# A neat experiment concerning quantum jumps. Also, an update on the data science side.

1. A new paper on quantum jumps:

This post has a reference to a paper published yesterday in Nature by Z. K. Minev and pals [^]; h/t Ash Joglekar’s twitter feed (he finds this paper “fascinating”). The abstract follows; the emphasis in bold is mine.

In quantum physics, measurements can fundamentally yield discrete and random results. Emblematic of this feature is Bohr’s 1913 proposal of quantum jumps between two discrete energy levels of an atom[1]. Experimentally, quantum jumps were first observed in an atomic ion driven by a weak deterministic force while under strong continuous energy measurement[2,3,4]. The times at which the discontinuous jump transitions occur are reputed to be fundamentally unpredictable. Despite the non-deterministic character of quantum physics, is it possible to know if a quantum jump is about to occur? Here we answer this question affirmatively: we experimentally demonstrate that the jump from the ground state to an excited state of a superconducting artificial three-level atom can be tracked as it follows a predictable ‘flight’, by monitoring the population of an auxiliary energy level coupled to the ground state. The experimental results demonstrate that the evolution of each completed jump is continuous, coherent and deterministic. We exploit these features, using real-time monitoring and feedback, to catch and reverse quantum jumps mid-flight—thus deterministically preventing their completion. Our findings, which agree with theoretical predictions essentially without adjustable parameters, support the modern quantum trajectory theory[5,6,7,8,9] and should provide new ground for the exploration of real-time intervention techniques in the control of quantum systems, such as the early detection of error syndromes in quantum error correction.

Since the paper was behind the paywall, I quickly did a bit of googling and then (very) rapidly browsed through the following three: [^], [^] and [(PDF) ^].

Since I didn’t find the words “modern quantum trajectory theory” explained in simple enough terms in these references, I did some further googling on “quantum trajectory theory”, high-speed browsed through them a bit, in the process browsing jumping through [^], [^], and landed first at [^], then at the BKS paper [(PDF) ^]. Then, after further googling on “H. J. Carmichael”, I high-speed browsed through the Wiki on Prof. Carmichael [^], and from there, through the abstract of his paper [^], and finally took the link to [^] and to [^].

My initial and rapid judgment:

Ummm… Minev and pals might have concluded that their experimental work lends “support” to “the modern quantum trajectory theory” [MQTT for short.] However, unfortunately, MQTT itself is not sufficiently deep a theory.

…  As an important aside, despite the word “trajectory,” thankfully, MQTT is, as far as I gather it, not Bohmian in nature either. [Lets out a sigh of relief!]

Still, neither is MQTT deep enough. And quite naturally so… After all, MQTT is a theory that focuses only on the optical phenomena. However, IMO, a proper quantum mechanical ontology would have the photon as a derived object—i.e., a higher-level abstraction of an object. This is precisely the position I adopted in my Outline document as well [^].

Realize, there  can be no light in an isolated system if there are no atoms in it. Light is always emitted from, and absorbed in, some or the other atoms—by phenomena that are centered around nuclei, basically. However, there can always be atoms in an isolated system even if there never occurs any light in it—e.g., in an extremely rare gas of inert gas atoms, each of which is in the ground state (kept in an isolated system, to repeat).

Naturally, photons are the derived or higher-level objects. And that’s why, any optical theory would have to assume some theory of electrons lying at even deeper a level. That’s the reason why MQTT cannot be at the deepest level.

So, my overall judgment is that, yes, Minev and pals’ work is interesting. Most important, they don’t take Bohr’s quantum jumps as being in principle un-analyzable, and this part is absolutely delightful. Still, if you ask me, for the reasons given above, this work also does not deal with the quantum mechanical reality at its deepest possible level. …

So, in that sense, it’s not as fascinating as it sounds on the first reading. … Sorry, Ash, but that’s how the things are here!

…Today was the first time in a couple of weeks or so that I read anything regarding QM. And, after this brief rendezvous with it in this post, I am once again choosing to close that subject right here. … In the absence of people interacting with me on QM (computational QChem, really speaking), and having already reached a very definite point of development concerning my new approach, I don’t find QM to be all that interesting these days.

