# “Measure for Measure”—a pop-sci video on QM

This post is about a video on QM for the layman. The title of the video is: “Measure for Measure: Quantum Physics and Reality” [^]. It is also available on YouTube, here [^].

I don’t recall precisely where on the ‘net I saw the video being mentioned. Anyway, even though its running time is 01:38:43 (i.e. 1 hour, 38 minutes, making it something like a full-length feature film), I still went ahead, downloaded it and watched it in full. (Yes, I am that interested in QM!)

The video was shot live at an event called “World Science Festival.” I didn’t know about it beforehand, but here is the Wiki on the festival [^], and here is the organizer’s site [^].

The event in the video is something like a panel discussion done on stage, in front of a live audience, by four professors of physics/philosophy. … Actually five, including the moderator.

Brian Greene of Columbia [^] is the moderator. (Apparently, he co-founded the World Science Festival.) The discussion panel itself consists of: (i) David Albert of Columbia [^]. He speaks like a philosopher but seems inclined towards a specific speculative theory of QM, viz. the GRW theory. (He has that peculiar, nasal, New York accent… Reminds you of Dr. Harry Binswanger—I mean, by the accent.) (ii) Sheldon Goldstein of Rutgers [^]. He is a Bohmian, out and out. (iii) Sean Carroll of CalTech [^]. At least in the branch of the infinity of the universes in which this video unfolds, he acts 100% deterministically as an Everettian. (iv) Ruediger Schack of Royal Holloway (the spelling is correct) [^]. I perceive him as a QBist; guess you would, too.

Though the video is something like a panel discussion, it does not begin right away with dudes sitting on chairs and talking to each other. Even before the panel itself assembles on the stage, there is a racy introduction to the quantum riddles, mainly on the wave-particle duality, presented by the moderator himself. (Prof. Greene would easily make for a competent TV evangelist.) This part runs for some 20 minutes or so. Then, even once the panel discussion is in progress, it is sometimes interwoven with a few short visualizations/animations that try to convey the essential ideas of each of the above viewpoints.

I of course don’t agree with any one of these approaches—but then, that is an entirely different story.

Coming back to the video, yes, I do want to recommend it to you. The individual presentations as well as the panel discussions (and comments) are done pretty well, in an engaging and informal way. I did enjoy watching it.

The parts which I perhaps appreciated the most were (i) the comment (near the end) by David Albert, between 01:24:19–01:28:02, esp. near 1:27:20 (“small potatoes”) and, (ii) soon later, another question by Brian Greene and another answer by David Albert, between 01:33:26–01:34:30.

In this second comment, David Albert notes that “the serious discussions of [the foundational issues of QM] … only got started 20 years ago,” even though the questions themselves do go back to about 100 years ago.

That is so true.

The video was recorded recently. About 20 years ago means: from about mid-1990s onwards. Thus, it is only from mid-1990s, Albert observes, that the research atmosphere concerning the foundational issues of QM has changed—he means for the better. I think that is true. Very true.

For instance, when I was in UAB (1990–93), the resistance to attempting even just a small variation to the entrenched mainstream view (which means, the Copenhagen interpretation (CI for short)) was so enormous and all pervading, I mean even in the US/Europe, that I was dead sure that a graduate student like me would never be able to get his nascent ideas on QM published, ever. It therefore came as a big (and a very joyous) surprise to me when my papers on QM actually got accepted (in 2005). … Yes, the attitudes of physicists have changed. Anyway, my point here is, the mainstream view used to be so entrenched back then—just about 20 years ago. The Copenhagen interpretation still was the ruling dogma, those days. Therefore, that remark by Prof. Albert does carry some definite truth.

Prof. Albert’s observation also prompts me to pose a question to you.

What could be the broad social, cultural, technological, economic, or philosophic reasons behind the fact that people (researchers, graduate students) these days don’t feel the same kind of pressure in pursuing new ideas in the field of Foundations of QM? Is the relatively greater ease of publishing papers in foundations of QM, in your opinion, an indication of some negative trends in the culture? Does it show a lowering of the editorial standards? Or is there something positive about this change? Why has it become easier to discuss foundations of QM? What do you think?

I do have my own guess about it, and I would sure like to share it with you. But before I do that, I would very much like to hear from you.

