QM: The physical view it takes—1

So, what exactly is quantum physics like? What is the QM theory all about?

You can approach this question at many levels and from many angles. However, if an engineer were to ask me this question (i.e., an engineer with sufficiently good grasp of mathematics such as differential equations and linear algebra), today, I would answer it in the following way. (I mean only the non-relativistic QM here; relativistic QM is totally beyond me, at least as of today):

Each physics theory takes a certain physical view of the universe, and unless that view can be spelt out in a brief and illuminating manner, anything else that you talk about it (e.g. the maths of the theory) tends to become floating, even meaningless.

So, when we speak of QM, we have to look for a physical view that is at once both sufficiently accurate and highly meaningful intuitively.

But what do I mean by a physical view? Let me spell it out first in the context of classical mechanics so that you get a sense of that term.

Personally, I like to think of separate stages even within classical mechanics.

Consider first the Newtonian mechanics. We can say that the Newtonian mechanics is all about matter and motion. (Maxwell it was, I think, who characterized it in this beautifully illuminating a way.) Newton’s original mechanics was all about the classical bodies. These were primarily discrete—not quite point particles, but finite ones, with each body confined to a finite and isolated region of space. They had no electrical attributes or features (such as charge, current, or magnetic field strength). But they did possess certain dynamical properties, e.g., location, size, density, mass, speed, and most importantly, momentum—which was, using modern terminology, a vector quantity. The continuum (e.g. a fluid) was seen as an extension of the idea of the discrete bodies, and could be studied by regarding an infinitesimal part of the continuum as if it were a discrete body. The freshly invented tools of calculus allowed Newton to take the transition from the discrete bodies (billiard balls) to both: the point-particles (via the shells-argument) as well as to the continuum (e.g. the drag force on a submerged body.)

The next stage was the Euler-Lagrange mechanics. This stage represents no new physics—only a new physical view. The E-L mechanics essentially was about the same kind of physical bodies, but now a number (often somewhat wrongly called a scalar) called energy being taken as the truly fundamental dynamical attribute. The maths involved the so-called variations in a global integral expression involving an energy-function (or other expressions similar to energy), but the crucial dynamic variable in the end would be a mere number; the number would be the outcome of evaluating a definite integral. (Historically, the formalism was developed and applied decades before the term energy could be rigorously isolated, and so, the original writings don’t use the expression “energy-function.” In fact, even today, the general practice is to put the theory using only the mathematical and abstract terms of the “Lagrangian” or the “Hamiltonian.”) While Newton’s own mechanics was necessarily about two (or more) discrete bodies locally interacting with each other (think collisions, friction), the Euler-Lagrange mechanics now was about one discrete body interacting with a global field. This global field could be taken to be mass-less. The idea of a global something (it only later on came to be called a field) was already a sharp departure from the original Newtonian mechanics. The motion of the massive body could be predicted using this kind of a formalism—a formalism that probed certain hypothetical variations in the global field (or, more accurately, in the interactions that the global field had with the given body). The body itself was, however, exactly as in the original Newtonian mechanics: discrete (or spread over definite and delimited region of space), massive, and without any electrical attributes or features.

The next stage, that of the classical electrodynamics, was about the Newtonian massive bodies but now these were also seen as endowed with the electrical attributes in addition to the older dynamical attributes of momentum or energy. The global field now became more complicated than the older gravitational field. The magnetic features, initially regarded as attributes primarily different from the electrical ones, later on came to be understood as a mere consequence of the electrical ones. The field concept was now firmly entrenched in physics, even though not always very well understood for what it actually was: as a mathematical abstraction. Hence the proliferation in the number of physical aethers. People rightly sought the physical referents for the mathematical abstraction of the field, but they wrongly made hasty concretizations, and that’s how there was a number of aethers: an aether of light, an aether of heat, an aether of EM, and so on. Eventually, when the contradictions inherent in the hasty concretizations became apparent, people threw the baby with the water, and it was not long before Einstein (and perhaps Poincare before him) would wrongly declare the universe to be devoid of any form of aether.

I need to check the original writings by Newton, but from whatever I gather (or compile, perhaps erroneously), I think that Newton had no idea of the field. He did originate the idea of the universal gravitation, but not that of the field of gravity. I think he would have always taken gravity to be a force that was directly operating between two discrete massive bodies, in isolation to anything else—i.e., without anything intervening between them (including any kind of a field). Gravity, a force (instantaneously) operating at a distance, would be regarded as a mere extension of the idea of the force by the direct physical contact. Gravity thus would be an effect of some sort of a stretched spring to Newton, a linear element that existed and operated between only two bodies at its two ends. (The idea of a linear element would become explicit in the lines of force in Faraday’s theorization.) It was just that with gravity, the line-like spring was to be taken as invisible. I don’t know, but that seems like a reasonable implicit view that Newton must have adopted. Thus, the idea of the field, even in its most rudimentary form, probably began only with the advent of the Euler-Lagrange mechanics. It anyway reached its full development in Maxwell’s synthesis of electricity and magnetism into electromagnetism. Remove the notion of the field from Maxwell’s theory, and it is impossible for the theory to even get going. Maxwellian EM cannot at all operate without having a field as an intermediate agency transmitting forces between the interacting massive bodies. On the other hand, Newtonian gravity (at least in its original form and at least for simpler problems) can. In Maxwellian EM, if two bodies suddenly change their relative positions, the rest of the universe comes to feel the change because the field which connects them all has changed. In Newtonian gravity, if two bodies suddenly change their relative positions, each of the other bodies in the universe comes to feel it only because its distances from the two bodies have changed—not because there is a field to mediate that change. Thus, there occurs a very definite change in the underlying physical view in this progression from Newton’s mechanics to Euler-Lagrange-Hamilton’s to Maxwell’s.