For some good pop. sci-level coverage of the paper, see Chris Lee’s post at his ArsTechnica blog [^], and Phillip Ball’s story at the Quanta Magazine [^].

2. An update on the Data Science side:

As you know, these days, I have been pursuing data science full-time.

Earlier, in the second half of 2018, I had gone through Michael Nielsen’s online book on ANNs and DL [^]. At that time, I had also posted a few entries here on this blog concerning ANNs and DL [^]. For instance, see my post explaining, with real-time visualization, why deep learning is hard [^].

Now, in the more recent times, I have been focusing more on the other (“canonical”) machine learning techniques in general—things like (to list in a more or less random an order) regression, classification, clustering, dimensionality reduction, etc. It’s been fun. In particular, I have come to love scikit-learn. It’s a neat library. More about it all, later—may be I should post some of the toy Python scripts which I tried.

… BTW, I am also searching for one or two good, “industrial scale” projects from data science. So, if you are from industry and are looking for some data-science related help, then feel free to get in touch. If the project is of the right kind, I may even work on it on a pro-bono basis.

… Yes, the fact is that I am actively looking out for a job in data science. (Have uploaded my resume at naukri.com too.) However, at the same time, if a topic is interesting enough, I don’t mind lending some help on a pro bono basis either.

The project topic could be anything from applications in manufacturing engineering (e.g. NDT techniques like radiography, ultrasonics, eddy current, etc.) to financial time-series predictions, to some recommendation problem, to… I am open for virtually anything in data science. It’s just that I have to find the project to be interesting enough, that’s all… So, feel free to get in touch.

… Anyway, it’s time to wrap up. … So, take care and bye for now.

A song I like

(Western, pop) “Money, money, money…”
Band: ABBA

/

# Wrapping up my research on QM—without having to give up on it

Guess I am more or less ready to wrap up my research on QM. Here is the exact status as of today.

1. The status today:

I have convinced myself that my approach (viz. the idea of singular potentials anchored into electronic positions, and with a $3D$ wave-field) is entirely correct, as far as QM of non-interacting particles is concerned. That is to say, as far as the abstract case of two particles in a $0$-potential $1D$ box, or a less abstract but still hypothetical case of two non-interacting electrons in the helium atom, and similar cases are concerned. (A side note: I have worked exclusively with the spinless electrons. I don’t plan to include spin right away in my development—not even in my first paper on it. Other physicists are welcome to include it, if they wish to, any time they like.)

As to the actual case of two interacting particles (i.e., the interaction term in the Hamiltonian for the helium atom), I think that my approach should come to reproduce the same results as those obtained using the perturbation theory or the variational approach. However, I need to verify this part via discussions with physicists.

All in all, I do think that the task which I had intended to complete (and to cross-check) before this month-end, is already over—and I find that I don’t have to give up on QM (as suspected earlier [^]), because I don’t have to abandon my new approach in the first place.

2. A clarification on what had to be worked out and what had to be left alone:

To me, the crucial part at this stage (i.e., for the second-half of March) was verifying whether working with the two ideas of (i) a $3D$ wavefield, and (ii) electrons as “particles” having definite positions (or more correctly, as points of singularities in the potential field), still leads to the same mathematical description as in the mainstream (linear) quantum mechanics or not.

I now find that my new approach leads to the same maths—at least for the QM of the non-interacting particles. And further, I also have very definite grounds to believe that my new approach should also work out for two interacting particles (as in the He atom).

The crucial part at this stage (i.e., for the second half of March) didn’t have so much to do with the specific non-linearity which I have proposed earlier, or the details of the measurement process which it implies. Working out the details of these ideas would have been impossible—certainly beyond the capacities of any single physicist, and over such a short period. An entire team of PhD physicists would be needed to tackle the issues arising in pursuing this new approach, and to conduct the simulations to verify it.

BTW, in this context, I do have some definite ideas regarding how to hasten this process of unraveling the many particular aspects of the measurement process. I would share them once physicists show readiness to pursue this new approach. [Just in case I forget about it in future, let me note just a single cue-word for myself: “DFT”.]

3. Regarding revising the Outline document issued earlier:

Of course, the Outline document (which was earlier uploaded at iMechanica, on 11th February 2019) [^] needs to be revised extensively. A good deal of corrections and modifications are in order, and so are quite a few additions to be made too—especially in the sections on ontology and entanglement.