Any guesses? What could be the reason(s) why the serious discussions on foundations of QM might have begun to occur much more freely only after mid-1990s—even though the questions had been raised as early as in 1920s (or earlier)?

Over to you.

Greetings in advance for the Republic Day. I [^] am still jobless.

[E&OE]

# Why is the physical space 3-dimensional?

Why I write on this topic?

Well, it so happened that recently (about a month ago) I realized that I didn’t quite understand matrices. I mean, at least not as well as I should. … I was getting interested in the Data Science, browsing through a few books and Web sites on the topic, and soon enough realized that before going further, first, it would be better if I could systematically write down a short summary of the relevant mathematics, starting with the topic of matrices (and probability theory and regression analysis and the lot).

So, immediately, I fired TeXMaker, and started writing an “article” on matrices. But as is my habit, once I began actually typing, slowly, I also began to go meandering—pursuing just this one aside, and then just that one aside, and then just this one footnote, and then just that one end-note… The end product quickly became… unusable. Which means, it was useless. To any one. Including me.

So, after typing in a goodly amount, may be some 4–5 pages, I deleted that document, and began afresh.

This time round, I wrote only the abstract for a “future” document, and that too only in a point-by-point manner—you know, the way they specify those course syllabi? This strategy did help. In doing that, I realized that I still had quite a few issues to get straightened out well. For instance, the concept of the dual space [^][^].

After pursuing this activity very enthusiastically for something like a couple of days or so, my attention, naturally, once again got diverted to something else. And then, to something else. And then, to something else again… And soon enough, I came to even completely forget the original topic—I mean matrices. … Until in my random walk, I hit it once again, which was this week.

Once the orientation of my inspiration thus got once again aligned to “matrices” last week (I came back via eigen-values of differential operators), I now decided to first check out Prof. Zhigang Suo’s notes on Linear Algebra [^].

Yes! Zhigang’s notes are excellent! Very highly recommended! I like the way he builds topics: very carefully, and yet, very informally, with tons of common-sense examples to illustrate conceptual points. And in a very neat order. A lot of the initially stuff is accessible to even high-school students.

Now, what I wanted here was a single and concise document. So, I decided to take notes from his notes, and thereby make a shorter document that emphasized my own personal needs. Immediately thereafter, I found myself being engaged into that activity. I have already finished the first two chapters of his notes.

Then, the inevitable happened. Yes, you guessed it right: my attention (once again) got diverted.

What happened was that I ran into Prof. Scott Aaronson’s latest blog post [^], which is actually a transcript of an informal talk he gave recently. The topic of this post doesn’t really interest me, but there is an offhand (in fact a parenthetical) remark Scott makes which caught my eye and got me thinking. Let me quote here the culprit passage:

“The more general idea that I was groping toward or reinventing here is called a hidden-variable theory, of which the most famous example is Bohmian mechanics. Again, though, Bohmian mechanics has the defect that it’s only formulated for some exotic state space that the physicists care about for some reason—a space involving pointlike objects called “particles” that move around in 3 Euclidean dimensions (why 3? why not 17?).”

Hmmm, indeed… Why 3? Why not 17?

Knowing Scott, it was clear (to me) that he meant this remark not quite in the sense of a simple and straight-forward question (to be taken up for answering in detail), but more or less fully in the sense of challenging the common-sense assumption that the physical space is 3-dimensional.

One major reason why modern physicists don’t like Bohm’s theory is precisely because its physics occurs in the common-sense 3 dimensions, even though, I think, they don’t know that they hate him also because of this reason. (See my 2013 post here [^].)

But should you challenge an assumption just for the sake of challenging one? …

It’s true that modern physicists routinely do that—challenging assumptions just for the sake of challenging them.

Well, that way, this attitude is not bad by itself; it can potentially open doorways to new modes of thinking, even discoveries. But where they—the physicists and mathematicians—go wrong is: in not understanding the nature of their challenges themselves, well enough. In other words, questioning is good, but modern physicists fail to get what the question itself is, or even means (even if they themselves have posed the question out of a desire to challenge every thing and even everything). And yet—even if they don’t get even their own questions right—they do begin to blabber, all the same. Not just on arXiv but also in journal papers. The result is the epistemological jungle that is in the plain sight. The layman gets (or more accurately, is deliberately kept) confused.

