Can You Prove Two Particles Are Identical?

This post is part of the Quan­tum Physics Se­quence.
Fol­lowup to: Where Philos­o­phy Meets Science, Joint Configurations

Be­hold, I pre­sent you with two elec­trons. They have the same mass. They have the same charge. In ev­ery way that we’ve tested them so far, they seem to be­have the same way.

But is there any way we can know that the two elec­trons are re­ally, truly, en­tirely in­dis­t­in­guish­able?

The one who is wise in philos­o­phy but not in physics will snort dis­mis­sal, say­ing, “Of course not. You haven’t found an ex­per­i­ment yet that dis­t­in­guishes these two elec­trons. But who knows, you might find a new ex­per­i­ment to­mor­row that does.”

Just be­cause your cur­rent model of re­al­ity files all ob­served elec­trons in the same men­tal bucket, doesn’t mean that to­mor­row’s physics will do the same. That’s mix­ing up the map with the ter­ri­tory. Right?

It took a while to dis­cover atomic iso­topes. Maybe some­day we’ll dis­cover elec­tron iso­topes whose masses are differ­ent in the 20th dec­i­mal place. In fact, for all we know, the elec­tron has a tiny lit­tle tag on it, too small for your cur­rent micro­scopes to see, read­ing ‘This is elec­tron #7,234,982,023,348...’ So that you could in prin­ci­ple toss this one elec­tron into a bath­tub full of elec­trons, and then fish it out again later. Maybe there’s some way to know in prin­ci­ple, maybe not—but for now, surely, this is one of those things that sci­ence just doesn’t know.

That’s what you would think, if you were wise in philos­o­phy but not in physics.

But what kind of uni­verse could you pos­si­bly live in, where a sim­ple ex­per­i­ment can tell you whether it’s pos­si­ble in prin­ci­ple to tell two things apart?

Maybe aliens gave you a tiny lit­tle de­vice with two tiny lit­tle boxes, and a tiny lit­tle light that goes on when you put two iden­ti­cal things into the boxes?

But how do you know that’s what the de­vice re­ally does? Maybe the de­vice was just built with mea­sur­ing in­stru­ments that go to the 10th dec­i­mal place but not any fur­ther.

Imag­ine that we take this prob­lem to an an­a­lytic philoso­pher named Bob, and Bob says:

“Well, for one thing, you can’t even get ab­solute proof that the two par­ti­cles ac­tu­ally ex­ist, as op­posed to be­ing some kind of hal­lu­ci­na­tion cre­ated in you by the Dark Lords of the Ma­trix. We call it ‘the prob­lem of in­duc­tion’.”

Yes, we’ve heard of the prob­lem of in­duc­tion. Though the Sun has risen on billions of suc­ces­sive morn­ings, we can’t know with ab­solute cer­tainty that, to­mor­row, the Sun will not trans­form into a gi­ant choco­late cake. But for the Sun to trans­form to choco­late cake re­quires more than an unan­ti­ci­pated dis­cov­ery in physics. It re­quires the ob­served uni­verse to be a lie. Can any ex­per­i­ment give us an equally strong level of as­surance that two par­ti­cles are iden­ti­cal?

“Well, I Am Not A Physi­cist,” says Bob, “but ob­vi­ously, the an­swer is no.”

Why?

“I already told you why: No mat­ter how many ex­per­i­ments show that two par­ti­cles are similar, to­mor­row you might dis­cover an ex­per­i­ment that dis­t­in­guishes be­tween them.”

Oh, but Bob, now you’re just tak­ing your con­clu­sion as a premise. What you said is ex­actly what we want to know. Is there some achiev­able state of ev­i­dence, some se­quence of dis­cov­er­ies, from within which you can le­gi­t­i­mately ex­pect never to dis­cover a fu­ture ex­per­i­ment that dis­t­in­guishes be­tween two par­ti­cles?

“I don’t be­lieve my logic is cir­cu­lar. But, since you challenge me, I’ll for­mal­ize the rea­son­ing.

