# A little more on my research on the diffusion equation

Alright. Here we go again. … In my last post, I had mentioned a bit about how attending the recent ISTAM conference at Pune, had helped me recall my thoughts on the diffusion equation and all. In particular, I had mentioned in that post how I had discovered a Berkeley professor’s paper only after publishing my own paper (in ISTAM, 2006, held at Vishakhapattanam), and that I would revise my ISTAM paper and send it over to a journal.

The gentleman in question is Prof. T. N. Narasimhan. Unfortunately, I gather, he has passed away in 2011 [^][^]. Here is the group of his relevant papers:

Narasimhan, T. N. (1999) “Fourier’s heat conduction equation: history, influence, and connections,” Reviews of Geophysics, vol. 37, no. 1, pp. 151–172

Narasimhan, T. N. (2009) “The dichotomous history of diffusion,” Physics Today, July 2009, pp. 48–53

Narasimhan, T. N. (2009) “Laplace, Fourier, and stochastic diffusion,” arXiv:0912.2798, 13 Dec 2009

Narasimhan, T. N. (2010) “On physical diffusion and stochastic diffusion,” Current Science, vol. 98, no. 1, 10 January 2010, pp. 23–26

All these papers are available somewhere or the other on the ‘net. (Copy-paste the paper titles in a Google Scholar search, and you should get to the PDF files.)

The first paper is the most comprehensive among them. In this paper, Prof. Narasimhan discusses the historical context of the development of Fourier’s theory and its ramifications. The paper even gives a very neat (and highly comprehensive) table of the chronology of the developments related to the diffusion equation.

In this paper, unlike in so many others on the diffusion equation, he explicitly (even if only passingly) looks into the issue of the action-at-a-distance. However, the way he discusses this issue, it seems to me, he perhaps had entirely missed the crucial objections that are to be made against the idea of IAD—the same basic things which, in yet another context, lead people into believing in quantum entanglement in the sense they do. (And, how!) In fact, in this paper, Narasimhan does not really discuss any of those basic considerations concerning IAD. If so, then what is it that he discusses?

Narasimhan uses the term “action at a distance” (AD), and not the more clarifying instantaneous action at a distance (IAD). However, such a usage hardly matters. It’s true that if it’s just the AD that you take up for discussion, then it’s the issue of the absence of a mediating agency or a medium (or the premise of a contact-less transmission of momenta/forces) that you highlight, and not so much the instantaneity of the transmission, even if the latter has always been implied in any such a discussion. As people had observed right in the time of Newton himself, the assumption of instantaneity, of IAD, was there, built right at the base of his theory of gravitation, even if it was billed only as an AD theory back then. The difference between IAD and AD is more terminological in nature.

But then, Narasimhan is not even very explicit in his positions with respect to AD either. To get to his rather indirect remarks on the AD issue, first see his discussion related to Biot and the particles approach that he was trying (pp. 154, the 1999 paper).

Today, i.e., after the existence and acceptance of the kinetic molecular theory for more than a century, a modern reader would expect the author to say that someone who believes in, or at least is influenced by, a particles-based approach would naturally be following a local approach, and hence should be found on the side of denying the IAD. Instead, though the author does not explicitly take any position, from the way he phrases his lines, he seems to suggest that he thinks that someone who adopts a particles-based approach would have found AD to be natural. This contradiction was what I had found intriguing initially.

But then, soon enough, I figured out a plausible way in which the author’s thought-train might have progressed. He must have taken the gravitational interaction between n number of bodies as the paradigm of every AD theory, and therefore, must have come to associate (I)AD with any particles-based approach—in exact opposition to the local nature of the particles-based theories of the 19th century and the later techniques (e.g. LBM, SPH, etc.) of the 20th century. That can only be the reason why Narasimhan makes the AD-related comments the way he makes them, especially in reference to Biot’s work (pp. 154).

I would try to gather more historical material, and in any case, address this issue in my forthcoming paper. That’s what I meant when I said I would revise my paper and send it to a journal. I didn’t mean to say that I would be revising my position—I would be only clarifying it to a greater detail. My position is that a kinetic theoretical model i.e. a particles-based approach, the default way people interpret it, does not involve IAD.

Anyway, back to Narasimhan’s paper. Further on this issue, on page 155 of the same paper, the author states the following:

“Essentially, Fourier moved away from discontinuous bodies and towards continuous bodies. Instead of starting with the basic equations of action at a distance, Fourier took an empirical, observational approach to idealize how matter behaved macroscopically.”

[Bold emphasis mine]

In this passage, to be historically accurate, in place of: “action at a distance” the author should have said: “a discrete/particles-based approach;” and in place of: “an empirical, observational” he should have said: “a continuum-based” approach. After all, none, to my knowledge, has ever empirically or experimentally observed an infinite speed of heat transmission—none possibly could.

Now, of course, the Fourier theory does not really acquire its IAD nature because it’s a continuum theory. The reasons are different; however, that’s yet completely different point. See my ISTAM paper for more details.

Coming back to Narasimhan’s paper, of course, the above-mentioned flaws present in it are wholly minor. On the other hand, his paper carries excellent and comprehensive commentary on so many other important aspects, including the historical ones. Indeed, he is to be lauded and thanked for at least including the (I)AD issue in a paper on diffusion, despite being at Berkeley. …

These days, given the attitudes of the people at places like Berkeley, Stanford, MIT, etc., they would seem to carry this attitude towards my paper on diffusion: “Uh. But it all was already known; wasn’t it?” No, it was not. That precisely is (and has been) the point. Unless you have read my paper, what goes by being “known” would squarely consist of something like the following:

“There is IAD in Newton’s theory of gravity. And also, in the Fourier theory, though its effects are quantitatively negligible, and so, we can always neglect it in analysis and interpretation. So, there is this IAD in the partial differential diffusion equation, and this fact has always been known. And, lately, some attempts have been made to rectify this situation, e.g. the relativistic heat equation, but with limited success.”

The correct statement is:

“There is IAD in Newton’s theory of gravity. There also is IAD in the Fourier theory. But there is no IAD in the diffusion equation itself. Following the commonly accepted way of taking it, the kinetic molecular theory may be taken not to have any IAD in it. However, it would be easily possible to introduce IAD also into it. The relativistic wave equation is not at all relevant to this set of basic observations.”

There is quite a difference between the two sets of statements.

I am still going through Narasimhan’s other papers, but at least after a cursory look, these seem more like just restatements being made to different audiences of what essentially are the same basic positions.

Apart from it all, here are a few other papers, now on the Brownian movement side of it:

Hanggi, Peter and Marchesoni, Fabio (2005) “100 years of Brownian motion,” arXiv:cond-mat/0502053v1. 2 Feb. 2005

Chowdhury, Debashish (2005) “100 years of Einstein’s theory of Brownian motion: from pollen grains to protein trains,” arXiv:cond-mat/0504610v1. 24 April 2005

Gillespie, Daniel T. (1996) “The mathematics of Brownian motion and Johnson noise,” American Journal of Physics, vol. 64, no. 3, pp. 225–240

As you can see by browsing through all these papers, few people seem to have appreciated the aspect of IAD or locality, in these two theories. Perhaps, that’s a part of the reason why quantum conundrums continue to flourish. Yet, it is important to isolate this particular aspect, if we are to be clear concerning our fundamentals. As someone said, Fourier’s theory has by now become a part of the very culture of science. So deep is its influence. It’s time we stopped being nonchalant about it, and began re-examining its premises and implications.

Ok. Enough for today. If you are interested, go through these papers, and I will be back with some further comments on them. Hopefully soon. I anyway need to finish this paper. Without getting a couple of papers or so published in journals, I cannot guide PhD students. But, that doesn’t mean I will deliberately send this diffusion paper to a sub-standard or even a low impact journal. I will try to get it published in as high quality (but fitting sort of) a journal as possible. And, if you have any suggestions as to which journal I should send this diffusion paper, please do not hesitate in dropping me a line.

