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.”


“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

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