# Talking of my attitudes…

No one asked [^] me. But I want to tell you, anyway!

1. What is your opinion about the randomness of individual quantum events (such as the decay of a radioactive nuclei)?

• The randomness is only apparent
• There is a hidden determinism
• The randomness cannot be removed from any physical theory
• Randomness is a fundamental concept of nature

Nearest option(s): The randomness is only apparent.

Comments: We have to make the laws-vs-systems distinction [^]. The fundamental laws of physics are always deterministic; e.g., Newton’s law of gravity forces on objects. The behaviour of a system, composed of many objects, may or may not be deterministic.

For example, the problem of determining trajectories in a $2$-body system exchanging Newtonian gravity forces, is deterministic. A similar problem but with $3$ or more bodies shows sensitive dependence on initial conditions. The fact that the law is expressed as a differential equation requires us to specify the initial condition exactly. The sensitive dependence on IC makes the integration procedure useless—even a small change in IC gives a wildly different prediction. Thus, the behaviour of the system becomes indistinguishable from an indeterministic system, even if the law is deterministic.

Individual quantum events still involve a large number of objects.

There can be hidden mechanisms, but that will not guarantee determinism for all systems.

2. Do you believe that physical objects have their properties well defined prior to and independent of measurement?

• Yes in all cases
• Yes in some cases
• No
• I am undecided

Nearest option(s): Yes in all cases.

Comments: The nature is not in a metaphysical flux.

3. How would you respond to the question “Where exactly in the orbital of a hydrogen atom is the electron prior to a measurement?”

• It is everywhere in its orbital
• It is not possible to know with our current understanding
• It is impossible to know
• The question is meaningless

Nearest option(s): It is not possible to know with our current understanding.

Comments: The fact that it can be anywhere does not mean that it is everywhere (cf. Wheeler, Feynman).

My new approach should give a better level clarity on this question.

4. Superpositions of macroscopically distinct states, e.g. a current loop in a superposition of two magnetic fluxes,…

• are in principle possible
• will eventually be realized experimentally
• are in principle impossible
• are impossible due to collapse theory

Nearest option(s): are in principle possible.

Clarifications/Comments: The question is fuzzy/not well-defined. If a buckyball undergoes diffraction, does it qualify to be called a macroscopically distinct state? Could Wigner’s friends be such that they exhibit QM behaviour but also are distinct enough to be called macroscopically different entities? I think yes. In such a case, their superpositions are of course possible.

5. In your opinion the observer

• is a complex quantum system
• should play no fundamental role whatsoever
• plays a fundamental role in the application of the formalism, but plays no distinguished physical role
• plays a distinguished physical role

Nearest option(s): should play no fundamental role whatsoever.

Comments: The question itself is vague or ill-posed. The term “observer” is not clear.

In the ordinary sense of the term: All observers have both consciousness and body. The body, like any other physical object, is a complex quantum system. The consciousness must not play any fundamental role in any theory of physics—QM or otherwise. (It can play a role in a theory of mind-body integration, which will be a different field. It will have to be compatible with both physics and what we know about consciousness. But you couldn’t call it a sub-field of physics or psychology. It would be an inter-disciplinary field.)

If we allow an inanimate object (like a detector) to be called an “observer”: Then the nearest option would be different. It would be: The observer “is a complex system”

6. How do you understand the measurement problem?

• It is a pseudo-problem
• It is solved by decoherence
• It is solved/will be solved in some other way
• It is a severe difficulty threatening quantum mechanics
• I don’t know the problem well enough to have formed an opinion

Nearest option(s): It is solved/will be solved in some other way.

Comments: I think my new approach solves it correctly. For the time being, see the Outline document here  (PDF) [^]. It is a real problem; decoherence doesn’t solve it; but I don’t think that the absence of a solution to it has been a source of severe difficulty threatening QM itself. The absence of a good solution does make the theory incomplete. The irrational/bad philosophies brought into the field, under the disguise of solving this problem, do threaten QM (and many other areas of physics). But leaving aside the role of irrational/bad philosophies, an unsolved problem never really speaking threaten an area of physics. The same applies here.

7. What is the message of the observed violations of Bell’s inequality?

• Hidden variables are impossible
• Some notion of non-locality
• Action-at-a-distance in the physical world
• I don’t know the inequality well enough to have formed an opinion

Nearest options(s): Some notion of non-locality. Also: Unperformed measurements have no results.

Comments: The violations don’t by themselves justify (instantaneous) action at a distance, IAD for short, in the physical world; but they are compatible with IAD.

I simulate my new approach using Fourier’s theory, which means, using IAD. However, conceptually, it could easily incorporate very high but finite speeds.

8. If two physical theories give the same predictions, what properties would make you support one over the other? (you can check more than one box)

• Simplicity—simple over complex
• Determinism—deterministic over indeterministic
• Ontic—describes nature not just our knowledge of it
• Chronology—The theory that was established first

Nearest option(s): Ontic—describes nature not just our knowledge of it. Also, as implications, and in this order (of decreasing relevance): Consistency. Simplicity. Determinism. Chronology.

9. Do physicists need an interpretation of quantum mechanics?

• Yes, it helps us understand how nature behaves
• Yes, it is important for pedagogical reasons
• No, it is irrelevant as long as quantum mechanics provides us with correct predictions/results
• No, it is entirely based on personal beliefs

Nearest option(s): Yes, it helps us understand how nature behaves.

Comments: My opinion is that those who say “no” shouldn’t even call themselves physicists. But far more important point is this: The very terms of the question assume (and exhibit) something deeply wrong.

You don’t invent a theory of physics out of the blue, say just using “math”, any more than you claim that you it was revealed to you mystically. You don’t thus formulate a theory (especially a “complete” theory like QM), and then go looking for its “interpretation” in the physical world.

You start with observations, organize your knowledge of phenomena into a consistent conceptual context, and derive or apply some good ontology—i.e., spell out what kind of objects you are assuming to exist in the reality out there, their nature (the causes), the nature of their actions (the effects), etc. On this basis, you form some hypotheses, and perform experiments—i.e. controlled observations. You then integrate the experimental findings with the pre-existing knowledge, all using the conceptually consistent phenomenological-ontological context we had talked about.

On this basis, you go about proposing what forms of equations there should be, why, and then derive the exact equations (together with units that avowed Platonists hate!). If necessary, you invent maths—whether to form equations or for finding methods to solve them. You then also show how to apply the quantitative theory.

If you follow this order, there never is an occasion to separately go hunting for “interpretations” at any stage.

In QM, physicists were not idiots—they actually were geniuses. But they also brought in bad philosophy in such a manner that one part of what they produced was indistinguishable from what an idiot would propose. To be fair, if they were to have the context of certain later developments like computational modelling, non-linear dynamics and catastrophe theory, chances are somewhat bright that they might have not said so many irrational things.

I say “somewhat” bright, and not “absolutely very bright”. The reason? von Neumann did have access to computers, did work out early computational modelling theory, but nevertheless, ended up positing the collapse postulate and axiomatizing a linear theory for QM at the same time. There won’t be many people as brilliant as him. He still did this stupid thing. Similarly, Heisenberg should have been aware of the role of non-linearities. His PhD thesis was on turbulence in fluids. He even said to the effect that QM mysteries would be solved easier than the mysteries of turbulence.