However, I will edit this document at my leisure later; I will not allocate a continuous stretch of time exclusively for this task any more.

In fact, a good idea here would be to abandon that Outline document as is, and to issue a fresh document that deals with only the linear aspects of the theory—with just a sketchy conceptual idea of how the measurement process is supposed to progress in a broad background context. Such a document then could be converted as a good contribution to a good journal like Nature, Science, or PRL.

4. The initial skepticism of the physicists:

Coming to the skepticism shown by the couple of physicists (with whom I had had some discussions by emails), I think that, regardless of their objections (hollers, really speaking!), my main thesis still does hold. It’s they who don’t understand the quantum theory—and let me hasten to add that by the words “quantum theory,” here I emphatically mean the mainstream quantum theory.

It is the mainstream QM which they themselves don’t understood as well as they should. What my new approach then does is to merely uncover some of these weaknesses, that’s all. … Their weakness pertains to a lack of understanding of the $3D \Leftrightarrow 3ND$ correspondence in general, for any kind of physics: classical or quantum. … Why, I even doubt whether they understand even just the classical vibrations themselves right or not—coupled vibrations under variable potentials, that is—to the extent and depth to which they should.

In short, it is now easy for me to leave their skepticism alone, because I can now clearly see where they failed to get the physics right.

5. Next action-item:

In the near future, I would like to make short trips to some Institutes nearby (viz., in no particular order, one or more of the following: IIT Bombay, IISER Pune, IUCAA Pune, and TIFR Mumbai). I would like to have some face-to-face discussions with physicists on this one single topic: the interaction term in the Hamiltonian for the helium atom. The discussions will be held strictly in the context that is common to us, i.e., in reference to the higher-dimensional Hilbert space of the mainstream QM.

In case no one from these Institutes responds to my requests, I plan to go and see the heads of these Institutes (i.e. Deans and Directors)—in person, if necessary. I might also undertake other action items. However, I also sincerely hope and think that such things would not at all be necessary. There is a reason why I think so. Professors may or may not respond to an outsider’s emails, but they do entertain you if you just show up in their cabin—and if you yourself are smart, courteous, direct, and well… also experienced enough. And if you are capable of holding discussions on the “common” grounds alone, viz. in terms of the linear, mainstream QM as formulated in the higher-dimensional spaces (I gather it’s John von Neumann’s formulation), that is to say, the “Copenhagen interpretation.” (After doing all my studies—and, crucially, after the development of what to me is a satisfactory new approach—I now find that I no longer am as against the Copenhagen interpretation as some of the physicists seem to be.) … All in all, I do hope and think that seeing Diro’s and all won’t be necessary.

I also equally sincerely hope that my approach comes out unscathed during / after these discussions. … Though the discussions externally would be held in terms of mainstream QM, I would also be simultaneously running a second movie of my approach, in my mind alone, cross-checking whether it holds or not. (No, they wouldn’t even suspect that I was doing precisely that.)

I will be able to undertake editing of the Outline document (or leaving it as is and issuing a fresh document) only after these discussions.

6. The bottom-line:

The bottom-line is that my main conceptual development regarding QM is more or less over now, though further developments, discussions, simulations, paper-writing and all can always go on forever—there is never an end to it.

7. Data Science!

So, I now declare that I am free to turn my main focus to the other thing that interests me, viz., Data Science.

I already have a few projects in mind, and would like to initiate work on them right away. One of the “projects” I would like to undertake in the near future is: writing very brief notes, written mainly for myself, regarding the mathematical techniques used in data science. Another one is regarding applying ML techniques to NDT (nondestructive testing). Stay tuned.

A song I like:

(Western, instrumental) “Lara’s theme” (Doctor Zhivago)
Composer: Maurice Jarre

/

# Should I give up on QM?

After further and deeper studies of the Schrodinger formalism, I have now come to understand the exact position from which the physicists must be coming (I mean the couple of physicists with who I discussed the ideas of my new approach, as mentioned here [^])—why they must be raising their objections. I came to really understand their positions only now. Here is how it happened.