So, that’s what I mean by the term: a physical view. It is a view of what kind of objects and interactions are first assumed to exist in the universe, before a physics theory can even begin to describe them—i.e., before any postulates can even begin to be formulated. Let me hasten to add that it is a physical view, and not a philosophical view, even though physicists, and worse, mathematicians, often do confuse the issue and call it a (mere) philosophical discussion (if not a digression). (What better can you expect from mathematicians anyway? Or even from physicists?)

Now, what about quantum mechanics? What kind of objects does it deal with, and what kind of a physical view is required in order to appreciate the theory best?

What kind of objects does QM deal with?

QM once again deals with bodies that do have electromagnetic attributes or features—not just the dynamical ones. However, it now seeks to understand and explain how these features come to operate so that certain experimentally observed phenomena such as the cavity radiation and the gas spectra (i.e., the atomic absorption- and emission-spectra) can be predicted with a quantitative accuracy. In the process, QM keeps the idea of the field more or less intact. (No, strictly speaking it doesn’t, but that’s what physicists think anyway). However, the development of the theory was such that it had to bring the idea of the spatially delimited massive body, occupying a definite place and traveling via definite paths, into question. (In fact, quantum physicists went overboard and threw it out quite gleefully, without a thought.) So, that is the kind of “objects” it must assume before its theorization can at all begin. Physicists didn’t exactly understand what they were dealing with, and that’s how arose all its mysteries.

Now, how about its physical view?

In my (by now revised) opinion, quantum mechanics basically is all about the electronic orbitals and their evolutions (i.e., changes in the orbitals, with time).

(I am deliberately using the term “electronic” orbital, and not “atomic” orbital. When you say “atom,” you must mean something that is localized—else, you couldn’t possibly distinguish this object from that at the gross scale. But not so when it is the electronic orbitals. The atomic nucleus, at least in the non-relativistic QM, can be taken to be a localized and discrete “particle,” but the orbitals cannot be. Since the orbitals are necessarily global, since they are necessarily spread everywhere, there is no point in associating something local with them, something like the atom. Hence the usage: electronic orbitals, not atomic orbitals.)

The electronic orbital is a field whose governing equation is the second-order linear PDE that is Schrodinger’s equation, and the problems in the theory involve the usual kind of IVBV problems. But a further complexity arises in QM, because the real-valued orbital density isn’t the primary unknown in Schrodinger’s equation; the primary unknown is the complex-valued wavefunction.

The Schrodinger equation itself is basically like the diffusion equation, but since the primary unknown is complex-valued, it ends up showing some of the features of the wave equation. (That’s one reason. The other reason is, the presence of the potential term. But then, the potential here is the electric potential, and so, once again, indirectly, it has got to do with the complex nature of the wavefunction.) Hence the name “wave equation,” and the term “wavefunction.” (The “wavefunction” could very well have been called the “diffusionfunction,” but Schrodinger chose to call it the wavefunction, anyway.) Check it out:

Here is the diffusion equation:

$\dfrac{\partial}{\partial t} \phi = D \nabla^2 \phi$
Here is the Schrodinger equation:
$\dfrac{\partial}{\partial t} \Psi = \dfrac{i\hbar}{2\mu} \nabla^2 \Psi + V \Psi$

You can always work with two coupled real-valued equations instead of the single, complex-valued, Schrodinger’s equation, but it is mathematically more convenient to deal with it in the complex-valued form. If you were instead to work with the two coupled real-valued  equations, they would still end up giving you exactly the same results as the Schrodinger equation. You will still get the Maxwellian EM after conducting suitable grossing out processes. Yes, Schrodinger’s equation must give rise to the Maxwell’s equations. The two coupled real-valued equations would give you that (and also everything else that the complex-valued Schrodinger’s equation does). Now, Maxwell’s equations do have an inherent  coupling between the electric and magnetic fields. This, incidentally, is the simplest way to understand why the wavefunction must be complex-valued. [From now on, don’t entertain the descriptions like: “Why do the amplitudes have to be complex? I don’t know. No one knows. No one can know.” etc.]

But yes, speaking in overall terms, QM is, basically, all about the electronic orbitals and the changes in them. That is the physical view QM takes.

Hold that line in your mind any time you hit QM, and it will save you a lot of trouble.

When it comes to the basics or the core (or the “heart”) of QM, physicists will never give you the above answer. They will give you a lot many other answers, but never this one. For instance, Richard Feynman thought that the wave-particle duality (as illustrated by the single-particle double-slit interference arrangement) was the real key to understanding the QM theory. Bohr and Heisenberg instead believed that the primacy of the observables and the principle of the uncertainty formed the necessary key. Einstein believed that entanglement was the key—and therefore spent his time using this feature of the QM to deny completeness to the QM theory. (He was right; QM is not complete. He was not on the target, however; entanglement is merely an outcome, not a primary feature of the QM theory.)

They were all (at least partly) correct, but none of their approaches is truly illuminating—not to an engineer anyway.

They were correct in the sense, these indeed are valid features of QM—and they do form some of the most mystifying aspects of the theory. But they are mystifying only to an intuition that is developed in the classical mechanical mould. In any case, don’t mistake these mystifying features for the basic nature of the core of the theory. Discussions couched in terms of the more mysterious-appearing features in fact have come to complicate the quantum story unnecessarily; not helped simplify it. The actual nature of the theory is much more simple than what physicists have told you.

