8.4 A story by Wheeler

Consider a simple question. Why are all electrons so absolutely equal? Would it not be a lot less boring if they had a range of masses and charges? As in “I found a really big electron this morning, with an unbelievable charge!” It does not happen.

And it is in fact far, far, worse than that. In quantum mechanics electrons are absolutely identical. If you really write the correct (classical) wave function for an hydrogen atom following the rules of quantum mechanics, then in principle you must include every electron in the universe as being present, in part, on the atom. Electrons are so equal that one cannot be present on a hydrogen atom unless every electron in the universe is.

There is a simple explanation that the famous physicist Wheeler gave to his talented graduate student Richard Feynman. In Feynman’s words:

“As a by-product of this same view, I received a telephone call one day at the graduate college at Princeton from Professor Wheeler, in which he said, ‘Feynman, I know why all electrons have the same charge and the same mass’ Why? Because, they are all the same electron! And, then he explained on the telephone, ...” [Richard P. Feynman (1965) Nobel prize lecture. [[5]]]

Figure 8.5: The space-time diagram of Wheeler’s single electron.

What Professor Wheeler explained on the phone is sketched in the space-time diagram figure 8.5. The world-line of the only electron there is is constantly traveling back and forwards between the past and the future. At any given time, like today, this single electron can be observed at countless different locations. At the locations where the electron is traveling to the future it behaves like a normal electron. And Wheeler recognized that where the electron is traveling towards the past, it behaves like a positively charged electron, called a positron. The mystery of all those countless identical electrons was explained.

What had Feynman to say about that!? Again in his words:

But, Professor, I said, ‘there aren't as many positrons as electrons.’ ‘Well, maybe they are hidden in the protons or something,’ he said. I did not take the idea that all the electrons were the same one from him as seriously as I took the observation that positrons could simply be represented as electrons going from the future to the past in a back section of their world lines. That, I stole!” [Richard P. Feynman (1965) Nobel prize lecture. [[5]]]

And there are other problems, like that electrons can be created or destroyed in weak interactions.

But without doubt, if this was art instead of quantum mechanics, Wheeler’s proposal would be considered one of the greatest works of all time. It is stunning in both its utter simplicity and its inconceivable scope.

There is a place for esthetics in quantum mechanics, as the Dirac equation illustrates. Therefore this section will take a very biased look at whether the idea is really so truly inconceivable as it might appear. To do so, only positrons and electrons will be considered, with their attendant photons. Shape-shifting electrons are a major additional complication. And recall classical mechanics. Some of the most esthetical results of classical mechanics are the laws of conservation of energy and momentum. Relativity and then quantum mechanics eventually found that classical mechanics is fundamentally completely wrong. But did conservation of energy and momentum disappear? Quite the contrary. They took on an even deeper and more esthetically gratifying role in those theories.

With other particles shoved out of the way, the obvious question is the one of Feynman. Where are all the positrons? One idea is that they ended up in some other part of space. But that seems to be hard to reconcile with the fact that space seems quite similar in all directions. The positrons will still have to be around us. So why do we not see them? Recall that the model considered here has no protons for positrons to hide in.

Obviously, if the positrons have nowhere to hide, they must be in plain view. That seems theoretically possible if it is assumed that the positron quantum wave functions are delocalized on a gigantic scale. Note that astronomy is short of a large amount of mass in the universe one way or the other. Delocalized antimatter to the tune of the visible matter would be just a drop in the bucket.

A bit of mathematical trickery called the Cauchy-Schwartz inequality can be used to illustrate the idea. Consider a universe of volume ${\cal V}$. For simplicity, assume that there is just one electron and one positron in this universe. More does not seem to make a fundamental difference, at least not in a simplistic model. The electron has wave function $\psi_1$ and the positron $\psi_2$. The Cauchy-Schwartz inequality says that:

\left\vert\int_{\cal V}\psi_1^*\psi_2{\,\rm d}^2{\skew0\ve...
...r}\int_{\cal V}\vert\psi_2\vert^2{\,\rm d}^2{\skew0\vec r}= 1

Take the left hand side as representative for the interaction rate between electrons and positrons. Then if the wave functions of both electrons and positrons are completely delocalized, the interaction rate is 1. However, if only the positrons are completely delocalized, it is much smaller. Suppose the electron is localized within a volume $\varepsilon{\cal V}$ with $\varepsilon$ a very small number. Then the interaction rate is reduced from 1 to $\varepsilon$. If both electrons and positrons are localized within volumes of size $\varepsilon{\cal V}$ it gets messier. If the electron and positron move completely randomly and quickly through the volume, the average interaction rate would still be $\varepsilon$. But electrons and positrons attract each other through their electric charges, and on a large scale also through gravity. That could increase the interaction rate greatly.

The obvious next question is then, how come that positrons are delocalized and electrons are not? The simple answer to that is: because electrons come to us from the compact Big Bang stages of the universe. The positrons come to us from the final stages of the evolution of the universe where it has expanded beyond limit.

Unfortunately, that answer, while simple, is not satisfactory. Motion in quantum mechanics is essentially time reversible. And that means that you should be able to explain the evolution of both electrons and positrons coming out of the initial Big Bang universe. Going forward in time.

A more reasonable idea is that the other options do not produce stable situations. Consider a localized positron in an early universe that by random chance happens to have more localized electrons than positrons. Because of attraction effects, such a positron is likely to find a localized electron to annihilate with. That is one less localized positron out of an already reduced population. A delocalized positron could interact similarly with a delocalized electron, but there are less of these. The reverse situation holds for electrons. So you could imagine a runaway process where the positron population evolves to delocalized states and the electrons to localized ones.

Another way to look at it is to consider how wave functions get localized in the first place. The wave function of a localized isolated particle wants to disperse out over time. Cosmic expansion would only add to that. In the orthodox view, particles get localized because they are measured. The basics of this process, as described by another graduate student of Wheeler, Everett III, are in section 8.6. Unfortunately, the process remains poorly understood. But suppose, say, that matter localizes matter but delocalizes antimatter, and vice-versa. In that case a slight dominance of matter over antimatter could conceivably lead to a run-away situation where the matter gets localized and the antimatter delocalized.

Among all the exotic sources that have been proposed for the dark matter in the universe, delocalized antimatter does not seem to get mentioned. So probably someone has already solidly shown that it is impossible.

But that does not invalidate Wheeler’s basic idea, of course. As Wheeler himself suggested, the positrons could in fact be hiding inside the protons through the weak-force mechanism. Then of course, you need to explain how the positrons came to be hiding inside the protons. Why not the electrons inside the antiprotons? That would be messier, but it does not mean it could not be true. In fact, it is one of the surprises of advanced particle physics that the entire lepton-quark family seems to be one inseparable multi-component particle, [27, p. 210]. It seems only fair to say that Wheeler’s idea predicted this. For clearly, the electron could not maintain its unmutable identity if repeatedly changed into particles with a separate and independent identity. So Wheeler’s idea may not be so crazy after all, looking at the facts. It provides a real explanation why identical particles are so perfectly identical. And it predicted something that would only be observed well into the future.

Still, the bottom line remains the beauty of the idea. As the mathematician Weyl noted, unfazed after Einstein shot down an idea of his:

“When there is a conflict between beauty and truth, I choose beauty.”