I was pursuing finding correspondence between the $3ND$ configuration space of the Schrodinger formalism on the one hand and the $3D$ physical space on the other, when I run into this subtle point which made everything look completely different. That point is the following:

Textbooks (or lecture notes, or lecturers) don’t ever highlight this point (in fact, indirectly, they actually obfuscate it), but I came to realize that even in the $1D$ cases like the QM harmonic oscillator (QHO), the Schrodinger formalism itself remains defined only on an abstract hyperspace—it’s just that in the case of the QHO, this hyperspace happens to be $1D$ in nature, that’s all.

I came to realize that, even in the simplest $1D$ case like the QHO the $x$ variable which appears in the Schrodinger equation does not directly refer to the physical space. In case of QHO, it refers to the change in the equilibrium separation between the centers of the two atoms.

Physicists and textbooks don’t mention this point, and in fact, the way they present QM, they make it look as if $x$ is the simple position variable. But in reality, no it is not. It can be made to look like a position variable (and not a change-in-the-interatomic-distance variable) by fixing the coordinate system to one of the two atoms (i.e. by making it a moving or Lagrangian coordinate system). But doing so leads to losing the symmetry in the motion of the two atoms, and more important, it further results in an obfuscation of the real nature of the issue. Mind you, textbook authors are trying to be helpful here. But unwittingly, they end up actually obfuscating the real story.

So, the $x$ variable whose Laplacian you take for the kinetic energy term also does not represent the physical space—not even in the simplest $1D$ cases like the QHO.

This insight, which I gained only now, has made me realize that I need to rethink through the whole thing once again.

In other words, my understanding of QM turned out to have been faulty—though the fault is much more on the part of the textbook authors (and lecturers) than on the part of someone like me—one who has learnt QM only through self-studies.

One implication of this better understanding now is that the new approach as stated in the Outline document isn’t going to work out. Even if there are a lot of good ideas in it (Only the Coulomb potentials, the specific nonlinearity proposed in the potential energy term, the ideas concerning measurements, etc.), there are several other ideas in that document which are just so weak that I will have to completely revise my entire approach once again.

Can I do that—take up a complete rethinking once again, and still hope to succeed?

Frankly, I don’t know. Not at this point of time anyway.

I still have not given up. But a sense of tiredness has crept in now. It now seems possible—very easily possible—that QM will end up defeating me, too.

But before outright leaving the fight, I would like to give it just one more try. One last try.

So, I have decided that I will “work” on this issue for just a little while more. May be a couple of weeks or so. Say until the month-end (March 2019-end). Unless I make some clearing, some breaththrough, I will not pursue QM beyond this time-frame.

What is going to be my strategy?

The only way an enterprise like mine can work out is if the connection between the $3D$ world of observations and the hyperspace formalism can be put in some kind of a valid conceptual correspondence. (That is to say, not just the measurement postulate but something deeper than that, something right at the level of the basic conceptual correspondence itself).

The only strategy that I will now pursue (before giving up on QM) is this: The Schrodinger formalism is based on the higher-dimensional configuration space not because a physicist like him would go specifically hunting for a higher-dimensional space, but primarily because the formulation of Schrodinger’s theory is based on the ideas from the energetics program, viz., the Leibniz-Lagrange-Euler-Hamilton program, their line(s) of thought.

The one possible opening I can think of as of today is this: The energetics program necessarily implies hyperspaces. However, at least in the classical mechanics, there always is a $1:1$ correspondence between such hyperspaces on the one hand and the $3D$ space on the other. Why should QM be any different? … As far as I am concerned, all the mystification they effected for QM over all these decades still does not supply any reason to believe that QM should necessarily be very different. After all, QM does make predictions about real world as described in $3D$! Why, even the position vectors that go into the potential energy operator $\hat{V}$ are defined only in the $3D$ space. …

… So, naturally, it seems that I just have to understand the nature of the correspondence between the Lagrangian mechanics and the $3D$ mechanics better. There must be some opening in there, based on this idea. In fact my suspicion is stronger: If at all there is a real opening to be found, if at all there is any real way to crack this nutty problem, then its key has to be lying somewhere in this correspondence.

So, I have decided to work on seeing if pursuing this line of thought yields something definitive or not. If it doesn’t, right within the next couple of weeks or so, I think I better throw in the towel and declare defeat.