Just the way the field in the EM theory is not exactly the same kind of a continuum as in the original Newtonian mechanics (e.g., in EM it is mass-less, unlike water), similarly, neither the field nor the massive object of the QM is exactly as in their classical EM descriptions. It can’t be expected to be.

QM is about some new kinds of the ultimate theoretical objects (or building blocks) that especially (but not exclusively) make their peculiarities felt at the microscopic (or atomic) scale. These theoretical objects carry certain properties such that the theoretical objects go on to constitute the observed classical bodies, and their interactions go on to produce the observed classical EM phenomena. However, the new theoretical objects are such that they themselves do not (and cannot be expected to) possess all the features of the classical objects. These new theoretical objects are to be taken as more fundamental than the objects theorized in the classical mechanics. (The physical entities in the classical mechanics are: the classical massive objects and the classical EM field).

Now, this description is quite handful; it’s not easy to keep in mind. One needs a simpler view so that it can be held and recalled easily. And that simpler view is what I’ve told you already:

To repeat: QM is all about the electronic orbital and the changes it undergoes over time.

Today, most any physics professor would find this view objectionable. He would feel that it is not even a physics-based view, it is a chemistry-based one, even if the unsteady or the transient aspect is present in the formulation. He would feel that the unsteady aspect in the formulation is artificial; it is more or less slapped externally on to the picture of the steady-state orbitals given in the chemistry textbooks, almost as an afterthought of sorts. In any case, it is not physics—that’s what he would be sure of. By that, he would also be sure to mean that this view is not sufficiently mathematical. He might even find it amusing that a physical view of QM can be this intuitively understandable. And then, if you ask him for a sufficiently physics-like view of QM, he would tell you that a certain set of postulates is what constitutes the real core of the QM theory.

Well, the QM postulates indeed are the starting points of QM theory. But they are too abstract to give you an overall feel for what the theory is about. I assert that keeping the orbitals always at the back of your mind helps give you that necessary physical feel.

OK, so, keeping orbitals at the back of the mind, how do we now explain the wave-particle duality in the single-photon double-slit interference experiment?

Let me stop here for this post; I will open my next post on this topic precisely with that question.

A Song I Like:

(Hindi) “ik ajeeb udaasi hai, meraa man_ banawaasi hai…”
Music: Salil Chowdhury
Singer: Sayontoni Mazumdar
Lyrics: (??)

[No, you (very probably) never heard this song before. It comes not from a regular film, but supposedly from a tele-film that goes by the name “Vijaya,” which was produced/directed by one Krishna Raaghav. (I haven’t seen it, but gather that it was based on a novel of the same name by Sharat Chandra Chattopadhyaya. (Bongs, I think, over-estimate this novelist. His other novel is Devadaas. Yes, Devadaas. … Now you know. About the Chattopadhyaya.)) Anyway, as to this song itself, well, Salil-daa’s stamp is absolutely unmistakable. (If the Marathi listener feels that the flute piece appearing at the very beginning somehow sounds familiar, and then recalls the flute in Hridayanath Mangeshkar’s “mogaraa phulalaa,” then I want to point out that it was Hridayanath who once assisted Salil-daa, not the other way around.) IMO, this song is just great. The tune may perhaps sound like the usual ghazal-like tune, but the orchestration—it’s just extraordinary, sensitive, and overall, absolutely superb. This song in fact is one of Salil-daa’s all-time bests, IMO. … I don’t know who penned the lyrics, but they too are great. … Hint: Listen to this song on high-quality head-phones, not on the loud-speakers, and only when you are all alone, all by yourself—and especially as you are nursing your favorite Sundowner—and especially during the times when you are going jobless. … Try it, some such a time…. Take care, and bye for now]

[E&OE]

/

Even while enjoying my writer’s block, I still won’t disappoint you. … My browsing has yielded some material, and I am going to share it with you.

It all began with googling for some notes on CFD. One thing led to another, and soon enough, I was at this page [^] maintained by Prof. Praveen Chandrashekhar of TIFR Bangalore.

Do go through the aforementioned link; highly recommended. It tells you about the nature of my trade [CFD]…

As that page notes, this article had first appeared in the AIAA Student Journal. Looking at the particulars of the anachronisms, I wanted to know the precise date of the writing. Googling on the title of the article led me to a PDF document which was hidden under a “webpage-old” sub-directory, for the web pages for the ME608 course offered by Prof. Jayathi Murthy at Purdue [^]. At the bottom of this PDF document is a note that the AIAA article had appeared in the Summer of 1985. … Hmm…. Sounds right.

If you enjoy your writer’s block [the way I do], one sure way to continue having it intact is to continue googling. You are guaranteed never to come out it. I mean to say, at least as far as I know, there is no equivalent of Godwin’s law [^] on the browsing side.

Anyway, so, what I next googled on was: “wind tunnels.” I was expecting to see the Wright brothers as the inventors of the idea. Well, I was proved wrong. The history section on the Wiki page [^] mentions Benjamin Robins and his “whirling arm” apparatus to determine drag. The reference for this fact goes to a book bearing the title “Mathematical Tracts of the late Benjamin Robins, Esq,” published, I gathered, in 1761. The description of the reference adds the sub-title (or the chapter title): “An account of the experiments, relating to the resistance of the air, exhibited at different times before the Royal Society, in the year 1746.” [The emphasis in the italics is mine, of course! [Couldn’t you have just guessed it?]]