* * * * *   * * * * *   * * * * *

I still remain jobless.

However, on a whim, for this post, I decided to add the “A Song I Like” section.

The selection, as usual, is more or less completely random. How random? Well, here is the story about the selection of this particular song, this time around…

As it so happens, sometimes, you “get” or “catch” some tune right in the morning, and then it stays with you for the whole day. The harder you try to get it out of your mind, the more lightly but more firmly it keeps returning to you throughout the day. It doesn’t even have to be a good tune; it simply keeps returning back. That’s what happened with this song, though, the song happens to be a better one. The song happened to so fleetingly alight on my mind in a recent short journey, that I had not realized that I was silently humming it almost halfway down in that journey. (I was not playing any song/music at that time.) So, even though there is another Lata number (“yeh kaun aayaa”) from the same movie (“saathee”) which I perhaps would have chosen if I were deliberately to make a selection, in view of the lightness with which it had come to me—almost as if entirely by itself—I decided to keep this particular song (“mere jeevan saathee”). (BTW, I haven’t seen the movie, and as usual, the video and other aspects don’t count.) Another point. Neither of these two songs looks like it was composed by Naushad—and, in my books, that’s a plus. When truly in his elements, Naushad feels—to me at least—too traditional, perhaps a bit too melancholic, and, what’s the word… too conforming and invention-less?… Yes, that’s it. He feels too much of a conformist and too invention-less, as far as I am concerned. (Even if some tunes of his might have actually been inventive or of high quality, they follow the groove of the traditional song composition, the traditional guideposts so faithfully, that upon listening to the song, he feels invention-less, anyway. And then, I have my own doubts as to how many times he actually was being inventive, anyway!) Alright. Here is that song—an exception for Naushad, as far as I am concerned. And then, Lata, as usual, takes what is only a first class tune, and manages to take it to an altogether different, higher plane, imparting it with, say, a distinction class:

A Song I Like:
(Hindi) “mere jeevan saathee, kalee thee main to pyaasee…”
Singer: Lata Mangeshkar
Lyrics: Majrooh Sultanpuri

[May be, a minor editing is due, though I would not spend much time on it when I return.]

[E&OE]

# An idea for the final year student projects in CS departments

In this post, let me jot down an idea that is suitable for a final year project by the CS/IT undergraduates (or, in a suitably expanded form, also by the CS/IT post-graduates) in engineering schools. Or, students from other similar courses like the MCA, MSc (CS) etc. The idea is (to me) neat and useful, and not at all difficult to implement. Do consider it in place of those usual topics like railway reservation/airline booking system, payroll system, etc.

The idea is to create an online searchable database of the history of science topics, that gives out customized timelines as query outputs or reports over the ‘net.

The idea occurred to me while doing literature search for my book. (BTW, idiots in those humanities departments degrade the word “research,” by using it for such things as plain literature search. The management and marketing folks do worse: they call activities like the Gallup polls, “research.”)

So long as a time-line involves only tens of items, it is easy to manage its contents manually. As an example of a small time-line, see my previous blog post. (In fact, I got the items in that time-line in an almost fully ready-made form from Manjit Kumar’s book: “Quantum.”)

However, while preparing for the pre-requisites part of my planned QM book, I found that no suitable time-line existed in a ready-made form for those classical physics topics.

One reason is that the pre-requisites of QM are so very many and so very diverse. For a neat visualization of physics as it existed some 70 years ago, see the “1939 map of physics”[^].

Another reason is that the sources of the detailed historical accounts are, again, so very numerous, and not all of them are equally accurate, authentic, or detailed.

While accuracy and authenticity are important, as far as the design work for this particular project goes, I think that the issue of the non-uniformity of the available details is of much greater importance.  The original sources of information themselves are not equally detailed.  For instance, see the Wiki timeline on the luminiferous aether [^], and ask yourself: would any one of the Encyclopedia Britannica’s timelines concerning the aether ever be as detailed as this one?

Of course, the administrators of the database may not rely only on the existing timelines. They may consult books to feed the data to create more detailed timelines. Once again, issues like the unevenness of details and the matter of ascribing proper dates at a detailed level, come up.

For instance, consider the Fourier analysis. Most books mention 1822 as the date of its beginning. It’s wrong.

We don’t have any specific record to answer the question of the precise time when Fourier first thought of this idea of spectral decomposition. However, we do have some evidence which indicates that he began working on the problem of heat propagation in solid bodies as early as 1804, and that he had presented a paper on this topic to the Paris Institute as early as on 21st December 1807. Though Fourier wanted to publish the paper in print, he didn’t (or perhaps couldn’t) do so, because it ran into some criticism. (Off hand, I would suppose, from Poisson.) Fourier had only the sines, not also the cosines, in that first 1807 paper of his. Though important, relatively speaking, it’s a minor omission. What is more important is the fact that he had already had the big idea right, viz. that any arbitrary function could be expanded as (possibly an infinite) series. Now the funny thing is this. While Fourier didn’t know the necessity to include the cosines, neither did his critics. They were rather against the completeness aspect of his idea, viz., that any arbitrary function could be expanded in this way, even the discontinuous functions. On that count, he was on the right track. However, it was hard to convince the established critics. One way or the other, the paper didn’t get published that same year. He sent a revised draft to a competition of the French Academy in 1811, and won it.  The critics, however, still were not satisfied. Fourier ultimately wrote and published the further revised account some 15 years later, in 1822.

Most books will tell you that the Fourier analysis began in 1822. However, unless you take into account the 1804–1807 genesis of this branch of analysis, the overall historical progression does not appear as smooth as it should be.

The question then becomes, in a time line, what date would you include for reporting? 1804? 1807? 1811? 1822? All the four?

Some cases are even more intriguing, and possibly of more than trivial interest to a researcher. For instance, consider Schrodinger’s formulation of quntum mechanics. His very first paper (written in January 1926) is concerned about the famous non-relativistic wave equation, now bearing his name. Yet, even before that very first paper was written, in fact barely just a month before, in December 1925, Schrodinger had initially worked on a relativistic wave equation. It doesn’t work fully satisfactorily, and so Schrodinger worked out the non-relativistic version before sending it out for publication. But wouldn’t the fact that Schrodinger had those relativistic ideas right in the last month of 1925, be important while tracing the development of the relativistic QM? After all, physicists in that era were unusually open in their communications and collaborations. For example, Pauli had begun working on his paper merely on the basis of a draft of Heisenberg’s paper. (I mean the manual draft as well as detailed physics notes Heisenberg exchanged with him—not even the printer’s proofs let alone what today we would call the preprint.) Similarly, Pauli’s (1924?) idea regarding a two-valued quantum number was known to Goudsmit and Uhlenbeck when they proposed the electronic spin (October, 1925).

Should such details be entered into the database? Should there be an attribute specifying the granularity of information that is being carried by a database entry? Should the queries allow various levels of granularities to be included in the search results?

And, going back to Fourier’s case, suppose that your database initially includes an entry for only one of the three dates (1804—1807, 1811 and 1822). Someone points out the other dates to you (i.e. to the system administrators populating the database) only later on. How would you allow for the conciliations of data items like these, in a mechanical sort of a way? What software mechanism would you provide?

Would you handle it via some attributes of the individual entries  in the database? Would these attributes be finalized beforehand, say a pre-determined set like: “Initial Idea/Presentation/Publication/Revision” etc? Or would it be possible and desirable to keep the set of such attributes extensible—say, on lines similar to how you apply attributes or tags to your blog posts, complete with an intellisense feature that shows the already applied attributes? Can the overall structure thus be kept a bit fluid? Is a relational database the optimal data structure, given such objectives? How about the queries on it? Should they be sensitive only to the attributes, or to the “full text”?

I think these are just some of the questions that the students will themselves have to handle. Here, all that I am now going to do is to give some idea of how the usage scenarios would look like. In my description here, I will use the terms like database etc., but only in a broad sense—it doesn’t necessarily mean a traditional data structure like the relational database.

