# 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

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# Why is the research on the foundations of QM necessary?

Why is the research on the foundations of QM necessary? … This post is meant to hold together some useful links touching on various aspects of this question.

Sabine Hossenfelder:

See her blog post: “Good Problems in the Foundations of Physics” [^]. Go through the entirety of the first half of the post, and then make sure to check out the paragraph of the title “The Measurement Problem” from her list.

Not to be missed: Do check out the comment by Peter Shor, here [^], and Hossenfelder’s reply to it, here [^]. … If you are familiar with the outline of my new approach [^], then it would be very easy to see why I must have instantaneously found her answer to be so absolutely wonderful! … Being a reply to a comment, she must have written it much on the fly. Even then, she not only correctly points out the fact that the measurement process must be nonlinear in nature, she also mentions that you have to give a “bottom-up” model for the Instrument. …Wow! Simply, wow!!

Lee Smolin:

Here is one of the most lucid and essence-capturing accounts concerning this topic that I have ever run into [^]. Smolin wrote it in response to the Edge Question, 2013 edition. It wonderfully captures the very essence of the confusions which were created and / or faced by all the leading mainstream physicists of the past—the confusions which none of them could get rid of—with the list including even such Nobel-laureates as Bohr, Einstein, Heisenberg, Pauli, de Broglie, Schrodinger, Dirac, and others. [Yes, in case you read the names too rapidly: this list does include Einstein too!]

Sean Carroll:

He explains at his blog how a lack of good answers on the foundational issues in QM leads to “the most embarrassing graph in modern physics” [^]. This post was further discussed in several other posts in the blogosphere. The survey paper which prompted Carroll’s post can be found at arXiv, here [^]. Check out the concept maps given in the paper, too. Phillip Ball’s coverage in the Nature News of this same paper can be found here [^].

See his pop-sci level paper “Quantum theory’s reality problem,” at arXiv [^]. He originally wrote it for Aeon in 2014, and then revised it in 2018 while posting at arXiv. Also notable is his c. 2000 paper: “Night thoughts of a quantum physicist,” Phil. Trans. R. Soc. Lond. A, vol. 358, 75–87. As to the fifth section (“Postscript”) of this second paper, I am fully confident that no one would have to wait either until the year 2999, or for any one of those imagined extraterrestrial colleagues to arrive on the scene. Further, I am also fully confident that no mechanical “colleagues” are ever going to be around.

…What Else?:

What else but the Wiki!… See here [^], and then, also here [^].

OK. This all should make for an adequate response, at least for the time being, to those physicists (or physics professors) who tend to think that the foundational issues do not make for “real” physics, that it is a non-issue. … However, for obvious reasons, this post will also remain permanently under updates…

Revision History:

2019.04.15: First published
2019.04.16: Some editing/streamlining
2019.05.05: Added the paper by Prof. Kent.

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# The rule of omitting the self-field in calculations—and whether potentials have an objective existence or not

There was an issue concerning the strictly classical, non-relativistic electricity which I was (once again) confronted with, during my continuing preoccupation with quantum mechanics.

Actually, a small part of this issue had occurred to me earlier too, and I had worked through it back then.

However, the overall issue had never occurred to me with as much of scope, generality and force as it did last evening. And I could not immediately resolve it. So, for a while, especially last night, I unexpectedly found myself to have become very confused, even discouraged.

Then, this morning, after a good night’s rest, everything became clear right while sipping my morning cup of tea. Things came together literally within a span of just a few minutes. I want to share the issue and its resolution with you.

The question in question (!) is the following.

Consider 2 (or $N$) number of point-charges, say electrons. Each electron sets up an electrostatic (Coulombic) potential everywhere in space, for the other electrons to “feel”.

As you know, the potential set up by the $i$-th electron is:
$V_i(\vec{r}_i, \vec{r}) = \dfrac{1}{4 \pi \epsilon_0} \dfrac{Q_i}{|\vec{r} - \vec{r}_i|}$
where $\vec{r}_i$ is the position vector of the $i$-th electron, $\vec{r}$ is any arbitrary point in space, and $Q_i$ is the charge of the $i$-th electron.

The potential energy associated with some other ($j$-th) electron being at the position $\vec{r}_j$ (i.e. the energy that the system acquires in bringing the two electrons from $\infty$ to their respective positions some finite distance apart), is then given as:
$U_{ij}(\vec{r}_i, \vec{r}_j) = \dfrac{1}{4 \pi \epsilon_0} \dfrac{Q_i\,Q_j}{|\vec{r}_j - \vec{r}_i|}$

The notation followed here is the following: In $U_{ij}$, the potential field is produced by the $i$-th electron, and the work is done by the $j$-th electron against the $i$-th electron.