Last year, I had written a post about what physicists mean by “higher-dimensional reality.” In fact, in 2013, I had also written a series of posts on the topic of space—which was more from a philosophical view, but unfortunately not yet completed. Check out my writings on space by hitting the tag “space” on my blog [^].

My last year’s post on the multi-dimensional reality [^] did address the issue of the $n > 3$ dimensions, but the writing in a way was geared more towards understanding what the term “dimension” itself means (to physicists).

In contrast, the aspect which now caught my attention was slightly different; it was this question:

Just how would you know if the physical space that you see around you is indeed was 3-, 4-, or 17-dimensional? What method would you use to positively assert the exact dimensionality of space? using what kind of an experiment? (Here, the experiment is to be taken in the sense of a thought experiment.)

I found an answer this question, too. Let me give you here some indication of it.

First, why, in our day-to-day life (and in most of engineering), do we take the physical space to be 3-dimensional?

The question is understood better if it is put more accurately:

What precisely do we mean when we say that the physical space is 3-dimensional? How do we validate that statement?

Mark a fixed point on the ground. Then, starting from that fixed point, walk down some distance $x$ in the East direction, then move some distance $y$ in the North direction, and then climb some distance $z$ vertically straight up. Now, from that point, travel further by respectively the same distances along the three axes, but in the exactly opposite directions. (You can change the order in which you travel along the three axes, but the distance along a given axis for both the to- and the fro-travels must remain the same—it’s just that the directions have to be opposite.)

What happens if you actually do something like this in the physical reality?

You don’t have to leave your favorite arm-chair; just trace your finger along the edges of your laptop—making sure that the laptop’s screen remains at exactly 90 degrees to the plane of the keyboard.

If you actually undertake this strenuous an activity in the physical reality, you will find that, in physical reality, a “magic” happens: You come back exactly to the same point from where you had begun your journey.

That’s an important point. A very obvious point, but also, in a way, an important one. There are other crucially important points too. For instance, this observation. (Note, it is a physical observation, and not an arbitrary mathematical assumption):

No matter where you stop during the process of going in, say the East direction, you will find that you have not traveled even an inch in the North direction. Ditto, for the vertical axis. (It is to ensure this part that we keep the laptop screen at exactly 90 degrees to the keyboard.)

Thus, your $x$, $y$ and $z$ readings are completely independent of each other. No matter how hard you slog along, say the $x$-direction, it yields no fruit at all along the $y$– or $z$– directions.

It’s something like this: Suppose there is a girl that you really, really like. After a lot of hard-work, suppose you somehow manage to impress her. But then, at the end of it, you come to realize that all that hard work has done you no good as far as impressing her father is concerned. And then, even if you somehow manage to win her father on your side, there still remains her mother!

To say that the physical space is 3-dimensional is a positive statement, a statement of an experimentally measured fact (and not an arbitrary “geometrical” assertion which you accept only because Euclid said so). It consists of two parts:

The first part is this:

Using the travels along only 3 mutually independent directions (the position and the orientation of the coordinate frame being arbitrary), you can in principle reach any other point in the space.

If some region of space were to remain unreachable this way, if there were to be any “gaps” left in the space which you could not reach using this procedure, then it would imply either (i) that the procedure itself isn’t appropriate to establish the dimensionality of the space, or (ii) that it is, but the space itself may have more than 3 dimensions.

Assuming that the procedure itself is good enough, for a space to have more than 3 dimensions, the “unreachable region” doesn’t have to be a volume. The “gaps” in question may be limited to just isolated points here and there. In fact, logically speaking, there needs to be just one single (isolated) point which remains in principle unreachable by the procedure. Find just one such a point—and the dimensionality of the space would come in question. (Think: The Aunt! (The assumption here is that aunts aren’t gentlemen [^].))

Now what we do find in practice is that any point in the actual physical space indeed is in principle reachable via the above-mentioned procedure (of altering $x$, $y$ and $z$ values). It is in part for this reason that we say that the actual physical space is 3-D.

The second part is this:

We have to also prove, via observations, that fewer than 3 dimensions do fall short. (I told you: there was the mother!) Staircases and lifts (Americans call them elevators) are necessary in real life.