First, there will be a set of users who will populate the database using appropriate sources. They will have the authority to edit the database, and may be called system administrators. The database may sit on a remote secure Web server. The sys admins should be able to use, say, secure http-based (i.e. https-based) forms to edit the database, i.e. to add/modify/delete the entries.

Then there will be the end-users. The end-users should be able to access the database using plain http. Typically, they will use a user-friendly form to send their queries to the server over plain http, and get answers as reports consisting of Web pages. [More on this, in my next post.]

The database will, at the minimum, have information on the following attributes. (If this were an RDBMS, it will have the following as fields or columns): date, scientist, work, remarks, source reference (for the database entry—not the citation record itself), subject areas to which the entry belongs. Many of these attributes are tricky.

There are multiple meanings of “date” in a context like this. Some historical sources give dates only in years, others also in months, still other right up to the day and the time. For some advances, the dates are known only vaguely. For example: “He began working on it during the late 1820s, and continued up to 1826-end” or “fl. 1810s”, or “circa 3rd quarter of 1925”. The calender systems may have changed. Then, there are those BC dates. (Not all database software handle all the dates very smoothly anyway.) Sometimes, the date a scientist sent his manuscript may be more important than when it was published, but not always. Sometimes, the published accounts may be somewhat misleading: Newton did get the basic ideas of calculus right during that plague-related vacation, around 1666. Yet, it also is a fact that he still was working on many of the important calculus ideas in the few years before the publication in 1687 of the Principia. Newton didn’t have everything ready right in 1666, contrary to many stories you might have heard. (There has been a tendency to ascribe too much of a genius to the years of the early youth; wit a dim-wit like Hammings—dim-wit, when it comes to the age at which scientists created their best original ideas.)

The database must store the precise way the date was stated in the original sources. Yet, when a user sends in a query, regardless of the specific original recording, it should still be possible to sort out all the entries, with a certain smartness built into the system. The student will have to resolve the issue of how to put all the entries in (some sort of an) order—vague entries like “in the decade prior to the I WW” together with the more precise entries like “in the afternoon, after lunch, on October 7, 1900.” (Anyone knows what’s special with that time? I would be delighted to know from you!)

As far as this problem of sorting of entries of varying degrees of precision of time-specification is concerned, I do have an idea how it could be done. However, I would rather let the students try to work on it. [OK, I promise to give some hint in my next post on the topic.]

The “scientist” field may undergo changes. A plain guy like Thomson is the same as (the first Baron/Lord) Kelvin (after whom the temperature scale is named). “Snellius” is the same guy as Snell (of the laws of optics). A real gauntlet to us in the 21st century was thrown by a single Swiss family: There is a Daniel who, being one of a family, carries the same last name as Jacob, who is the same as Jakob is the same as Jacques is the same as James.  It doesn’t end there. Johann, is the same as John is the same as Jean, but different from James. And then, there are Johann-II and also Johann-III. Worse: Sometimes they had a feud in the open. The famous brachistochrone problem was posed because one brother tried to show the other brother down. (The other brother succeeded, together with Lebniz and his credits-wise adversary, the “lion’s paws”-bearing Newton.) There has been a son in that family who thought his father had taken credit really due to him. The point? Your user should not get confused, and should be given all the relevant information as to which Bernoulli it was.

Some users may be interested in getting a time-line of all the works of just “Newton.” Others may wish to access a similar information for “Isaac Newton, Jr.” … Yes, the falling apple guy was named after his father, though few people know it, and even fewer (if ever) add the “Jr.” suffix to his name. A more likely search string, especially if originating from a tiny island lying above France and to the west of, say, Finland, might be: “Sir Isaac Newton.”

The “work” field needs to have both succinct information (so that the idea of a time-line or an Excel spreadsheet like report makes sense), but also a place for additional informative details. For instance, less known or less advertised factoids like the fact that Leibniz didn’t originate the idea of the “vis viva,” but was inspired by its descriptions in Huygens’ writings. The additional details should perhaps be supplied only in a more verbose report, but the point is that the database itself should be well-designed that there is a systematic and unambiguous way to tell such things if the need be.

The subjects attributes is important. It identifies all the subject areas in which a given advance of science falls. It will enable the end-user to run subject-specific queries. For example, a query like: “Give me a time-line of the kinetic theory,” which is slightly different from “Give me a time-line of the statistical mechancis,” which is slightly different from “Give me a time-line of the thermodynamics.” Contrary to the current pedagogy followed in the engineering schools, development in statistical mechanics was almost concurrent to thermodynamics. … Incidentally, thermodynamics didn’t exactly begin with Joule. It had sprung into action right after Watt’s engine, with Lazare Carnot, the father of Sadi Carnot, having a lot of thoughts related to the energetics program in physics. In case you have ever wondered how come the Second Law of Thermo came on the scene before the First one did, such a bit might be of interest to you.

Thus, the subject attributes may be both coarse-grained or fine-grained, and they may have multiple abstract hierarchies among them. The system administrators may invite subject experts (e.g. professors) to apply these attributes. This task, though complex and time-consuming, should not too difficult in a way: the attributes to use may come from certain standardized classification schemes such as the PACS classification [^], the AMS classifications [^], etc. What is more important for this project on the CS side is this requirement: the application mechanisms of subject attributes should be sufficiently neat that the application people—the subject experts—should find it very easy to find all the relevant attributes suggested to them in a handy way, when they apply these attributes to the individual entries.

As indicated above, one real tricky point is this: Some discoveries/inventions span across fine-grained descriptions. Another tricky point is the following: Some discoveries span across many fine-grained description only in the context of a later discovery—not otherwise. Therefore, they should not get applied for reports-generation unless the later discovery also is included in the search results. For instance, consider the topic of Fermat’s principle. This is a from classical, geometric, optics. It would not have been a potentially very important candidate for inclusion in mechanics-related searches until Schrodinger made it impossible to avoid it. … Actually, the idea of looking at Fermat’s principle as a mechanical principle goes centuries back; off-hand, I suppose it was Huygens (again), right in the 17th century, who first linked the mechanics of transfer of momentum (he would call it “the motion”) and Fermat’s principle. Then, in the 19th century, there was Hamilton who got inspired by the same conceptual link. But the point here is, before Schrodinger, not identifying the topic as a mechanical one, could, perhaps, have been an excusable omission. After Schrodinger, it is entirely inexcusable.

Each entry should clearly and explicitly identify the source(s) of the information on which it is based. For instance, McTutor’s, Encl. Brit., a certain book, a certain paper, etc. There could be multiple original sources objectively required for the same entry. For instance, when a subject expert rightly ascribes to Euler the impetus provided for development of the calculus of variation (CoV), he would need to take it away from Maupertuis, and for the same reason, he would need original sources authored by both.

BTW, this entry (on the impetus to CoV) would be different from another entry that is concerned with the first correct formalization of the CoV, which would involve a 19-year old Lagrange. And, both these entries would be different form the first-ever problem of CoV correctly posed and applied with correct a correct solution approach, which would be a credit due in (off-hand speaking, a 50+ year-old) Newton’s account (more than a century before Lagrange’s work).

Thus, the individual database entries should carry links to other, related, entries. Such things may come in the Remarks section, or there could be a special “See Also” section.

On the second thoughts, there could an extra attribute (to be carried by a database entry) for specifying the set of pre-requisite entries for it.

Further on the second thoughts, there could also be another, separate attribute that specifies a linkage to the other entries immediately preceding it and constituting the “prior development” for a given entry. (The prior development is not necessarily the same as a pre-requisite. For instance, the caloric fluid theory of heat is a prior development to the kinetic theory, but it’s not a pre-requisite.)

All the report fields may carry hyperlinks to the resources on the ‘net.