Now, understanding the energetics program better meant opening up once again the books. But given my style, you know, it couldn’t possibly be the maths books—but only the conceptual ones.

So, this morning, I spent some time opening a couple of the movers-and-packers boxes (in which stuff was still lying as I mentioned before [^]), and also made some space in my room (somehow) by shoving the boxes a bit away to open the wall-cupboard, and brought out a few books I wanted to read  / browse through. Here they are.

The one shown opened is what I had mentioned as “the energetics book” in the background material document (see this link [^] in this post [^]). I am going to begin my last shot at QM—the understanding of the $3ND$$3D$ issue, starting with this book. The others may or may not be helpful, but I wanted to boast that they are just a part of personal library too!

Wish me luck!

(And suggest me a job in Data Science all the same! [Not having a job is the only thing that gets me (really) angry these days—and it does. So there.])

BTW, I really LOL on the Record of 17 off 71. (Just think what happened in 204!)

A song I like:

(Hindi) “O mere dil ke chain…”
Singer: Kishor Kumar
Music: R. D. Burman
Lyrics: Majrooh Sultanpuri

Minor editing to be done and a song to be added, tomorrow. But feel free to read the post right starting today.

Song added on 2019.03.10 12.09 AM IST. Subject to change if I have run it already.

/

# Absolutely Random Notings on QM—Part 3: Links to some (really) interesting material, with my comments

The “pride of place” for this post goes to a link to this book:

Norsen, Travis (2017) “Foundations of Quantum Mechanics: An Exploration of the Physical Meaning of Quantum Theory,” Springer

This book is (i) the best supplementary book for a self-study of QM, and simultaneously, also (ii) the best text-book on a supplementary course on QM, both at the better-prepared UG / beginning PG level.

A bit expensive though, but extensive preview is available on Google books, here [^]. (I plan to buy it once I land a job.)

I was interested in the material from the first three chapters only, more or less. It was a delight even just browsing through these chapters. I intend to read it more carefully soon enough. But even on the first, rapid browsing, I noticed that several pieces of understanding that I had so painstakingly come to develop (over a period of years) are given quite straight-forwardly here, as if they were a matter of well known facts—even if other QM text-books only cursorily mention them, if at all.

For instance, see the explanation of entanglement here. Norsen begins by identifying that there is a single wavefunction, always—even for a multi-particle system. Then after some explanation, he states: “But, as usual in quantum mechanics, these states do not exhaust the possibilities—instead, they merely form a basis for the space of all possible wave functions. …”… Note the emphasis on the word “basis” which Norsen helpfully puts.

Putting this point (which Norsen discusses with a concrete example), but in my words: There is always a single wavefunction, and for a multi-particle system, its basis is bigger; it consists of the components of the tensor product (formed from the components of the basis of the constituent systems). Sometimes, the single wavefunction for the multi-particle system can be expressed as a result of a single tensor-product (in which case it’s a separable state), and at all other times, only as an algebraic sum of the results of many such tensor-products (in which case they all are entangled states).

Notice how there is no false start of going from two separate systems, and then attempting to forge a single system out of them. Notice how, therefore, there is no hand-waving at one electron being in one galaxy, and another electron in another galaxy, and so on, as if to apologize for the very idea of the separable states. Norsen achieves the correct effect by beginning on the right note: the emphasis on the single wavefunction for the system as a whole to begin with, and then clarifying, at the right place, that what the tensor product gives you is only the basis set for the composite wavefunction.

There are many neat passages like this in the text.

I was about to say that Norsen’s book is the Resnick and Halliday of QM, but then came to hesitate saying so, because I noticed something odd even if my browsing of the book was rapid and brief.

Then I ran into

Ian Durham’s review of Norsen’s book, at the FQXi blog,

which is our link # 2 for this post [^].

Durham helpfully brings out the following two points (which I then verified during a second visit to Norsen’s book): (i) Norsen’s book is not exactly at the UG level, and (ii) the book is a bit partial to Bell’s characterization of the quantum riddles as well as to the Bohmian approach for their resolution.

