# Some comments on QM and CM—Part 2: Without ontologies, “classical” mechanics can get very unintuitive too. (Also, a short update.)

We continue from the last post. If you haven’t read and understood it, it can be guaranteed that you won’t understand anything from this one! [And yes, this post is not only long but also a bit philosophical.]

The last time, I gave you a minimal list of different ontologies for physics theories. I also shared a snap of my hurriedly jotted (hand-written) note. In this post, I will come to explain what I meant by that note.

1. In the real world, you never get to see the objects of “classical” mechanics:

OK, let’s first take a couple of ideas from Newtonian mechanics.

1.1. Point-particles:

The Newtonian theory uses a point particle. But your perceptual field never holds the evidence for any such an object. The point particle is an abstraction. It’s an idealized (conceptual-level) description of a physical object, a description that uses the preceding mathematical ideas of limits (in particular, the idea of the vanishingly small size).

The important point to understand here isn’t that the point-particle is not visible. The crucial point here is: it cannot be visible (or even made visible, using any instrument) because it does not exist as a metaphysically separate object in the first place!

1.2. Rigid bodies:

It might come as a surprise to many, esp. to mechanical engineers, but something similar can also be said for the rigid body. A rigid body is a finite-sized object that doesn’t deform (and unless otherwise specified, doesn’t change any of its internal fields like density or chemical composition). Further, it never breaks, and all its parts react instantaneously to any forces exerted on any part of it. Etc.

When you calculate the parabolic trajectory of a cricket ball (neglecting the air resistance), you are not working with any entity that can ever be seen/ touched etc.—in principle. In your calculations, in your theory, you are only working with an idea, an abstraction—that of a rigid body having a center of mass.

Now, it just so happens that the concepts from the Newtonian ontologies are so close to what is evident to you in your perceptual field, that you don’t even notice that you are dealing with any abstractions of perceptions. But this fact does not mean that they cease to be abstract ideas.

2. Metaphysical locus of physics abstractions, and epistemology of how you use them:

2.1. Abstractions do exist—but only in the mind:

In general, what’s the metaphysics of abstractions? What is the metaphysical locus of its existence?

An abstraction exists as a unit of mental integration—as a concept. It exists in your mind. A concept doesn’t have an existence apart from, or independent of, the men who know and hold that concept. A mental abstraction doesn’t exist in physical reality. It has no color, length, weight, temperature, location, speed, momentum, energy, etc. It is a non-material entity. But it still exists. It’s just that it exists in your mind.

In contrast, the physical objects to which the abstractions of objects make a reference, do exist in the physical reality out there.

2.2. Two complementary procedures (or conceptual processings):

Since the metaphysical locus of the physical objects and the concepts referring to them are different, there have to be two complementary and separate procedures, before a concept of physics (like the ideal rigid body) can be made operational, say in a physics calculation:

2.2.1. Forming the abstraction:

First, you have to come to know that concept—you either learn it, or if you are an original scientist, you discover/invent it. Next, you have to hold this knowledge, and also be able recall and use it as a part of any mental processing related to that concept. Now, since the concept of the rigid body belongs to the science of physics, its referents must be part of the physical aspects of existents.

2.2.2. Applying the abstraction in a real-world situation:

In using a concept, then, you have to be able to consider a perceptual concrete (like a real cricket ball) as an appropriate instance of the already formed concept. Taking this step means: even if a real ball is deformable or breakable, you silently announce to yourself that in situations where such things can occur, you are not going to apply the idea of the rigid body.

The key phrases here are: “inasmuch as,” “to that extent,” and “is a.” The mental operation of regarding a concrete object as an instance of a concept necessarily involves you silently assuming this position: “inasmuch as this actual object (from the perceptual field) shows the same characteristics, in the same range of “sizes”, as for what I already understand by the concept XYZ, therefore, to that extent, this actual object “is a” XYZ.

2.2.3. Manipulation of concepts at a purely abstract level is possible (and efficient!):

As the next step, you have to be able to directly manipulate the concept as a mere unit from some higher-level conceptual perspective. For example, as in applying the techniques of integration using Newton’s second law, etc.

At this stage, your mind isn’t explicitly going over the defining characteristics of the concept, its relation to perceptual concretes, its relation to other concepts, etc.

Without all such knowledge at the center of your direct awareness, you still are able to retain a background sense of all the essential properties of the objects subsumed by the concept you are using. Such a background sense also includes the ideas, conditions, qualifications, etc., governing its proper usage. That’s the mental faculty automatically working for you when you are born a human.

You only have to will, and the automatic aspects of your mind get running. (More accurately: Something or the other is always automatically present at the background of your mind; you are born with such a faculty. But it begins serving your purpose when you begin addressing some specific problem.)

All in all: You do have to direct the faculty which supplies you the background context, but you can do it very easily, just by willing that way. You actually begin thinking on something, and the related conceptual “material” is there in the background. So, free will is all that it takes to get the automatic sense working for you!

2.2.4. Translating the result of a calculation into physical reality:

Next, once you are done with working ideas at the higher-level conceptual level, you have to be able to “translate the result back to reality”. You have to be able to see what perceptual-level concretes are denoted by the concepts related to the result of calculation, its size, its units, etc. The key phrase here again are: “inasmuch as” and “to that extent”.