But none of them took a path to the kind of natural nonlinearity which I have proposed.

Yes, bad philosophy, and the practice of deification of towering figures/groupism/ridiculing the outsiders, etc., definitely have significant effects—even on the kind of physics you propose and/or defend.

10. What characterizes the Copenhagen interpretation of quantum mechanics? (you can check multiple boxes)

• Collapse of the wavefunction upon measurement
• Indeterminism—Results are not completely specified by initial conditions
• Nonlocality, i.e. action-at-a-distance
• Quantum mechanics works well, but does not describe nature as it really is
• The correspondence principle—quantum mechanics reproduces classic physics in the limit of high quantum numbers
• The principle of complementarity—objects have complementary properties which cannot be observed or measured at the same time

Nearest option(s): In the order of importance or how core it is to the Copenhagen interpretation: 1. Correspondence principle. 2. Indeterminism. 3. Complementarity principle.

Comments: The collapse postulate was formulated by von Neumann, and not by Bohr, Heisenberg, and the other, original, advocates of the Copenhagen interpretation (CI). However, the collapse postulate only makes explicit what was already implicit in the CI. So, if the collapse postulate to be included in the CI, then its relative importance position, IMO, would be at no. 3. (Thus, the order would be: Correspondence, Indeterminism, Collapse, Complementarity.)

Indeterminism was more important to Heisenberg than to Bohr. Yes, Bohr accepted it, and advocated it too. But he belonged to an earlier, better, generation. He had arrived at the Correspondence  principle years before 1925, and insisted on it all his life.

11. What characterizes the many worlds interpretation of quantum mechanics? (you can check multiple boxes)

• The existence of multiple parallel worlds
• The existence of multiple minds belonging to one person
• Locality, i.e no action-at-a-distance
• The observer is treated as a physical system
• No wave function collapse
• Determinism—Evolution of universal wavefunction is completely governed by the wave equation
• I don’t know the interpretation well enough to have formed an opinion

Nearest option(s): In the order of importance: 1. The existence of multiple parallel worlds. Also: 2. No wave function collapse. 3. The observer is treated as a physical system.

12. What characterizes De Broglie – Bohm pilot wave interpretation of quantum mechanics? (you can check multiple boxes)

• Hidden variables in form of the particles exact positions and momenta
• Nonlocality
• Determinism—Events are completely specified by initial conditions
• Possibility of deriving Born’s Rule
• Wave function collapse
• Quantum potential—each particle has an associated potential that guides the particle
• I don’t know the interpretation well enough to have formed an opinion

Nearest option(s): In the order of importance, both of these at no. 1: (1a) Hidden variables in form of the particles exact positions and momenta. (1b) Quantum potential—each particle has an associated potential that guides the particle. Also, 2. Nonlocality.

13. What is your favourite interpretation of quantum mechanics?

• Consistent Histories
• Copenhagen
• de Broglie-Bohm
• Everett (many worlds and/or many minds)
• Information-based / information-theoretical
• Modal interpretation
• Objective collapse (e.g., GRW, Penrose)
• Quantum Bayesianism
• Statistical (ensemble) interpretation
• Transactional interpretation
• Other
• I have no preferred interpretation of quantum mechanics

Nearest option(s): Other.

Comments: By “Other” I mean: my new approach, even though, I think, what I am doing is much more than just a new “interpretation”. Yes, I do think that, eventually, my approach should also qualify to be called a new theory of (non-relativistic) quantum phenomena. (After all, I have proposed a new form of nonlinearity—i.e. new “math” too!)

If I were not to have my new approach, I would have probably said: 1. Copenhagen and 2. de Broglie-Bohm, in that order—but I would have hastened to add the necessary qualification that these interpretations, though I used them more preferentially or more often, couldn’t possibly be called my “favourites”.

14. What are your reasons for NOT favoring the Copenhagen interpretation? (you can check multiple boxes)

• The role the observer plays in determining the physical state is too important
• The paradoxes that arise on the macroscopic scale, e.g. Scrödinger’s cat and Wigner’s friend
• Nonlocality
• Quantum mechanics describes nature as it really is
• Other

Nearest option(s): Other.

Comments: Again, by “Other”, I mean: my new approach. If I were not to have my new approach, I would’ve picked 2 options: “the paradoxes”, as well as “Other” (citing the basis of the Copenhagen interpretation in poor/irrational/inconsistent/unworkable philosophy).

15. What are your reasons for NOT favoring the many worlds interpretation? (you can check multiple boxes)

• The notion of multiple worlds seems too far-fetched
• The notion of multiple minds seems too far-fetched
• The interpretation is too complex compared to others—i.e. Ockham’s razor
• The interpretation is unable to explain the Born rule
• It can never be corroborated experimentally
• Other

Nearest option(s): Other.

Comments: By “Other”, again, I mean: my new approach.

Now that I have my approach, I could stop by just saying “Other”.

If I were not to have this new approach, however, I would’ve said: Nothing about it (MWI) is worth commenting upon, except for emphatically noting that all that this “development” actually demonstrates is an absolute lack of even the most basic philosophical sense concerning the term: “universe.”

On the question of whether there can be many universes, my position is this: To say that there is only one universe is, in a vague sense, true. But a more basic truth here is this: You cannot in principle assign any numbers/quantitative measures to the concept of universe—not if, by that concept, you mean “everything that exists/has ever existed/will ever exist”. Taken in this later sense, the universe becomes an axiomatic concept of physics. And, as in any field of knowledge, you can’t quantify axiomatic concepts. Quantification is possible only when you have two or more existents to compare in a size-wise manner. But there is nothing else with which the universe can at all be compared, because the concept, by its very definition, includes everything there is.

Accordingly, the only reasonable thing I have ever found about this whole “MWI interpretation” business here is that anecdotal story about its genesis. I mean, the story that they had “copious” amounts of sherry that night. I don’t know if this story is true. If yes, I would really wonder if they had stopped only at the sherry, for their “copious” amounts. But yes, the story does make sense. Everything else about the “theory” is senseless.

16. What are your reasons for NOT favoring De Broglie-Bohm theory? (you can check multiple boxes)

• It is too complex compared to other interpretations—i.e. Ockhams razor
• It has hidden variables, which makes the theory untenable according to Bell’s inequality
• Nonlocality
• The notion of all particles possessing a quantum potential that guides them seems too far-fetched
• Other

Closest option(s): Other.

Comments: Again, I mean: my new approach. But even if I were not to have my new approach, I still would’ve picked up many other reasons for not finding this theory as acceptable or satisfactory. If you want, I will write a bit more, in a separate post. (As stated later, I may continue this post a bit further.)

17. How often have you switched to a different interpretation?

• Never
• Once
• Several times
• I have no preferred interpretation of quantum mechanics

Closest option(s): Once.

Comments: Actually, I didn’t have any preferred interpretation of QM. For instance, Einstein found the Copenhagen interpretation (CI) unsatisfactory. But since CI (plus what others like Neumann added to it) is what they mostly follow in writing text-books (or even in pop-sci books, when they show you the lay of the land), you might say that I was following CI, but without preferring it over others. Then, during my PhD times, I tried to develop a theory for the propagation of photons based on Einstein’s idea of the spatially discrete photon. So, for photons, I could be said to have been following his “interpretation” (which I no longer believe in). Etc. In answering this question I picked up “once” mainly to indicate that I have abandoned not just the CI and Einstein’s ideas, but all others too. Instead, I am now developing a completely new approach. I find it consistent and satisfactory—at least so far.