It might be desirable to include other information like the biographical details for scientists: dates (!) of birth and death, places thereof, nationalities (e.g. Prussia, West Germany, United Germany, etc!), educational institutions attended and degrees, if any, received, the career-wise affiliating institutes for each database entry.  (Consider: would the work as a patent clerk qualify as an affiliating institute for a discovery in physics?) Details like these are not so important, and may be taken up during the version 2.0 of the project.

Finally, I would like to jot down a couple of resources. Informations from multiple sources like these needs to come together in a single, easily accessible database:

The McTutor Chronology [^]
The Wiki List of Timelines [^] (We need only the S&T related timelines. For version 1.0, focus only on physics and mathematics. For version 2.0, add: inventions/patents, engineering and technology. For version 3.0, add: biological sciences. For history of the humanities kind—who killed who to grab which power when and had how many wives—my advice to you would be: don’t bother.)

Now, a few things of business, if you are interested in picking this up as a project.

Don’t get in touch with me simply to ask if you can use this idea. Of course you can, so long as you acknowledge the intellectual credit due to me, and so long as you don’t release this idea into the Open Source/CopyLeft sort of a domain, and so long as you are not going to make any money out of it—I reserve all the rights to this idea in the first place.

However, do drop me a line before you begin any work on this idea. To repeat, I do reserve all the rights to this idea in the first place.

Also, if you are a student, don’t get in touch with me to ask if I could be become a guide to you. I won’t. What I have done here is to give you the basic idea in general (but in enough details). The particular fleshing out and the particular implementation is basically between you and your official college guide. This much should have been obvious, but the politics junkies and/or IB and/or CIA etc. have used a trick in the past, simply to harass me. For instance, when I was applying for a job as a professor to a college in Pune, someone from Gadchiroli or Ichalkaranji, claiming to be an undergraduate student, would call me, ask me if I can go to their place to be a guide—but could not name the name of his college principal.

However, once your official guide himself approaches me—which he must, if you are going to work on this project idea (see the above point again)—at that point, if he himself requests me to be an official co-guide, I wouldn’t necessarily mind—I will think about it, and let him know regarding my decision in this respect.

Finally, if you are a student, don’t get in touch with me to ask me if I could implement this project for some money from you. I won’t. It is true that as of today I am jobless, and that, even otherwise, I am always looking for ways to make money. But, I have not, don’t, would not ever sell finished (or even “half”-finished) projects to students for a charge.

I would rather write blogs and go further and harass the Maharashtra CM (any one occupying that post), possibly to (his or her) death, (correctly) blaming him (or her or they) for my joblessness, than start selling student projects for a charge. The first should come easier to me. That’s why.

Now, see if you wish to pick it up as a project. It will be useful. To a great degree. (And, perhaps, for this same reason, it won’t be suitable as a JPBTI project. Though, I couldn’t care less if IIT students pick it up as their project topic.)

As I said, there are a few more things about this project that I could say. I will write another blog post about it.

* * * * *   * * * * *   * * * * *

No “A Song I Like” section, once again. I still go jobless. Keep that in mind—if your mind is not so small as not to have the capacity to carry this factoid.

Prithviraj (the BITS Pilani- and Berkeley-trained CM)? Jairam (the IIT Bombay- and MIT-trained JPBTI)? Others like you? Ashamed of the fact that I still don’t have a job? Or not so? Or, perhaps, not at all so? What say? Not even on a Gandhi Jayanti day? Have borrowed that convenient “maun vrat” from a certain someone from Ralegan Siddhi? Really?

[This is revision 1, published on October 2, 2012, 12:35 PM IST. Initial draft posted on October 1, 2012, 4:10 PM, IST. I think I am done with the additions/clarifications for this post, and probably won’t come back to further update this post (unless a grammatical error is too glaring). The promised related post should appear in a few days’ time.]
[E&OE]

If you are a working/budding physicist, in all likelihood, you have been taught your quantum physics using books that mangle the historical order of development. Indeed, even McQuarrie’s book on quantum chemistry gives only a sketchy idea about the actual order in which the subject actually got developed. (IMO, the quantum chemistry books are better suited for a self-study of quantum physics. Among them, McQuarrie’s is the easiest to follow, though Levine’s has some topics better covered.)

The conceptual confusion that results out of abandoning the proper hierarchical order is just too huge. For instance, here is a quick question: Every one knows that Bohr’s model came on the scene before the real QM did. But the question is: When did the correspondence principle arrive? During those Bohr-Einstein debates? Or earlier?

Answer: Earlier. Right in 1913, when Bohr put forth his model. The Bohr-Einstein debates, in contrast, came much later, around the 1927 times, i.e., after all the essential principles of QM had already been discovered. And, BTW, it’s the complementarity principle which came during these latter times of the Bohr-Einstein debates.

Interesting? Ready for another question? Ok. Here we go.

Identify which development came first: (i) Dirac’s use of the Poisson brackets in quantum theory, (ii) The application of the matrix mechanics to the calculation of the hydrogen atom spectrum, (iii) The probability interpretation of the wave function?

If you are like 99% of others, you will say: In the order: (iii), (ii) and (i). The correct answer is: (i), (ii) and (iii), precisely in that order!

Don’t let yourself think that such questions are good for those fun quiz competitions or for the generally satisfying trivia. There is a very simple but very profound truth hidden in here: If Pauli could work out the hydrogen atom spectrum before anyone had even an inkling of a probability interpretation, what it obviously means is that there is some way that Pauli used, which is (implicitly or explicitly) more fundamental than some formal system that posits “probability currents” as the first axiom of QM.

More generally, if the historically less-progressed context (i.e. knowledge available by a certain year X) was factually enough (or sufficient) for someone to think of a great new idea, then, among all the conceivable or proposed ordering of topics or contexts that can be taken as foundational to explain that novel idea, the historically least progressed context is the only one that is necessary.

All the rest of the conceivable schemes are either after-thoughts, or organizational devices like mnemonics, or mere deductive tricks, or worse: mere cognitive burden on anyone who takes them seriously as a hierarchically proper scheme.

Having said that, now, pick up any of the introductory textbooks on quantum theory, carefully check out the order in which the topics are progressed in that book, and then ask yourself: How much of an unnecessary, useless cognitive burden is this particular author (i.e. an influential physicist) thrusting on your mind? How much lighter, better, would you feel if the order were something like the following? (The dates in parentheses follow the YYYY/MM format):

• Planck (1900/10): The quantization of energy of the electromagnetic oscillators in the walls of a light-radiating cavity
• Einstein(1905/06): The explanation of the photoelectric effect by quantizing the light radiation itself
• Einstein(1906/12): The first quantum theory of the specific heat of solids
• Bohr(1913/02–09): An explanation of the pattern of the discrete lines in the atomic spectra
• Bohr(1913/02–09): The correspondence principle
• Sommerfeld (1916–1920): Corrections to the Bohr model, introducing additional quantum numbers
• Compton (1923/05): A light scattering experiment, which confirms the quantum nature of light
• de Broglie (1923/09): The hypothesis of the matter waves, with a view to extend the wave-particle duality of light to matter as well
• Pauli (1925/01): The discovery of the exclusion principle, for the electrons in atoms
• Heisenberg (1925/06): The invention of the arrays of observables, to explain the atomic spectra
• Born and Jordon (1925/09): The first physical law stated using non-commuting symbols: $pq - qp = i\hbar I$
• Goudsmit and Uhlenbeck (1925/10): The experimental discovery of the electron spin
• Pauli (1925/11): The first success in applying the matrix mechanics to the line spectrum of hydrogen, including the Stark effect
• Dirac (1925/11): The identification of a Poisson brackets structure in Heisenberg’s analysis.
• Born, Jordon and Heisenberg (1925/11): The “three-man paper” on the mathematics of matrix mechanics submitted for publication (9 days after Dirac’s above paper)
• Schrodinger (1925/12): Formulation of the first ideas of his wave mechanics
• Schrodinger (1926/01): Successful application of his wave equation to the hydrogen atom
• Schrodinger (1926/03): Demonstration of the mathematical equivalence of the matrix mechanics and the wave mechanics
• Born (1926/07): The probability interpretation of the wave function
• Dirac (1926/09): The transformation theory—the wave and matrix mechanics as special cases
• Davisson and Germer (1927/01): Experimental confirmation of diffraction of electrons by a crystal lattice
• Heisenberg (1927/02): Formulation of the uncertainty principle
• Bohr (1927/09): Formulation of the complementarity principle and the Copenhagen interpretation
• Thomson (1927/11): Another experiment which confirms that matter diffracts
• Dirac (1930/05): Publication of the first edition of his book (having its “first chapter missing”)
• Dirac (1939): The third edition of his book introduces the bra-ket notation—the starting point for today’s “Alice and Bob”-obsessed idiots

As you probably know, I have been trying to follow the historical sequence in writing my book. So, in a way, I have been looking a bit carefully into the historical order in which things happened. Still, I had a few surprises in store even for me when I really sat down to compile the above list. Here they are: (i) Einstein’s 1906 paper (which I used to put somewhere in the late teens), and, (ii) Dirac’s 1925 paper.