Since I didn’t know anything about the “whirling arm,” and since the Wiki article didn’t explain it either, a continuation of googling was entirely in order. [The other reason was what I’ve told you already: I was enjoying my writer’s block, and didn’t want it to go away—not so soon, anyway.] The fallout of the search was one k-12 level page maintained by NASA [^]. Typical of the government-run NASA, there was no diagram to illustrate the text. … So I quickly closed the tab, came back to the next entries in the search results, and landed on this blog post [^] by “Gina.” The name of the blog was “Fluids in motion.”

… Interesting…. You know, I knew about, you know, “Fuck Yeah Fluid Dynamics” [^] (which is a major time- and bandwidth-sink) but not about “Fluids in motion.” So I had to browse the new blog, too. [As to the FYFD, I only today discovered the origin of the peculiar name; it is given in the Science mag story here [^].]

Anyway, coming back to Gina’s blog, I then clicked on the “fluids” category, and landed here [^]… Turns out that Gina’s is a less demanding on the bandwidth, as compared to FYFD. [… I happen to have nearly exhausted my monthly data limit of 10 GB, and the monthly renewal is on the 5th June. …. Sigh!…]

Anyway, so here I was, at Gina’s blog, and the first post in the “fluids” category was on “murmuration of starlings,” [^]. There was a link to a video… Video… Video? … Intermediate Conclusion: Writer’s blocks are costly. … Soon after, a quiet temptation thought: I must get to know what the phrase “murmuration of starlings” means. … A weighing in of the options, and the final conclusion: what the hell! [what else], I will buy an extra 1 or 2 GB add-on pack, but I gotta see that video. [Writer’s block, I told you, is enjoyable.] … Anyway, go, watch that video. It’s awesome. Also, Gina’s book “Modeling Ships and Space Craft.” It too seems to be awesome: [^] and [^].

The only way to avoid further spending on the bandwidth was to get out of my writer’s block. Somehow.

So, I browsed a bit on the term [^], and took the links on the first page of this search. To my dismay, I found that not even a single piece was helpful to me, because none was relevant to my situation: every piece of advice there was obviously written only after assuming that you are not enjoying your writer’s block. But what if you do? …

Anyway, I had to avoid any further expenditure on the bandwidth—my expenditure—and so, I had to get out of my writer’s block.

So, I wrote something—this post!

[Blogging will continue to remain sparse. … Humor apart, I am in the middle of writing some C++ code, and it is enjoyable but demanding on my time. I will remain busy with this code until at least the middle of June. So, expect the next post only around that time.]

[May be one more editing pass tomorrow… Done.]

[E&OE]

/

The anti-, an anti-anti-, my negativism, and miscellaneous

Prologue:

A better title could very well have been “What I am up against.” However, that title, I thought, would be misleading. … I really am up against many things which I am going to touch on, in this post. But the point is, these are not the only things that I am up against, and so, that title would therefore be too general.

Part I: The Anti-

First, of course, comes the anti.

I stumbled across W. E. Lamb, Jr.’s excellent paper: “Anti-photon” (1995) Appl. Phys. B, vol. 60, p. 77–84. Here is the abstract:

“It should be apparent from the title of this article that the author does not like the use of the word “photon”, which dates from 1926. In his view, there is no such thing as a photon. Only a comedy of errors and historical accidents led to its popularity among physicists and optical scientists. I admit that the word is short and convenient. Its use is also habit forming. Similarly, one might find it convenient to speak of the “aether” or “vacuum” to stand for empty space, even if no such thing existed. There are very good substitute words for “photon”, (e.g., “radiation” or “light”), and for “photonics” (e.g., “optics” or “quantum optics”). Similar objections are possible to use of the word “phonon”, which dates from 1932. Objects like electrons, neutrinos of finite rest mass, or helium atoms can, under suitable conditions, be considered to be particles, since their theories then have viable non-relativistic and non-quantum limits. This paper outlines the main features of the quantum theory of radiation and indicates how they can be used to treat problems in quantum optics.”

BTW, in case you don’t know, W. E. Lamb, Jr., was an American, who received a Nobel in physics, for his work related to the fine structure of hydrogen [^].

So, that’s the first bit of what I am up against.

Also in case you didn’t notice, the initials are important; this isn’t (Sir) Horace Lamb (who, in case you don’t know, was that late 19th–early 20th century British guy who wrote books on hydrodynamics and acoustics that people like me still occasionally refer to [^]. (Lamb and Love continue to remain in circulation (even if a low circulation) among mechanicians even today. (Love, who? … That’s an exercise left for the reader…)))

Oh, BTW, talking of very good books that now have come in the public domain, and (the preparation required for) QM, and all the anti- and un- things, note that Professor Howard Georgi [^]’s excellent book on waves has by now come in the public domain [^].

(Even if only parenthetically, I have to note: I am anti-diversity, too. … This anti thing simply doesn’t leave me alone, though I will try to minimize its usage. Starting right now. … Georgi was born in California. He also maintains a page about women in physics [^].)

… Ummm, I’d better wrap up this part, and so…

… All in all, you can see that I don’t seem to be taking my opposition very seriously, though I admit I should start doing so some day. But the paper is great. (We were talking about the anti-photon paper, remember?) Here is an excerpt in case I haven’t already succeeded in persuading you to go through it, immediately:

“During my eight years in Berkeley, I had just one conversation with Lewis, in 1937, when he called me into his office to give some advice. It was: “When a theorist does not know what to do next, he is useless. An experimental scientist can always go into his laboratory and “polish up the brass”.”

This is the same Lewis who coined the word “photon.” … Now it convinces you to go through the paper, doesn’t it? (The paper is by Lamb; W. E. Lamb.)

[… On a more serious note, this paper has very good notings regarding the history of the idea of the photon.]