OK. That’s where the questions in the linked paper end. In the coming weeks or so, I may provide some further, more detailed comments concerning why I chose one option vs. others, provided I find enough time to do that. (See the next section.) … Further, I have also found 2–3 other, similar, surveys, with some new questions not covered in the linked paper. I may provide my answers to these additional question via a post later on.

An update regarding development of my new approach:

Some further conceptual issues have come up.

In particular, last week, I found that a significant part of the development I did in this July (perhaps going back up to this June or so) was, conceptually wrong. Plain wrong!

Some of it might work in some scenarios, but not in general. Some other suppositions/assumptions I had made (and even implemented in code) were plain wrong. So, a lot of code written earlier has to be simply junked!

However, in the process, I also found ways to tackle these issues. I’ve found that I don’t have to give up my basic ontological ideas. My basic ontology still holds—and in fact, it also shows the way to the correct generalizations (including correct quantitative predictions). OTOH, if the basic ontology (as indicated in the Outline document and in the Ontologies series of posts here) were to be outright erroneous at its core, then it would have spelt the end of my new approach. But that’s not at all the case here.

But yes, all in all, this “discovery” of where I was going wrong, has only added to the work that still needs to be done. First of all, I have to re-work through many things. Only then can I go over to the topics that were scheduled for the current week.

However, finishing up this whole enterprise (I mean, spelling out the basic essentials of the new approach, together with at least a few minimal simulations) still does seem achievable in a time-frame that’s not extended in too far a future. One or two weeks might get added to the schedule, but the overall task, I find, is still pretty much doable in the near future.

I have changed my work-flow. I am now writing down everything (whether in LaTeX or by hand, on paper) before starting writing any code, so that some simplifications introduced just for the sake of implementing some concrete simulation, don’t end up clouding my thinking about the underlying physics too. That’s what had actually happened. (In retrospect, I think I tried to rush through the things. That’s why I didn’t notice the places where I was going wrong even conceptually.)

This change in the work-flow has already proved effective. (Writing everything down isn’t always necessary. There is a value to trying some things by immediately writing some rough-and-ready simulations too. All that I am saying here is that, for my particular problem and the particular stage that I am already at, this change in the work-flow is turning out to be effective—for the time being.)

Another point. As you know, I had plans of simulating the H atom (or two interacting electrons) in a box with PBCs. I was going to take some short-cuts for evolving the dynamics. (I think I had tweeted to that effect a while ago.) Now, I’ve found that with the new mechanisms to be implemented, there no longer is any particular advantage left with those short-cuts. So, I may or may not use those short-cuts. In case you are wondering a lot: The short-cuts involved making use of the fact that what probability represents is the ideal time-fraction that a particle spends in an infinitesimally small CV (Eulerian/fixed control volume) of the domain. I had noticed this fact on my own; had never seen it being mentioned in the QM literature. (Notice, the idea makes use of the fact that a particle does have a definite position at each time instant. Thus, we are not referring to the position as measured by the position measurement apparatus.) … Anyway, these short-cuts themselves, I think, are a pretty good set of ideas; they may come in handy some time later on.

Anyway, to sum up and conclude: I found some mistakes in my conceptual development related to my new approach to QM, and therefore also in a significant part of the code that I had implemented over the last month or so. However, I’ve also have found what I believe is the correct solution for these problems. I mean to say, I now have even better (more detailed) description of the underlying physical mechanisms.  All in all, I hope to wrap up everything by, say, by September-end. May be before that.

A song I like:

(Hindi) हे, नीले गगन के तले, धरती का प्यार पले (“hey, neele gagan ke tale, dharati kaa pyaar pale”)
Singer: Mahendra Kapoor
Music: Ravi
Lyrics: Sahir Ludhianvi

[The original song (1967) is here [^].

Also check out the live version here [^]. Notice how effortlessly Mahendra Kapoor sings here, even in his mature age….

PS: Comments on the page for the second link above say that a Thai song has priority; it came out in 1965. Out of curiousity, I checked out the link [^]. Yes, the tune is absolutely the same! Obviously, Ravi picked it up either from this song, or from some other song that preceded them both!

The sound quality for this Thai video isn’t so great, though one can make out that, what we in Hindi call तान “taan” was rendered superlatively in it! A matter of pleasure!

… Another comment linked to a more recent Thai performance of the same (Thai) song, here [^]. This looks like a “cover” version. Yes, likable, but a few pauses overemphasize the rhythm, in the process losing the fluidity of the original.

…I was definitely reminded of any number of “covers” of the old Hindi songs that I’ve tried recently. I had to discard almost all of them (even those from the Sony Jam Room), simply because all these young singers seem to follow a very weird kind of a voice-culture and a way of rendering that, as a rule, ends up killing the very soul of the original. There may be exceptions here and there, but I was talking of the rule … Mind you, I don’t have a problem with singing an old Hindi song in some “Western” style as such—certainly not when it’s done on experimental or innovative basis. I am all for “fusion” too. But the style of singing these young Indian idiots follow is not at all in any of the better Western traditions either. It is some Indian-perceived version of the Western. It’s pathetic. Their careless yoddlings and predictable but pathetic stretchings of the original tune is such that one wonders if they weren’t trying to match the experiences of a drug addict or so.

… Anyway, coming back to this new Thai song by some young singer (I mean this one [^]): The “taan” which I spoke about, now appears in a slightly modern (Westernized) form of rendering here; check out at around 2:15. Yes, the song style definitely seems to have been broken at a couple of places, but I did like the “taan” portion of it. … Anyway, enough of comments! ]

History:
— 2020.08.23 17:03 IST: First published
— 2020.08.23 20:19 IST: Revision. Added several comments; expanded a bit some existing comments. Almost 3,000 words by now! Must now leave this post in whatever condition it is in.
— 2020.08.24 12:00 IST: Added further comments. 4,000+ words now. Must now leave this post in whatever condition it is in.

# String theory of engineers, for physicists and mathematicians

A Special note for the Potential Employers from the Data Science field:

Recently, in April 2020, I achieved a World Rank # 5 on the MNIST problem. The initial announcement can be found here [^], and a further status update, here [^].

All my data science-related posts can always be found here [^]

1. You know the classical wave equation:

You know the classical wave equation, right?

What’s there in it? Just:

$u(t) = \sin( \omega t )$!

or, OK, to make it more general…

$u(t) = A \cos( \omega t ) + B \sin( \omega t )$

Something like that might have passed in your mind, first. When someone says “wave”, people think the water waves, say the ocean waves. Or, they think of the light or sound waves, the interference experiments, even the wave-particle duality. Yet, curiously, when you say “wave equation”, people tend to think of the SHM (simple harmonic motion)—the oscillations of a point-mass, but not the waves in continua.

So, to make it clear, suppose I ask you:

Ah, I see what you mean. Pretty simple, too. But now it makes sense to get into a little bit of the complex algebra:

$u(x,t) = A e^{i( \vec{k}\;\cdot\;\vec{x} - \omega\,t)}$

You are likely to continue…

…Remember the Euler identity? The minus sign, because we want to have a wave that travels to the right? Oops, in the positive $x$-direction…

Ummm…

You know,

I would have to say at this juncture,

the wave equation? I mean, the differential equation. The linear one!

