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“Even in principle, one cannot demand an alibi of an electron!”
Hermann Weyl, The Theory of Groups and Quantum Mechanics (1950)
Have you ever heard the story of Martin Guerre?
He lived with his bride and newborn son in Artigat, a small village in the Pyrenees foothills of Southwestern France. In 1548, at the age of 24, after being accused by his own parents of theft, Martin Guerre disappeared, leaving his family behind. Eight years later, after his parents had passed away, Guerre returned home, reuniting with his wife, son, and fellow villagers.
Over the next three years, Guerre and his wife, Bertrande, had two more children. All was going swimmingly until a foreign soldier came through town and claimed that the man who had returned was not the real Martin Guerre, but an imposter named Arnault du Tilh. The accuser claimed to have fought alongside Guerre in the Spanish army, and said that Guerre had lost a leg in battle. Bertrande ignored the accusation, certain that the man with whom she was living was, and had always been, her husband. But soon Guerre’s uncle and Bertrande’s stepfather joined the foreign soldier in accusing the man of forging Guerre’s identity, and took him to trial.
If electrons were distinguishable, all would be chaos.
It is a story that has persisted in our collective imagination—having been made into a movie, a musical, a historical novel, a TV series, and a Simpsons episode—because it strikes an ever-raw nerve: There is a sense in which our notions of identity are insecure. How can we be sure who someone is, even someone close? How can we be sure who we are, or that we are? What can identity mean in a world that is constantly changing?
The early vitalist philosophers had a ready answer: Each of us is distinguished by a divine soul, our physical bodies mere puppets animated by our invisible selves. But science has eroded this answer, and sought identity in the physical body itself: At a microscopic level, promises the reductionist dream, there must be something to distinguish each one of us from another. A hard-nosed foundation for our identity, one made of molecules and atoms.
That path, however, is far shakier than it might seem. Cast your gaze on Guerre, standing there in the courtroom. Zoom in. Look closer at his face, his skin, his pores. Burrow down to his most basic constituents. There: See that electron? It’s a building block of Guerre himself. But what if we were to put the electron on trial instead of Guerre?
Yes, we can all giggle at the oddity of trying an elementary particle to the full extent of the law. Yes, we can all expect some cringe-worthy puns. Let’s get them out of the way now, shall we? The air in the room is electric. The defendant is charged with the serious crime of identity fraud. Good. Let’s move on.
The judge raps his gavel to call the courtroom to order. Twelve jurors sit at attention. The defendant squirms in its seat, to the chagrin of its attorneys and the frustration of the sketch artist.
An electron—any electron—is an elementary particle, which is to say it has no known substructure. Guerre was made of molecules, molecules are made of atoms, atoms are made of elementary particles, but elementary particles are the end of the line. They are made of nothing, being, as they are, the most basic building blocks of the material world. An electron is a point, taking up quite literally no space at all. Every electron is defined solely in terms of its mass (tiny), its spin (1/2) and its charge (negative). Those three features comprise in toto the complete and comprehensive identity of the electron, as its want for spatial extent bears no room to house any further attributes.
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 like Guerre—or anything else in our everyday experience—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.
Experiment confirms that the microworld obeys the 1-in-3 statistics. Swap out the accused with another of its kind—the universe would register no difference, and neither would we.
Score one for the defense. To drive the point home, the defense attorney calls to the stand one Frank Wilczek, a theoretical physicist from the Massachusetts Institute of Technology, for expert testimony. To establish his status as an expert, counsel recites Wilczek’s credentials into the record: countless books and scientific papers published, a protracted list of awards won. “Oh,” the attorney smiles, “and a Nobel Prize.” The prosecutor looks begrudgingly impressed.
“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.
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.
One can’t help but wonder whether it is the electrons who are so devious or rather space itself.
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.
Their identicalness means that while we can speak of electrons generally, we are barred from making any statements about any particular electron. “If you have two electrons and later observe two electrons, but you didn’t watch them at intermediate stages, you cannot say which was originally which,” says Wilczek. “It’s not just that you are confused—it is impossible in principle to tell which was which.”
Peter Pesic, a physicist, historian, and musician at St. John’s College in Santa Fe, New Mexico, puts it this way: “We could make a statement like, ‘There are five electrons.’ We can give them a cardinal number. But we couldn’t give them ordinal numbers.” Cardinal numbers are counting numbers. Five electrons. Ordinal numbers are ordering numbers—the first, the second, the third, the fourth, the fifth. To say that we can have cardinals without ordinals is to say that we can attach labels to the group but not to any of its individual members—which is itself to say that its members are not individuals at all. “That’s very surprising,” Pesic continued, “because we think of cardinal and ordinal numbers as if they always both apply. At the microscopic level, that’s not true. You have one without the other.”