I am sure that things like the following would come as surprises to many of you: (i) Dirac’s transformation theory being formulated before either the uncertainty principle or the complementarity principle was, (ii) Pauli working out the hydrogen atom line spectrum using the matrix mechanics barely within 2 months of the writing of Heisenberg’s first paper, and in fact before Heisenberg himself could succeed doing so, (iii) Born identifying the matrix nature of the Heisenberg’s non-commutative arrays and Jordon working out the derivations of the mathematics involved in it.

I also think that the help that was both required and received by Heisenberg, might have come as a surprise to many of you, esp. when contrasted with Schrodinger’s single-handed development of all the fundamentals of the wave mechanics—except, of course, the probability interpretation of the wave function, which was supplied by Born.

Most importantly, I think, quite a few must have been shocked to find that Dirac could work out his theory, even predict the existence of anti-matter, without explicitly using the bra-ket notation itself. It has become a fashion to explain this notation right in chapter 1 (though, thankfully, not right in the preface—not yet, anyway).

While writing on the heuristics that he follows while deciding whether a paper on the P-vs-NP issue is worth reading or not, Prof. Scott Aaronson has indicated that any paper not written in LaTeX is suspect.

I have a similar test for books, papers, tutorials etc. on quantum physics, especially the introductory or foundational ones. (Seriously. I have actually followed it over quite a few years in the recent past, and very successfully, too.)

I don’t take any paper/notes/tutorials/book on the foundations of quantum physics for a serious consideration (i.e., I don’t even browse or flip through its abstract) if it has “Alice” and “Bob” written anywhere within it.

Ditto, for any textbook on quantum physics, if it has those two words appearing within the first 90% of the real text matter.

So, hey physicists! Revise your books to follow the kind of an order I have given above!!

Why?

Because, I say so. That’s why.

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No “A Song I Like” section, once again. I still go jobless. Keep that in mind.

[This is initial draft, published on September 26, 2012, 8:07 PM, IST. May be I will make some minor corrections/updates later on.]
[E&OE]

This year seems to be a bit unusual. I have not one, but two very strong recommendations for your holiday reading list.

The first book, of course, was David Harriman’s “The Logical Leap: Induction in Physics,” which formed the topic of my last blog post [^]. Now I am very pleased to bring another great book to your notice: Manjit Kumar’s “Quantum: Einstein, Bohr and the Great Debate on the Nature of Reality” [^].

I am in a hurry. So, please do not regard this post even as an attempt to provide a proper review. I am just going to jot down a few points that occur to me on the fly, after having finished reading the book a week ago or so.

Here is my overall opinion of Kumar’s book: This book is the most outstanding account of quantum mechanics meant for the layman that I have ever read in print, period.

Now, that is saying a lot, and yes, I do mean it. … I have read a lot of pop-science type of material on this topic. Throw in also those philosophically oriented popular accounts. I am not exaggerating when I say that Kumar’s book is the best. Yes, Kumar’s book easily beats, just for example, both of Gribbin’s books: “In Search of Schrodinger’s Cat” [^] and its sequel “Schrodinger’s Kitten and the Search for Reality” [^], even Feynman’s books (Lectures and QED), not to mention Alastair I. M. Rae’s “Quantum Physics: Illusion or Reality?”[^].

Compared to Gribbin’s books, Kumar’s is a far more balanced, accurate, and a detailed account, with a lot of material concerning the personalities of the early quantum founders (even though Kumar does not cover the later period in much detail). Compared to Feynman’s accounts, Kumar’s book has one outstanding virtue: it is historically well-ordered, which by itself takes care of streamlining so much understanding in such a subtle manner. And, compared to Rae’s book, Kumar’s book does not care to dwell on the subtleties of the philosophical nonsense. It is a book that manages to keep physics at its focus through and through.

Kumar begins right from Planck’s hesitant acceptance of the quantum nature of emission and absorption of the cavity radiation, and then carefully goes through the evolution of ideas at the hands of Einstein (quantization of radiation itself), Rutherford and Bohr (atomic structure), and then all the European men, young and not so young, who discovered one facet after another in a breathtakingly rapid manner: Heisenberg, Pauli, Born, Jordon, Kronig, Uhlenbeck, Goudsmit, Schrodigner, et al. For each physicist, before describing his work, Kumar first provides an essentialized bio of the man: his family, inclinations in the school, academic work, supervisor and his personality, the ideas already “in the air” at that time, the initial faltering steps and false starts, and even the frustrations before the solution was discovered and the personality clashes after it was. And, Kumar also manages to place all these things in a broader cultural and historical contexts—the demands of the industry and governments to solve particular problems, the two world wars, the antisemitism, the reactions of the academic community to the evils surrounding them, etc.

Even before starting reading this book, I knew that these early quantum physicists worked in a very closely networked manner. But I had hardly realized the deep personal respect they gave each other and the deep feelings of friendship they enjoyed. It is easy to over-dramatize the tension between Bohr and Einstein; so many popular accounts have so routinely done so to such an extent that one might imagine as if Bohr and Einstein were at least playing a turf-war of sorts if not reaching for each others’ throats. You have to read this book to realize how far away such recent depictions have gone from the facts of the matter. Kumar possesses just the right sensitivity to the culture of those times to present just those essential facts with which, after finishing this book, one does come out enlightened about the way this extraordinarily brilliant chapter in the entire history of physics had actually unfolded. And, how cultured and intelligent its main actors were—regardless of whatever errors, even blunders—concerning physics or philosophy—that they might have committed.

This book has helped me correct many of the misleading or wrong impressions that I unwittingly had happened to gather, regarding many founders of QM—and even concerning the contents of their ideas.

Kumar covers the material roughly in the same sequence in which the development actually occurred. In following this policy, it is obvious that it is the author who has work harder, thereby lessening the burden of integration on the reader’s part. In this task, Kumar has succeeded brilliantly. To appreciate the sheer volume of reference material the author must have dug through, just take a look at Mehra’s volumes alone! And, yet, for many controversial ideas, Kumar manages both to present all the relevant material to the reader and and yet also to allow the reader the room to let him think about that issue and come to his own conclusions.

One also comes to develop an appreciation for the subtle nuances in the differences of the ideas held by the founders of QM, and the roles they played. Here, a few things stand out: (i) The way Heisenberg evolved matrix mechanics and the roles played by Born, Jordan, and Pauli, in developing it. (ii) How the idea of spin was so simple—and so much required in the order of development—that it could get introduced way before the 1927 Solvay conference. (iii) How close to the eventual formalism were certain nascent ideas held by the less celebrated physicists. (iv) The roles played by extraordinary mentors like Rutherford, Bohr, Sommerfeld, and others. (v) And yes, the special circumstantial reasons why the Copenhagen dogma could get established so easily in the academia.