The second point—viz., Norsen’s fascination for / inclination towards Bell and Bohm (B&B for short)—becomes important only because the book is, otherwise, so good: it carries so many points that are not even passingly mentioned in other QM books, is well written (in a conversational style, as if a speech-to-text translator were skillfully employed), easy to understand, thorough, and overall (though I haven’t read even 25% of it, from whatever I have browsed), it otherwise seems fairly well balanced.

It is precisely because of these virtues that you might come out giving more weightage to the B&B company than is actually due to them.

Keep that warning somewhere at the back of your mind, but do go through the book anyway. It’s excellent.

At Amazon, it has got 5 reader reviews, all with 5 stars. If I were to bother doing a review there, I too perhaps would give it 5 stars—despite its shortcomings/weaknesses. OK. At least 4 stars. But mostly 5 though. … I am in an indeterminate state of their superposition.

… But mark my words. This book will have come to shape (or at least to influence) every good exposition of (i.e. introduction to) the area of the Foundations of QM, in the years to come. [I say that, because I honestly don’t expect a better book on this topic to arrive on the scene all that soon.]

Which brings us to someone who wouldn’t assign the $|4\rangle + |5\rangle$ stars to this book. Namely, Lubos Motl.

If Norsen has moved in the Objectivist circles, and is partial to the B&B company, Motl has worked in the string theory, and is not just partial to it but even today defends it very vigorously—and oddly enough, also looks at that “supersymmetric world from a conservative viewpoint.” More relevant to us: Motl is not partial to the Copenhagen interpretation; he is all the way into it. … Anyway, being merely partial is something you wouldn’t expect from Motl, would you?

But, of course, Motl also has a very strong grasp of QM, and he displays it well (even powerfully) when he writes a post of the title:

“Postulates of quantum mechanics almost directly follow from experiments.” [^]

Err… Why “almost,” Lubos? 🙂

… Anyway, go through Motl’s post, even if you don’t like the author’s style or some of his expressions. It has a lot of educational material packed in it. Chances are, going through Motl’s posts (like the present one) will come to improve your understanding—even if you don’t share his position.

As to me: No, speaking from the new understanding which I have come to develop regarding the foundations of QM [^] and [^], I don’t think that all of Motl’s objections would carry. Even then, just for the sake of witnessing the tight weaving-in of the arguments, do go through Motl’s post.

Finally, a post at the SciAm blog:

“Coming to grips with the implications of quantum mechanics,” by Bernardo Kastrup, Henry P. Stapp, and Menas C. Kafatos, [^].

The authors say:

“… Taken together, these experiments [which validate the maths of QM] indicate that the everyday world we perceive does not exist until observed, which in turn suggests—as we shall argue in this essay—a primary role for mind in nature.”

No, it didn’t give me shivers or something. Hey, this is QM and its foundations, right? I am quite used to reading such declarations.

Except that, as I noted a few years ago on Scott Aaronson’s blog [I need to dig up and insert the link here], and then, recently, also at

Roger Schlafly’s blog [^],

you don’t need QM in order to commit the error of inserting consciousness into a physical theory. You can accomplish exactly the same thing also by using just the Newtonian particle mechanics in your philosophical arguments. Really.

Yes, I need to take that reply (at Schlafly’s blog), edit it a bit and post it as a separate entry at this blog. … Some other time.

For now, I have to run. I have to continue working on my approach so that I am able to answer the questions raised and discussed by people such as those mentioned in the links. But before that, let me jot down a general update.

A general update:

Oh, BTW, I have taken my previous QM-related post off the top spot.

That doesn’t mean anything. In particular, it doesn’t mean that after reading into materials such as that mentioned here, I have found some error in my approach or something like that. No. Not at all.

All it means is that I made it once again an ordinary post, not a sticky post. I am thinking of altering the layout of this blog, by creating a page that highlights that post, as well as some other posts.

But coming back to my approach: As a matter of fact, I have also written emails to a couple of physicists, one from IIT Bombay, and another from IISER Pune. However, things have not worked out yet—things like arranging for an informal seminar to be delivered by me to their students, or collaborating on some QM-related simulations together. (I could do the simulations on my own, but for the seminar, I would need an audience! One of them did reply, but we still have to shake our hands in the second round.)

