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If it is true that an electron can be anywhere in the cosmos at any given time, then is it even theoretically possible that there is only one electron, instead of multiple electrons in the cosmos? If it isn't, I'd appreciate it if someone could point out where I may have misunderstood something as far as the fundamental properties of electrons are concerned.

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If it is true that an electron can be anywhere in the cosmos at any given time,

No it is not a true statement. The true quantum mechanical statement is that

there exists a probability that an electron can be anywhere in the cosmos at any given time

This probability is infinitesimally small, and thus the probability that atoms could evolve is practically zero. So it is not even theoretically possible . Your misunderstanding comes from ignoring probabilities coming from the quantum mechanical basic framework.

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It is improbable that there is only 1 electron since there are many protons that require several electrons to neutralize the charge on the molecules.

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    $\begingroup$ See this: One Electron Universe $\endgroup$ – Gigi Butbaia May 6 '14 at 4:28
  • $\begingroup$ @GigiButbaia That's his theory, I have mine. $\endgroup$ – LDC3 May 6 '14 at 4:59
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The idea that there could be just one electron originated when John Wheeler pointed out that a positron is the same as an electron moving backwards in time. All the electrons and positrons could be a single electron zigzagging backwards and forwards in time.

However this idea preceded the development of quantum electrodynamics, and QED completely changed the way we think about electrons. In QED there is an electron quantum field that permeates all of spacetime, and electrons are just excitations in this field. If we add a quantum of energy to the electron field it appears as a particle, that is an electron. Add a second quantum of energy and we have two electrons, and so on. Likewise, take a quantum of energy out of the field and an electron disappears.

This process allows electrons, and other particles, to be created and destroyed. For example every time a carbon 14 nucleus decays it creates an electron (and an anti-neutrino that is required to conserve lepton number) and the number of electrons in the universe increases by one. Conversely every time a magnesium 23 nucleus decays it creates a positron, and that positron can annihilate an electron and reduce the number of electrons in the universe by one.

So the fundamental object is now the electron quantum field rather than the electrons, which are just excitations in it. It no longer makes sense to argue there might be just a single electron, because the number of electrons is not a conserved quantity.

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