For example: “Inasmuch as the actual cricket ball is a rigid body, after being subjected to so much force, by the laws governing rigid bodies (because the laws concern themselves only with the rigid bodies, not with cricket balls), a rigid body should be precisely at 100.0 meter after so much time. Inasmuch as the cricket ball can also be said to have an exact initial position (as for a rigid body used in the calculations), its final position should be exactly 100 meter away. Inasmuch as a point on the ground can be regarded as being exactly 100 meter away (in the right direction), the actual ball can also be expected, to that extent, to be at [directly pointing out] that particular spot after that much time. Etc.

2.3: A key take-away:

So, an intermediate but big point I’ve made is:

Any theory of classical mechanics too makes use of abstractions. You have to undertake procedures involving the mappings between concretes and abstractions, in classical mechanics too.

2.4. Polemics:

You don’t see a rigid body. You see only a ball. You imagine a rigid body in the place of the given ball, and then decide to do the intermediate steps only with this instance of the imagination. Only then can you invoke the physics theory of Newtonian mechanics. Thus, the theory works purely at the mental abstractions level.

A theory of physics is not an album of photographs; an observation being integrated in a theory is not just a photograph. On the other hand, a sight of a ball is not an abstraction; it is just a concretely real object in your perceptual field. It’s your mind that makes the connection between the two. Only can then any conceptual knowledge be acquired or put to use. Acquisition of knowledge and application of knowledge are two sides of the same coin. Both involve seeing a concrete entity as an instance subsumed under a concept or a mental perspective.

2.5. These ideas have more general applicability:

What we discussed thus far is true for any physics theory: whether “classical” mechanics (CM) or quantum mechanics (QM).

It’s just that the first three ontologies from the last post (i.e. the three ontologies with “Newtonian” in their name) have such abstractions that it’s very easy to establish the concretes-to-abstractions correspondence for them.

These theories have become, from a hindsight of two/three centuries and absorption of its crucial integrative elements into the very culture of ours, so easy for us to handle, they seem to be so close to “the ground” that we have to think almost nothing to regard a cricket ball as a rigid body. Doesn’t matter. The requirement of you willingly having to establish the correspondenc between the concretes and abstractions (and vice versa) still exists.

Another thing: The typical application of all the five pre-quantum ontologies also typically fall in the limited perceptual range of man, though this cannot be regarded as the distinguishing point of “classical” mechanics. This is an important point so let me spend a little time on it.

Trouble begins right from Fourier’s theory.

3. “Classical” mechanics is not without tricky issues:

3.1. Phenomenological context for the Fourier theory is all “classical”:

In its original form, Fourier’s theory dealt with very macroscopic or “every day” kind of objects. The phenomenological context which gave rise to Fourier’s theory was: the transmission of heat from the Sun by diffusion into the subterranean layers of the earth, making it warm. That was the theoretical problem which Fourier was trying to solve, when he invented the theory that goes by his name.

Actually, that was a bit more complicated problem. A simpler formulation of the same problem would be: quantitatively relating the thermal resistance offered by wood vs. metal, etc. The big point I want to note here is: All these (the earth, a piece of wood or metal) are very, very “everyday” objects. You wouldn’t hesitate saying that they are objects of “classical” physics.

3.2. But the Fourier theory makes weird predictions in all classical physics too:

But no matter how classical these objects look, an implication is this:

The Fourier theory ends up predicting infinite velocity for signal propagation for “classical” objects too.

This is a momentous implication. Make sure you understand it right. Pop-sci writers never highlight this point. But it’s crucial. The better you understand it, the less mysterious QM gets!

In concrete terms, what the Fourier theory says is this:

If you pour a cup of warm water on ground at the North pole, no doubt the place will get warmer for some time. But this is not the only effect your action would have. Precisely and exactly at the same instant, the South pole must also get warmer, albeit to a very small extent. Not only the South Pole, every object at every place on the earth, including the cell phone of your friend sitting in some remote city also must get warmer. [Stretching the logic, and according a conduction mode also to the intergalactic dust: Not just that, every part of the most distant galaxies too must get warmer—in the same instant.] Yes, the warming at remote places might be negligibly small. But in principle, it is not zero.

And that’s classical physics of ordinary heat conduction for you.

3.3. Quantum entanglement and Heisenberg’s uncertainty principle are direct consequences of the same theory:

Now, tell me, how intuitive was Fourier’s predictions?

My answer: Exactly as unintuitive as is the phenomenon of quantum entanglement—and, essentially, for exactly the same ontological-physical-mathematical reasons!

Quantum entanglement is nothing but just another application of the Fourier theory. And so is Heisenberg’s uncertainty principle. It too is a direct consequence of the Fourier theory.

3.4. Another key take-away:

So, the lesson is:

Not all of “classical” mechanics is as “intuitive” as you were led to believe.

3.5. Why doesn’t any one complain?

If classical physics too is that unintuitive, then how come that no one goes around complaining about it?

The reason is this:

Classical mechanics involves and integrates a conceptually smaller range of phenomena. Most of its application scenarios too are well understood—even if not by you, and then at least by some learned people, and they have taken care to explain all these scenarios to you.

For instance, if I ask you to work out how the Coriolis force works for two guys sitting diametrically opposite on a rotating disco floor and throwing balls at each other, I am willing to take a good bet that you won’t be able to work out everything on your own using vector analysis and Newton’s laws. So, this situation should actually be non-intuitive to you. It in fact is: Without searching on the ‘net, be quick and tell me whether the ball veers in the direction of rotation or opposite it? See? It’s just that no pop-sci authors highlight issues like this, and so, no philosophers take notice. (And, as usual, engineers don’t go about mystifying anything.)