To which, you are likely to retort back

What a ridiculous question! Of course I know it!

OK, it goes like this…

You might then proceed to jot down the following equation in a hurried manner, more or less to get done and be over with my questioning:

$\dfrac{\partial^2 u}{\partial x^2} = \dfrac{1}{c^2} \dfrac{\partial^2 u}{\partial t^2}$

Yeah, of course, so you do seem to know it. That’s what I was saying!

You studied the topic as early as in XI or XII standard (if not in your high-school). You had mastered it—right back then. You aced your exams, always. You then went to a great engineering school, and studied waves that were a lot more complicated. Like, may be, the EM waves radiated by a radio antenna, or may be, the vibrations in the machinery and cars, whatever …. You have even mastered the simulation techniques for them. Not just FDM but also FEM, BEM, pseudo-spectral methods, and all that.

Or, may be, you weren’t driven by the lowly commercial considerations. You were really interested in the fundamentals. So, you were interested in physics.

“Fundamentals”, you remember you had said some time ago in a distant past, as if to just once again re-affirm your conviction, all in the silence of your mind. And so, obviously, it would have to be physics! It couldn’t possibly have been chemistry for you! And that’s how, you went ahead and attended a great university. Just to pursue physics.

You calculated a lot of quantum wavefunctions but only while you were in UG years—and only in order to clear those stupid exams. But you already knew that fundamental physics is where your focus really was. Real physics. Mathematical physics. Maths!

That’s why, you zipped past that ridiculously simple stage of those mere wavefunctions. You still remember that way before your formal coursework covered it, you had mastered the Dirac notation, the Heisenberg formulation (where operators are time-dependent, not the stupid wavefunction, you had announced to your stupid class-mates), the Uncertainty Principle (uh!), the Poisson brackets, and all that… You had studied it all completely on your own. Then, you had gone into the relativistic QM as well—the Klein-Gordon equation, Dirac’s equation, Feynman’s path integral formulation… All of that. Also GR. Even QFT… May be you even landed into the string theory right while you still were a high-school or UG student.

… It was long ago that you had left those idiotic wavefunctions and all way behind you. They were best left for others to look after, you just knew. That’s what you had thought, and that’s how you’d come to that conclusion.

2. Will you be able to explain its derivation, now?:

So, whether you are an engineer or a physicist, now, it indeed seems that it’s been a long time since you studied the wave equation. That’s why, if someone now asks you to explain the derivation of the wave equation, you might perhaps narrow your eyes a bit. The reason is, unless you’ve been teaching courses to UG students in the recent times, you may not be able to do it immediately. You may have to take a look at the text-book, perhaps just the Wiki? … The Wiki may not be reliable, but since your grasp has been so solid, it wouldn’t take much to mentally go on correctingt the Wiki even as you are reading through it. …Yes, it might take a little bit of time now, but not much. May be a few minutes? Half an hour at the most? May be. But that’s only because you are going to explain it to someone else…

All the same, you are super-duper-damn confident that given the derivation in the text-books (those XII standard or UG level text-books), you are going to zip through it.

Given a brilliant school-kid, you would obviously be able to explain him the derivation all the way through: each and every step of it, and all the assumptions behind them, and even the mathematical reasonability of all those assumptions, too, in turn. You could easily get it all back right in a moment—or half an hour. … “It’s high-school classical physics, damnit”—that’s what you are likely to exclaim! And, following Feynman, you think you are going to enjoy it too…

You are right, of course. After all, it’s been more than 200 years that the $1D$ wave equation was first formulated and solved. It has become an inseparable part of the very intuition of the physicist. The great physicists of the day like d’Alembert and Euler were involved in it—in analyzing the wave phenomena, formulating the equation and inventing the solution techniques. Their thought processes were, say, a cut above the rest. They couldn’t overlook something non-trivial, could they? especially Euler? Wasn’t he the one who had first written down that neat identity which goes by his name? one of the most beautiful equations ever?

That’s what you think.

Euler, Lagrange, Hamilton, … , Morse and Feschback, Feynman…

They all said the same thing, and they all couldn’t possibly be careless. And you had fully understood their derivations once upon a time.

So, the derivation is going to be a cake-walk for you now. Each and every part of it.

Well, someone did decide to take a second look at it—the derivation of the classical wave equation. Then, the following is what unfolded.

3. A second look at the derivation. Then the third. Then the fourth. …:

3.1. Lior Burko (University of Alabama at Huntsville, AL, USA) found some problems with the derivation of the transverse wave equation. So, he wrote a paper:

Burko, Lior M. (2010), “Energy in one-dimensional linear waves in a string,” European Journal of Physics, Volume 31, Number 5. doi: [^]. PDF pre-print here [^].

Abstract: “We consider the energy density and energy transfer in small amplitude, one-dimensional waves on a string and find that the common expressions used in textbooks for the introductory physics with calculus course give wrong results for some cases, including standing waves. We discuss the origin of the problem, and how it can be corrected in a way appropriate for the introductory calculus-based physics course.”

In this abstract and all the ones which follow, the emphasis in italicized bold is mine.

3.2. Eugene Butikov (St. Petersburg State University, St. Petersburg, Russia) found issues with Burko’s arguments. So, he wrote a paper (a communication) by way of a reply in the same journal:

Butikov, Eugene I. (2011) “Comment on Energy in one-dimensional linear waves in a string’,” European Journal of Physics, Volume 32, Number 6. doi: [^] . PDF e-print available here [^].

Abstract: “In this communication we comment on numerous erroneous statements in a recent letter to this journal by Burko (Eur. J. Phys. 2010 31 L71–7) concerning the energy transferred by transverse waves in a stretched string.”

3.3. C. E. Repetto, A. Roatta, and R. J. Welti (Vibration and Wave Laboratory, Physics Department, Faculty of Exact Sciences, Engineering and Surveying [per Google Translate] (UNR), Rosario Santa Fe, Argentina, and Institute of Physics, Rosario, Argentina) also found issues with Burko’s paper, and so, they too wrote another paper, which appeared in the same issue as Butikov’s:

Repetto, Roatta and Welti (2011), “Energy in one-dimensional linear waves,” European Journal of Physics, Volume 32, Number 6. doi: [^] . PDF available here [^].

Abstract: “This work is based on propagation phenomena that conform to the classical wave equation. General expressions of power, the energy conservation equation in continuous media and densities of the kinetic and potential energies are presented. As an example, we study the waves in a string and focused attention on the case of standing waves. The treatment is applicable to introductory science textbooks.”

Though they didn’t mention Burko’s paper in the abstract, the opening line made it clear that this was a comment on the latter.

3.4. Burko, the original author, replied back to both these comments. All the three were published in the same issue of the same journal:

Burko, Lior M. (2011) “Reply to comments on Energy in one-dimensional linear waves in a string’,” European Journal of Physics, Volume 31, Number 6. doi: [^]. PDF eprint available here [^].

Abstract: “In this reply we respond to comments made by Repetto et al and by Butokov on our letter (Burko 2010 Eur. J. Phys. 31 L71–7), in which we discussed two different results for the elastic potential energy of a string element. One derived from the restoring force on a stretched string element and the other from the work done to bring the string to a certain distorted configuration. We argue that one cannot prefer from fundamental principles the former over the latter (or vice versa), and therefore one may apply either expression to situations in which their use contributes to insight. The two expressions are different by a boundary term which has a clear physical interpretation. For the case of standing waves, we argue that the latter approach has conceptual clarity that may contribute to physical understanding.”