“Sup­pose there are par­ti­cles {P1, P2, …} and a suite of ex­per­i­men­tal tests {E1, E2, …} Each of these ex­per­i­men­tal tests, ac­cord­ing to our best cur­rent model of the world, has a causal de­pen­dency on as­pects {A1, A2...} of the par­ti­cles P, where an as­pect might be some­thing like ‘mass’ or ‘elec­tric charge’.

“Now these ex­per­i­men­tal tests can es­tab­lish very re­li­ably—to the limit of our be­lief that the uni­verse is not out­right ly­ing to us—that the de­pended-on as­pects of the par­ti­cles are similar, up to some limit of mea­surable pre­ci­sion.

“But we can always imag­ine an ad­di­tional as­pect A0 that is not de­pended-on by any of our ex­per­i­men­tal mea­sures. Per­haps even an epiphe­nom­e­nal as­pect. Some philoso­phers will ar­gue over whether an epiphe­nom­e­nal as­pect can be truly real, but just be­cause we can’t le­gi­t­i­mately know about some­thing’s ex­is­tence doesn’t mean it’s not there. Alter­na­tively, we can always imag­ine an ex­per­i­men­tal differ­ence in any quan­ti­ta­tive as­pect, such as mass, that is too small to de­tect, but real.

“Th­ese ex­tra prop­er­ties or marginally differ­ent prop­er­ties are con­ceiv­able, there­fore log­i­cally pos­si­ble. This shows you need ad­di­tional in­for­ma­tion, not pre­sent in the ex­per­i­ments, to definitely con­clude the par­ti­cles are iden­ti­cal.”

That’s an in­ter­est­ing ar­gu­ment, Bob, but you say you haven’t stud­ied physics.

“No, not re­ally.”

Maybe you shouldn’t be do­ing all this philo­soph­i­cal anal­y­sis be­fore you’ve stud­ied physics. Maybe you should beg off the ques­tion, and let a philoso­pher who’s stud­ied physics take over.

“Would you care to point out a par­tic­u­lar flaw in my logic?”

Oh… not at the mo­ment. We’re just say­ing, You Are Not A Physi­cist. Maybe you shouldn’t be so glib when it comes to say­ing what physi­cists can or can’t know.

“They can’t know two par­ti­cles are perfectly iden­ti­cal. It is not pos­si­ble to imag­ine an ex­per­i­ment that proves two par­ti­cles are perfectly iden­ti­cal.”

Im­pos­si­ble to imag­ine? You don’t know that. You just know you haven’t imag­ined such an ex­per­i­ment yet. But per­haps that sim­ply demon­strates a limit on your imag­i­na­tion, rather than demon­strat­ing a limit on phys­i­cal pos­si­bil­ity. Maybe if you knew a lit­tle more physics, you would be able to con­ceive of such an ex­per­i­ment?

“I’m sorry, this isn’t a ques­tion of physics, it’s a ques­tion of episte­mol­ogy. To be­lieve that all as­pects of two par­ti­cles are perfectly iden­ti­cal, re­quires a differ­ent sort of as­surance than any ex­per­i­men­tal test can provide. Ex­per­i­men­tal tests only fail to es­tab­lish a differ­ence; they do not prove iden­tity. What par­tic­u­lar physics ex­per­i­ments you can do, is a physics ques­tion, and I don’t claim to know that. But what ex­per­i­ments can jus­tify be­liev­ing is an episte­molog­i­cal ques­tion, and I am a pro­fes­sional philoso­pher; I ex­pect to un­der­stand that ques­tion bet­ter than any physi­cist who hasn’t stud­ied for­mal episte­mol­ogy.”

And of course, Bob is wrong.

Bob isn’t be­ing stupid. He’d be right in any clas­si­cal uni­verse. But we don’t live in a clas­si­cal uni­verse.

Our abil­ity to perform an ex­per­i­ment that tells us pos­i­tively that two par­ti­cles are en­tirely iden­ti­cal, goes right to the heart of what dis­t­in­guishes the quan­tum from the clas­si­cal; the core of what sep­a­rates the way re­al­ity ac­tu­ally works, from any­thing any pre-20th-cen­tury hu­man ever imag­ined about how re­al­ity might work.

