Yo—5: Giving thanks to the Fourier transform

Every year, at the time of thanksgiving, the CalTech physicist (and author of popular science books) Sean Carroll picks up a technique, principle, or theory of physics (or mathematics), for giving his thanks. Following this tradition (of some 8 years, I gather), Carroll has, for this year, picked up the Fourier transform as the recipient of his thanks. [^]

That way, it’s quite a good choice, if you ask me. …

…Though, of course, as soon as I began reading Carroll’s post, a certain thing to immediately cross my mind was what someone had said concerning Fourier’s theory.

Fourier’s is the most widely used theory in the entire history of physics, he had said, as well as the most abused one . … The words may not be exact, but that was the sense of what had been said. Someone respectable had said it, though I can’t any longer recall exactly who. (Perhaps, an engineer, not a physicist.)

The Fourier theory has fascinated me for long; I have published not just a paper on it but also quite a few blog posts.

To cut a long story short, I would pick out (i) the Lagrangian program (including what is known as the Lagrangian mechanics as well as the calculus of variations, the stationarity/minimum/maximum/action etc. principles, the Hamiltonian mechanics, etc.) and (ii) the Fourier theory, as the two basic “pillars” over which every modern quantum-mechanical riddle rests.

Yes, including wave-particle duality, quantum entanglement, EPR, Bell’s inequalities,  whatnot….

As I have been pointing out, the biggest good point that both these theories have in common is that they allow us to at all perform at least some kind of a mathematical calculation of the analytical kind—even if, often times, only in a physically approximate sense—in situations where none would otherwise be possible.

The bad point goes with the good point.

The biggest bad point common to both of them is that they both take some physics that actually occurs only locally (say the classical Newtonian mechanics) and smear it onto a supposedly equivalent “world”—an imaginary non-entity serving as a substitute for the actually existing physical world. And, this non-entity, in both theories (Lagrangian and Fourier’s) is global in nature.

The substitution of the global mathematics in place of the local physics is the sin common to the abuse of both the theories.

Think of the brachistochrone problem, for instance [^]. The original Newtonian approach of working with the local forces using \vec{F} = d\vec{p}/dt (including their reactions), is in principle applicable also in this situation. The trouble is, both the gravitational potential field and the constraints are continuous in nature, not discrete. As the bead descends on the curve, it undergoes an infinity of collisions, and so, as far as performing calculations go, the vector approach can’t be put to use in a direct manner here: you can’t possibly calculate an infinity of forces, or reactions to them, or use them to incrementally calculate the changes in velocities that these come to enforce. Thus, it is the complexity of the constraints (or the “boundary conditions”)—though not the inapplicability of the basic governing physical laws—that make Newton’s original approach impracticable in situations like the brachistochrone. The Lagrangian approach allows us to approach the same problem in a mathematically far simpler manner. [Newton himself was one of the very first to solve this problem using this alternative approach which, later on, to be formalized by Lagrange. (Look up the “lion’s paws” story.)]

Something similar happens also with the Fourier analysis. Even if a phenomenon is decidedly local, like diffusion of the physically distinct material particles (or parcels) from one place to another, the Fourier theory takes these distinct (spatially definite) particles, and then replaces them by positing a global non-entity that is spread everywhere in the universe, but with some peak coinciding with where the actual particles physically are. The so-smeared non-entity is the place-holder [!] for the spatially delimited particles, in Fourier’s theory. The globally spread-out entity is not just an abstraction, but, really speaking, also an approximation—a mathematical approximation. And as far as the inaccuracies in the calculations go, it turns out, this approximation does work out very well in practice. (The reason is not mystical. It is simply that the diffusing particles (atoms/molecules) are so small and so numerous in the physically existing universe.) But if you therefore commit the error of substituting this approximate mathematical abstraction in place of the exact physical reality, you directly end up having the riddles of QM.

If you are interested in pursuing this matter further, you should see my conference paper, first. (Drop me a line if you haven’t already downloaded it when it was available off my Web site, or can’t locate it any other way.) … Though I have also written quite a few posts on the topic, they don’t make for the best material—they are far too informally written (meaning: written completely on the fly and without any previously thought out structure at all). They also too lengthy, and often dwell on technical aspects that are too detailed.