Part II: The Anti- Equals the Anti-Anti-

There is no typo here.

Even as I was recoiling off the glow (I won’t use “radiation” or “light”) of [the physics Nobel laureate] Lamb’s reputation, I began wondering precisely how I would counter his anti-photon argument. I even thought of doing a blog post about it. (After all, recently, Roger Schlafly has been hinting at that same idea, too. [May be TBD: insert links])

However, a better sense prevailed, and I did a Google search. I found a good blog post that gives a good rejoinder to the anti-photon arguments. The post is written in simple enough language that any one could understand. … But should I recommend it to you?… The thing is: It comes from a physicist who is reputed to have attempted teaching quantum physics to dogs. Or, at least, teaching people how to teach quantum physics, to dogs.

But of course, in physics, personalities don’t count, and neither do, you know, sort of like, “insults.” [I am also anti-animal rights, BTW [though all in favor of dogs].] And so, let me lead you to the relevant post.

The quantum physics-loving folks would have guessed the man by now (and every one, the fact that the author must be a man, not a woman). So the only remaining part would be which post by Chad Orzel. Here it is [^]. Once you finish reading it (including the comments on the post), then, also go through these couple of others posts by him touching on the same topic [^] [^] (and their blog comments). And, a great post (at wired.com!) by Rhett Allain [^] on the anti-photon side, to which Orzel makes a reference.

Orzel’s basic argument is that anti-bunching equals anti-anti-photon.

That explains the second part of the title.

But, before wrapping up this part, just a word on the PhD guides on the “polishing brass” side, and Indians. The anti-bunching experiments were done by Leonard Mandel [^], who among other things also guided Rupamanjari Ghosh’s PhD thesis. … Rupa…, who? I will save you the trouble of googling; see here: [^ (I am anti-government in education and science, too)] and here [^ (oh well, this post is getting just too long)].

Part III: My Negativism

Roger Schlafly has just recently written an interestingly long post on quantum entanglement. (Very long, by his standards.) In that post [^], he identifies himself as a logical positivist. This isn’t the first time that he has attributed logical positivism to his intellectual positions. Schlafly’s recent post is written, as usual, with good/great clarity

Now consider the premises, this time three, instead of the usual two: (i) Schlafly identifies himself as a logical positivist, (ii) I don’t agree with some part of his positions, and (iii) logic is logic—it cares for completeness.

Ergo, I must be a logical negativist.

That explains the third part of the title.

Some day I plan to write a post on the triplet and singlet states, and quantum entanglement.

Some still later day, I plan to explain how QM is incomplete, by pointing out how it can be made complete. … That is too big a goal to keep, you say?

Well, I do plan to at least explain in simpler terms the phenomenon of quantum entanglement, but only in reference to the text-book treatments. … That should be doable, what say?

… Don’t hold me responsible etc. on this promise; I am careless etc.; and so,  it might very well be in mid-2016 when I might actually deliver on it. … So, for the time being, make do with my logical negativism.

Part IV: Miscellaneous

M1: The preface to Georgi’s book notes the help he received while writing the book inter alia from (the same) Griffiths (as the one who has written very popular undergraduate text-books on electrodynamics and QM). (Griffiths studied at Harvard where Georgi has been a professor, though chances are they were contemporaries.) (No, this Griffiths isn’t the same as the Griffiths of fracture mechanics [^].) (Yes, this Georgi is the same as the one who has advocated the unparticle mechanics [^]. (But why didn’t he use the anti- prefix here?))

M2: The most succinct (and as far as I can make out, correct) treatment of the meaning of “hidden variables” has been not in the recent Internet writings on Bell’s inequalities but in Griffith’s undergraduate text-book on QM.

Why I mention this bit… That’s because, recently, the MIT professor Scott Aaronson had a field day about hidden variables (notably with Travis Norsen) [^], though since then he seems to have moved on to some other things related to theoretical computational complexity, e.g. this graph isomorphism-related thingie [^].

But, no, if you want to know about the so-called hidden variables well (and don’t have my “approach” or at least my “confidence”), then don’t look up the material on the ‘net or blog posts, esp. those by CS folks or complexity theorists. Instead, hit Griffith’s (text-)book.

M3: However, I am unhappy about Griffith’s treatment of the quantum postulates—he (like QChem and most all UG QM books) has only the usual $\Psi$ and doesn’t include the spinor function right while discussing the state definition. Indeed, he continues implicitly treating the two in a somewhat disjoint manner even afterwards (exactly like all UG text-books do). Separable doesn’t mean disjointed.

I am also unhappy about Griffith’s (and every other QM text-book’s) treatment of the basic ideas of identical particles and their states—the treatments are just not conceptually clarifying enough. … May I assist you rewriting this topic, Professor Griffiths? … Oh well… Before I actually make that offer to him, I will try my hand at the task, at this blog…. Sometime in/after mid-2016. (Hopefully earlier.)

But, yes, if you ask me, it’s only the spin and identical particles that still remain truly nebulous topics for the student, today. With single-particle interference experiments and the ubiquity of simulations, one wouldn’t think that people would have too much difficulty with wave-particle duality or interference etc.

Contrast staring at one or two manually drawn static graphs in a book/paper, and imagining how things would change with time, under different governing equations and different boundary conditions, vs. going through simulations on your smartphone, adjusting FPS, changing boundary conditions with the flick of a button… Students (like me) must be having it exponentially easier to learn QM these days, as compared to those hapless 20th century guys.

