## Is there only one electron in the Universe?

Posted in The Universe and Stuff with tags , , , , , , , , on February 1, 2012 by telescoper

I started teaching Nuclear and Particle Physics to the 3rd year Physics students today. I decided to warm up with a few basics about elementary particles and their properties – all pretty standard stuff and no hairy mathematics. Cue pretty picture:

This doesn’t show the whole picture, of course, because for every particle there is an antiparticle, so there are antiquarks and antileptons. The existence of these was first suggested by Paul Dirac in 1928 based on his investigations into relativistic quantum theory, basically because invariance of special relativity is compatible with the existence of both positive and negative energy states, i.e.

$E^2 = p^2c^2 +m^2 c^4$

has two sets of solutions, one with $E>0$ and the other with $E<0$. Instead of simply assuming the latter set were physically unrealistic, Dirac postulated that they might be real, but completely filled in “empty” space; these filled negative-energy states are usually called the “Dirac Sea”. Injection of an appropriate amount of energy can promote something from a negative state into a positive one, leaving behind a kind of hole (very similar to what  happens in the case of semiconductor). This process creates a pair consisting of a (positive energy) particle and a (negative energy) antiparticle (i.e. a hole in the Dirac Sea). In the case of electrons, the hole is called a positron.

The alternative, and even wackier, explanation of antimatter I usually mention in these lectures derives, I think, from Feynam who noted that in quantum (wave) mechanics the time evolution of particles involves things like

$\exp(i\omega t)=\exp(i Et/\hbar),$

which have the property that changing $E$ into $-E$ has the same effect as changing $t$ into $-t$. This is, in essence, the reason why, in Feynman diagrams, antiparticles are usually represented as particles travelling backwards in time…

This is a useful convention from the point-of-view of using such diagrams in calculations, but it allows one also to raise the wacky bar to a higher level still, to a suggestion that, coincidentally, was  doing the rounds very recently – namely whether it is possible that there may really be only one electron in the entire Universe:

….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, “suppose that the world lines which we were ordinarily considering before in time and space—instead of only going up in time were a tremendous knot, and then, when we cut through the knot, by the plane corresponding to a fixed time, we would see many, many world lines and that would represent many electrons, except for one thing. If in one section this is an ordinary electron world line, in the section in which it reversed itself and is coming back from the future we have the wrong sign to the proper time—to the proper four velocities—and that’s equivalent to changing the sign of the charge, and, therefore, that part of a path would act like a positron.”
—Feynman, Richard, Nobel Lecture December 11, 1965

In other words, a single electron can appear in many different places simultaneously if it is allowed to travel backwards and forwards in time…

I think this is a brilliant idea, especially if you like science fiction stories, but there’s a tiny problem with it in terms of science fact. In order for it to work there should be as many positrons in the Universe as there are electrons. Where are they?