And, that way, they don’t have much mathematical depth, anyway.

But since I seem to be the only person in the entire world who has ever thought along these lines (and one who continues to care), you may want to have a look at myQ detailed musings, too: [^] [^] [^][^].

(… And, no, as far as this issue goes, by no means am I done. I would continue exploring this topic further in my research, also in the future. Though, let me wind it all up for now… This was supposed to be a short and sweet post—a “Yo” post!)

* * * * *   * * * * *   * * * * *

A Song I Like:

(Marathi) “ekaTyaane ekaTe gardeet chaalaave”
Singer: Avadhoot Gupte
Lyrics: Mangesh Padgaonkar
Music: Shreedhar Phadake

[May be I should post a translation of this song some time later. … Also, of that another Marathi song which I have run just a few posts ago, viz., “man pisaaT majhe…” As to that song (“man pisaaT”) I know for a fact that a lot of Marathi-“knowing” people have never bothered to carefully go through the actual words, they have never tried to put them in some kind of a context, and thus, paying only a fragmentary attention here and there, they have come to associate something of a too abstract and weird (or “artsy”) kind of a sense to it. Their appreciation of that song rests mostly on the musical tune and the singer’s rendition, but their sense of the lyrics seems to be quite off the mark. The actual song isn’t of a meaningless “artsy” kind, and I hope to bring out what I think is the original sense of that song, too. And, as far as the present song goes, there isn’t just an  innovative sort of tune and a wonderful rendering by the singer. There also is a very beautiful piece of poetry lying underneath. … It’s a young new song (it came out only in 2010), but with an obvious touch of class to it. The original CD is just Rs. 100. … Enjoy…. More, later]




3 thoughts on “Yo—5: Giving thanks to the Fourier transform

  1. Hi Ajit. I don’t know if you know, but people say that something like an optical Fourier transform* is performed when you detect a photon. It has an E=hf wave nature. It takes many paths like a seismic wave takes many paths, and it goes through both slits. But when you detect it at one slit, it is converted into a dot, and goes through that slit only. Then when you detect it at the screen, it is converted into a dot. No multiverses are required.

    * See Steven Lehar’s webpage for this: http://cns-alumni.bu.edu/~slehar/fourier/fourier.html

    • Hi John,

      Thanks for your interesting comment. I am sorry for the delay in running and replying it. The reason is that I receive so few comments at this blog that I check the pending comments-queue only infrequently. Really sorry for that. (I will check if there is a wordpress feature to generate an email to me as soon as someone makes a comment that is not identified by wordpress as a spam.)

      Coming back to the physical description of the single-photon interference experiment that you mention, broadly speaking, it does seem right. However, this description would have to be supplemented with many crucial additional details that are missing from it—and, these details in fact form many of the valid issues of contention in the foundations theory.

      For instance, what is the nature of the mechanism that converts the posited quantum wave (which presumably is spread all over the domain) into a dot (which must be quite localized)? Can it be captured via a quantitative model (say, via a differential or integral equation)?

      For simplicity, assume here that the experiment is performed with a monochromatic radiation, say, with a sequence of parametrically down-converted (PDC) single photons.

      Now, a reasonable assumption might be that the quantum wave has precisely the same value of total energy/momentum as that for the detected “dot.” This is a lower-bound assumption, in the sense, you could possibly also have descriptions in which the total wave energy is actually greater than what is experimentally observed in a single detection event, and thus, the detection event does not completely exhaust the disturbance energy (in which case, the residual energy would be an additional factor in determining the spot of occurrence of the next detection event).

      Now, think again, for the PDC photons: What precisely is the physical mechanism that converts the wave into the “dot”?

      And, many subsidiary questions: Why does the mechanism convert the wave into only a single dot of a precisely reproducible energy/momentum? (where “precisely” is to be taken in the sense: “within the best experimental means available today”). What prevents it from generating multiple dots at different points in space (not necessarily simultaneously)? Or is the mechanism broad enough to accomodate also this possibility? If so, what are its particular features?

