8.3 Global Symmetrization

When computing, say a hydrogen molecule, it is all nice and well to say that the wave function must be antisymmetric with respect to exchange of the two electrons 1 and 2, so the spin state of the molecule must be the singlet one. But what about, say, electron 3 in figure 8.1, which can with 50% chance be found on Mars and otherwise on Venus? Should not the wave function also be antisymmetric, for example, with respect to exchange of this electron 3 in one of two places in space with electron 1 on the hydrogen molecule on Earth? And would this not locate electron 3 in space also in part on the hydrogen molecule, and electron 1 also partly in space?

The answer is: absolutely. Nature treats all electrons as one big connected bunch. The given solution for the hydrogen molecule is not correct; it should have included every electron in the universe, not just two of them. Every electron in the universe is just as much present on this single hydrogen molecule as the assumed two.

From the difficulty in describing the 33 electrons of the arsenic atom, imagine having to describe all electrons in the universe at the same time! If the universe is truly flat, this number would not even be finite. Fortunately, it turns out that the observed quantities can be correctly predicted pretending there are only two electrons involved. Antisymmetrization with far-away electrons does not change the properties of the local solution.

If you are thinking that more advanced quantum theories will eventually do away with the preposterous notion that all electrons are present everywhere, do not be too confident. As mentioned in addendum {A.15.1}, the idea has become a fundamental tenet in quantum field theory.