3.5. David Rowland (University of Queensland, Brisbane, Australia) also wrote a reply, which too was published in the same issue of the same journal.

Rowland, David R. (2011) “The potential energy density in transverse string waves depends critically on longitudinal motion,” European Journal of Physics, Volume 31, Number 6. doi: [^]. The author’s pre-print (pre-publication version) is available here, [^].

Abstract: “The question of the correct formula for the potential energy density in transverse waves on a taut string continues to attract attention (e.g. Burko 2010 Eur. J. Phys. 31 L71), and at least three different formulae can be found in the literature, with the classic text by Morse and Feshbach (Methods of Theoretical Physics pp 126–127) stating that the formula is inherently ambiguous. The purpose of this paper is to demonstrate that neither the standard expression nor the alternative proposed by Burko can be considered to be physically consistent, and that to obtain a formula free of physical inconsistencies and which also removes the ambiguity of Morse and Feshbach, the longitudinal motion of elements of the string needs to be taken into account,even though such motion can be neglected when deriving the linear transverse wave equation. Two derivations of the correct formula are sketched, one proceeding from a consideration of the amount of energy required to stretch a small segment of string when longitudinal displacements are considered, and the other from the full wave equation. The limits of the validity of the derived formulae are also discussed in detail.”

3.6. Butikov wrote another paper, a year later, now in Physica Scripta.

Butikov, Eugene I. (2012) ”Misconceptions about the energy of waves in a strained string,” Physica Scripta, Vol. 86, Number 3, p. 035403. doi: [^]. PDF ePrint available here [^]:

Abstract: “The localization of the elastic potential energy associated with transverse and longitudinal waves in a stretched string is discussed. Some misunderstandings about different expressions for the density of potential energy encountered in the literature are clarified. The widespread opinion regarding the inherent ambiguity of the density of elastic potential energy is criticized.

3.7. Rowland, too, seems to have continued with the topic even after the initial bout of papers. He published another paper in 2013, continuing in the same journal where earlier papers had appeared:

Rowland, David R. (2013) “Small amplitude transverse waves on taut strings: exploring the significant effects of longitudinal motion on wave energy location and propagation,” European Journal of Physics, Volume 34, Number 2. doi: [^] . PDF ePrint is available here [^].

Abstract: “Introductory discussions of energy transport due to transverse waves on taut strings universally assume that the effects of longitudinal motion can be neglected, but this assumption is not even approximately valid unless the string is idealized to have a zero relaxed length, a requirement approximately met by the slinky spring. While making this additional idealization is probably the best approach to take when discussing waves on strings at the introductory level, for intermediate to advanced undergraduate classes in continuum mechanics and general wave phenomena where somewhat more realistic models of strings can be investigated, this paper makes the following contributions. First, various approaches to deriving the general energy continuity equation are critiqued and it is argued that the standard continuum mechanics approach to deriving such equations is the best because it leads to a conceptually clear, relatively simple derivation which provides a unique answer of greatest generality. In addition, a straightforward algorithm for calculating the transverse and longitudinal waves generated when a string is driven at one end is presented and used to investigate a $cos^2$ transverse pulse. This example illustrates much important physics regarding energy transport in strings and allows the `attack waves’ observed when strings in musical instruments are struck or plucked to be approximately modelled and analysed algebraically. Regarding the ongoing debate as to whether the potential energy density in a string can be uniquely defined, it is shown by coupling an external energy source to a string that a suggested alternative formula for potential energy density requires an unphysical potential energy to be ascribed to the source for overall energy to be conserved and so cannot be considered to be physically valid.

3.8. Caamaño-Withall and Krysl (University of California, San Diego, CA, USA) aimed for settling everything. They brought in a computational engineer’s perspective too:

Caamaño-Withall, Zach and Krysl, Petr (2016) “Taut string model: getting the right energy versus getting the energy the right way,” World Journal of Mechanics, Volume 6, Number 2. doi: [^]. This being an open-access article, the PDF is available right from the doi.

Abstract: “The initial boundary value problem of the transverse vibration of a taut stringisa classic that can be found in many vibration and acoustics textbooks. It is often used as the basis for derivations of elementary numerical models, for instance finite element or finite difference schemes. The model of axial vibration of a prismatic elastic baralso serves in this capacity, often times side-by-side with the first model. The stored (potential) energy for these two models is derived in the literature in two distinct ways. We find the potential energy in the taut string model to be derived from a second-order expression of the change of the length of the string. This is very different in nature from the corresponding expression for the elastic bar, which is predictably based on the work of the internal forces. The two models are mathematically equivalentin that the equations of one can be obtained from the equations of the other by substitution of symbols such as the primary variable, the resisting force and the coefficient of the stiffness. The solutions also have equivalent meanings, such as propagation of waves and standing waves of free vibration. Consequently, the analogy between the two models can and should be exploited, which the present paper successfully undertakes. The potential energy of deformation of the string was attributed to the seminal work of Morse and Feshbachof 1953. This book was also the source of a misunderstanding as to the correct expression for the density of the energy of deformation. The present paper strives to settle this question.”

4. A standard reference:

Oh, BTW, for a mainstream view prevalent before Burko’s paper, check out a c. 1985 paper by Mathews, Jr. (Georgetown University):

Mathews Jr., W. N. (1985) “Energy in a one‐dimensional small amplitude mechanical wave,” American Journal of Physics, Volume 53, 974. doi: [^].

Abstract: We present a discussion of the energy associated with a one‐dimensional mechanical wave which has a small amplitude but is otherwise general. We consider the kinetic energy only briefly because the standard treatments are adequate. However, our treatment of the potential energy is substantially more general and complete than the treatments which appear in introductory and intermediate undergraduate level physics textbooks. Specifically, we present three different derivations of the potential energy density associated with a one‐dimensional, small amplitude mechanical wave. The first is based on the ‘‘virtual displacement’’ concept. The second is based on the ideas of stress and strain as they are generally used in dealing with the macroscopic elastic properties of matter. The third is based on the principle of conservation of energy, and also leads to an expression for the energy flux of the wave. We also present an intuitive and physical discussion based on the analogy between our system and a spring.

I could not access it, but it was quoted by most (all?) of the papers cited above (which I could).

5. Is it a settled matter, now?:

Have these last few papers settled all the issues that were raised?

Ummm… Why don’t you read the papers and decide by yourself?

6. Why bother?

“But why did you get into all this exasperating thing / stupidity / mess, when all engineers have anyway been using the wave equation to design everything from radios, TVs, Internet router hardware to cars, washing machines, and what not?”

Many of you are likely to phrase your question that way.

My answer is: Well, simply because I ran into these papers while thinking something else about the wave equation and waves. I got puzzled a bit about one very simple and stupid physical idea that had struck me. Far, far simpler than what’s discussed in the above papers. Even just a conceptual analysis of my stupid-simple idea seemed pretty funny to me. So, I’d googled on the related topics just in order to know if any one had thought of along the same lines. Which then led me to the above selection of papers.

What was that idea?

Not very important. Let me mention it some other time. I think there is much more than enough material already in this post!