The prosecutor paces slowly back and forth in front of the witness stand, considering his cross-examination. Perhaps, he suggests, we can distinguish electrons not by their inherent features, but by their location in space. Even if two electrons are in every way identical, surely we can tell them apart by the mere fact that one is here and the other there, can we not?
Wilczek’s reply is a simple “no.” While corpuscular particles appear to occupy specific points in space, waves, by their very nature, do not. And so when particles like our electron go unwatched, they become diffuse and placeless. Their wavefunctions, though concentrated in particular regions of space, stretch out to infinity, so that there is always a small but non-zero chance that their particulate nature should manifest itself anywhere, should someone decide to go looking for it.
When no one is looking, the electron isn’t anywhere, but merely has the probability to be found in various locations—a fact so strange that one can’t help but wonder whether it is the electrons who are so devious or rather space itself. What happens to space when we’re not looking? Does it simply fall away?
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.
The identicalness of electrons not only cripples the concept of a thing, but also the concept of space.
In the old way of looking at things, we had space and then we put things in it. In the quantum view, we have things—like electrons—and space emerges as a way to describe the complex set of relations and interdependencies among them, with localized points like “here” and “there” peeking through like the tips of icebergs.
When two particles are entangled, their properties—their identities—reside in neither particle individually but in the relationship between them, a relationship that flaunts the usual constraints of space, bypassing it with what Einstein called “spooky action at a distance.” “We’ve got this problem that particles of matter are often entangled so,” says philosopher James Ladyman of the University of Bristol. “The state of the world can’t be written as the state of all the particles separately. They’re all tied together.”
The identicalness of electrons—of all particles—not only cripples the concept of a thing, but also the concept of space, revealing them to be opposite sides of the same feeble coin. It is a clue that there is something wrong with the way that we cut the world into parts. A clue to a kind of holism, an underlying oneness.
One what?
Some, like Wilczek, say one field. It is no mystery that all electrons look alike, he 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. Gottfried Leibniz, the 17th-century philosopher, put forth the Principle of the Identity of Indiscernibles, which said if you can’t tell two things apart, they are not two things. On one hand, electrons appear to refute the principle. On the other, perhaps the multiplicity of particles—or the multiplicity of the world—is a kind of funhouse illusion.
Time, it has been said, is what keeps everything from happening at once. In the same vein, space is what keeps everything from being one—or in Wheeler’s words, “what prevents everything from happening to me.” But in the realm of the quantum, the scaffolding provided by space recedes, and with it all notions of identity and thingness, and with them, the plurality of existence. The electron is everywhere, the electron is nowhere. A fugitive without a form. An outlaw sans alibi.
And, clearly, innocent of identity fraud—by definition. But what about the human being that it builds?
Zoom out.
Bertrande, Guerre’s wife, had always refused to believe that her husband was a fraud. But during his trial she changed her mind, deciding that although the man claiming to be Martin Guerre knew many intimate details of their early relationship, he was not the man she’d married. And yet, when the disputed Guerre turned to her and wagered that if she were to swear that he wasn’t her betrothed, he would happily agree to his own execution Bertrande remained silent. Martin Guerre, now judged to be Arnault du Tilh, was convicted and sentenced to death by beheading.
The doomed man appealed his case in Toulouse, insisting that he was in fact the true Martin Guerre. He made such a compelling case that the judges of the appellate court were ready to acquit him when, to everyone’s amazement, a man appeared in the courthouse claiming to be the real Martin Guerre. He looked very much like the accused, except that he walked with a wooden leg. Although this Martin Guerre was unable to recall many convincing private details about his earlier marriage, Guerre’s family and townsfolk were immediately convinced: This was Martin Guerre. The man who stood trial was hauled off for beheading while Bertrande begged her husband for forgiveness.
In its decision, the court decided that du Tilh was not Martin Guerre. But what does it mean to be Martin Guerre? It is an act of persistence. It is to connect by some smooth and seamless trajectory to the Martin Guerre of every other moment in space and time, following what Einstein called a worldline with perfect fidelity and without deviation.
Zoom in again. Guerre consists of fundamental particles, but their wordlines are not lines at all, just a series of points separated by strange and sundered lacunae. The elecron’s worldline is, in the words of Wheeler, a great smoky dragon with a well-defined head, a distinct tail and nothing but vapor in between. “What we call reality,” Wheeler said, “consists of a few iron posts of observation between which we fill in by an elaborate papier mache construction of imagination and theory.”