Symmetrically, the potential energy for this configuration can also be expressed as:
$U_{ji}(\vec{r}_j, \vec{r}_i) = \dfrac{1}{4 \pi \epsilon_0} \dfrac{Q_j\,Q_i}{|\vec{r}_i - \vec{r}_j|}$

If a system has only two charges, then its total potential energy $U$ can be expressed either as $U_{ji}$ or as $U_{ij}$. Thus,
$U = U_{ji} = U_{ij}$

Similarly, for any pair of charges in an $N$-particle system, too. Therefore, the total energy of an $N$-particle system is given as:
$U = \sum\limits_{i}^{N} \sum\limits_{j = i+1}^{N} U_{ij}$

The issue now is this: Can we say that the total potential energy $U$ has an objective existence in the physical world? Or is it just a device of calculations that we have invented, just a concept from maths that has no meaningful physical counterpart?

(A side remark: Energy may perhaps exist as an attribute or property of something else, and not necessarily as a separate physical object by itself. However, existence as an attribute still is an objective existence.)

The reason to raise this doubt is the following.

When calculating the motion of the $i$-th charge, we consider only the potentials $V_j$ produced by the other charges, not the potential produced by the given charge $V_i$ itself.

Now, if the potential produced by the given charge ($V_i$) also exists at every point in space, then why does it not enter the calculations? How does its physical efficacy get evaporated away? And, symmetrically: The motion of the $j$-th charge occurs as if $V_j$ had physically evaporated away.

The issue generalizes in a straight-forward manner. If there are $N$ number of charges, then for calculating the motion of a given $i$-th charge, the potential fields of all other charges are considered operative. But not its own field.

How can motion become sensitive to only a part of the total potential energy existing at a point even if the other part also exists at the same point? That is the question.

This circumstance seems to indicate as if there is subjectivity built deep into the very fabric of classical mechanics. It is as if the universe just knows what a subject is going to calculate, and accordingly, it just makes the corresponding field mystically go away. The universe—the physical universe—acts as if it were changing in response to what we choose to do in our mind. Mind you, the universe seems to change in response to not just our observations (as in QM), but even as we merely proceed to do calculations. How does that come to happen?… May be the whole physical universe exists only in our imagination?

Got the point?

No, my confusion was not as pathetic as that in the previous paragraph. But I still found myself being confused about how to account for the fact that an electron’s own field does not enter the calculations.

But it was not all. A non-clarity on this issue also meant that there was another confusing issue which also raised its head. This secondary issue arises out of the fact that the Coulombic potential set up by any point-charge is singular in nature (or at least approximately so).

If the electron is a point-particle and if its own potential “is” $\infty$ at its position, then why does it at all get influenced by the finite potential of any other charge? That is the question.

Notice, the second issue is most acute when the potentials in question are singular in nature. But even if you arbitrarily remove the singularity by declaring (say by fiat) a finite size for the electron, thereby making its own field only finitely large (and not infinite), the above-mentioned issue still remains. So long as its own field is finite but much, much larger than the potential of any other charge, the effects due to the other charges should become comparatively less significant, perhaps even negligibly small. Why does this not happen? Why does the rule instead go exactly the other way around, and makes those much smaller effects due to other charges count, but not the self-field of the very electron in question?

While thinking about QM, there was a certain point where this entire gamut of issues became important—whether the potential has an objective existence or not, the rule of omitting the self-field while calculating motions of particles, the singular potential, etc.

The specific issue I was trying to think through was: two interacting particles (e.g. the two electrons in the helium atom). It was while thinking on this problem that this problem occurred to me. And then, it also led me to wonder: what if some intellectual goon in the guise of a physicist comes along, and says that my proposal isn’t valid because there is this element of subjectivity to it? This thought occurred to me with all its force only last night. (Or so I think.) And I could not recall seeing a ready-made answer in a text-book or so. Nor could I figure it out immediately, at night, after a whole day’s work. And as I failed to resolve the anticipated objection, I progressively got more and more confused last night, even discouraged.

However, this morning, it all got resolved in a jiffy.

Would you like to give it a try? Why is it that while calculating the motion of the $i$-th charge, you consider the potentials set up by all the rest of the charges, but not its own potential field? Why this rule? Get this part right, and all the philosophical humbug mentioned earlier just evaporates away too.

I would wait for a couple of days or so before coming back and providing you with the answer I found. May be I will write another post about it.

Update on 2019.03.16 20:14 IST: Corrected the statement concerning the total energy of a two-electron system. Also simplified the further discussion by couching it preferably in terms of potentials rather than energies (as in the first published version), because a Coulombic potential always remains anchored in the given charge—it doesn’t additionally depend on the other charges the way energy does. Modified the notation to reflect the emphasis on the potentials rather than energy.

A song I like:

[What else? [… see the songs section in the last post.]]
(Hindi) “woh dil kahaan se laaoon…”
Singer: Lata Mangeshkar
Music: Ravi
Lyrics: Rajinder Kishen

A bit of a conjecture as to why Ravi’s songs tend to be so hummable, of a certain simplicity, especially, almost always based on a very simple rhythm. My conjecture is that because Ravi grew up in an atmosphere of “bhajan”-singing.

Observe that it is in the very nature of music that it puts your mind into an abstract frame of mind. Observe any singer, especially the non-professional ones (or the ones who are not very highly experienced in controlling their body-language while singing, as happens to singers who participate in college events or talent shows).