In the meanwhile, I go jobless, but I keep myself busy. I am preparing a shortish set of write-ups / notes which could be used as a background material when (at some vague time in future) I go and talk to some students, say at IIT Bombay/IISER Pune. It won’t be comprehensive. It will be a little more than just a white-paper, but you couldn’t possibly call it even just the preliminary notes for my new approach. Such preliminary notes would come out only after I deliver a seminar or two, to physics professors + students.

At the time of delivering my proposed seminar, links like those I have given above, esp. Travis Norsen’s book, also should prove a lot useful.

But no, I haven’t seen something like my approach being covered anywhere, so far, not even Norsen’s book. There was a vague mention of just a preliminary part of it somewhere on Roger Schlafly’s blog several years ago, only once or so, but I can definitely say that I had already had grasped even that point on my own before Schlafly’s post came. And, as far as I know, Schlafly hasn’t come to pursue that thread at all, any time later…

But speaking overall, at least as of today, I think I am the only one who has pursued this (my) line of thought to the extent I have [^].

So, there. Bye for now.

I Song I Like:
(Hindi) “suno gajar kya gaaye…”
Singer: Geeta Dutt
Music: S. D. Burman
Lyrics: Sahir Ludhianvi
[There are two Geeta’s here, and both are very fascinating: Geeta Dutt in the audio, and Geeta Bali in the video. Go watch it; even the video is recommended.]

As usual, some editing after even posting, would be inevitable.

Some updates made and some streamlining done on 30 July 2018, 09:10 hrs IST.

Here are a few interesting links I browsed recently, listed in no particular order:

“Mathematicians Tame Turbulence in Flattened Fluids” [^].

The operative word here, of course, is: “flattened.” But even then, it’s an interesting read. Another thing: though the essay is pop-sci, the author gives the Navier-Stokes equations, complete with fairly OK explanatory remarks about each term in the equation.

(But I don’t understand why every pop-sci write-up gives the NS equations only in the Lagrangian form, never Eulerian.)

“A Twisted Path to Equation-Free Prediction” [^]. …

“Empirical dynamic modeling.” Hmmm….

“Machine Learning’s `Amazing’ Ability to Predict Chaos” [^].

Click-bait: They use data science ideas to predict chaos!

8 Lyapunov times is impressive. But ignore the other, usual kind of hype: “…the computer tunes its own formulas in response to data until the formulas replicate the system’s dynamics. ” [italics added.]

“Your Simple (Yes, Simple) Guide to Quantum Entanglement” [^].

Click-bait: “Entanglement is often regarded as a uniquely quantum-mechanical phenomenon, but it is not. In fact, it is enlightening, though somewhat unconventional, to consider a simple non-quantum (or “classical”) version of entanglement first. This enables us to pry the subtlety of entanglement itself apart from the general oddity of quantum theory.”

Don’t dismiss the description in the essay as being too simplistic; the author is Frank Wilczek.

“A theoretical physics FAQ” [^].

Click-bait: Check your answers with those given by an expert! … Do spend some time here…

Tensor product versus Cartesian product.

If you are engineer and if you get interested in quantum entanglement, beware of the easily confusing terms: The tensor product and the Cartesian product.

The tensor product, you might think, is like the Cartesian product. But it is not. See mathematicians’ explanations. Essentially, the basis sets (and the operations) are different. [^] [^].

But what the mathematicians don’t do is to take some simple but non-trivial examples, and actually work everything out in detail. Instead, they just jump from this definition to that definition. For example, see: “How to conquer tensorphobia” [^] and “Tensorphobia and the outer product”[^]. Read any of these last two articles. Any one is sufficient to give you tensorphobia even if you never had it!

You will never run into a mathematician who explains the difference between the two concepts by first directly giving you a vague feel: by directly giving you a good worked out example in the context of finite sets (including enumeration of all the set elements) that illustrates the key difference, i.e. the addition vs. the multiplication of the unit vectors (aka members of basis sets).

A third-class epistemology when it comes to explaining, mathematicians typically have.

A Song I Like:

(Marathi) “he gard niLe megha…”
Singers: Shailendra Singh, Anuradha Paudwal
Music: Rushiraj
Lyrics: Muralidhar Gode

[As usual, a little streamlining may occur later on.]