The points where today’s students are likely to falter would be a bit more advanced ones, like angular momentum. In fact, today’s students don’t know angular momentum well even in the classical mechanics settings. (Ask yourself: how clear and confident are you about, say, Coriolis forces, say, as covered in Shames, or in Timoshenko and Young?).

So, to wrap up, it has to be identical particles and spin that still remain the really difficult topics. Now, it so happens that it is these concepts that underlie popular expositions of entanglement. Little surprise that people never get the confidence that they would be able to deal with entanglement right.

(Focusing on “just” two states of the spin up- and down-, and therefore treating the phenomenon via an abstract two component vector, and then thinking that starting a discussion with this “simple” vector, is a very bad idea, epistemologically speaking. … Yes, I am anti-Susskind’s “theoretical minimum,” too. And yes, Griffiths is right in choosing the traditional way (of the sequence in which to present the QM spin). It’s just that he needs to explain it in (even) better manner, that’s all….)

M4: The day before yesterday was the first time this year that I happened to finally sense that wonderful winter-time air of Pune’s, while returning in the evening from our college. (Monday was a working day for us; no continuous 9-day patch of a vacation.)

It still doesn’t feel like the Diwali air this year in Pune, but it’s getting close: I spotted some nice fog/mist on the nearby nallah (i.e. a small stream) and a nearby canal, a couple of times. …

This has been a year of (heavy) drought. And anyway, these days, there is virtually no difference between the Diwali days and the rest of the year. … Shopping malls are fully Diwali-like at any time of the year for those who have the money, and most women—whether working or otherwise—these days outsource their (Marathi) “chakalee”-making anyway—even during Diwali. So, there isn’t much of a difference between the Diwali days and the other days. Except for the weather. Weather still continues to change in a distinctly perceptible way sometime around Diwali. … So, that’s about all what Diwali means to me, this year.

And, of course, some memories of the magical Diwalis that I have spent in my childhood… Many of these were spent (at least for the (Marathi) “bhau-beej” day and a couple of days more) at my maternal uncle’s place (a very small town, a sub taluka-level place). … As far as I am concerned, those Diwali’s are still real; they would easily remain that way throughout my life.

PS: Having written the post, I just stepped into the kitchen to make me a cup of tea, and that’s when father told me that home-made (Marathi) “chakalee”s had arrived from our family friends just last evening; I didn’t know about it.

Instantaneously, my song-selection collapsed into an anti-previously measured state. (It happens. Real life is more weird than QM.)

Epilogue:

Happy Diwali!

PS (also) to Epilogue:

Excuse me for a couple of weeks now. I will continue studying QM (from text-books), but I will also have to be taking out my notes for an undergraduate course on CFD (computational fluid dynamics, in case you didn’t know) that I should be teaching the next semester—which begins right in mid-December. (In India, we don’t always follow the Christmas–New Year’s–Next Term sequence.) I anyway will also be traveling a bit (just short distances like Mumbai and Nasik or so) over the next couple of weeks. So, I don’t think I will have the time to write a post. (That, in fact, was the reason why I threw in a lot of stuff right in this post.)

… So, there… Take care, and best wishes, once again, for a bright and happy Diwali (and to those of you who start a new year in Diwali, best wishes for a happy and prosperous new year too.)

A Song I Like

(Marathi) “tabakaamadhye ithe tevatee…”  (search on the transcriptionally incorrect “divya divyanchi jyot”)
Singer: Asha Bhosale
Lyrics: Ravindra Bhat

[PS: I kept on adding material after publication of post, and now it has become some 1.5 times the original one. Sorry about that (though I did all the revisions right within 18 hours of publication), but now I am going stop editing any further. Put up with my grammatical mistakes and awkward constructions, as usual. And, if in doubt, ask me! Bye for now.]

[E&OE]

/

Mathematics—Historic, Contemporary, and Its Relation to Physics

The title of this post does look very ambitious, but in fact the post itself isn’t. I mean, I am not going to even attempt to integrate these diverse threads at all. Instead, I am going to either just jot down a few links, or copy-paste my replies (with a bit editing) that I had made at some other blogs.

1. About (not so) ancient mathematics:

1.1 Concerning calculus: It was something of a goose-bumps moment for me to realize that the historic Indians had very definitely gotten to that branch of mathematics which is known as calculus. You have to understand the context behind it.

Some three centuries ago, there were priority battles concerning invention of calculus (started by Newton, and joined by Liebniz and his supporters). Echoes of these arguments could still be heard in popular science writings as recently as when I was a young man, about three decades ago.

Against this backdrop, it was particularly wonderful that an Indian mathematician as early as some eight centuries ago had gotten to the basic idea of calculus.

The issue was highlighted by Prof. Abinandanan at the blog nanpolitan, here [^]. It was based on an article by Prof. Biman Nath that had appeared in the magazine Frontline [^]. My replies can be found at Abi’s post. I am copy-pasting my replies here. I am also taking the opportunity to rectify a mistake—somehow, I thought that Nath’s article appeared in the Hindu newspaper, and not in the Frontline magazine. My comment (now edited just so slightly):

0. Based on my earlier readings of the subject matter (and I have never been too interested in the topic, and so, it was generally pretty much a casual reading), I used to believe that the Indians had not reached that certain abstract point which would allow us to say that they had got to calculus. They had something of a pre-calculus, I thought.

Based (purely) on Prof. Nath’s article, I have now changed my opinion.

Here are a few points to note:

1. How “jyaa” turned to “sine” makes for a fascinating story. Thanks for its inclusion, Prof. Nath.

2. Aaryabhata didn’t have calculus. Neither did Bramhagupta [my spelling is correct]. But if you wonder why the latter might have laid such an emphasis on the zero about the same time that he tried taking Aaryabhata’s invention further, chances are, there might have been some churning in Bramhagupta’s mind regarding the abstraction of the infinitesimal, though, with the evidence available, he didn’t reach it.