Putting it all together:

If $n =3$ does cover all the points in space, and if $n > 3$ isn’t necessary to reach every point in space, and if $n < 3$ falls short, then the inevitable conclusion is: $n = 3$ indeed is the exact dimensionality of the physical space.

QED?

Well, both yes and no.

Yes, because that’s what we have always observed.

No, because all physics knowledge has a certain definite scope and a definite context—it is “bounded” by the inductive context of the physical observations.

For fundamental physics theories, we often don’t exactly know the bounds. That’s OK. The most typical way in which the bounds get discovered is by “lying” to ourselves that no such bounds exist, and then experimentally discovering a new phenomena or a new range in which the current theory fails, and a new theory—which merely extends and subsumes the current theory—is validated.

Applied to our current problem, we can say that we know that the physical space is exactly three-dimensional—within the context of our present knowledge. However, it also is true that we don’t know what exactly the conceptual or “logical” boundaries of this physical conclusion are. One way to find them is to lie to ourselves that there are no such bounds, and continue investigating nature, and hope to find a phenomenon or something that helps find these bounds.

If tomorrow we discover a principle which implies that a certain region of space (or even just one single isolated point in it) remains in principle unreachable using just three dimensions, then we would have to abandon the idea that $n = 3$, that the physical space is 3-dimensional.

Thus far, not a single soul has been able to do that—Einstein, Minkowski or Poincare included.

No one has spelt out a single physically established principle using which a spatial gap (a region unreachable by the linear combination procedure) may become possible, even if only in principle.

So, it is 3, not 17.

QED.

All the same, it is not ridiculous to think whether there can be 4 or more number of dimensions—I mean for the physical space alone, not counting time. I could explain how. However, I have got too tired typing this post, and so, I am going to just jot down some indicative essentials.

Essentially, the argument rests on the idea that a physical “travel” (rigorously: a physical displacement of a physical object) isn’t the only physical process that may be used in establishing the dimensionality of the physical space.

Any other physical process, if it is sufficiently fundamental and sufficiently “capable,” could in principle be used. The requirements, I think, would be: (i) that the process must be able to generate certain physical effects which involve some changes in their spatial measurements, (ii) that it must be capable of producing any amount of a spatial change, and (iii) that it must allow fixing of an origin.

There would be the other usual requirements such as reproducibility etc., though the homogeneity wouldn’t be a requirement. Also observe Ayn Rand’s “some-but-any” principle [^] at work here.

So long as such requirements are met (I thought of it on the fly, but I think I got it fairly well), the physically occurring process (and not some mathematically dreamt up procedure) is a valid candidate to establish the physically existing dimensionality of the space “out there.”

Here is a hypothetical example.

Suppose that there are three knobs, each with a pointer and a scale. Keeping the three knobs at three positions results in a certain point (and only that point) getting mysteriously lit up. Changing the knob positions then means changing which exact point is lit-up—this one or that one. In a way, it means: “moving” the lit-up point from here to there. Then, if to each point in space there exists a unique “permutation” of the three knob readings (and here, by “permutation,” we mean that the order of the readings at the three knobs is important), then the process of turning the knobs qualifies for establishing the dimensionality of the space.

Notice, this hypothetical process does produce a physical effect that involves changes in the spatial measurements, but it does not involve a physical displacement of a physical object. (It’s something like sending two laser beams in the night sky, and being able to focus the point of intersection of the two “rays” at any point in the physical space.)

No one has been able to find any such process which even if only in principle (or in just thought experiments) could go towards establishing a $4$-, $2$-, or any other number for the dimensionality of the physical space.

I don’t know if my above answer was already known to physicists or not. I think the situation is going to be like this:

If I say that this answer is new, then I am sure that at some “opportune” moment in future, some American is simply going to pop up from nowhere at a forum or so, and write something which implies (or more likely, merely hints) that “everybody knew” it.

But if I say that the answer is old and well-known, and then if some layman comes to me and asks me how come the physicists keep talking as if it can’t be proved whether the space we inhabit is 3-dimensional or not, I would be at a loss to explain it to him—I don’t know a good explanation or a reference that spells out the “well known” solution that “everybody knew already.”

In my (very) limited reading, I haven’t found the point made above; so it could be a new insight. Assuming it is new, what could be the reason that despite its simplicity, physicists didn’t get it so far?

