to be one of the hallmarks of modern physical theory, and when a new model is put forth without it, the model is often rejected as the work of a crank. Even though the ABC Preon Model does have math, the math is of the simple sort so far, since it consists pretty much of just addition and subtraction, as these simple tools are all that is needed to make numerous predictions. (All of which have been experimentally verified, by the way.) But not addressed yet in publication has been the problem of the force law between the preons, along with the ramifications that such a law would have.

 

Preonic Force 2. Before we get into the preonic force specifics, a few posts should be devoted to where things stand with the competing Standard Model. The standard model has a lot of complicated math, some of which has been featured on the TV comedy "The Big Bang Theory". I took most of the classes explaining the standard model's theory back in grad school about 30 years ago, and I was very disgusted with it at the time. In an earlier time period, physicists would relate math to underlying physical models. Matter was either thought to consist of a continuum of stuff, or it was thought to consist of elementary particles that actually existed and interacted with each other through forces. But in the new way of thinking the math became divorced from such underlying pictures and began to stand on its own.

 

Preonic Force 3. While the complicated math of the standard model does, more or less, correlate to many of the experimental results know to exist in high energy physics experiments, it doesn't do things very well. It was only on rare occasions that the math made any specific predictions of what the energy would be for various of its phenomena, and it couldn't really predict at which mass certain particles would appear. It did make predictions for things, such as predicting a top quark or the Higgs Boson, but it utterly failed at predicting what masses these particles would have once they were produced. (The simple math of the ABC Preon Model does predict masses, however.)

 

Preonic Force 4. Even beyond the problem of predicting certain experimental results, such as the top quark or Higgs mass, I was greatly disturbed 30 years ago at a more fundamental issue regarding the Standard Model's mathematical underpinning. The problem was that it couldn't even result in a calculation of meson masses! In the Standard Model, mesons are believed to consist of a quark and an anti-quark. That is, in the Standard Model, mesons are simple two body systems. Over the past several centuries, the two-body problem has been studied in great detail, with significant advances made in understanding it. The gravitational two body problem can be used to arrive at very accurate calculations for orbits of planets (with, for example, the planet and sun being the two bodies, and other planets perturbing the more simple two body solution). In atomic physics, the Hydrogen atom, consisting of a single proton and a single electron, was also a two body problem. The solution to the two body problem in the case of the non-relativistic, non-magnetic Hydrogen atom was solved to an exquisite detail, and the results verified by comparing it to spectroscopic data. It is a very beautiful theory. But despite the equally simple two body state of mesons, the Standard Model's theory of quantum chromodynamics has been an utter failure at describing mesonic matter.

 

Preonic Force 5. While the Standard Model has thus far failed to solve the problem of two-body mesonic states, the ABC Preon Model has its own similar failure, since at this point in time the ABC Preon Model has so far not been able to solve the problem of lepton masses. In the ABC Preon Model, massive leptons (the electron, muon and tauon) are all assumed to be an anit-A Preon bound to a B Preon by the newly proposed neutrinic force; that is, the leptons are simply two-body states. Since they are simple two body states, it should be possible to set an attraction constant, run the math, and arrive at a prediction of what the masses should be for the electron, muon and tauon. Unfortunately, the problem is far from simple.

 

Preonic Force 6. The big problem facing the calculation of the lepton masses within the ABC Preon Model is the fact that the problem concerns both quantum mechanics AND relativity. The standard Hydrogen atom calculation involves quantum mechanics, but the standard Hydrogen atom solution is non-relativistic. Schrodinger's Equation is the non-relativistic equation that is used in first order to calculate the spectroscopic evidence of Hydrogen atomic decays, and relativistic corrections are done through a perturbation series. A perturbation series is essentially a series of ever smaller corrections to an existing solution. This all works quite well for atomic Hydrogen, since the energy of the electron as it orbits the Hydrogen nucleus is non-relativistic. But for the ABC Preon Model we know that the electron's mass is far less than the mass of the preons, and hence we know that what we now need is a completely relativistic solution. If we can come up with a new treatment of the Hydrogen atom that is exact, then that treatment might be useful for the ABC Preon Model by simply changing the constants involved in the analogous forces. So at this point (about a year ago) my focus shifted to a study of the Hydrogen atom.

 

Preonic Force 7. As mentioned in an earlier post, the present Standard Model treatment of relativistic quantum mechanics is quite poor, as it results in a situation where it can't even calculate the simple problem of mesonic masses. About a year or two ago I was working with a new method for solving the relativistic quantum mechanical problem. After playing around with a few simple approximations, I came to the belief that not only must one include quantum mechanics and relativity in the solution, but one must also take into account the factor of the spin of the particles. Things were getting very complicated.

 

Preonic Force 8. In addition to relativistic corrections, the Hydrogen atom also has another perturbation that must be accounted for. Both the electron and the proton have an internal magnetic moment. It has been speculated that this magnetic moment results from the electron and proton spinning, since a spinning charge will produce a magnetic moment, but at this point in time we should consider that to be speculation. Nonetheless, there is indeed a magnetic effect within the Hydrogen atom, and this magnetic effect does lead to a separation of energy states. This effect is called hyperfine splitting, since it splits the spectroscopic lines by a very small amount. The calculation of hyperfine splitting is done similarly to the calculation of relativistic corrections. A perturbation analysis is performed by first calculating the states of hydrogen without the magnetic field, and then perturbing that solution with the magnetic effect. This perturbation analysis leads to a very accurate calculation for the hyperfine splitting. The relativistic perturbation increases the accuracy even further.