In the meanwhile, browse through these papers and see if you get all the subtle arguments—all of them being accessible to engineers too, not just to physicists or mathematicians.

Come to think of it, it might be a good idea to post a shortened version of this entry at iMechanica too. … May be a few days later…

In the meanwhile, take care and bye for now…

A song I like:

(Western, Pop): “karma chamelion”
Band: Culture Club

[A Western song that is also hummable! … As always, I couldn’t (and still can’t!) make out words, though today I did browse the lyrics [^] and the Wiki on the song [^]. Back in the 1980s, it used to be quite popular in Pune. Also in IIT Madras. … I like this song for its “hummability” / “musicality” / ”lyricality” / melody or so. Also, the “texture” of the sound—the bass and the rhythm blends really well with the voices and other instrumentals. A pretty neat listen…]

# The singularities closest to you

A Special note for the Potential Employers from the Data Science field:

Recently, in April 2020, I achieved a World Rank # 5 on the MNIST problem. The initial announcement can be found here [^], and a further status update, here [^].

All my data science-related posts can always be found here [^].

0. Preamble/Preface/Prologue/Preliminaries/Whatever Pr… (but neither probability nor public relations):

Natalie Wolchover writes an article in the Quanta Magazine: “Why gravity is not like the other forces” [^].

Motl mentions this piece in his, err.. “text” [^], and asks right in the first para.:

“…the first question should be whether gravity is different, not why [it] is different”

Great point, Lubos, err… Luboš!

Having said that, I haven’t studied relativity, and so, I only cursorily went through the rest of both these pieces.

But I want to add. (Hey, what else is a blog for?)

1. Singularities in classical mechanics:

1.1 Newtonian mechanics:

Singularity is present even in the Newtonian mechanics. If you consider the differential equation for gravity in Newtonian mechanics, it basically applies to point-particles, and so, there is a singularity in this 300+ years old theory too.

It’s a different matter that Newton got rid of the singularities by integrating gravity forces inside massive spheres (finite objects), using his shells-based argument. A very ingenious argument that never ceases to impress me. Anyway, this procedure, invented by Newton, is the reason why we tend to think that there were no singularities in his theory.

1.2 Electrostatics and electrodynamics:

Coulomb et al. couldn’t get rid of the point-ness of the point-charges the way Newton could, for gravity. No electrical phenomenon was found that changed the behaviour at experimentally accessible small enough separations between two charges. In electrostatics, the inverse-square law holds through and through—on the scales on which experiments have been performed. Naturally, the mathematical manner to capture this behaviour is to not be afraid of singularities, and to go ahead, incorporate them in the mathematical formulations of the physical theory. Remember, differential laws themselves are arrived at after applying suitable limiting processes.

So, electrostatics has point singularities in the electrostatic fields.

Ditto, for classical electro-dynamics (i.e. the Maxwellian EM, as recast by Hendrik A. Lorentz, the second Nobel laureate in physics).

Singularities exist at electric potential energy locations in all of classical EM.

Lesson: Singularities aren’t specific to general relativity. Singularities predate relativity by decades if not by centuries.

2. Singularities in quantum mechanics:

2.1 Non-relativistic quantum mechanics:

You might think that non-relativistic QM has no singularities, because the $\Psi$ field must be at least $C^0$ continuous everywhere, and also not infinite anywhere even within a finite domain—else, it wouldn’t be square-normalizable. (It’s worth reminding that even in infinite domains, Sommerfeld’s radiation condition still applies, and Dirac’s delta distribution most extremely violates this condition.)

Since wavefunctions cannot be infinite anywhere, you might think that any singularities present in the physics have been burnished off due to the use of the wavefunction formalism of quantum mechanics. But of course, you would be wrong!

What the super-smart MSQM folks never tell you is this part (and they don’t take care to highlight it to their own students either): The only way to calculate the $\Psi$ fields is by specifying a potential energy field (if you want to escape the trivial solution that all wavefunctions are zero everywhere), and crucially, in a fundamental quantum-mechanical description, the PE field to specify has to be that produced by the fundamental electric charges, first and foremost. (Any other description, even if it involves complex-valued wavefunctions, isn’t fundamental QM; it’s merely a workable approximation to the basic reality. For examples, even the models like PIB, and quantum harmonic oscillator aren’t fundamental descriptions. The easiest and fundamentally correct model is the hydrogen atom.)

Since the fundamental electric charges remain point-particles, the non-relativistic QM has not actually managed to get rid of the underlying electrical singularities.

It’s something like this. I sell you a piece of a land with a deep well. I have covered the entire field with a big sheet of green paper. I show you the photograph and claim that there is no well. Would you buy it—my argument?

The super-smart MSQM folks don’t actually make such a claim. They merely highlight the green paper so much that any mention of the well must get drowned out. That’s their trick.

2.2 OK, how about the relativistic QM?

No one agrees on what a theory of GR (General Relativity) + QM (Quantum Mechanics) looks like. Nothing is settled about this issue. In this piece let’s try to restrict ourselves to the settled science—things we know to be true.

So, what we can talk about is only this much: SR (Special Relativity) + QM. But before setting to marry them off, let’s look at the character of SR. (We already saw the character of QM above.)

3. Special relativity—its origins, scope, and nature:

3.1 SR is a mathematically repackaged classical EM:

SR is a mathematical reformulation of the classical EM, full-stop. Nothing more, nothing less—actually, something less. Let me explain. But before going to how SR is a bit “less” than classical EM, let me emphasize this point:

Just because SR begins to get taught in your Modern Physics courses, it doesn’t mean that by way of its actual roots, it’s a non-classical theory. Every bit of SR is fully rooted in the classical EM.

3.2 Classical EM has been formulated at two different levels: Fundamental, and Homogenized:

The laws of classical EM, at the most fundamental level, describe reality in terms of the fundamental massive charges. These are point-particles.

Then, classical EM also says that a very similar-looking set of differential equations applies to the “everyday” charges—you know, pieces of paper crowding near a charged comb, or paper-clips sticking to your fridge-door magnets, etc. This latter version of EM is not the most fundamental. It comes equipped with a lot of fudges, most of them having to do with the material (constitutive) properties.

3.3 Enter super-smart people:

Some smart people took this later version of the classical EM laws—let’s call it the homogenized continuum-based theory—and recast them to bring out certain mathematical properties which they exhibited. In particular, the Lorentz invariance.

Some super-smart people took the invariance-related implications of this (“homogenized continuum-based”) theory as the most distinguished character exhibited by… not the fudges-based theory, but by physical reality itself.

In short, they not only identified a certain validity (which is there) for a logical inversion which treats an implication (viz. the invariance) as the primary; they blithely also asserted that such an inverted conceptual view was to be regarded as more fundamental. Why? Because it was mathematically convenient.

These super-smart people were not concerned about the complex line of empirical and conceptual reasoning which was built patiently and integrated together into a coherent theory. They were not concerned with the physical roots. The EM theory had its roots in the early experiments on electricity, whose piece-by-piece conclusions finally came together in Maxwell’s mathematical synthesis thereof. The line culminated with Lorentz’s effecting a reduction in the entire cognitive load by reducing the number of sub-equations.

