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November 29, 2022

Spot the Difference to Reveal Exotic Particles
Questioning the symmetrization postulate of quantum mechanics and the notion that electrons are indistinguishable could reveal whether hypothetical new particles exist.
by Colin Stuart
FQXi Awardees: Philip Goyal
October 29, 2014
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Philip Goyal
University at Albany (SUNY)
As humans, we’re often told to celebrate our individuality and the unique features that distinguish us from each other. But what about elementary particles? Is one electron pretty much the same as the next?

It may seem like a frivolous question, but the answer has profound consequences for the foundations of quantum theory. When carrying out quantum calculations, physicists must assume that electrons are identical and indistinguishable, and they formalize this using a rule called the symmetrization postulate. So far, experiments seem to back this procedure. But Philip Goyal, a physicist at the University of Albany, New York, wants to know if the rule is just a mathematical trick needed to get the sums to work, or if it is a true feature of reality. What he discovers could have implications for the existence of unexpected new elementary particles.

To understand why physicists developed the view that electrons are indistinguishable, picture the helium atom, which is made up of a nucleus and two orbiting electrons. (To get the simplest model of what’s going on, it pays to ignore any interaction between the electrons and their internal magnetic properties, known as their "spins.") The normal way to work out how the atom changes and evolves over time is to write down mathematical expressions for the energy of each electron and plug these into an equation, developed by one of the founding fathers of quantum theory Erwin Schrödinger.

In classical physics, we would treat identical electrons as distinguishable; that is, we could imagine labeling one as "electron A" and the other as "electron B" and then tracking their separate journeys over time. If you make that same assumption in quantum theory, however, you encounter a roadblock. The answer you get shows that both electrons can simultaneously exist in the lowest energy "ground state" of the helium atom. The trouble is that this directly contradicts another fundamental rule of quantum mechanics, the Pauli Exclusion Principle, which dictates that no two electrons, with the same properties, can occupy the same quantum state. It is because of this principle that electrons are forced to stack up in different shells around an atomic nucleus, leading to the different chemical properties of the elements in the Periodic Table.

Early pioneers of quantum theory in the 1920s, including Paul Dirac, realized that the situation could be rescued by enforcing a cunning mathematical rule called the symmetrization postulate. This requires treating two electrons as indistinguishable. If that’s the case, the math needs adjusting. The upshot is that the probability of two spinless, non-interacting electrons occupying the ground state also goes to zero—it’s impossible. Very conveniently the problem is solved. "It preserves the Pauli Exclusion Principle," says Goyal. But does that mean that it is necessarily correct? "What I want to know is whether the symmetrization postulate is just a rule of thumb that happens to work, or whether it is something more fundamental," Goyal adds.

Shaky Ground

Goyal’s mission to unlock the puzzle of the symmetrization postulate began when he was a student. "When I first encountered it as undergraduate I just didn’t believe any of the textbook arguments for it," he says. "Later I discovered that it had been on shaky ground from the very beginning of quantum theory. I find it fascinating that no-one knows where it comes from." Goyal plans to use a $93,127 FQXi grant to get the bottom of the mystery.

Experiments have already been carried out to test whether electrons really are identical and indistinguishable. They involve two electrons fired from electrons guns, one positioned on the left and the other on the right. The fired electrons enter a collision area in the middle and either scatter up or down. So there are two possible scenarios: either the left electron goes up and the right goes down, or the left goes down and the right goes up.

It turns out that you can calculate the probability of observing each outcome based on the idea that electrons are distinguishable—that they have some individuality of their own. The experimental results do not fit with this calculation, however. The measured probabilities do fit with calculations carried out assuming that the electrons are truly indistinguishable. These experiments suggest that this feature is fundamental, rather than something that arises through an experimental limitation.