There can be scope for improvement. However, in a book like this, it is a secondary or an outright minor matter. In any case, I don’t have any specific suggestion right away—I will have to do a more careful second/third reading before coming up with any. Yet, I must add, there is something that I found something missing right on my first reading.

I would have so much appreciated it if the author could have traced the evolution of Dirac’s re-formulation of QM. This development happened early enough, and it is important enough. The author has rendered a great service by carefully isolating and demarcating how the progression of thoughts occurred in Heisenberg’s formulation—starting with all the relevant events and ideas before his visit to that island resort and going on to the subsequent developments. It would have been a great (and easy) learning for the reader if the author could have done the same for Dirac’s development. This topic is conspicuous by its absence. The absence is all the more remarkable given the fact that (i) Dirac was a British gentleman, just the way the author is, and (ii) the author does manage to touch on such later developments like the EPR controversy and Bell’s inequalities. It is my sincere hope that the author would consider adding a chapter in a future edition. Or, make it available at his blog.

Overall, this book indeed makes for both a great read and a valuable reference. It could even be made recommended supplemental reading for university courses.

All in all, very strongly recommended.

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I don’t expect to write another blog post this year, and so, let me say “Happy New Year!” to you right away!

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A Song I Like:

(Hindi) “yeh raat bheegee bheegee…”
Music: Shankar-Jaikishen
Singers: Manna Dey and Lata Mangeshkar
Lyrics: Hasrat Jaipuri

[E&OE]

# Wanted: Bean Counters…

Recently, I was searching for some key numbers concerning India’s economy, and Googling some related phrases didn’t help at all.

The data I wanted were mostly concerned with the annual government expenditure in India, as well as the total sizes of the public and private sectors in India.

For instance, I wanted data like:

• The size of the public sector in the years 1947, 1950, and for every year (or five year period) since then. Also, it would be helpful if also similar data for the years before independence are available. The size of the public sector is to be measured in terms of both total money spent on them (planned and unplanned expenditure, capital invested, the money spent to cover up the losses they made etc.), as well as the percentage of the annual GDP these organizations came to occupy. (It’s OK even if no data are made available on the pork portion of these numbers.)
• The relative size of the total government spending in India
• It should be available separately for the Center and for each of the States/UTs.
• The fraction of the total government expenditure in India that was spent in providing the three specific services of Defence, Police, and Courts (i.e. the entire judiciary), and for all the other activities of the government. Also, similar data, if available, for the money directly spent on the legislative branch.

Ideally (since I am so lazy) I would like it if the data have already been adjusted for inflation—any base year would do.

The data might be only rough (ballpark) estimates.

I would also like some important related data such as the number of people employed in each sector (and branch of government) are available, though some of these type of data are already available at Wikipedia here[^].

More than 25 years ago, I had already started believing in Capitalism. As a part of understanding the world around me better, I would go out and buy a small pocket book of the title (if I remember it right) “Statistical Outline of India”. It was a book that all socialists and academics but no businessman would buy. It used to be brought out, if I mistake not, by some concern of Tatas—possibly, the Tata Institute of Social Sciences, Bombay. I guess they used to issue a new edition every year or so, because I distinctly remember having bought a different version more than once.

If you know of any Internet links for the above kind of data, please drop a line to that effect.

And, if you fail to find any easily accessible links for even as important and salient data as these, think about the depths of socialism/statism to which this country has already plunged.

[E&OE]

–  –  –  –  –
A Song I Like:

(Marathi) “jay jay mahaaraashtra maajhaa…”
Music: “shreenivas khaLe”
Singer: “shaahir saabLe”

# Does QMC Have Priority over FAQ? + Miscellanea

One of my recent muses is the title question of this post. … Allow me to explain.

As I have pointed out a few times in the past in this blog and also elsewhere, I have discovered a way to resolve the quantum wave-particle paradox. See my slides and papers here. Soon later on, I began calling the new approach by the name “FAQ,” which is short for: fields as quanta. That is what the “FAQ” in the title of this post refers to.

Before making my claim, of course, I had done an extensive literature search, almost none of which was cited in the abovementioned papers, simply out of the space limitations of a conference paper. But the search was there. One of the things I had quickly browsed through, during this search, was the literature on QMC—short for Quantum Monte Carlo.

In recent weeks, I decided to search once again. It must have been the n-th occasion that I was searching thus. This time round, I was reading more closely the papers, and so ran into some interesting passages in a few early papers (circa 1975) by Professor James B. Anderson of Penn State (USA). In some of these papers, Professor Anderson indicates that Metropolis, Ulam and John von Neumann had stated in one of their early papers (in late 1940s) that the idea for something like QMC had already occurred to the great physicist and Nobel laureate Enrico Fermi.

Curious, I immediately wrote him an email, and promptly received the directions to look up his new book: “Quantum Monte Carlo: Origins, Developments and Applications”. The relevant pages of this book can be browsed at Google books [here]. This book is a researcher’s dream come true. Original papers tracing the development are hand selected, and brief introductions to each provided. (These introductions written with expertise and yet remain accessible to a “lay reader” like me.)

It’s enough to browse the first few pages of Anderson’s book to realize that Schrodinger himself was right up there, thinking about these ideas (and also publishing them in journal papers) right in 1935.

Thus, QMC has… what’s the word here? precedents? antecedents? … Whatever. I will use the term “precedents” here. So, QMC  has two independent lines of precedents, both traceable to famous people (Nobel laureates), one going as far back as to 1935, to Schrodinger himself. The other line goes back to late 1930s and early 1940s to Fermi. (I forgot my tracks on the Internet or in the books/papers here, but will add links to them later on.)

Very deeply interesting, this all is. Also, even satisfying in a way!

However, all my reading of all such material tells me that all the precedents to the modeling of Schrodinger’s Equation (SE) using ideas such as random walks or Monte Carlo involve the imaginary time—not the real time. (Refer to Anderson’s excellent book and papers to know what this means.)

My approach, in contrast, involves the real time (and the real space—not a configuration space).

It might seem amazing that Einstein worked on both the photoelectric effect and the Brownian motion in the same year 1905, and yet didn’t think of extending the second to explain the first. Even more amazing is the fact that both Schrodinger and Fermi thought of using the second to model quanta, but never thought of doing so in the real time.

It is for these reasons that I conclude that QMC cannot be said to have the priority over FAQ. In other words, my claims are valid.

This post is to bring the matter to your careful and serious attention. If you have any [proper] evidence contrary to my conclusion/claim, then kindly do drop me a line or provide the links.

[BTW, note, links like this may be good otherwise, but are not detailed enough to be of help in this matter.]

– – – – –

A few clarifications/asides:

I am not at all interested in any sort of a priority battle. But one likes to be as direct as is possible in communications. The purposes this policy serves are things such as: precision and economy in thought; propriety in the allocation of intellectual credit where one is due.

Interestingly, after submitting an abstract to an upcoming international conferences in India, the reviewing committee noted to the effect that “direct claims” such as what I was making could not be entertained.

To say that I was surprised would be an understatement. It was nothing less than shocking. … Of course one is supposed to be as direct as possible in all communications of this kind. Indirectness might have its allure in poetry, esp. of the romantic sort. Consider here the beauty of: <Hindi>”Kyaa Kehnaa Hai, Kyaa Sunanaa Hai…”</Hindi> … You know what I mean—literally…. But trying to use it in science/research? (LOL!)

So, one tries to be direct. And, one remains open and available (i.e. active-minded) to correct oneself—if a correction is necessary. It is in this spirit that I make all my claims. After a conscientious and as wide a literature search as possible. But directly, as directly as possible, thereafter. … Sigh… Not all folks in India know or understand or support this way of approaching science!

Anyway, to return to more interesting matters than them (and their science/engineering), let me know if I am understanding QMC in a wrong way and/or making a wrong claim somewhere in my research. I would appreciate being kept corrected—if one is necessary.