Answer to that question, in essential terms (I’ve really got too tired today) is this:

They define the very idea of space itself via spanning; they don’t first define the concept of space independently of any operation such as spanning, and only then see whether the space is closed under a given spanning operation or not.

In other words, effectively, what they do is to assign the concept of dimensionality to the spanning operation, and not to the space itself.

It is for this reason that discussions on the dimensionality of space remain confused and confusing.

Food for thought:

What does a $2.5$-dimensional space mean? Hint: Lookup any book on fractals.

Why didn’t we consider such a procedure here? (We in fact don’t admit it as a proper procedure) Hint: We required that it must be possible to conduct the process in the physical reality—which means: the process must come to a completion—which means: it can’t be an infinite (indefinitely long or interminable) process—which means, it can’t be merely mathematical.

[Now you know why I hate mathematicians. They are the “gap” in our ability to convince someone else. You can convince laymen, engineers and programmers. (You can even convince the girl, the father and the mother.) But mathematicians? Oh God!…]

A Song I Like:

(English) “When she was just seventeen, you know what I mean…”
Band: Beatles

[May be an editing pass tomorrow? Too tired today.]

[E&OE]

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

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

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

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

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

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

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

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

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

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

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

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

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

So, I wrote something—this post!

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

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

[E&OE]

/

# The indistinguishability of the indistinguishable particles is the problem

For many of you (and all of you in the Western world), these would be the times of the Christmas vacations.

For us, the Diwali vacations are over, and, in fact, the new term has already begun. To be honest, classes are not yet going on in full swing. (Many students are still visiting home after their examinations for the last term—which occurred after Diwali.) Yet, the buzz is in the air, and in fact, for an upcoming accreditation visit the next month, we are once again back to working also on week-ends.

Therefore, I don’t (and for a month or so, won’t be able to) find the time to do any significant blogging.

Yes, I do have a few things lined up for blogging—in my mind. On the physical plane, there simply is no time. Still, rather than going on cribbing about lack of time, let me give you something more substantial to chew on, in the meanwhile. It’s one of the things lined up, anyway.

Check out this piece [^] in Nautilus by Amanda Gefter [^]; H/T to Roger Schlafly [^].

Let me reproduce the paragraph that Roger did, because it really touches on the central argument by Frank Wilczek [^][^]. In the Nautilus piece, Amanda Gefter puts him in a hypothetical court scene:

“Dr. Wilczek,” the defense attorney begins. “You have stated what you believe to be the single most profound result of quantum field theory. Can you repeat for the court what that is?”

The physicist leans in toward the microphone. “That two electrons are indistinguishable,” he says.

Dude, get it right. It’s not the uncertainty principle. It’s not the wave-particle duality. It’s not even the spooky action-at-a-distance and entanglement. It is indistinguishability. Amanda Gefter helps us understand the physics Nobel laureate’s viewpoint

The smoking gun for indistinguishability, and a direct result of the 1-in-3 statistics, is interference. Interference betrays the secret life of the electron, explains Wilczek. On observation, we will invariably find the electron to be a corpuscular particle, but when we are not looking at it, the electron bears the properties of a wave. When two waves overlap, they interfere—adding and amplifying in the places where their phases align—peaks with peaks, troughs with troughs—and canceling and obliterating where they find themselves out of sync. These interfering waves are not physical waves undulating through a material medium, but mathematical waves called wavefunctions. Where physical waves carry energy in their amplitudes, wavefunctions carry probability. So although we never observe these waves directly, the result of their interference is easily seen in how it affects probability and the statistical outcomes of experiment. All we need to do is count.

The crucial point is that only truly identical, indistinguishable things interfere. The moment we find a way to distinguish between them—be they particles, paths, or processes—the interference vanishes, and the hidden wave suddenly appears in its particle guise. If two particles show interference, we can know with absolute certainty that they are identical. Sure enough, experiment after experiment has proven it beyond a doubt: electrons interfere. Identical they are—not for stupidity or poor eyesight but because they are deeply, profoundly, inherently indistinguishable, every last one.

This is no minor technicality. It is the core difference between the bizarre world of the quantum and the ordinary world of our experience. The indistinguishability of the electron is “what makes chemistry possible,” says Wilczek. “It’s what allows for the reproducible behavior of matter.” If electrons were distinguishable, varying continuously by minute differences, all would be chaos. It is their discrete, definite, digital nature that renders them error-tolerant in an erroneous world.