So, what happens in CM is that some expert works out the actual solution, explains to you. You then snatch some bits and pieces, may be just a few clues from his explanation, and memorize them. Slowly, as the number of such use-cases increases, you get comfortable enough with CM. Then you begin to think that CM is intuitive. And then, the next time when your grandma asks you how come that motorcyclist spinning inside the vertical well doesn’t fall off, you say that he sticks to the wall due to the centrifugal force. Very intuitive! [Hint, hint: Is it actually centrifugal or centripetal?]

OK, now let’s go over to QM.

4. The abstract-to-concretes mappings are much more trickier when it comes to QM:

4.1. The two-fold trouble:

The trouble with QM is two-fold.

First of all, the range of observations (or of phenomenology) underlying it is not just a superset of CM, it’s a much bigger superset.

Second: Physicists have not been able to work out a consistent ontology for QM. (Most often, they have not even bothered to do that. But I was talking about reaching an implicit understanding to that effect.)

So, they are reduced to learning (and then teaching) QM in reference to mathematical quantities and equations as the primary touch-stones.

4.2. Mathematical objects refer to abstract mental processes alone, not to physical objects:

Now, mathematical concepts have this difference. They are not only higher-level abstractions (on top of physical concepts), but their referents too in themselves are invented and not discovered. So, it’s all in the mind!

It’s true that physics abstractions, qua mental entities, don’t exist in physical reality. However, it also is true that the objects (including their properties/characteristics/attributes/acctions) subsumed under physics concepts do have a physical existence in the physical world out there.

For instance, a rigid body does not exist physically. But highly rigid things like stones and highly pliable or easily deformable things like a piece of jelly or an easily fluttering piece of cloth, do exist physically. So, observing them all, we can draw the conclusion that stones have much higher rigidity than the fluttering flag. Then, according an imaginary zero deformability to an imaginary object, we reach the abstraction of the perfectly rigid body. So, while the rigid body itself does not exist, rigidity as such definitely is part of the natural world (I mean, of its physical aspects).

But not so with the mathematical abstractions. You can say that two (or three or $n$ number of) stones exist in a heap. But what actually exists are only stones, not the number $2$, $3$, or $n$. You can say that a wire-frame has edges. But you don’t thereby mean that its edges are geometrical lines, i.e., objects with only length and no thickness.

4.3. Consequence: How physicists hold, and work with, their knowledge of the QM phenomena:

Since physicists could not work out a satisfactory ontology for QM, and since concepts of maths do not have direct referents in the physical reality as apart from the human consciousness processing it size-wise, their understanding of QM does tend to be a lot more shaky (the comparison being with their understanding of the pre-quantum physics, esp. the first three ontologies).

As a result, physicists have to develop their understanding of QM via a rather indirect route: by applying the maths to even more number of concrete cases of application, verifying that the solutions are borne out by the experiments (and noting in what sense they are borne out), and then trying to develop some indirect kind of a intuitive feel, somehow—even if the objects that do the quantum mechanical actions aren’t clear to them.

So, in a certain sense, the most talented quantum physicists (including Noble laureates) use exactly the same method as you and me use when we are confronted with the Coriolios forces. That, more or less, is the situation they find themselves in.

The absence of a satisfactory ontology has been the first and foremost reason why QM is so extraordinarily unintuitive.

It also is the reason why it’s difficult to see CM as an abstraction from QM. Ask any BS in physics. Chances are 9 out of 10 that he will quote something like Planck’s constant going to zero or so. Not quite.

4.4. But why didn’t any one work out an ontology for QM?

But what were the reasons that physicists could not develop a consistent ontology when it came to QM?

Ah. That’s too complicated. At least 10 times more complicated than all the epistemology and physics I’ve dumped on you so far. That’s because, now we get into pure philosophy. And you know where the philosophers sit? They all sit on the Humanities side of the campus!

But to cut a long story short, very short, so short that it’s just a collage-like thingie: There are two reasons for that. One simple and one complicated.

4.4.1. The simple reason is this: If you don’t bother with ontologies, and then, if you dismiss ideas like the aether, and go free-floating towards ever higher and still higher abstractions (especially with maths), then you won’t be able to get even EM right. The issue of extracting the “classical” mechanical attributes, variables, quantities, etc. from the QM theory simply cannot arise in such a case.

Indeed, physicists don’t recognize the very fact that ontologies are more basic to physics theories. Instead, they whole-heartedly accept and vigorously teach and profess the exact opposite: They say that maths is most fundamental, even more fundamental than physics.

Now, since QM maths is already available, they argue, it’s only a question of going about looking for a correct “interpretation” for this maths. But since things cannot be very clear with such an approach, they have ended up proposing some 14+ (more than fourteen) different interpretations. None works fully satisfactorily. But some then say that the whole discussion about interpretation is bogus. In effect, as Prof. David Mermin characterized it: “Shut up and calculate!”

That was the simple reason.

4.4.2. The complicated reason is this:

The nature of the measurement problem itself is like that.

Now, here, I find myself in a tricky position. I think I’ve cracked this problem. So, even if I think it was a very difficult problem to crack, please allow me to not talk a lot more about it here; else, doing so runs the risk of looking like blowing your own tiny piece of work out of all proportion.

So, to appreciate why the measurement problem is complex, refer to what others have said about this problem. Coleman’s paper gives some of the most important references too (e.g., von Neumann’s process 1 vs. process 2 description) though he doesn’t cover the older references like the 1927 Bohr-Einstein debates etc.

