An update on my research

28th February is the National Science Day in India.

The story goes that it was on this day (in 1928) that C. V. Raman discovered the effect known by his name.

I don’t believe that great discoveries like that are made in just one single day. There is a whole sequence of many crucially important days involved in them.

Yes, on this day, Raman might have achieved a certain milestone or made a key finding regarding his discovery. However, even if true in this case (which I very much doubt), it’s not true in general. Great discoveries are not made in a single day; they are usually spread over much longer span of time. A particular instant or a day has more of just a symbolic value—no matter how sudden the discovery might have looked to someone, including to the discoverer.

There of course was a distinguished moment when Kekule, in his famous dream, saw a snake swallowing its own tail. However, therefore to say that he made the discovery concerning the ring structure of the benzene molecule, just in a single moment, or in a single flash of imagination, is quite a bit of a stretch.

Try it out yourself. Think of a one-line statement that encapsulates the findings of a discovery made by a single man. Compare it with another statement which encapsulates any of the previous views regarding the same matter (i.e., before this discovery came along). This way, you can isolate the contributions of a single individual. Then analyze those contributions. You would invariably find that there are several different bits of progress that the discovery connected together, and these bits themselves (i.e., the contributions made individually by the discoverer himself) were not all discovered on the same day. Even if a day or an hour is truly distinctive in terms of the extent of progress made, it invariably has the character of taking an already ongoing process to a state of completion—but not of conducting that entire process. Mystical revelation is never a good metaphor to employ in any context—not even in the spiritual matters, let alone in the scientific ones.

Anyway, it’s nice that they didn’t choose Raman’s birth-day for this Day, but instead chose a day that was related to his most famous work in science. Good sense! And easy to remember too: 28-02-’28.

Let me celebrate this year’s Science Day in my own, small, personal way. Let me note down a bit of an update on my research.


1. I have had a bit of a correspondence, regarding my new approach, with a couple of physicists. Several objections were made by them, but to cut a long story short, neither seemed to know how to get into that mode kind of thinking which most naturally leads to my main thesis, and hence helps understand it.

The typical thought process both these physicists displayed was the one which is required in finding analytical solutions of problems of a certain kind, using an analysis of a specific kind. But it is not the kind of thought process which is typically required in the computational modeling of complex phenomena. Let me remind you that my theory is nonlinear in nature. Nonlinearity, in particular, is best approached only computationally—you would be hopelessly out of your wits if you try to find analytical solutions to a nonlinear system. What you should instead pursue is: thinking in terms of the following ingredients: certain objects, an algorithm to manipulate their states, and tracing the run-time evolution of the system. You try this algorithmic way of thinking, and the whole thing (I mean understanding the nature of a nonlinear system) becomes easy. Otherwise, it looks hopelessly complicated, incomprehensible, and therefore, deeply suspicious, if not outright wrong. Both the physicists with who I interacted seemed to be thinking in terms of the linear theory of QM, thereby restricting their thought modes to only the traditional formalism based on the abstract Hilbert-spaces and linear Hermitian operators. Uh oh! Not good. QM is fundamentally nonlinear; the linear formulations of QM are merely approximations to its true nature. No matter how analytically rigorous you can get in the traditional QM, it’s not going to help you understand the true nature of quantum phenomena, simply because a linear system is incapable of throwing much light on the nonlinear system of which it is an approximation.

I believe it was out of this reason—their continuing to think in terms of linear systems defined over hyperspaces and the operator algebra—that one of them raised the objection that if \Psi in MSQM (mainstream QM) is defined on a 3ND configuration space, how come my \Psi(x,t) could be defined over the physical 3D space. He didn’t realize, even after I supplied the example of the classical N-particle molecular dynamics (MD) simulations, that using an abstract higher-dimensional space isn’t the only viable manner in which you can capture the physics of a situation. (And I had indicated right in the Outline document too, that you first try to understand how a Newtonian evolution would work for multiple, charged, point-particles as in classical physics, and only then modify this evolution by introducing the system wavefunction.)

I came to gather that apparently, some people (who follow the Bohmian mechanics doctrine) have tried to find a 3ND \leftrightarrow 3D correspondence for a decade, if not more. Apparently, they didn’t succeed. I wonder why, because doing so should be so damn straight-forward (even if it would not be easy). You only have to realize that a configuration space refers to all possible configurations, whereas what an evolution over a 3D physical space directly deals with is only one initial configuration at a time. That is what specifying the ICs and the BCs does for you.

In case of MD simulations, you don’t define a function over the entire 3ND configuration space in the first place. You don’t try to produce an evolution equation which relies on only those kinds of operators which modify all parts of the entire hyperspace-function in one shot, simultaneously. Since you don’t think in such hyperspace terms in the first place, you also don’t have to think in terms of the projection operators bringing the system dynamics down to 3D in particular cases either. You don’t do that in the context of MD simulations, and you don’t do it in the context of my approach either.

This physicist also didn’t want me to say something using analogies and metaphors, and so I didn’t mention it to him, but I guess I can use an analogy here. It will allow even a layman to get a sense of the issue right.

This physicist was insisting on having a map of an entire territory, and was more or less completely dismissing my approach on the grounds that I only supply the surveying instruments like the theodolite and the triangulation algorithm. He expected to see the map—even when a theory is at a fledgling stage. He nevertheless was confident that I was wrong because I was insisting that each physical object in the actual territory is only at one place at any given instant, that it is not spread all over the map. This analogy is not exact, but it is helpful: it does bring out the difference of focusing on only the actually followed trajectory in the configuration space, vs. an insistence on using the entirety of the configuration space for any description of an evolution. But that guy didn’t get this point either. And he wanted equations, not analogies or metaphors.

