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?

Enter: Ladyman, a philosopher:

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.

Ladyman tutoring Amanda, that was.

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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