If you have a par­ti­cle P1 and a par­ti­cle P2, and it’s pos­si­ble in the ex­per­i­ment for both P1 and P2 to end up in ei­ther of two pos­si­ble lo­ca­tions L1 or L2, then the ob­served dis­tri­bu­tion of re­sults will de­pend on whether “P1 at L1, P2 at L2” and “P1 at L2, P2 at L1″ is the same con­figu­ra­tion, or two dis­tinct con­figu­ra­tions. If they’re the same con­figu­ra­tion, we add up the am­pli­tudes flow­ing in, then take the squared mod­u­lus. If they’re differ­ent con­figu­ra­tions, we keep the am­pli­tudes sep­a­rate, take the squared mod­uli sep­a­rately, then add the re­sult­ing prob­a­bil­ities. As (1 + 1)2 != (12 + 12), it’s not hard to dis­t­in­guish the ex­per­i­men­tal re­sults af­ter a few tri­als.

(Yes, half-in­te­ger spin changes this pic­ture slightly. Which I’m not go­ing into in this se­ries of blog posts. If all episte­molog­i­cal con­fu­sions are re­solved, half-in­te­ger spin is a difficulty of mere math­e­mat­ics, so the is­sue doesn’t be­long here. Half-in­te­ger spin doesn’t change the ex­per­i­men­tal testa­bil­ity of par­ti­cle equiv­alences, or al­ter the fact that par­ti­cles have no in­di­vi­d­ual iden­tities.)

And the flaw in Bob’s logic? It was a fun­da­men­tal as­sump­tion that Bob couldn’t even see, be­cause he had no al­ter­na­tive con­cept for con­trast. Bob talked about par­ti­cles P1 and P2 as if they were in­di­vi­d­u­ally real and in­de­pen­dently real. This turns out to as­sume that which is to be proven. In our uni­verse, the in­di­vi­d­u­ally and fun­da­men­tally real en­tities are con­figu­ra­tions of mul­ti­ple par­ti­cles, and the am­pli­tude flows be­tween them. Bob failed to imag­ine the se­quence of ex­per­i­men­tal re­sults which es­tab­lished to physi­cists that this was, in fact, how re­al­ity worked.

Bob failed to imag­ine the ev­i­dence which falsified his ba­sic and in­visi­bly as­sumed on­tol­ogy—the dis­cov­er­ies that changed the whole na­ture of the game; from a world that was the sum of in­di­vi­d­ual par­ti­cles, to a world that was the sum of am­pli­tude flows be­tween multi-par­ti­cle con­figu­ra­tions.

And so Bob’s care­ful philo­soph­i­cal rea­son­ing ended up around as use­ful as Kant’s con­clu­sion that space, by its very na­ture, was flat. Turned out, Kant was just re­pro­duc­ing an in­visi­ble as­sump­tion built into how his pari­etal cor­tex was mod­el­ing space. Kant’s imag­in­ings were ev­i­dence only about his imag­i­na­tion—grist for cog­ni­tive sci­ence, not physics.

Be care­ful not to un­der­es­ti­mate, through benefit of hind­sight, how sur­pris­ing it would seem, a pri­ori, that you could perfectly iden­tify two par­ti­cles through ex­per­i­ment. Be care­ful not to un­der­es­ti­mate how en­tirely and perfectly rea­son­able Bob’s anal­y­sis would have seemed, if you didn’t have quan­tum as­sump­tions to con­trast to clas­si­cal ones.

Ex­per­i­ments tell us things about the na­ture of re­al­ity which you just plain wouldn’t ex­pect from a pri­ori rea­son­ing. Ex­per­i­ments falsify as­sump­tions we can’t even see. Ex­per­i­ments tell us how to do things that seem log­i­cally im­pos­si­ble. Ex­per­i­ments de­liver sur­prises from blind spots we don’t even know ex­ist.

Bear this in mind, the next time you’re won­der­ing whether mere em­piri­cal sci­ence might have some­thing to­tally un­ex­pected to say about some im­pos­si­ble-seem­ing philo­soph­i­cal ques­tion.

Part of The Quan­tum Physics Sequence

Next post: “Clas­si­cal Con­figu­ra­tion Spaces

Pre­vi­ous post: “Where Philos­o­phy Meets Science