 

Preonic Force 9. Unfortunately for the ABC Preon Model, I could not use a simple non-relativistic equation that neglects magnetic effects to get at anything resembling a first pass solution. The problem for the binding within the ABC Preon Model is that the effects of relativity and magnetism are far too great to ignore; even dominant. For that reason, I needed a solution to the Hydrogen atom that doesn't treat relativity and magnetism as small perturbations to an otherwise exact solution for the non-relativistic, non-magnetic electric-only case. Rather, what I needed was a treatment that includes relativity, magnetism and electricity all on an equal footing from the beginning.

 

Preonic Force 10. About a year ago, as I began my work on a new approach to the relativistic, hyperfine, quantum mechanical treatment of the Hydrogen atom, I recalled that I might have started down this path 30 years ago. And I recalled at the time being told that "You can't do that!". Fellow graduate students explained something about commutators, and how I was violating some or another important principle. But I went on then (only to fail) and I will tarry on now as well. The one thing I know now that I did not know then, and something that is not taught at all, is that all this business about what you can and can't do with the math only arises because someone long ago went down a path that led to that. But what caused those results to be accepted by others was the fact that they agreed with the experimental results! It was not God handing them tablets on a mountain. So we'll next review what experimental data led to what things, and see if we can go down a different path to get to the same results for the non-relativistic case (we can) and then try the more generalized case.

 

 

Preonic Force 12. The Electric Force law is quite well established experimentally. Early on, it was noticed that if you rub a couple of substances together you can see some effect. For instance, you can rub a baloon against your hair. If you take the baloon away, your hair might stand on end. You can stick the baloon to a wall. Over time, it became realized that what was happening was that something was being transferred from one substance to the other. Then, it became possible to more closely specify how much of that substance was transferred, and a force law for this physical effect could be proposed.

 

 

Preonic Force 14. In addition to the electric force, another force of nature was also descovered. Some of the earliest discoveries of this force involved lodestones - stones that magically seemed to attract or repel each other, depending upon how they were aligned. Another force was observed to exist whenever charges were in motion through wires. Wires that had currents flowing through them in the same direction would attract each other. Eventually experiments were done to refine this force, which was called the magnetic force. The magnetic force is quite complicated, but fortunately some simplification is possible when we look at the potential energy.

 

 

 

 

 

 

 

Preonic Force 21. Before moving on to our goal of a relativistic quantum mechanics, at this point it is good to review what has been described in the most recent posts. We’ve now derived Schrodinger’s Equation by starting with simple observations: the energy is proportional to frequency and the momentum is inversely proportional to wavelength. We then recognized that these relations are consistent with an underlying wave description for the photon. We then took partial derivatives of that wave equation to arrive at expressions for the energy and momentum in terms of those derivatives. We then postulated that those same derivative relationships could be used for matter waves, and that resulted in Schrodinger’s Equation. It was as simple as those few steps. I believe this is more or less how mankind’s advance into quantum mechanics was made. Specifically, certain empirical truths where found (relating energy to frequency, for example) and then a guess was made as to the nature of matter leading to such relationships. But by the time I took quantum mechanics in school, it was about 50 years old. Teaching shifted to emphasizing mathematical underpinnings rather than physical ones. And that is what led to statements about “you can’t do that!” if I deviated off of the presently approved mathematical derivations. But of course we can deviate from them. We just did. And we will stubbornly follow this approach next, as we work toward a non-perturbative, hyperfine-inclusive, relativistic treatment for the Hydrogen atom.

 

 

 

 

 

 

 

 

 

Preonic Force 30. At this point in our development, it is time to step back and recall where we are and how we got here. Over the past dozen or so posts, the math has gotten quite hairy. But the physics behind it all has been very simple. All that has been done is to take observations for energy, momentum, electric forces, and magnetic forces and assume there is some underlying wave that is consistent with those observations at the quantum level. With some basic assumptions for that wave, assumptions that are empirically valid in the non-relativistic limit, we then arrived at our equation for the complete equation for the Hydrogen atom. So we see that as complex as the math has become, the underlying physics is very simple in this approach. And while others search for mathematical simplicity, it has long been my view that it will be nature that is the simple thing; our mathematical analysis of it may not be. So that is where we are now. But now we need to solve that monstrous equation to determine what the first few wave functions are!

 

 

 

 

 

Preonic Force 35. Conclusion. In Post 34, we got to where I am today in my attempt to solve the exact hydrogen atom quantum mechanical equation. As discussed some there, things are rather unruly. The present thoughts concerning this involve the fact that one can, in principle, reduce terms that are products of trig functions to linear trig functions. This would linearize the problem, which might make it tractable. Also, one could, in principle, turn things over to a computer once the linearization is complete. Usually in physics when you evaluate the highest frequency terms (high n here) the terms rapidly go to zero. This is due to the fact that extremely variable terms tend not to match closely to the underlying physical entities. So perhaps one could replace the infinite sums by sums that only involve the first ten or twenty terms, and maybe it is possible to unwind this that way. It will take a lot of effort to do, and once done there is no guarantee that it will agree with the experimental results, as it is possible the treatment here missed something in the derivation that the contemporary approach does in a superior fashion. In the past week I've also given some more thought about a direct integration of the equation, without using trial solutions. That approach also has rather nasty problems. So that's where I am now - quite a bit of progress made, but quite a way from a solution!