Electron Identity
In 2013, the orbital structure of a hydrogen atom, with a single electron was imaged using
photoionization microscopy (Phys. Rev. Lett. 110, 21300). But are all electrons indistinguishable?
Yet, these experiments do not fully address Goyal’s concerns about the validity of the symmetrization postulate on a fundamental level. According to quantum mechanics, before they are observed, the exact location and state of quantum particles cannot be specified with precision. Instead, a particle, such as an electron is described mathematically by a "wavefunction"—an expression that captures the range of properties that can be attributed to the particle. The symmetrization postulate provides a detailed mathematical prescription for combining the wavefunctions of two electrons in Schrödinger’s equation. "It’s not clear how you get that mathematical rule from the bare statement that two electrons are indistinguishable and identical," Goyal says.

Dirac did present a logical argument for such a derivation in his original 1926 work on the subject; however, a flaw in his work was pointed out to him soon afterwards. Many others have since tried to bridge the gap between the abstract idea of indistinguishability and the symmetrization postulate. One of Goyal’s goals is to analyze and carefully critique those arguments. He also hopes to reconcile them as they lead to very different conclusions about the quantum nature of identical particles.

The lack of a rigorous grounding of the symmetrization postulate in fundamental physics opens the door to a tantalizing possibility. This hinges on the different rules needed for combining particles of different types. Electrons are part of a larger family of particles called "fermions." According to the postulate, to combine the wavefunctions of two fermions in order to carry out further quantum calculations, you first swap the two electrons over and subtract the resulting wavefunction from the original one. This is called "anti-symmetrizing" the wavefunction. By contrast, if you add the swapped wavefunctions, then you are "symmetrizing" the wavefunction, which is the process required when describing the family of particles called "bosons" (which includes photons and the famous Higgs particle).

What this means is that without proof that either option isn’t just a cute mathematical trick, you can’t rule out other types of as-yet-undiscovered particle that violate the symmetrization postulate.

Indeed, one of the most influential approaches that attempts to derive the symmetrization postulate from scratch, proposed in 1977 by Leinaas & Myrheim (Il Nuovo Cimento, 37B, N. 1, 1977) predicts that identical particles confined to two dimensions will generically behave as so-called "anyons," hypothetical particles that have yet to be observed.


There have been hints of anyonic behaviour seen in the experimental phenomenon known as the "fractional quantum Hall effect." In these experiments, an electron gas, confined to two dimensions, is subjected to a magnetic field. Under these conditions, the gas appears to behave as though it is made up of a new kind of "quasiparticle" that conducts in such a way that it appears to have a fractional charge. The 1998 Nobel prize in physics was awarded to the physicists who discovered and described this effect.

If we didn’t have the symmetrization
postulate, we wouldn’t understand
basic chemistry or star formation.
- Yasser Omar
But Goyal’s own work calls the relationship between the symmetrization postulate and the proposed anyons into question. Goyal’s main concern about previous attempts to understand the symmetrization postulate is that researchers had to theoretically label the individual electrons to keep track of them. The problem with doing that is that the electrons are supposed to be identical and indistinguishable, so labeling them differently seems to run counter to that initial assumption.

In September 2013, Goyal published a paper in which he describes a way to derive the symmetrization postulate using an information-theoretic approach that does not involve labeling the electrons ( arXiv:1309.0478). Crucially, his derivation implies that identical particles do not generically exhibit anyonic behaviour in two dimensions. Goyal plans to use his FQXi grant to understand how to square this with the conclusion of Leinaas & Myrheim’s 1977 argument.

Yasser Omar at the Instituto de Telecomunicações in Lisbon, Portugal finds Goyal’s work to be "very intuitive and elegant." "I give him great credit for tackling a problem that a lot of people assume is solved when it isn’t," he says. "If we didn’t have the symmetrization postulate, we wouldn’t understand basic chemistry or star formation, so it is great to see a new take on the problem of where it comes from." However, Omar believes that it important to investigate how Goyal’s new work fits with the fractional quantum Hall effect experiments.

"I’m glad this fundamental research is being funded. It really touches the heart of quantum physics," says Guglielmo Tino, a quantum physicist from the University of Florence, Italy. "There have been many attempts to find a good justification and formal treatment" of the symmetrization postulate, "but there is still a lot of room for new ideas."

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