Then there are others who say that the measurement problem does not exist; that we have to just accept a probabilistic OS at the firmware level by postulation. How to answer them? That’s a homework left for you.

5. A word about Prof. Coleman’s lecture:

If Prof. Coleman’s lecture led you to conclude that everything was fine with QM, you got it wrong. In case this was his own position, then, IMO, he too got it wrong. But no, his lecture was not worthless. It had a very valuable point.

If Coleman were conversant with the ontological and epistemological points we touched on (or hinted at), then he would have said something to the following effect:

All physics theories presuppose a certain kind of ontology. An ontology formulates and explains the broad nature of objects that must be assumed to exist. It also puts forth the broad nature of causality (objects-identities-actions relations) that must be assumed to be operative in nature. The physics theory then makes detailed, quantitative, statements about how such objects act and interact.

In nature, physical phenomena differ very radically. Accordingly, the phenomenological contexts assumed in different physical theories also are radically different. Their radical distinctiveness also get reflected in the respective ontologies. For instance, you can’t explain the electromagnetic phenomena using the pre-EM ontologies; you have to formulate an entirely new ontology for the EM phenomena. Then, you may also show how the Newtonian descriptions may be regarded as abstractions from the EM descriptions.

Similarly, we must assume an entirely new kind of ontological nature for the objects if the maths of QM is to make sense. Trying to explain QM phenomena in terms of pre-quantum ontological ideas is futile. On the other hand, if you have a right ontological description for QM, then with its help, pre-QM physics may be shown as being a higher-level, more abstract, description of reality, with the most basic level description being in terms of QM ontology and physics.

Of course, Coleman wasn’t conversant with philosophical and ontological issues. So, he made pretty vague statements.

6. Update on the progress in my new approach. But RSI keeps getting back again and again!

I am by now more confident than ever that my new approach is going to work out.

Of course, I still haven’t conducted simulations, and this caveat is going to be there until I conduct them. A simulation is a great way to expose the holes in your understanding.

So take my claim with a pinch of salt, though I must also hasten to note that with each passing fortnight (if not week), the quantity of the salt which you will have to take has been, pleasantly enough (at least for me), decreasing monotonically (even if not necessarily always exponentially).

I had written a preliminary draft for this post about 10 days ago, right when I wrote my last post. RSI had seemed to have gone away at that time. I had also typed a list of topics (sections) to write to cover my new approach. It carried some 35+ sections.

However, soon after posting the last blog entry here, RSI began growing back again. So, I have not been able to make any substantial progress since the last post. About the only things I could add were: some 10–15 more section or topic names.

The list of sections/topics includes programs too. However, let me hasten to add: Programs can’t be written in ink—not as of today, anyway. They have to be typed in. So, the progress is going to be slow. (RSI.)

All in all, I expect to have some programs and documentation ready by the time Q1 of 2021 gets over. If the RSI keeps hitting back (as it did the last week), then make it end-Q2 2021.

OK. Enough for this time round.

A song I like:

[When it comes to certain music directors, esp. from Hindi film music, I don’t like the music they composed when they were in their elements. For example, Naushad. For example, consider the song: मोहे पनघट पे (“mohe panghat pe”). I can sometimes appreciate the typical music such composers have produced, but only at a somewhat abstract level—it never quite feels like “my kind of music” to me. Something similar, for the songs that Madan Mohan is most famous for. Mohan was a perfectionist, and unlike Naushad, IMO, he does show originality too. But, somehow, his sense of life feels like too sad/ wistful/ even fatalistic to me. Sadness is OK, but a sense of inevitability (or at least irromovability) of suffering is what gets in the way. There are exceptions of course. Like, the present song by Naushad. And in fact, all songs from this move, viz. साथी (“saathi”). These are so unlike Naushad!

I have run another song from this movie a while ago (viz. मेरे जीवन साथी, कली थी मै तो प्यासी (“mere jeevan saathee, kalee thee main to pyaasee”).

That song had actually struck me after a gap of years (may be even a decade or two), when I was driving my old car on the Mumbai-Pune expressway. The air-conditioner of my car is almost never functional (because I almost never have the money to get it repaired). In any case, the a/c was neither working nor even necessary, on that particular day late in the year. So, the car windows were down. It was pretty early in the morning; there wasn’t much traffic on the expressway; not much wind either. The sound of the new tires made a nice background rhythm of sorts. The sound was very periodic, because of the regularity of the waviness that comes to occur on cement-concrete roads after a while.

That waviness? It’s an interesting problem from mechanics. Take a photo of a long section of the railway tracks while standing in the middle, especially when the sun is rising or setting, and you see the waviness that has developed on the rail-tracks too—they go up and down. The same phenomenon is at work in both cases. Broadly, it’s due to vibrations—a nonlinear interaction between the vehicle, the road and the foundation layers underneath. (If I recall it right, in India, IIT Kanpur had done some sponsored research on this problem (and on the related NDT issues) for Indian Railways.)

So, anyway, to return to the song, it was that rhythmical sound of the new tires on the Mumbai-Pune Expressway which prompted something in my mind, and I suddenly recalled the above mentioned song (viz. मेरे जीवन साथी, कली थी मै तो प्यासी (“mere jeevan saathee, kalee thee main to pyaasee”). Some time later, I ran it here on this blog. (PS: My God! The whole thing was in 2012! See the songs section, and my the then comments on Naushad, here [^])

OK, cutting back to the present: Recently, I recalled the songs from this movie, and began wondering about the twin questions: (1) How come I did end up liking anything by Naushad, and (2) How could Naushad compose anything that was so much out of his box (actually, the box of all his traditional classical music teachers). Then, a quick glance at the comments section of some song from the same film enlightened me. (I mean at YouTube.) I came to know a new name: “Kersi Lord,” and made a quick search on it.