3. Bhaaskara II, if the evidence in the article is correct, clearly did reach calculus. No doubt about it.

He did not only reach a more abstract level, he even finished the concept by giving it a name: “taatkaalik.” Epistemologically speaking, the concept formation was complete.

I wonder why Prof. Nath, writing for the Frontline, didn’t allocate a separate section to Bhaaskara II. The “giant leap” richly deserved it.

And, he even got to the max-min problem by setting the derivative to zero. IMO, this is a second giant leap. Conceptually, it is so distinctive to calculus that even just a fleeting mention of it would be enough to permanently settle the issue.

You can say that Aaryabhata and Bramhagupta had some definite anticipation of calculus. And you can’t possible much more further about Archimedes’ method of exhaustion either. But, as a sum total, I think, they still missed calculus per say.

But with this double whammy (or, more accurately, the one-two punch), Bhaaskara II clearly had got the calculus.

Yes, it would have been nice if he could have left for the posterity a mention of the limit. But writing down the process of reaching the invention has always been so unlike the ancient Indians. Philosophically, the atmosphere would generally be antithetical to such an idea; the scientist, esp. the mathematician, may then be excused.

But then, if mathematicians had already been playing with infinite series with ease, and were already performing the calculus of finite differences in the context of these infinite series, even explicitly composing verses about their results, then they can be excused for not having conceptualized limits.

After all, even Newton initially worked only with the fluxion and Leibniz with the infinitesimal. The modern epsilon-delta definition still was some one–two centuries (in the three–four centuries of modern science) in the coming.

But when you explicitly say “instantaneous,” (i.e. after spelling out the correct thought process leading to it), there is no way one can say that some distance had yet to be travelled to reach calculus. The destination was already there.

And as if to remove any doubt still lingering, when it comes to the min-max condition, no amount of merely geometric thinking would get you there. Reaching of that conclusion means that the train had not already left the first station after entering the calculus territory, but also that it had in fact gone past the second or the third station as well. Complete with an application from astronomy—the first branch of physics.

I would like to know if there are any counter-arguments to the new view I now take of this matter, as spelt out above.

4. Maadhava missed it. The 1/4 vs. 1/6 is not hair-splitting. It is a very direct indication of the fact that either Maadhava did a “typo” (not at all possible, considering that these were verses to be by-hearted by repetition by the student body), or, obviously, he missed the idea of the repeated integration (which in turn requires considering a progressively greater domain even if only infinitesimally). Now this latter idea is at the very basis of the modern Taylor series. If Maadhava were to perform that repeated integration (and he would be a capable mathematical technician to be able to do that should the idea have struck him), then he would surely get 1/6. He would get that number, even if he were not to know anything about the factorial idea. And, if he could not get to 1/6, it’s impossible that he would get the idea of the entire infinite series i.e. the Taylor series, right.

5. Going by the content of the article, Prof. Nath’s conclusion in the last paragraph is, as indicated above, in part, non-sequitur.

6. But yes, I, too, very eagerly look forward to what Prof. Nath has to say subsequently on this and related issues.

But as far as the issues such as the existence of progress only in fits here and there, and indeed the absence of a generally monotonously increasing build-up of knowledge (observe the partial regression in Bramhagupta from Aaryabhat, or in Maadhav from Bhaaskar II), I think that philosophy as the fundamental factor in human condition, is relevant.

7. And, oh, BTW, is “Matteo Ricci” a corrupt form of the original “Mahadeva Rishi” [or “Maadhav Rishi”] or some such a thing? … May Internet battles ensue!

1.2 Concerning “vimaan-shaastra” and estimating $\pi$: Once again, this was a comment that I made at Abi’s blog, in response to his post on the claims concerning “vimaan-shaastra” and all, here[^]. Go through that post, to know the context in which I wrote the following comment (reproduced here with a bit of copy-editing):

I tend not to out of hand dismiss claims about the ancient Indian tradition. However, this one about the “Vimaan”s and all does seem to exceed even my limits.

But, still, I do believe that it can also be very easy to dismiss such claims without giving them due consideration. Yes, so many of them are ridiculous. But not all. Indeed, as a less noted fact, some of the defenders themselves do contradict each other, but never do notice this fact.

Let me give you an example. I am unlike some who would accept a claim only if there is a direct archaeological evidence for it. IMO, theirs is a materialistic position, and materialism is a false premise; it’s the body of the mind-body dichotomy (in Ayn Rand’s sense of the terms). And, so, I am willing to consider the astronomical references contained in the ancient verses as an evidence. So, in that sense, I don’t dismiss a 10,000+ old history of India; I don’t mindlessly accept 600 BC or so as the starting point of civilization and culture, a date so convenient to the missionaries of the Abrahamic traditions. IMO, not every influential commentator to come from the folds of the Western culture can be safely assumed to have attained the levels obtained by the best among the Greek or enlightenment thinkers.

And, so, I am OK if someone shows, based on the astronomical methods, the existence of the Indian culture, say, 5000+ years ago.

Yet, there are two notable facts here. (i) The findings of different proponents of this astronomical method of dating of the past events (say the dates of events mentioned in RaamaayaNa or Mahaabhaarata) don’t always agree with each other. And, more worrisome is the fact that (ii) despite Internet, they never even notice each other, let alone debate the soundness of their own approaches. All that they—and their supporters—do is to pick out Internet (or TED etc.) battles against the materialists.