      Is there a contribution that the detector material makes to this wave-to-dot localization process? What precisely is its nature? Can the interaction of the wave with the detector material be captured in a governing differential equation and with some additional explanation for localization, such as, say, the chaos theory (viz., the fact that the statistical noise in the detector material varies from point to point and the localization process is extremely sensitive to this variation)? After all, bifurcation points indeed are unalterable singular points (given a certain scaling), just the way the experimentally observed quantum levels are. So, is it useful to pursue this thread?

      Given a wave of energy E_1 = hf_1, what prevents the quantum wave (which is taken as being spread over the entire domain) from not generating multiple dots such that their sum: hf_2 + h_f3 + hf_4 + … is still equal to E_1? Why does the transmission of the PDC photons occurs in such a way that only a single frequency f_1 is observed for all the detected dots?

      And the complementary question, now on the absorption side: Does the mechanism operate in a way that the detection event occurs as a result of many prior emitted waves—in the sense, does a given dot get generated because it has been accumulating some part of the hf_2 + h_f3 + hf_4 + … energies of the earlier emitted waves?

      Another crucially important detail to be supplied would be: What is the time-wise behavior for the transmission of the energy/momentum via such waves? Do the signals travel instantaneously? Or is this quantum wave very much like the ordinary seismic wave, in the sense that a finite interval of time does elapse between the generation of wave and its conversion into a (single) dot? The answer has a direct implication for entanglement, EPR, Bell’s inequalities, etc.

      A closely related point, concerning the complementary aspect: What is the mechanism for the generation of the wave? Does the process of generation of wave itself impose a restriction on the possible energy levels that the wave carries? Does the emission side, too, involve a more basic nonlinear mechanics involving the chaos theory?

      As you can see, there are many, many details still to be supplied, and these do involve many issues of contention; they touch on issues like instantaneous action at a distance (IAD), entanglement, etc.

      Sure, the explanation you mentiond does deny multiverses.

      But then, as I have pointed out several times earlier, as far as I am concerned, multiverses is too stupid an idea to be even mentioned in any serious thought.

      Objectivity requires the physical existence to be one and the same as is perceived. Please read that again. The physical existence has to be one. Actually, a sum totality is not a variable or a sequence: it is a certain result of a certain mental processing of certain existents, and therefore, uniquely identifiable. Now, if a quantitative statement must be made about this result, then it may be said that a sum totality always is “one.” The physical existence in theory must be the sum totality of that which is perceived. It therefore must be one.

      To even just think of alternatives to the uniqueness of a specific sum totality is to invalidate the entire process that leads to that sum totality. This principle, as applied to physical existence, by positing many physical universes, is to invalidate the sum totality that is the physical existence. But existence, in fact, cannot be invalidated; any attempt to do so only invalidates the consciousness that undertakes the enterprise. To deny existence is to eject oneself—or at least the theories one puts forth—in principle out of existence. To deny the uniqueness of the sum totality is to eject the theory out of existence. As a direct implication, such theories are nothing, literally speaking.

      In contrast, in life and in science, we think of something, not of a (literal) nothing.

      The responsibility to establish the connection of “multiverses” with the actually existing physical world lies squarely with those who posit that idea (of “multiverses”).

      And, they can’t deliver on that responsibility. In fact, they don’t even acknowledge the existence of this requirement. And the reason for both is because, having ejected the contents of their minds out of existence, i.e., having emptied their mind, they are rendered too “stupid” to actually think of anything at all: either the physical world or the objective requirements of the theory.

      That’s about all I can “devote” towards their ideas, in my entire life-time. Indeed, what I have done above is not to think of the multiverses idea, but to simply repeat my earlier stated position. All that any one can in cognitive propriety at all do with these so thoroughly “stupid” ideas (the way I have done above) is not to think of them, but only about them (if at all one picks them for consideration). You cannot think of nothing.

      Therefore, all that one can do with multiverses is just to circumscribe these ideas and then identify the bundle, by the force of their own explicit argumentation, as being out of existence. No one can (actually) think of such ideas: certainly not them, and for that matter, not even me!

      But a rejection of multiverses still does not mean that a quantum theory, in all the required details, has thereby been put forth. All the actual issues still remain to be settled; some of these, I have indicated above.



  2. Pingback: Yo—7: Multiverses, and the best that I can do | Ajit Jadhav's Weblog

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