We want to believe that a thing is somehow more than the sum of its parts. That if we removed an electron’s charge, its mass, its spin, there would be something leftover, a bald electron, a haecceity, as the philosophers say, a primitive thisness. We want to believe that there is something that it means to be this electron rather than that, even if no observation, experiment, or statistic could ever reveal it. We want to believe in a primitive thisness because we want to believe in a primitive ourness—that should we one day meet our double, a perfect clone down to every detail, every dream, utterly indistinguishable to even the most discerning observer, that still there would be, here on the inside, something that it feels like to be us and not our double, a difference invisible and ineffable but true. That had there been no difference at all between the two Martin Guerres, one would still smile to himself in the secret knowledge that he was the real one.
We want to believe it, but quantum mechanics doesn’t let us. “We are fooled into thinking that our distinguishability inheres in our material substance, but that’s just a big misunderstanding on our part,” says Pesic. What becomes of the electron’s haecceity when it interferes with another’s, its primitive thisness muddled with thatness? Epistemology dictates ontology. And so it seems ever more likely that haecceity is a kind of philosopher’s rendition of the soul, a comfort, an illusion. In mythology, in religion, we seek oneness. Just not so much of it that we disappear.
So if the elementary particles of which we are made don’t really exist as objects, how do we exist?
“I think in the end,” says Ladyman, “it may well be that the world isn’t made of anything.”
“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. “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.”
“That’s a weird and beautiful idea,” Pesic continues. “Not one of our components—no electron, no proton—has any kind of stamp on it. But together they exist in a state that becomes sufficiently complex that it can then be distinguished from the state of every other person who’s composed of the same indistinguishable electrons and protons.”
“My thingness is in how I’m organized, not what I’m made of,” says Ladyman. “But of course we know that anyway, because we know that the cells in our bodies are getting replaced all the time. Functional organization of structure, not the matter it’s made of, is what counts.”
Yes we know this, that we are entities in physical flux, our bodies ships of Theseus, passing like paradoxes in the night. And yet we tend to believe that if we take a snapshot of ourselves at any given moment we will find that we are made of something—something that will pass, something that will change, but something.
But the jury declares: No. There is no there there.
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.”
Consider a cat. In computational terms, we can represent a cat using a bit map, a verbatim description down to the finest grain. Alternatively, we can render the cat in a coarser grain by ignoring its microscopic details and calling it simply “cat.” In the first case, we’d have to use many bits and great computational resources to describe how each bit individually changes its position over time. In the second, it takes barely a breath to achieve the same feat with the statement, “The cat walked across the room.” A cat, then, is a real pattern—a genuine ontological article of a mind-independent universe—because it’s computationally efficient.
Now consider a non-genuine object. “Don Ross gave this example of his left earlobe, the largest elephant in Namibia and Miles Davis’ last solo,” Ladyman says. “Imagine a composite object of those three things. You don’t get any reduction in the computational complexity of tracking the world by qualifying over those three things because they don’t form a real pattern. That collection doesn’t figure in any projectable generalizations. The parts of you do. You’re a real pattern, over and above all your individual body parts, because we can just talk about you and you drag all your bits around with you.”
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.”
We are nothing but fleeting patterns, signals in the noise. Drill down and the appearance of materiality gives way; underneath it, nothing. “I think in the end,” says Ladyman, “it may well be that the world isn’t made of anything.”
Even so, we can point to patterns, and assign names. The more complex the pattern, the more we have to potentially gain by compressing its microscopic description, and the greater the case for identity. Consider a brain—with as many neurons as stars in the galaxy linked together through trillions of connections it’s the most complex object in the known universe. Try to compress it. Call it by just two words. Call it Martin Guerre. Push further. A single word, a single letter.
Call it “I.”
Amanda Gefter is a physics writer and author of Trespassing on Einstein’s Lawn: A father, a daughter, the meaning of nothing and the beginning of everything. She lives in Cambridge, Massachusetts.
Additional Reading
Ladyman, J. & Ross, D. Every Thing Must Go: Metaphysics Naturalized Oxford University Press, Oxford, U.K. (2007).
Pesic, P. Seeing Double: Shared Identities in Physics, Philosophy and Literature MIT Press, Cambridge, MA (2002).
Wilczek, F. & Devine, B. Longing for the Harmonies: Themes and Variations from Modern Physics W.W. Norton & Co., New York, NY (1987).
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