A far deeper thinking is required to even just approach these (and such) issues. But the proponents don’t show the required maturity.

It is far too easy to jump to conclusions and blindly assert that there were material “Vimaana”s; that “puShpak” etc. were neither a valid description of a spiritual/psychic phenomenon nor a result of a vivid poetic imagination. It is much more difficult, comparatively speaking, to think of a later date insertion into a text. It is most difficult to be judicious in ascertaining which part of which verse of which book, can be reliably taken as of ancient origin, which one is a later-date interpolation or commentary, and which one is a mischievous recent insertion.

Earlier (i.e. decades earlier, while a school-boy or an undergrad in college etc.), I tended to think the very last possibility as not at all possible. Enough people couldn’t possibly have had enough mastery of Sanskrit, practically speaking, to fool enough honest Sanskrit-knowing people, I thought.

Over the decades, guess, I have become wiser. Not only have I understood the possibilities of the human nature better on the up side, but also on the down side. For instance, one of my colleagues, an engineer, an IITian who lived abroad, could himself compose poetry in Sanskrit very easily, I learnt. No, he wouldn’t do a forgery, sure. But could one say the same for every one who had a mastery of Sanskrit, without being too naive?

And, while on this topic, if someone knows the exact reference from which this verse quoted on Ramesh Raskar’s earlier page comes, and drops a line to me, I would be grateful. http://www.cs.unc.edu/~raskar/ . As usual, when I first read it, I was impressed a great deal. Until, of course, other possibilities struck me later. (It took years for me to think of these other possibilities.)

But, in case you missed it, I do want to highlight my question again: Do you know the reference from which this verse quoted by Ramesh Raskar (now a professor at MIT Media Lab) comes? If yes, please do drop me a line.

2. An inspiring tale of a contemporary mathematician:

Here is an inspiring story of a Chinese-born mathematician who beat all the odds to achieve absolutely first-rank success.

I can’t resist the temptation to insert my trailer: As a boy, Yitang Zhang could not even attend school because he was forced into manual labor on vegetable-growing farms—he lived in the Communist China. As a young PhD graduate, he could not get a proper academic job in the USA—even if he got his PhD there. He then worked as an accountant of sorts, and still went on to solve one of mathematics’ most difficult problems.

Alec Wilkinson writes insightfully, beautifully, and with an authentic kind of admiration for man the heroic, for The New Yorker, here [^]. (H/T to Prof. Phanish Suryanarayana of GeorgiaTech, who highlighted this article at iMechanica [^].)

3. FQXi Essay Contest 2015:

(Hindi) “Picture abhi baaki nahin hai, dost! Picture to khatam ho gai” … Or, welcome back to the “everyday” reality of the modern day—modern day physics, modern day mathematics, and modern day questions concerning the relation between the two.

In other words, they still don’t get it—the relation between mathematics and physics. That’s why FQXi [^] has got an essay contest about it. They even call it “mysterious.” More details here [^]. (H/T to Roger Schlafly [^].)

Though this last link looks like a Web page of some government lab (American government, not Indian), do check out the second section on that same page: “II Evaluation Criteria.” The main problem description appears in this section. Let me quote the main problem description right in this post:

The theme for this Essay Contest is: “Trick or Truth: the Mysterious Connection Between Physics and Mathematics”.

In many ways, physics has developed hand-in-hand with mathematics. It seems almost impossible to imagine physics without a mathematical framework; at the same time, questions in physics have inspired so many discoveries in mathematics. But does physics simply wear mathematics like a costume, or is math a fundamental part of physical reality?

Why does mathematics seem so “unreasonably” effective in fundamental physics, especially compared to math’s impact in other scientific disciplines? Or does it? How deeply does mathematics inform physics, and physics mathematics? What are the tensions between them — the subtleties, ambiguities, hidden assumptions, or even contradictions and paradoxes at the intersection of formal mathematics and the physics of the real world?

This essay contest will probe the mysterious relationship between physics and mathematics.

Further, this section actually carries a bunch of thought-provocative questions to get you going in your essay writing. … And, yes, the important dates are here [^].

Is this issue interesting enough? Yes.

Will I write an essay? No.

Why? Because I haven’t yet put my thoughts in a sufficiently coherent form.

However, I notice that the contest announcement itself includes so many questions that are worth attempting. And so, I will think of jotting down my answers to these questions, even if in a bit of a hurry.

However, I will neither further forge the answers together in a single coherent essay, nor will I participate in the contest.

And even if I were to participate… Well, let me put it this way. Going by Max Tegmark’s and others’ inclinations, I (sort of) “know” that anyone with my kind of answers would stand a very slim chance of actually landing the prize. … That’s another important reason for me not even to try.

But, yes, at least this time round, many of the detailed questions themselves are both valid and interesting. And so, it should be worth your while addressing them (or at least knowing what you think of them for your answers). …

As far as I am concerned, the only issue is time. … Given my habits, writing about such things—the deep and philosophical, and therefore fascinating things, the things that are interesting by themselves—have a way of totally getting out of control. That is, even if you know you aren’t going to interact with anyone else. And, mandatory interaction, incidentally, is another FQXi requirement that discourages me from participating.

So, as the bottom-line: no definitive promises, but let me see if I can write a post or a document by just straight-forwardly jotting down my answers to those detailed questions, without bothering to explain myself much, and without bothering to tie my answers together into a coherent whole.

Ok. Enough is enough. Bye for now.

[May be I will come back and add the “A Song I Like” section or so. Not sure. May be I will; may be I won’t. Bye.]

[E&OE]

/