– – – – –

And, while quoting Javed Saheb’s poetry (which also is a song), it occurred to me that it might be a good idea to start jotting down a few songs that I like after every post I make. (This idea traces its origins to Jean Moroney Binswanger’s advice [here] to everyday jot down three good things—major or minor—that happened to you on that day…. Well, three per day is too much, but three per post isn’t a bad idea to implement).

Accordingly, here I begin, right away. Three poems/songs/tunes/musical compositions I like (of whatever type/genre, for whatever reasons, from whatever language I know, etc., more or less at random, out of hundreds of such):

1. <Hindi>ye raate ye mausam nadi kaa kinaaraa…</Hindi> (Kishore Kumar)
2. <Marathi>jan paL bhar mhaNatil haay haay…</Marathi> (Lata, Bhaa. Raa. Taambe.)
3. (Words not necessarily exact)<Hindi>chhaayee barakhaa bahaar, kare jiyaraa pukaar</Hindi> (Lata)

An important note about the third song: As far as I can make out, (and I am confident about it), this song is in the “raag” “bhairavee.” The reason I am so confident is because I once heard an unforgettable “jugalbandi” of Pandit Bismilla Khan on “shehnai” and Mrs. Rajam on violin in “bharavee”. It was an especially memorable performance because it was the first time ever that I had really appreciated a piece of the Indian classical music. (As a rule, I find it boring—but always with notable exceptions. (Also Western classical—most of it, too, is boring.)) They had announced the “raag” at that time, it was “bhairavee.” I had observed, right then, that this song of Lata (which I quote above) was exactly like that “raag” and vice versa.  (The venue was the Open Air Theatre of IIT Madras, when I was a master’s student there at that time.)… The reason to share this all side information is to emphasize doubly and triply that the song I have in mind here is not the “aayi barkhaa bahaar…” by Lata and Madan Mohan. Neither is it any of the other “aayi barkhaa bahaars…” that are listed on Google within the first 30 pages. And of course, it is not that Salil Choudhary’s unforgettable “o sajanaa, barakhaa bahaar aayee” (which is not in “bhairavee” anyway.)

… Yes, the song I have in mind is for real. It exists. But I don’t recall any of its other credits (like the film, the lyricist, the music director, etc.) except for the fact that it’s been sung by Lata and that it is in “bhairavee.” (And yes, at least one member of our family could distinctly remember that there is such a song (despite the fact that I am not a good singer), though they too can’t recall its film etc. … Please do let me know if you find it.

# A small document on the history of the calculus of variations

I have begun preparing a small document on the history of the calculus of variations. The document is still in progress, and likely will remain that way for quite some time to come. The current version (updated without notice) may be downloaded from here. Corrections and thoughtful comments are welcome.

# “Think It Over, and Then, Program!”… And, also, a bit on the calculus of variations…

0. Introductory:

In my last post (below), I talked about changing the way my study room is set up—removing the computer network from the desk-top and making room for writing in long hand. Now, I will talk something about a small sign-board (for reminding me) that I am going to put up in my room.

The sign-board will carry a simple command:

“Think It Over, and Then, Program!”

It will have the sub-title:

“Can you make a C++ program out of this?”

(The slogan obviously is paraphrasing of sorts of Dirac’s famous commandment: “Shut up and calculate!”.

Update on 07 August 2018: The line comes neither from Feynman nor from Dirac, but was cooked up by David Mermin, in order to characterize the typical attitude the Copenhagen interpretation takes.)

The thing is that I am reading a lot on and about classical mathematics lately. I got into it when I was preparing for my forthcoming FEM course. Well, it at least began that way. But then, soon, it slipped into quite general readings on the history of calculus of variations, too. And further thereafter, it began getting transformed into finding roots of so many subtle assumptions that we so blithely make today in the areas of mathematical physics, engineering mechanics, elasticity, computational mechanics, etc.

For instance, consider this question: Why is quantum theory linear?

1. Nature of theories from mathematical physics:

In mathematics, there are no monolithic blocks of periods such as a period of “classical physics,” or one of “analysis,” etc.; there often are many different streams of thought existing simultaneously at any given time. The emphasis may differ, but the strains continue to exist.

Consider here, that there is this basic idea that physical systems should be described, not using Newtonian ideas of spatially delineated agents such as particles and the forces they enact and react to or the changes in velocities or momenta they suffer, but using an alternate set of ideas such as the potential of a field, a scalar quantity called energy, the idea of action, etc. That is, the ideas originated and propagated by Leibniz, Lagrange, Hamilton, et al.

I am not sure if I am getting the division right, but at least at a cursory glance, it seems that there are these two camps: (i) Fermat, Leibniz, Bernoulli, Euler, Lagrange, Hamilton, et al all taken together on one hand, and, (ii) in his majestic and towering isolation, Newton, alone, on the other hand.

But I am not sure if I am getting the “camping” right.

The reason I am so tentative about this two-camps “theory” of mine is because I am not quite sure how to place three/four other names within the confines of my scheme; they disturb my “theorization”; they are:

d’Alembert, Fourier, Gauss, and von Helmholtz (and perhaps one or two other people).

Sure, d’Alembert and Fourier were responsible for creating those basic tools which are sympathetic to the ideations of the former camp (and to the action-at-a-distance viewpoint), the tools such as the technique of separation of variables (and the sin of making that appear static which actually is dynamic), and the idea of modeling using the spectral analysis, respectively. But still, somehow, I carry the impression that on the whole, these two Frenchmen never quite fully belonged to the former camp. Not as fully as Leibniz and Hamilton (and their modern-day followers in the area of quantum confusion mechanics) do. The two Germans, Gauss and von Helmholtz, particularly the latter, were just too good by way of their approach and work to belong to the former camp.

Just for the record, I, of course, am on the side of Newton. I never did like the idea of CoV. (Update on 07 August 2018: My opinion has changed considerably, even though the points I make next still remain appealing to me.)

It has always been a very competent mathematics and still, a bad (or very bad) modeling idea—that’s my opinion of it. I hate to ascribe to space what properly belongs to entities.

It does no good to say that you can always get force by taking a gradient of a scalar and how it reduces the labor from three component equations to a single scalar equation, but, in the process of saying so, outright evade the issue of how anyone on earth is going to know what specific potential to use. Is divination or day-dreaming the recommended manner? And if not, who is going to calculate the effort in getting to the right potential?

Overall, I think it doesn’t really simplify the problem but only changes its appearance and shifts the points of inconvenience, hiding them behind a nice “global” theory… Field theory may have its technical advantages, but the manner of advocacy—the underlying philosophy—of the field theoretic camp is often pretty bad, despite its popularity today, and needs to be exposed.

In any case, I am sure you are convinced that I am reading about a lot of classical mathematical ideas these days. Now ideas are fine as far as reading and thinking goes, but, as an engineer, one also has to get something real out of them. Here, given my strengths and inclinations, I have decided that I would rather create programs out of this kind of thinking of mine, rather than writing or solving analytical mathematical problems. But no, I do not thereby mean to imply that I don’t understand analysis or cannot deal with it… Below, I let me give some concretes of what I mean:

2. Comments on the calculus of variations:

Consider the basic or starting ideas in the calculus of variations. How do people state it? How does the discussion begin?

The way they state it is, first of all, via an equation:

$I[y(x)] = \int_{x_1}^{x_2} F[ x, y(x), dy/dx ] dx$

It is as if they would rather be found dead than not write down an equation. But more on this, just a short while later, i.e. right in this post, but after a short while. For the time being, look at that equation again. … Try to think like a fresh student.

The first challenge with the above equation is to decide precisely what is the unknown in or about it.

Naturally, all your learning cries out at you that it has got to be $I$, whatever it may be.

Wrong. Plain wrong.

In the great weird world of thinking along the CoV lines, the great unknown, of course, as you know, turns out to be the $y(x)$. The $I$ is just a silent spectator, so to speak. The unknown, or the problem, is $y(x)$, even if it sits deeply nested inside an integral on the right hand side of the equation.