When they sing, their eyes seem to roll in a very peculiar manner. It seems random but it isn’t. It’s as if the eyes involuntarily get set in the motions of searching for something definite to be found somewhere, as if the thing to be found would be in the concrete physical space outside, but within a split-second, the eyes again move as if the person has realized that nothing corresponding is to be found in the world out there. That’s why the eyes “roll away.” The same thing goes on repeating, as the singer passes over various words, points of pauses, nuances, or musical phrases.

The involuntary motions of the eyes of the singer provide a window into his experience of music. It’s as if his consciousness was again and again going on registering a sequence of two very fleeting experiences: (i) a search for something in the outside world corresponding to an inner experience felt in the present, and immediately later, (ii) a realization (and therefore the turning away of the eyes from an initially picked up tentative direction) that nothing in the outside world would match what was being searched for.

The experience of music necessarily makes you realize the abstractness of itself. It tends to make you realize that the root-referents of your musical experience lie not in a specific object or phenomenon in the physical world, but in the inner realm, that of your own emotions, judgments, self-reflections, etc.

This nature of music makes it ideally suited to let you turn your attention away from the outside world, and has the capacity or potential to induce a kind of a quiet self-reflection in you.

But the switch from the experience of frustrated searches into the outside world to a quiet self-reflection within oneself is not the only option available here. Music can also induce in you a transitioning from those unfulfilled searches to a frantic kind of an activity: screams, frantic shouting, random gyrations, and what not. In evidence, observe any piece of modern American / Western pop-music.

However, when done right, music can also induce a state of self-reflection, and by evoking certain kind of emotions, it can even lead to a sense of orderliness, peace, serenity. To make this part effective, such a music has to be simple enough, and orderly enough. That’s why devotional music in the refined cultural traditions is, as a rule, of a certain kind of simplicity.

The experience of music isn’t the highest possible spiritual experience. But if done right, it can make your transition from the ordinary experience to a deep, profound spiritual experience easy. And doing it right involves certain orderliness, simplicity in all respects: tune, tone, singing style, rhythm, instrumental sections, transitions between phrases, etc.

If you grow up listening to this kind of a music, your own music in your adult years tends to reflect the same qualities. The simplicity of rhythm. The alluringly simple tunes. The “hummability quotient.” (You don’t want to focus on intricate patterns of melody in devotional music; you want it to be so simple that minimal mental exertion is involved in rendering it, so that your mental energy can quietly transition towards your spiritual quest and experiences.) Etc.

I am not saying that the reason Ravi’s music is so great is because he listened his father sing “bhajan”s. If this were true, there would be tens of thousands of music composers having talents comparable to Ravi’s. But the fact is that Ravi was a genius—a self-taught genius, in fact. (He never received any formal training in music ever.) But what I am saying is that if you do have the musical ability, having this kind of a family environment would leave its mark. Definitely.

Of course, this all was just a conjecture. Check it out and see if it holds or not.

… May be I should convert this “note” in a separate post by itself. Would be easier to keep track of it. … Some other time. … I have to work on QM; after all, exactly only half the month remains now. … Bye for now. …

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# 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.

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# A preliminary document on my fresh new approach to QM

I have uploaded the outline document I had promised as an attachment to a blog post at iMechanica; see here [^]. I will copy-paste the text of that post below:

Hello, World!

Here is a document that jots down, in a brief, point-wise manner, the elements of my new approach to understanding quantum mechanics.

Please note that the writing is very much at a preliminary stage. It is very much a work in progress. However, it does jot down many essential ideas.

I am uploading the document at iMechanica just to have an externally verifiable time-stamp to it. Further versions will also be posted at this thread.

Comments are welcome. However, I may not be able to respond all of them immediately, because (i) I wish to immediately switch over to my studies of Data Science (ii) discussions on QM, especially on its foundations, tend to get meandering very fast.

Best,

–Ajit

It was only yesterday that I had said that preparing this document would take longer, may be a week or so. But soon later, I discarded the idea of revising the existing document (18 pages), and instead tried re-writing a separate summary for it completely afresh. Turns out that starting afresh all over again was a very good idea. Yesterday, I was at about 2 pages, and today, I finished jotting down all the ideas, at least in essence, right within 8 pages (+ 1 page of the reference section).

A song I like:

[It happens to be one of the songs which I first heard roughly around the same time that I got my own bicycle, and the times when I first came across the quantum mechanical riddles—which was during my X–XII standard days. I had liked it immediately—I mean the song. I don’t know why it doesn’t appear frequently enough on people’s lists of their favorites, but it has a very, very fresh feel to it. It anyway is a song of the teenage, of all those naive expectations and unspoiled optimism. Usha Mangeshkar, with her deft, light touch, somehow manages to bring that sense of life, that sense of freshness and confidence, fully alive in this song. The music and the lyrics are neat too… All in all, an absolutely wonderful song. … Perhaps also equally important, it is of great nostalgic value to me, too.

It used to be not easily available. So, let me give you the link to listening and buying it at Gaana.com; it’s song # 9 on this page [^] ]