The relativistic didn’t care for these roots. Indeed, sometimes, it appears as if many of them were gloating to cut off the maths from its physical grounding. It’s these super-smart people who put forth the arbitrary assertion that the relativistic viewpoint is more fundamental than the inductive base from which it was deduced.

3.4 What is implied when you assert fundamentality to the relativistic viewpoint?

To assert fundamentality to a relativistic description is to say that the following two premises hold true:

(i) The EM of homogenized continuaa (and not the EM of the fundamental point particles) is the simplest and hence most fundamental theory.

(ii) One logical way of putting it—in terms of invariance—is superior to the other logical way of putting it, which was: a presentation of the same set of facts via inductive reasoning.

The first premise is clearly a blatant violation of method of science. As people who have done work in multi-scale physics would know, you don’t grant greater fundamentality to a theory of a grossed out effect. Why?

Well, a description in terms of grossed out quantities might be fine in the sense the theory often becomes exponentially simpler to use (without an equal reduction in percentage accuracy). Who would advocate not using Hooke’s law as in the linear formulation of elasticity, but insist on computing motions of $10^23$ atoms?

However, a good multi-scaling engineer / physicist also has the sense to keep in mind that elasticity is not the final word; that there are layers and layers of rich phenomenology lying underneath it: at the meso-scale, micro-scale, nano-scale, and then, even at the atomic (or sub-atomic) scales. Schrodinger’s equation is more fundamental than Hooke’s law. Hooke’s law, projected back to the fine-grained scale, does not hold.

This situation is somewhat like this: Your $100 \times 100$ photograph does not show all the features of your face the way they come out in the original $4096 \times 4096$ image. The finer features remain lost even if you magnify the $100 \times 100$ image to the $4096 \times 4096$ size, and save it at that size. The fine-grained features remain lost. However, this does not mean that $100 \times 100$ is useless. A $28 \times 28$ pixels image is enough for the MNIST benchmark problem.

So, what is the intermediate conclusion? A “fudged” (homogenized) theory cannot be as fundamental—let alone be even more fundamental—as compared to the finer theory from which it was homogenized.

Poincaré must have thought otherwise. The available evidence anyway says that he said, wrote, and preached to the effect that a logical inversion of a homogenized theory was not only acceptable as an intellectually satisfying exercise, but that it must be seen as being a more fundamental description of physical reality.

Einstein, initially hesitant, later on bought this view hook, line and sinker. (Later on, he also became a superposition of an Isaac Asimov of the relativity theory, a Merilyn Monroe of the popular press, and a collage of the early 20th century Western intellectuals’ notions of an ancient sage. But this issue, seen in any basis—components-wise or in a new basis in which the superposition itself is a basis—takes us away from the issues at hand.)

The view promulgated by these super-smart people, however, cannot qualify to be called the most fundamental description.

3.5 Why is the usual idea of having to formulate a relativistic quantum mechanics theory a basic error?

It is an error to expect that the potential energy fields in the Schroedinger equation ought to obey the (special) relativistic limits.

The expectation rests on treating the magnetic field at a par with the static electric field.

However, there are no monopoles in the classical EM, and so, the electric charges enjoy a place of greater fundamentality. If you have kept your working epistemology untarnished by corrupt forms of methods and content, you should have no trouble seeing this point. It’s very simple.

It’s the electrons which produce the electric fields; every electric field that can at all exist in reality can always be expressed as a linear superposition of elementary fields each of which has a singularity in it—the point identified for the classical position of the electron.

We compress this complex line of thought by simply saying:

Point-particles of electrons produce electric fields, and this is the only way any electric field can at all be produced.

Naturally, electric fields don’t change anywhere at all, unless the electrons themselves move.

The only way a magnetic field can be had at any point in physical space is if the electric field at that point changes in time. Why do we say “the only way”? Because, there are no magnetic monopoles to create these magnetic fields.

So, the burden of creating any and every magnetic field completely rests on the motions of the electrons.

And, the electrons, being point particles, have singularities in them.

So, you see, in the most fundamental description, EM of finite objects is a multi-scaled theory of EM of point-charges. And, EM of finite objects was, historically, first formulated before people could plain grab the achievement, recast it into an alternative form (having a different look but the same physical scope), and then run naked in the streets shouting “Relativity!”, “Relativity!!”.

Another way to look at the conceptual hierarchy is this:

If you solve the problem of an electron in a magnetic field quantum mechanically, did you use the most basic QM? Or was it a multi-scale-wise grossed out (and approximate) QM description that you used?

Hint: The only way a magnetic field can at all come into existence is when some or the other electron is accelerating somewhere or the other in the universe.

For the layman: The situation here is like this: A man has a son. The son plays with another man, say the boy’s uncle. Can you now say that because there is an interaction between the nephew and the uncle, therefore, they are what all matters? that the man responsible for creating this relationship in the first place, namely, the son’s father cannot ever enter any fundamental or basic description?

Of course, this viewpoint also means that the only fundamentally valid relativistic QM would be one which is completely couched in terms of the electric fields only. No magnetic fields.

3.6. How to incorporate the magnetic fields in the most fundamental QM description?

I don’t know. (Neither do I much care—it’s not my research field.) But sure, I can put forth a hypothetical way of looking at it.

Think of the magnetic field as a quantum mechanical effect. That is to say, the electrostatic fields (which implies, the positions of electrons’ respective singularities) and the wavefunctions produced in the aether in correspondence with these electrostatic fields, together form a complete description. (Here, the wavefunction includes the spin.)

You can then abstractly encapsulate certain kinds of changes in these fundamental entities, and call the abstraction by the name of magnetic field.

You can then realize that the changes in magnetic and electric fields imply the $c$ constant, and then trace back the origins of the $c$ as being rooted in the kind of changes in the electrostatic fields (PE) and wavefunction fields (KE) which give rise to the higher-level of phenomenon of $c$.

But in no case can you have the hodge-podge favored by Einstein (and millions of his devotees).

To the layman: This hodge-podge consists of regarding the play (“interactions”) between the boy and the uncle as primary, without bothering about the father. You would avoid this kind of a hodge-podge if what you wanted was a basic consistency.

3.7 Singularities and the kind of relativistic QM which is needed:

So, you see, what is supposed to be the relativistic QM itself has to be reformulated. Then it would be easy to see that:

There are singularities of electric point-charges even in the relativistic QM.

In today’s formulation of relativistic QM, since it takes SR as if SR itself was the most basic ground truth (without looking into the conceptual bases of SR in the classical EM), it does take an extra special effort for you to realize that the most fundamental singularity in the relativistic QM is that of the electrons—and not of any relativistic spacetime contortions.

4. A word about putting quantum mechanics and gravity together:

Now, a word about QM and gravity—Wolchover’s concern for her abovementioned report. (Also, arguably, one of the concerns of the physicists she interviewed.)

Before we get going, a clarification is necessary—the one which concerns with mass of the electron.

4.1 Is charge a point-property in the classical EM? how about mass?

It might come as a surprise to you, but it’s a fact that in the fundamental classical EM, it does not matter whether you ascribe a specific location to the attribute of the electric charge, or not.

In particular, You may take the position (1) that the electric charge exists at the same point where the singularity in the electron’s field is. Or, alternatively, you may adopt the position (2) that the charge is actually distributed all over the space, wherever the electric field exists.

Realize that whether you take the first position or the second, it makes no difference whatsoever either to the concepts at the root of the EM laws or the associated calculation procedures associated with them.