(Here, most mathematicians would lovingly repeat that idiotic quote (from Boltzmann) about leaving elegance to, of all professionals, cobblers and tailors, and thereby evade all the further issues actually observed and raised thus far. For example, the misleading way of formulating the problem—there is direct evidence of confusion of thinking in there, don’t you think?)

The second difficulty: Next, notice that the above equation tells only an incomplete story. The real problem is not the above. The real problem is to find stationarity or “minimality”, i.e., to say something like:

$\delta \int_{x_1}^{x_2} F[ x, y(x), dy/dx ] dx = 0$

LOL! Can you make any C++ program out of what we have said so far? If not, then, consider whether you (really) understand the matter as well as you should, or not. [Update on 07 August 2018: Of course, you don’t!]

Third difficulty: The notation: Another minor point, concerning analysis and CoV. The inconvenient notation. Two sub-points here.

(i) Why include both $y(x)$ and $x$ in the definition of the functional? Wouldn’t the context make the chain-relationship clear? Or is the whole idea to confound the reader as much as possible? Doesn’t the notation betray the (lack of) thinking?

(ii) Why insist on using the form of an equation?

You see, the whole thing becomes inconvenient precisely because the overall idea here is that you have to express everything in the format of “LHS = RHS”.

But this kind of a format makes for a very bad notation when the fact to express is a “choice” or (added on 07 August 2018) a distinguished possibility, from amongst an infinity of alternatives!

Why not invent a new notation that clearly brings out the idea that there are an infinity of possible variations and that the one particular integral amongst them—the ultimate solution—is special and is to be singled out?

Fourth confusing point: Wrong choice of words: Indeed, why at all call the choices by the name “variations”? Doesn’t using that word implicitly assume that you possess the knowledge of the desired but unknown solution in advance? (Variation, on what?) The idea of teleology is entangled far too deep in this issue, even if I believe that it can be separated out.

But far more important: Why define terms in reference to an unknown? Isn’t it Platonic/Kantian/Hegelian kind of idealism oozing through here? Why not describe the issue straight-forwardly as an infinite set of definite integrals of a common definition and a common set of the specified boundary conditions? What wrong would that do?

Fifth bad point: Too abstract a description: When the notion of a function is introduced for the first time to kids, it is a normal practice to create two blobs (one each for the input and output sets i.e. the domain and the range sets) and connect them using one-way arrows (showing unique-valued correspondence). Everybody agrees that this kind of a diagram helps in rightly anchoring the idea in the mind.

But when it comes to functionals, however, they never give a nomogram kind of a visualization… Why not?

Do mathematicians fear losing abstractness of their definitions if they supply one? Oh yeah? Is that the motivation?

Or is the motivation to keep the definitions as far away from a concrete-reality-based understanding and as high floating in the air as possible?

What kind of motivation explains the complete lack of a good explanatory diagram (a concept map) for so basic concept as a functional, in all the 150 years of its history?

… Sometimes, at least, you have to think if there isn’t more than plain teaching incompetence at work here, i.e., if there isn’t a kind of an “ideological” stink involved in here..

Sixth bad point: starting with a bad example: But, coming back to the text-book writing… What do you think they suggest by way of a “motivating” example? The same stale stuff of the brachistochrone problem!!

Can’t you see that it is such an artificial problem, very specific to the constant gravitational field at the earth’s surface? That it cannot at all bring out the real essence of the idea….

Regardless, every textbook writer thinks it a terrible wrong if he doesn’t start CoV without (i) the brachistochrone (ii) the isoperimetrics (iii) the geodesics problems. (BTW, geodesics are always shown drawn on a neat sphere—not even on an ellipsoid, let alone on an arbitrary shaped surface.)

3. A good simulation example to get the ball rolling:

Here, I think, the command “Think It Over, and Then, Program!” would came in handy… In short, it’s high time we asked mathematicians to “Shut Up” asking us to “Calculate” all the time, and instead, think about the real world and the possible connections between their ideas and the real world…

When I recently deeply thought about how this thing could be better explained, I thought that the easiest way perhaps would be via computer simulation of a modification in the game of carrom.

What you can do is to simulate the game of carrom, but, say, with a soft iron striker (or a wooden striker affixed with a thin stamping of steel on top), and a few electromagnets placed on the sides (or underneath) the main surface of the carrom-board, so as to create a field inside the 2D domain.

In the simulation, you would let the user vary the strengths and placements of the electromagnets, as well as the initial velocity (speed and direction) of the striker. You could then ask them to predict in advance where the striker will end up. (We assume that $c >> v$ so that the situation is non-relativistic.) The simulation would then show them the actual trajectory. The software could ask them to choose the direction and the speed so that the striker itself would end up in a corner pocket.. The software could also allow them to manually modify the actual trajectory (i.e. introduce variations) and automatically compute the value of the “action” for each variation. The program could help visualize the potential by plotting its surface $z = f(x,y)$ in $3D$. … So on and so forth…

4. The bad ideals mathematicians are taught to keep:

It amazes me how stale analytical mathematicians can get.

… Incidentally, this word “stale” reminds me of a story about Hamilton that I read somewhere (in an authentic kind of a book on the history of mathematics) a long time back. The story goes thus: in the times that he developed the idea of quaternions and his grand version of mechanics, Hamilton had gone half-mad of sorts. They had recovered discarded bones left over after his meals together with Hamilton’s original (and seriously meant) handwritten papers. The papers (and the rest of the stuff) was found thrown all over the floor in his room.

Apparently, Hamilton was not so engrossed in his work that he would forget to eat his meals; apparently, he had become so careless that he would not distinguish between his serious work (the papers)—by his own proclamation, the finest and grandest among mechanical thoughts—and the garbage left over from his meals.

(In contrast, Newton was known to get so engrossed in his work as to forget taking his meals. But, despite his reputation as a sort of mad-man among quantum mechanicians of the 20th century, Newton never did get into any such mental states as Hamilton evidently did. Indeed, even when Newton ran the royal mint, it was with exemplary efficiency—not with a “mad”-ness as historians and quantum physicists and mathematicians have preferred to tell us.)

Apparently, the staleness which appears in the teaching of this kind of mathematicization—the case-studies and the ways of presentation of the mathematical ideas—comes about because our quantum mechanicians (and mathematicians and historians) take over into mathematics what were merely Hamilton’s personal habits. That, perhaps, could be the reason why, even today, we have to go through the same stale stuff of only the vertically oriented brachistochrones and the stupid but “mandatory” sequence involving the more or less completely useless (and yawn-inducing) isoperimetric problems. Such staleness!

5. What can we do?

Our students deserve better. … What I am doing (or, rather, proposing to do) is just a beginning. … You could do better than me. …

But yes, the reminder: “Think It Over, and Then, Program: Can you make a C++ program out of this?” is clearly useful in many different ways.

… I won’t go as far as to suggest that you should make a software program out of every little idea of mathematics. Or, more seriously (and importantly) that this kind of a “constructivist” approach is the only way to do good mathematics. Nope, it isn’t—though, it could very well be the only way to *validate* mathematical abstractions.

… Anyway, I haven’t thought about this matter a great deal. BTW, there is a proper movement of sorts in mathematics which goes by the name “constructivism.” I don’t mean to use this word “constructivist” in that technical sense of the term. In fact, I haven’t read enough about constructivism to be able to form a judgement about it. All that I mean to say here is that mathematical ideas cannot come from thin air, that one must know what the referents of any mathematical concepts are, and if one does, it is easy to construct the higher-level abstraction from the lower-lever ones…

But coming back to my main point here, clearly, thinking about how particular programs could be written, is a way that seems to encourage at least fresh, if not highly creative, thinking…. It certainly helps counter the menace of floating abstractions… [Update on 07 August 2018: My latest development related to QM stems precisely from pursuing thinking using such methods/metaphors.]

Think about it. … After all, not just our students’ but even our own minds deserve better—better than the kind of stale treatments we have been dished out thus far…