You have to read the entire article in order to understand what Amanda means when she says the “1-in-3 statistics.” Here are the relevant excerpts:

An electron—any electron—is an elementary particle, which is to say it has no known substructure.

[snip]

What does this mean? That every electron is the precise spitting image of every other electron, lacking, as it does, even the slightest leeway for even the most minuscule deviation. Unlike a composite, macroscopic object [snip] electrons are not merely similar, if uncannily so, but deeply, profoundly identical—interchangeable, fungible, mere placeholders, empty labels that read “electron” and nothing more.

This has some rather curious—and measurable—consequences. Consider the following example: We have two elementary particles, A and B, and two boxes, and we know each particle must be in one of the two boxes at any given time. Assuming that A and B are similar but distinct, the setup allows four possibilities: A is in Box 1 and B is in Box 2, A and B are both in Box 1, A and B are both in Box 2, or A is in Box 2 and B is in Box 1. The rules of probability tell us that there is a 1-in-4 chance of finding the two particles in each of these configurations.

If, on the other hand, particles A and B are truly identical, we must make a rather strange adjustment in our thinking, for in that case there is literally no difference between saying that A is in Box 1 and B in Box 2, or B is in Box 1 and A is in Box 2. Those scenarios, originally considered two distinct possibilities, are in fact precisely the same. In total, now, there are only three possible configurations, and probability assigns a 1-in-3 chance that we will discover the particles in any one of them.

Some time ago, I had mentioned how, during my text-book studies of QM, I had got stuck at the topic of spin and identical particles [^]. … Well, I didn’t have this in mind, but, yes, identical particles is the topic where I had got stuck anyway. (I still am, to some extent. However, since then, this article [^] by Joshua Samani did help in getting things clarified.)

Anyway, coming back to Wilczek and QM, Gefter reports:

Wilczek puts it this way: “Another aspect of quantum mechanics closely related to indistinguishability, and a competitor for its deepest aspect, is that if you want to describe the state of two electrons, it’s not that you have a wavefunction for one and a separate wavefunction for the other, each living in three-dimensional space. You really have a six-dimensional wavefunction that has two positions in it where you can fill in two electrons.” The six-dimensional wavefunction means that the probabilities for finding each electron at a particular location are not independent—that is, they are entangled.

It is no mystery that all electrons look alike, he [i.e. Wilczek] says, because they are all manifestations, temporary excitations of one and the same underlying electron field, which permeates all space, all time. Others, like physicist John Archibald Wheeler, say one particle. He suggested that perhaps electrons are indistinguishable because there’s only one, but it traces such convoluted paths through space and time that at any given moment it appears to be many.

Ummm. Not quite—this only one electron part. Wheeler never got “it” right, IMO. He also influenced Feynman and “won” him, but in the reverse order: he first got Feynman as a graduate student, and then, of course, influenced him. … BTW, how come Wheeler’s idea hasn’t been used to put forth monotheistic arguments? Any idea? As to me, I guess, two reasons: (i) the monotheistic people wouldn’t like their God doing this frenzied a running around in the material world, and (ii) the mainstream QM insists on the vagueness in the position of the quantum particle, so that its running from “here” to “there” itself is untenable. … Anyway, let’s continue with Amanda Gefter:

So if the elementary particles of which we are made don’t really exist as objects, how do we exist?

Good job, Amanda!

… Her search for the answer involves other renowned physicists, too; in particular, Peter Pesic [^]:

“When you have more and more electrons, the state that they together form starts to be more and more capable of being distinct,” Pesic said.

Only when you have “more and more” electrons?

“So the reason that you and I have some kind of identity is that we’re composed of so enormously many of these indistinguishable components. It’s our state that’s distinguishable, not our materiality.”

IMO, Pesic nearly got it—and then, just as easily, also lost it!

It has to be something to do with the state! After all, in QM, state defines everything. But you don’t really need the many here—there is no need for a “collective” approach like that, IMO. And, as to the state vs materiality distinction: The quantum mechanical state is supposed to describe each and every material aspect of every thing.

So, that’s a physicist thinking about QM(,) for you.