Turns out, Naushad was not alone in composing the music for this film: साथी (“saathee”). He had taken assistance from Kersi Lord, a musician who was quite well-versed with the Western classical and Western pop music. (Usual, for a Bawa from Bombay, those days!) The official credits don’t mention Kersi Lord’s name, but just a listen is enough to tell you how much he must have contributed to the songs of this collaboration (this movie). Yes, Naushad’s touch is definitely there. (Mentally isolate Lata’s voice and compare to मोहे पनघट पे (“mohe panghat pe”).) But the famous Naushad touch is so subdued here that I actually end up liking this song too!

So, here we go, without further ado (but with a heartfelt appreciation to Kersi Lord):

(Hindi) ये काैन आया, रोशन हो गयी (“yeh kaun aayaa, roshan ho gayee)
Singer: Lata Mangeshkar
Lyrics: Majrooh Sultanpuri

A good quality audio is here [^].

]

PS: May be one little editing pass tomorrow?

History:
— 2020.12.19 23:57 IST: First published
— 2020.12.20 19:50 IST and 2020.12.23 22:15 IST: Some very minor (almost insignificant) editing / changes to formatting. Done with this post now.

# Some comments on QM and CM—Part 1: Coleman’s talk. Necessity of ontologies.

Update on 2020.12.10 16:02 IST:

I’ve corrected the descriptions in the ontologies of the Newtonian rigid bodies, Newtonian gravity, and Newtonian deformable bodies. (In particular, the idea of the continuum goes back to Newton’s shells-based argument to make a point-particle out of a sphere.) I have also added considerably to all the five pre-quantum ontologies.

Before getting going, let me briefly mention an update concerning my RSI.

0. RSI:

Good news: After taking an almost complete break from typing for may be 3–4 weeks, my RSI seems to have subsided significantly.

Bad news: It still is palpably lurking in the background. If I type a bit, I don’t get pains as such. However, certain subtle but easily identifiable early warning signs do appear. Like, a bit of soreness or stiffness at the base of thumb/fingers or at the wrist, etc.

Current course: I do not type for more than 30–40 minutes at a stretch. I force myself a break as soon as I notice that the time is up. I take rest of at least one hour before returning to the keyboard.

Let’s see.

1. Professor Coleman’s talk:

Sidney Coleman was a professor of physics at Harvard. I came to know about him through other physicists talking / writing about him. I think that so far, I’ve watched just parts of one or two video lectures of his (the ones from a course on QFT).

In these videos, I think that he was assuming that he was talking to rather sharp people. He did seem to care quite a bit about making very careful statements. Yet at the same time, he also seemed to pull it off quite effortlessly. Actually, he seemed to carry a very informal air about him—i.e., even while being in the middle of a rigorous point. Also, a certain kind of spontaneity, and an in-built sense of humour. … If you knew the background of the topic, you could expect to remain hooked to the lecture all throughout, but you wouldn’t be quite spell-bound, really speaking: his very style of presentation would make sure that you would want to remain active. … Thus I gathered that there was ample truth to how other physicists were describing him. In a way, he came across as if he were the very idea of the physicist, personified. I had actually thought of that phrase before running into how the Wiki article [^] describes him:

He’s not a Stephen Hawking; he has virtually no visibility outside. But within the community of theoretical physicists, he’s kind of a major god. He is the physicist’s physicist.

That’s why, when Prof. Peter Woit (of Columbia Uni.) highlighted a line by Coleman in a recent post at his blog [^], I immediately went ahead and downloaded the paper [^]. Actually, it’s not a paper; it’s a transcript of Coleman’s Dirac Prize lecture. [PS: If you are into watching videos, it’s on YouTube, here [^].)

The problem is not the interpretation of quantum mechanics. That’s getting things just backwards. The problem is the interpretation of classical mechanics.

Aha!

2. One little entry from my research notebook:

It was just a few days ago that I was thinking about QM and CM, and had noted something in my handwritten “journal”. The entry was made on 03 December 2020, 10:58 IST; it was noted down in a very hurriedly manner, using a pencil. Here’s a snap-shot of the same. …I usually don’t share such things (even if I am not afraid of those “handwriting” experts), but here I am making an exception—more or less on a whim:

An entry from my handwritten research notebook/logbook/journal.

Inserting some parenthetical clarifications/addition in square brackets “[ ]”, the note reads:

“You never “see” a perfect circle, a singularity, an infinitely sharp boundary. Yet you use them [such concepts] in CM [classical mechanics].

The lesson [point] is: you don’t see CM abstractions. You only perceive *some* of the objects. You never perceive a field. Ever. You only perceive its effects on a massive charged body.

And, Stat. Mech. has randomness.

[Note completed at 11:01 IST [same day]].

My notes are almost always like that. They are scribbles, not notes. Written in a very hurried way, often without taking a hard thing for a pad underneath. My notes are basically meant only for me. They are just a means to jiggle my memory so that I don’t lose hold of some point that I notice is passing through my mind rather quickly. That’s why, they are not likely to make full sense to most anyone (and often don’t give my correct position either—they are just points noted).