Little wonder they have not been successful in finding out what logical connection there is between the abstract 3ND hyperspace on the one hand, and the 3D physical space on the other hand. Little wonder they don’t progress despite having worked on the problem for a decade or so (as this guy himself said).

Yeah, physicists, work harder, I say! [LOL!]


2. Apart from it all—I mean all those “discussions”—I have also realized that there are several errors or confusing explanations in the Outline document which I uploaded at iMechanica on 11th February 2019. Of course, these errors are more minor in nature. There are many, many really important ideas in that document which are not in error.

The crucially important and new ideas which are valid include, just to cite a few aspects: (i) my insistence on using only those potentials that are singularly anchored into the point-particle charges, (ii) the particular nonlinearity I have proposed for the system evolution, (iii) the idea that during a measurement it is the Instrument whose state undergoes a cascade of bifurcations or catastrophic changes, whereas the System state essentially remains the same (that there is no wavefunction collapse). And, many, many other ideas too. These ideas are not only crucial to my approach but they also are absolutely new and original. (Yes, you can be confident about this part, too—else, Americans would have pointed out the existing precedence by now. (They are just looking to find errors in what(ever) I say.)) All these ideas do remain intact. The confusing part or the one having erroneous statements indeed is more minor. It concerns more with how I tried to explain things. And I am working on removing these errors too.

I have also come to realize that I need to explicitly give a set of governing equations, as well as describe the algorithm that could be used in building the simulations. Yes, the physicist had asked me for an evolution equation. I thought that any one, given the Schrodinger equation and my further verbal additions / modifications to it, could easily “get” it. But apparently, he could not. So, yes, I will explicitly write down the evolution equation for my approach, as an equation that is separate from Schrodinger’s. In the next revision of the document (or addition to it) I will not rely on the only implicitly understood constraints or modifications to the TDSE.


3. There also are some other issues which I noticed entirely on my own, and I am working on them.

One such issue concerns the way the kinetic energy is captured in the MSQM vs. how my approach ought to handle and capture it.

In MSQM, the kinetic energy consists of a sum of 1-particle Laplacian operators that refer to particle coordinates. Given the fact that my approach has the wavefunction defined over the 3D space, how should this aspect be handled? … By the time I wrote my Outline document (version 11 February 2019), I had not thought a lot about the kinetic energy part. Now, I found out, I have to think really deep about it. May be, I will have to abandon the form of Schrodinger’s equation itself to a further extent. Of course, the energy analysis will progress on the same lines (total energy = kinetic + potential), and the de Broglie relations will have to be honored. But the form of the equation may turn out to be a bit different.

You see, what MSQM does is to represent the particles using only the \Psi(x,t) field. The potential energy sure can be constructed in reference to a set of discrete particle positions even in MSQM, but what the \hat{V} operator then yields is just a single number. (In case of time-dependent potentials, the value of this variable varies in time.) The multiplication by the hyperspace-function \Psi(x,t) then serves to distribute this much amount of energy (that single number) over the entire hyperspace. Now realize that |\Psi(x,t)|^2 gives the probability. So, in a way, indirectly, even if you can calculate / compute the potential energy of the system starting from a certain set of particle positions, in the MSQM, you then have to immediately abandon them—the idea of the discrete particles. The MSQM formalism doesn’t need it—the particle positions. You deal only with the hyperspace-occupying \Psi(x,t). The formulation of kinetic energy also refers to only the \Psi(x,t) field. Thus, in MSQM, particles are ultimately represented only via the \Psi(x,t) field. The \Psi(x,t) is the particles.

In contrast, in my approach, the particles are represented directly as point-phenomena, and their positions remain significant throughout. The \Psi(x,t) field of my approach connects, and causally interacts with, the particles. But it does not represent the particles. Ontologically, \Psi(x,t) is basically different from particles, even if the background object does interacts with the particles. Naturally, why should I represent their kinetic energies via the Laplacian terms? … Got the idea? The single number that is the kinetic energy of the particles, need not be regarded as being distributed over the 3D space at all, in my approach. But in 11th February version of the Outline document, I did say that the governing equation is only Schrodinger’s. The modifications required to be made to the TDSE on account of the kinetic energy term, is something I had not even thought of, because in writing that version, I was trying focusing on getting as many details regarding the potential energy out as possible. After all, the nonlinear nature of QM occurs due to the potential term, doesn’t it?

So, I need to get issues like these straightened out too.

… All in all, I guess I can say that I am more or less (but not completely) done with the development concerning the spin-less 1-particle systems, esp. the time-independent states. So far, it seems that my approach does work fine with them. Of course, new issues continue to strike me all the time, and I continue finding answers to them as well—as happens in any approach that is completely new. New, right from the stage of the very basic ideation¬† concerning what kind of objects there should be, in the theory.

I have just about begun looking into the (spin-less) multi-particle states. That is the natural order in which the theory should progress, and my work is tracing just this same path. But as I said, I might also be revising some parts of the earlier presented theory, as and when necessary.


4. I also realized on my own, but only after the interaction with the physicists was already over, that actually, I need not wait for the entire multi-particle theory to get developed before beginning with simulations. In fact, it should be possible to handle some simple 1-particle 1D cases like the particle in a box or the QHO (quantum-mechanical oscillator) right away.

I plan to pursue these simulations right in the near future. However, I will not be able to complete pursuing all their aspects in the near future—not even in the simple cases involving just 1D simulations. I plan to do a preliminary simulation or two, and then suspend this activity until the time that I land a well-paying job in data science in Pune.


No songs section this time because I happened to post several entries almost back to back here, and in the process, I seem to have used up all the songs that were both new (not run here before) and also on the top of my mind. … May be I will return later and add a song if one strikes me easily.

Bye for now, and have a happy Science Day!


Minor editing may be done later today. Done by 20:15 hrs the same day.

 

 

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