…Anyway, Amanda has a job to do, and she continues doing as best of it as she can:

Our identity is a state, but if it’s not a state of matter—not a state of individual physical objects, like quarks and electrons—then a state of what?

A state, perhaps, of information. Ladyman suggests that we can replace the notion of a “thing” with a “real pattern”—a concept first articulated by the philosopher Daniel Dennett and further developed by Ladyman and philosopher Don Ross. “Another way of articulating what you mean by an object is to talk about compression of information,” Ladyman says. “So you can claim that something’s real if there’s a reduction in the information-theoretic complexity of tracking the world if you include it in your description.”

There is more along this line:

Should such examples give the impression that the real patterns are patterns of particles, beware: Particles, like our electron, are real patterns themselves. “We’re using a particle-like description to keep track of the real patterns,” Ladyman says. “It’s real patterns all the way down.”

Honest, what I experienced when I first read this passage was: a very joyful moment!

We are nothing but fleeting patterns, signals in the noise. Drill down and the appearance of materiality gives way; underneath it, nothing.

Here is a conjecture about the path they trace together; the part in the square brackets [] is optional:

We (i.e. a physical object in this context)-> Fleeting Patterns -> Fleeting Patterns -> Signals in the Noise –> [We –>] Signals in the Noise –> Appearance of Materiality –> Appearance of Materiality –> Appearance –> Nothing.

Fascinating, these philosophers (really) are. Ladyman proves the point, once again:

“I think in the end,” says Ladyman, “it may well be that the world isn’t made of anything.”

You could tell how rapidly he would go from “may well be” to “is,” couldn’t you?

So, that is what I have picked up for thinking. I mean, the two issues raised by Wilczek.

(1) The first issue was about how the indistinguishability of the indistinguishable particles is a problem. I will come back at it some later time, but in the meanwhile, here is the answer in brief (and in the vague):

Electrons are identical because: (i) the only extent to which we can at all determine that they are identical is based on quantum-mechanical observations, and (ii) observables are operators in QM.

That much of an answer is enough, but just in case it doesn’t strike the right chord:

The fact that observables are operators means that they are mathematical processes. These processes operate on wavefunctions. They “bring out” a mathematical aspect of the wavefunction.

Even if electrons were not to be exactly identical in all respects, as long as the QM postulates remain valid—as long as observables must be represented via Hermitian operators so that only real eigenvalues can be had—you would have no way to tell in what micro-way they might actually be different.

If you must have a (rather bad) analogy, take two particles of sand of roughly the same size, and gently drop both of them in a jar of honey (or some suitable fluid) at the same time. Both will fall at the same rate (within the experimental margin), and if, somehow, classical mechanics were such that it was only the rate of falling that could at all be measured in experiment, or at least, if the rate of descent alone could tell you anything about the size (and shape) of the sand particle, then you would have to treat both the particles as exactly the same in all respects.

The analogy is bad because QM measurements involve eigenvalues, and, practically speaking, their measurements are more robust (involving less variability from one experiment to another) as compared to the rate of descent. Why? Simple. Because, no matter how limiting you might get, fluid dynamics equations are basically nonlinear; eigenvalue situations are basically linear. That’s why.

I don’t think this much of explanation is enough. It’s just that I haven’t the time either to think through my newer QM conjectures, or work out their maths, let alone write blog posts about them. The situation will continue definitely for at least a month or so (till the course and the labs and all settle down), perhaps also for the entire teaching term (about 4 months).

(2) The second issue was about how multi-dimensionality of the wavefunction implies entanglement of particles. As to entanglement, I will be able to come to it even later—i.e., after issue no. (1) here.

Regarding purely the multi-dimensionality part, however, I can already direct you to a recent post (by me), here [^]. (I think it can be improved—the distinction of embedded vs embedding space needs to be made more clear, and the aspect of “projection” needs to be looked into—but, once again: I’ve no time; so some time later!)

Bye for now.

A Song I Like:

(Marathi) “ashee nishaa punhaa kadhee disel kaa?”
Singers: Hridaynath Mangeshkar, Lata Mangeshkar
Music: Yashawant Deo
Lyrics: Yashawant Deo

[May be another pass tomorrow or so. I also am not sure whether I ran this song before or not. In case I did, I would come back and replace it with some other song.]

[E&OE]

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