[Parenthetical clarifications: Yes, I went to Marathi medium schools. No, they didn’t teach the cursive in Marathi medium schools. Yes, I taught the cursive handwriting myself. In my XI standard. I used the Barge Surekha slate (which was quite a new invention back then). It took me more than one year to get used to it. Yes, my handwriting varies a lot—much more than others’. No, I usually write in ink. Enough?]

Alright. Let me explain what I meant by the above note. In doing so, I am going to add a lot of background and explanatory material too. Indeed, as it so happens, I have to split this blog post into at least two parts, this being the first.

3. Starting point of a physics theory:

The starting point of a physics theory is not any of the following:

• Illustration of simple applications, say using sketches, photographs, simulations, or interactive media
• Definition of terms used in the fundamental laws
• Statements of the fundamental laws
• Notation being used
• Governing equations
• Proofs of the governing equations
• Description of experiments that led to the theory
• Etc.

The starting point of a physics theory is:

some object(s), including their actions, posited in it.

Any theory of physics describes actions of some or the other object(s).

In physics, the laws governing some phenomena are often stated quantitatively, via some equation. Thus, the laws are stated in terms of the sizes of causes and of effects. But causes and effects do not exist as disembodied entities. They directly or indirectly refer to some or the objects that undergo lawful changes.

The identity or the nature of an object is the cause, and the actions it takes are the effects. That’s causality at the most basic level for you. (Causality is not at all restricted to an orderly progression in time. For more on causality, see my earlier post here [^].)

So, objects are, logically, starting point of a physics theory—an already developed theory.

However, the process of development of a theory doesn’t start with well defined ideas regarding what objects it posits and uses. It begins only with some loosely organized body of knowledge, and makes a phenomenology out of them. Some degree of good phenomenology must exist before the activity of theory building proper can occur.

4. Phenomenological context:

Much before a physical theory is built, there already comes to be a large body of interrelated items of observational or phenomenological knowledge. Such prior observations include those that are made before any hypothesis has ever been formulated, before any experiment is ever designed or conducted.

The knowledge may not be well integrated, and its items may not be rigorously formulated. But it is knowledge, all the same.

The body of such pre-existing knowledge includes (but is not limited to): descriptions of arrangements or configurations of various kinds of objects, miscellaneous items of condensed descriptions, rules of thumb that seem to work, observations of regularities, and even items of mere imagination (e.g. conjectures).

So, it all is a big mess. But such a mess must exist before a neat theory can at all get built. Indeed, any one who intends to build a theory must make himself well conversant with (if not an expert of) all that mess. They call this mess “phenomenological context”.

Thus, there is a lot of background material before a theory begins to get built. Due to the nature of concept formation, a theory cannot make sense except in reference to such (logically and chronologically) preceding background material.

5. The Objects-Identity-Actions form:

If you decide to condense all the background material relating to some physical phenomenon, you will soon find that it inevitably acquires this form:

There are a certain kind of objects, and when subject to certain physical conditions (say of interactions with other objects that themselves may be only loosely specified), the objects under study do something peculiar/ interesting/ outstanding/ important.

Why does your description acquire the form of: objects-their identities-their actions? That’s a deeper philosophical point than we care to look into, right now. But it’s there. It’s a metaphysical-epistemological thing, not a thing of just linguistics. You must first assume that something exists, before you can make any statement (about any thing). The ontological idea of a “physical object” is rooted in that profound statement. Let’s leave it at that.

6. Astounding variety of physical phenomena:

Now, the next “obvious” point I want to make is this:

Physical phenomena differ very widely in their very nature.

Hot or cold weather affects objects, including your sense of temperature, in a radically different manner than the sound of a temple bell does, and both radically differ from how the birds fall silent during an eclipse. Phenomena differ radically. And, physical theories are ultimately nothing but pieces of knowledge that seek to explain phenomena in a lawful, causal, manner.

Since phenomena themselves are radically different from each other, physical theories covering them also end up making use of very different kinds of objects in their respective descriptions. The objects used in a theory of motions of objects are quite different from those in light, for example.

7. Identification of ontologies is a task neglected by physicists:

Now, it’s true that often, theorists don’t explicitly identify the deepest nature of the objects that appear in the statements of their laws.

Even in the earlier times, they didn’t always explicitly identify the ontological nature of the objects; rather, they often chose to rely on analogies. For example: Not knowing what the true nature of the static electricity was really like, some 18th century physicists posited an imaginary fluid for the electricity. So, a “special kind of a fluid” was the ontological object assumed in their theory. It must be assumed to exist, they thought, before Coulomb’s inverse-square law could make any causal sense.

In modern times, mere hints is all you get to hear from physicists—if at all. And then, at least in textbook presentations, a discussion of such ideas is completely left out. And that’s for a valid reason, too: There really is not enough time at the disposal of the teacher or the student. The interested student, therefore, must refer to other books like history of science, encyclopaediae, or even good pop-sci books.

So, the point for now is: Working physicists, even theory builders among them, tend to focus on the narrow technicalities of their speciality, the narrow range of the phenomena under study. In the pursuit of precision in their statement of the laws of physics, they tend to focus on formulating quantitative laws that work satisfactorily. However, in the process, they lose the precision in the ontological sense—and they don’t even notice this fact.

They don’t even use the word “ontology,” or, God forbid, “metaphysical nature.” The best of them say “qualitative nature”. However, with the advance of the catastrophe theory, this latter term has again acquired a rather narrow, mathematically defined, meaning.

