I've recently heard of Wheeler's one-electron universe idea. Personally, I find it very beautiful and elegant, but I'm aware that there are any number of problems with it.

However, I'm curious why it's restricted to electrons. Surely, an antiproton looks the same as a time-reversed proton, and all protons look the same. Also, it would be a bit weird to have a universe with just one electron but plenty of protons. So, why not a one-proton universe?

Has this idea ever been seriously contemplated? If not, then what is fundamentally different about the two?

(If the key problem is that a proton consists of quarks, then how about a one-(top quark) universe or one-muon universe?)

A one-particle universe idea would work with stable, color neutral elementary particles with a strict number conservation. No such particles exist. Electrons can turn to neutrinos via a W boson and disappear from the world line. Quarks exchange colors, and are no longer the same after that. Protons are not elementary particles and can become neutrons inside a nucleus in residual strong interactions mediated by pions. Muons and top quarks decay. The main point in Wheeler's idea was not a one-particle universe, but the fact that positrons can be viewed as electrons moving back in time. However, there is not enough positrons in the universe for the one-electron universe idea to work.

• there is not enough positrons in the universe - maybe they are hidden behind the cosmic horizon... – Christoph Oct 9 '17 at 6:43
• @Christoph If you follow FLRW that produces the horizon, then it requires the universe to be uniform. If they are not here - they are not anywhere. If you follow some different model, say, Mielne, then there is no horizon in the sense that all universe is observable. Well, Mielne is a special case of FLRW, but you know what I mean. – safesphere Oct 9 '17 at 7:51
• What we require is approximate uniformity of energy-momentum, and wasn't the early universe dominated by neutrinos and photons anyway? So I don't think this possibility can be immediately excluded without an argument that goes a bit deeper... – Christoph Oct 9 '17 at 12:03
• @Christoph Would positrons hidden behind the horizon be able to participate in the one-electron universe that assumes a single timelike world line for all electrons and positrons? Can matter cross the cosmic horizon? If so, then we either would have a massive invasion of anti-matter annihilating the visible universe or otherwise would have to take a trip to beyond to be annihilated there. Occam probably wouldn't approve :) – safesphere Oct 9 '17 at 17:50
• Can matter cross the cosmic horizon? Sure, but only in one direction – Christoph Oct 9 '17 at 21:21

In quantum systems, (especially quantum many-body systems), the fermionic system possesses fermionic number parity $\mathbb{Z}_2^f$ conservation (mod 2) and the fermionic number parity $\mathbb{Z}_2^f$ symmetry.

Say, the systems of any fermions (electrons, protons, quarks) obey the symmetry invariance: $$\psi_f \to -\psi_f,$$ for the ground state $|\Psi \rangle$ and the Hamiltonian $H$, so $$|\Psi \rangle \to |\Psi \rangle,$$ $$H \to H.$$

It looks to me that a closed quantum universe (or a closed quantum system) that contains only a single (or odd number of) fermions (e.g. electron, proton, quark) encounters the problem of the fermionic number parity $\mathbb{Z}_2^f$ symmetry is violated.

The global symmetry like this may be a strong constraint for a closed quantum system.