However, we may consider the fact that the singularity indeed is a very distinguished point. There is only one such a point associated with the interaction of a given electron with another given electron. Each electron sees one and only one singular point in the field produced by the other electron.

Each electron also has just one charge, which remains constant at all times. An electron or a proton does not possess two charges. They do not possess complex-valued charges.

So, based on this extraneous consideration (it’s not mandated by the basic concepts or laws), we may think of simplifying the matters, and say that

the charge of an electron (or the other fundamental particle, viz., proton) exists only at the singular point, and nowhere else.

All in all, we might adopt the position that the charge is where the singularity is—even if there is no positive evidence for the position.

Then, continuing on this conceptually alluring but not empirically necessitated viewpoint, we could also say that the electron’s mass is where its electrostatic singularity is.

Now, a relatively minor consideration here also is that ascribing the mass only to the point of singularity also suggests an easy analogue to the Newtonian particle-mechanics. I am not sure how advantageous this analogue is. Even if there is some advantage, it would still be a minor advantage. The reason is, the two theories (NM and EM) are, hierarchically, at highly unequal levels—and it is this fact which is far more important.

All in all, we can perhaps adopt this position:

With all the if’s and the but’s kept in the context, the mass and the charge may be regarded as not just multipliers in the field equations; they may be regarded to have a distinguished location in space too; that the charge and mass exist at one point and no other.

We could say that. There is no experiment which mandates that we adopt this viewpoint, but there also is no experiment—or conceptual consideration—which goes against it. And, it seems to be a bit easier on the mind.

4.2 How quantum gravity becomes ridiculous simple:

If we thus adopt the viewpoint that the mass is where the electrostatic singularity is, then the issue of quantum gravity becomes ridiculously simple… assuming that you have developed a theory to multi-scale-wise gross out classical magnetism from the more basic QM formalism, in the first place.

Why would it make the quantum gravity simple?

Gravity is just a force between two point particles of electrons (or protons), and, you could directly include it in your QM if your computer’s floating point arithmetic allows you to deal with it.

As an engineer, I wouldn’t bother.

But, basically, that’s the only physics-wise relevance of quantum gravity.

4.3 What is the real relevance of quantum gravity?

The real reason behind the attempts to build a theory of quantum gravity (by following the track of the usual kind of the relativistic QM theory) is not based in physics or nature of reality. The reasons are, say “social”.

The socially important reason to pursue quantum gravity is that it keeps physicists in employment.

Naturally, once they are employed, they talk. They publish papers. Give interviews to the media.

All this can be fine, so long as you bear in your mind the real reason at all times. A field such as quantum gravity was invented (i.e. not discovered) only in order to keep some physicists out of unemployment. There is no other reason.

Neither Wolchover nor Motl would tell you this part, but it is true.

5. So, what can we finally say regarding singularities?:

Simply this much:

Next time you run into the word “singularity,” think of those small pieces of paper and a plastic comb.

Don’t think of those advanced graphics depicting some interstellar space-ship orbiting around a black-hole, with a lot of gooey stuff going round and round around a half-risen sun or something like that. Don’t think of that.

Singularities is far more common-place than you’ve been led to think.

Your laptop or cell-phone has of the order of $10^23$ number of singularities, all happily running around mostly within that small volume, and acting together, effectively giving your laptop its shape, its solidity, its form. These singularities is what gives your laptop the ability to brighten the pixels too, and that’s what ultimately allows you to read this post.

Finally, remember the definition of singularity:

A singularity is a distinguished point in an otherwise finite field where the field-strength approaches (positive or negative) infinity.

This is a mathematical characterization. Given that infinities are involved, physics can in principle have no characterization of any singularity. It’s a point which “falls out of”, i.e., is in principle excluded from, the integrated body of knowledge that is physics. Singularity is defined not on the basis of its own positive merits, but by negation of what we know to be true. Physics deals only with that which is true.

It might turn out that there is perhaps nothing interesting to be eventually found at some point of some singularity in some physics theory—classical or quantum. Or, it could also turn out that the physics at some singularity is only very mildly interesting. There is no reason—not yet—to believe that there must be something fascinating going on at every point which is mathematically described by a singularity. Remember: Singularities exist only in the abstract (limiting processes-based) mathematical characterizations, and that these abstractions arise from the known physics of the situation around the so distinguished point.

We do know a fantastically great deal of physics that is implied by the physics theories which do have singularities. But we don’t know the physics at the singularity. We also know that so long as the concept involves infinities, it is not a piece of valid physics. The moment the physics of some kind of singularities is figured out, the field strengths there would be found to be not infinities.

So, what’s singularity? It’s those pieces of paper and the comb.

Even better:

You—your body—itself carries singularities. Approx. $100 \times 10^23$ number of them, in the least. You don’t have to go looking elsewhere for them. This is an established fact of physics.

Remember that bit.

6. To physics experts:

Yes, there can be a valid theory of non-relativistic quantum mechanics that incorporates gravity too.

It is known that such a theory would obviously give erroneous predictions. However, the point isn’t that. The point is simply this:

Gravity is not basically wedded to, let alone be an effect of, electromagnetism. That’s why, it simply cannot be an effect of the relativistic reformulations of the multi-scaled grossed out view of what actually is the fundamental theory of electromagnetism.

Gravity is basically an effect shown by massive objects.

Inasmuch as electrons have the property of mass, and inasmuch as mass can be thought of as existing at the distinguished point of electrostatic singularities, even a non-relativistic theory of quantum gravity is possible. It would be as simple as adding the Newtonian gravitational potential energy into the Hamiltonian for the non-relativistic quantum mechanics.

You are not impressed, I know. Doesn’t matter. My primary concern never was what you think; it always was (and is): what the truth is, and hence, also, what kind of valid conceptual structures there at all can be. This has not always been a concern common to both of us. Which fact does leave a bit of an impression about you in my mind, although it is negative-valued.

A song I like:

(Hindi) ओ मेरे दिल के चैन (“O mere, dil ke chain”)
Singer: Lata Mangeshkar
Music: R. D. Burman
Lyrics: Majrooh Sultanpuri

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I think I have run the original version by Kishore Kumar here in this section before. This time, it’s time for Lata’s version.

Lata’s version came as a big surprise to me; I “discovered” it only a month ago. I had heard other young girls’ versions on the YouTube, I think. But never Lata’s—even if, I now gather, it’s been around for some two decades by now. Shame on me!

To the $n$-th order approximation, I can’t tell whether I like Kishor’s version better or Lata’s, where $n$ can, of course, only be a finite number though it already is the case that $n > 5$.

… BTW, any time in the past (i.e., not just in my youth) I could have very easily betted a very good amount of money that no other singer would ever be able to sing this song. A female singer, in particular, wouldn’t be able to even begin singing this song. I would have been right. When it comes to the other singers, I don’t even complete their, err, renderings. For a popular case in point, take the link provided after this sentence, but don’t bother to return if you stay with it for more than, like, 30 seconds [^].

Earlier, I would’ve expected that even Lata is going to fail at the try.

But after listening to her version, I… I don’t know what to think, any more. May be it’s the aforementioned uncertainty which makes all thought cease! And thusly, I now (shamelessly and purely) enjoy Lata’s version, too. Suggestion: If you came back from the above link within 30 seconds, you follow me, too.

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