Further, the roots of the mathematical concepts being very hard to identify, physicists altogether leave the issue of explicitly identifying the ontological nature of the underlying objects they do assume to exist in their theories.

It is the task of ontology to identify, clarify and explain such ideas and issues as given below:

the basic types of objects used in a theory, their nature, the kind of actions they take, how they interact with other objects, the restrictions placed on their actions and interactions, etc.

Thus, ontology of physics identifies the broad pre-physical (or “metaphysical”) nature of the objects that are actually used in a theory of physics. Let’s take an example to make this point clear.

8. But what is the “ontological nature” of an object? An example:

The physics theory of EM assumes that there are “fields,” and it defines a field mathematically: as a function of the physical space coordinates. That’s all the statements of laws of EM make use of.

It’s the task of ontology of EM to say that there is an aether whose attribute an EM field is. It is the ontology that proceeds to explain the characteristic way in which the aether transmits forces (or energies) within itself.

Hint: The aether is not at all like how the continua of engineering sciences. The fluids and solids, as used in engineering sciences (and physics of massive continua) transmit forces across definite regions lying internally within a given continuum. The idea of the mathematical cut, and of control volumes, so as to transform a continuum into a collection of definite (bounded) objects that are in direct contact (so that the earlier techniques of Newtonian mechanics of discrete bodies can be applied to them), happens to be present in both EM and engineering fluid/solid mechanics. But there are certain crucial differences regarding the nature of what all properties the continuum exhibits too.

See my ontologies series [^] for a 2019-level explanation for the difference of the EM ontology from the Newtonian mechanics ontology! I am giving an updated list immediately below.

9. Ontologies in physics theories—a minimal list:

As of today, I think that we can say that the following is a minimal list of different ontologies required by the science of physics (excluding the relativity theory, and its relation to other physical theories including the QM). I will also give some randomly selected pointers to make their nature graspable; such description is by no means carefully stated let alone comprehensive; it’s more or less completely on the fly.

9.1. Newtonian mechanics of particles and of discrete rigid bodies:

This ontology talks of forces exchanged between two or more inertia-possessing objects via the direct contact during their collisions (i.e. only for the duration of time they are in a point- or surface-contact), but not otherwise.

It has a special object of the empty space that offers no resistance to the motion of particles or rigid bodies.

Particles are point-objects. Rigid bodies have finite sizes, but can be represented as point-masses via the idea of the centre of mass. However, their finite size comes into picture in problems like forces exchanged by finite-sized bricks in a static wall—the transmission of force occurs over a surface, not through a single point.

The key idea in tackling the rigid body is that of taking the imaginary cut, so as to make a collection of many discrete rigid bodies out of a single continuous body that has no physically separated parts. This idea began with Newton himself when he formulated the shells argument and arrived at the idea of the centre of mass. But I guess it was Cauchy who gave the procedure of the mathematical cut the definitive form in which we understand such analyses today. (However, I will have to check with Truesdell before I confirm it.) Also see the ontology of deformable bodies (sec. 9.3) below.

9.2. Newtonian theory of gravity:

The key word here is: The instantaneous transmission of force across the empty space (i.e. an instantaneous action at a distance, or IAD for short). Otherwise, this ontology continues to have the same discrete objects of the previous ontology (whether point particles or rigid bodies).

Newton himself had surmised “strings” i.e. $1D$ objects as the medium that conveys the force of gravity to planets; he did not think of any continuous and $3D$ space-filling fields. However, he then decided to refuse to take any definitive position for the existence or otherwise of these strings, for a lack of evidence. (One of the key skills in science is learning how to carefully delimit your claim. Newton showed a natural mastery of this skill when he said: “I feign no hypotheses.” He meant something different from what a modern reader might think he meant.)

9.3. Newtonian mechanics of deformable continua:

The key idea here is not that a body may be a continuum. Even the rigid body already is a kind of continuum. The key idea here is that the control volumes internal to the continuum (which are obtained by taking suitable mathematical cuts), are no longer rigid bodies, but can deform.

Deformation of a finite body is a change in its size and/or shape. If you apply limiting processes to the internal CVs, you get an infinity of point-particles that together make up the finite body. With this view, the deformable continua can be seen, using more modern terminology, as carrying a non-uniform field of displacement vectors within itself. However, notice, the idea of fields as ontologically important aspects of continua came to be explicitly recognized only after Faraday’s lines of force and their mathematization into the concept of fields at the hands of James Clerk Maxwell.

As to the deformable continua, Newton again seems to have been the first to correctly address their mechanics, when he defined an internal shear force, and used it in the definition of the internal friction (viscosity) in fluids.

9.4. Fourier-theoretical description of changes in continua:

Fourier was the first to describe the changes occurring inside a continuum, in terms of some globally acting “waves”.

These waves aren’t necessarily the familiar travelling waves (as on the surface of an ocean) or even the standing waves (as the motion of a guitar string shows). The purely spatial aspects of these waves denote static waves (as in the wavy patterns left in the sand by water at a beach, or by wind in a desert). With the passage of time, the global waves of the Fourier theory do undergo changes. Thus, the purely spatial part too changes. But these changes don’t necessarily follow the laws of travelling or standing waves. In diffusion, the changes occur via exponential decay (and not sinusoidal time factors).

The key idea here is that the causal physical agents are localized in the frequency space, and not in the physical space. The frequency spectrum could, however, be continuous.

Effectively, this description means: IAD (instantaneous action at a distance). However, unlike Newtonian gravity, this IAD is not restricted to interaction between two point-masses via a straight line of no thickness. Fourier-theoretical IAD occurs at every point (an infinity of them) inside a finite or even infinite domain. You can interpret this IAD to mean: There is a simultaneous exertion of “forces” at all points in a domain, even an infinite domain. However, such forces cannot of arbitrary sizes, because their actions are subject to the appropriate time-evolution laws imposed on them. (In detail, the matter is related to how the sine and cosine waves combine and evolve.)

9.5. Electromagnetic fields in the aether:

All the ontologies so far dealt with objects that were electrically neutral (i.e. electrically uncharged and hence electrically inactive). Once you allow the objects to be charged, the peculiar ontology behind the EM theory kicks in.

Yes, the other attributes like mass/density, and capacity to move in space, carry momentum/kinetic energy, exchange gravity forces, and undergo internal deformations all remain intact. However,  the existence of electrical charges implies the existence of electrical and magnetic fields.

The key word here is what all modern physicists hate and rebel against: The aether.

The EM ontology assumes that there is a certain unmoved and unmovable object called aether. It offers no resistance to the motion of massive objects except when they are electrically charged. Thus, it replaces the “empty” space of Newton’s descriptions. When electrically charged objects are present in the control volume under study, the aether in the domain comes to carry continuous force fields. Thus, fields are nothing but attributes of the aether.

The trick of the mathematical cut again applies, as in all continua: adjacent CV’s are supposed to “transmit” fields via the direct physical contact, with zero divergence.

Realize, the aether transmits forces without any part of it (i.e. any CV within it) undergoing any deformation. This single characteristic requires us to posit a new ontology for the EM phenomena. Not many people realized this point—which, after the hindsight of 100+ years, looks rather simple to grasp! The idea that the internal CVs must undergo deformation is what marks the so-called “mechanical” view for the aether. In contrast, the EM aether must be seen as the background object whose internal CVs need not at all undergo any displacements, let alone relative displacements (i.e. deformations), for it to be able to transmit forces.

Even Maxwell continued to think in terms of the mechanical aether. Lorentz seems to have been the first to indicate the correct approach. (However, I haven’t yet read either Whitaker or Lorentz’ original works. So, I can’t be sure if Lorentz had fully reached the transmission-without-local-deformations viewpoint. The reason for my hesitation: I know that Lorentz was thinking in terms of deformations of a charged body when he formulated the transformations that go by his name.

In any case, Lorentz was the first to realize how to ultimately connect the other attributes mentioned previously (like mass/density, accelerations under forces, etc.) with the specifically EM attribute of charge. This connection is known as Lorentz’ force law; it is the “fifth’ equation” that is required in order to make a complete system out of Maxwell’s “purely” EM equations (18 in his system, reduced to 4 by Heaviside).

9.6. Ontology of quantum mechanics:

What is the key idea here?

Well, you don’t expect a short-n-sweet description for QM, do you?

And, even if I were to give you that, would you (the physicist) understand anything? So there.

I’ve begun writing a new document that will replace the Outline document of 2019 [^]. It will have some description concerning what kind of ontology we must expect if the QM postulates are to work. However, the specifically ontological issues are going to be spread all over the planned document. But yes, I am confident that you will come to have a very good idea concerning this ontology.

10. Pre-Quantum $\neq$ “Classical”:

The list of ontologies given in the section 9. above supersedes my previous writings on the ontologies in physics [^]. (However, a lot of the points spelt out in that series, of course, continue to remain valid.)

I hope that you can now appreciate the fact that:

Clubbing everything pre-quantum into the “classical” is not a good idea. There are at least five different ontologies operative for the so-called “classical” physics.

However, I admit, even I myself am getting used to calling it “pre-quantum” physics.

All the same, remember that, in saying just “classical” physics, there are contexts in which you cannot hope to have both precision/unambiguity and generality to your statements.

In the next post, I will come to connecting the objects of the Newtonian mechanics with the concrete objects you perceive in your perceptual field. That will lay down the context for understanding the research notebook entry I just shared in this post (I mean the photograph). It will then become possible to make further comments on what Coleman indicated.

So, in the meanwhile, if you are like me, go through Coleman’s paper [^]. Otherwise, i.e. if you like videos better, then go through the video available on YouTube [^].

A song I like:

(Hindi) तुम को भी तो ऐसा ही कुछ होता होगा (“tum ko bhee to aisaa hee kuchha”)
Singers: Kishore Kumar, Lata Mangeshkar
Music: Laxmikant-Pyarelal
Lyrics: Anand Bakshi

[Another song from my high-school days. That way, this song is not much above the usual average/good songs of Hindi film music. On second/third listen, I think this song is much above average; it definitely is, IMO, very good, if you think about it. … Apart from the nostalgic value, it’s a song I like for the style of rendering. …

You know, just take any Pandit/Ustaad Utterly Boring Guy from the Indian classical music, especially one who you much admire (or perhaps are a भक्त (“bhakta”) of), and ask him to take Kishore Kumar’s place in this song. Just do that, and record the “performance” for posterity. If replaying, make sure to play that rendering on an awesome sound system too.

Then, just relax back and see if you can enjoy anything of it! (And, more or less, ditto for any Utterly Boring Gal of the Indian classical music.) …

A good quality audio is here [^]. ]

History:
— 2020.12.09 19:53 IST: Originally published.
— 2020.12.10 16:02 IST: A significant (though limited) update effected. Noted at the beginning of the post.

# 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…]