For it must needs be, that there is an opposition in all things. If not so, my first-born in the wilderness, righteousness could not be brought to pass, neither wickedness, neither holiness nor misery, neither good nor bad. Wherefore, all things must needs be a compound in one; wherefore, if it should be one body it must needs remain as dead, having no life neither death, nor corruption nor incorruption, happiness nor misery, neither sense nor insensibility.
– 2 Ne 2:11
If you swap the position of two electrons, you multiply the wave function by -1. If we have two electrons in the same place (remember, they act like waves: it’s like playing two notes at the same time) and we assume those two electrons are spinning the same direction, then we can’t distinguish the state before swapping their positions (since they’re the same position) from the state after—the wavefunction has to stay the same. The only function that stays the same when you multiply it by -1 is the constant function that maps every number to zero. That wavefunction describes a state with no electrons at all! So you can’t have two electrons spinning the same way and in the same place.
This is why there’s such a thing as chemistry. You can only fit two electrons close to a nucleus before all the states are taken: this is why we have two elements, hydrogen and helium, at the top of the periodic table. Out a little farther, there’s room for three more pairs: we get lithium, beryllium, boron, carbon, oxygen, fluorine, and neon on the next row.
Electrons can pair up into “Cooper pairs” when you lower the temperature enough. Swapping electrons two at a time multiplies by -1 twice—that is, it does nothing. So you can have as many Cooper pairs in the same state as you want. Materials in which Cooper pairs form are called superconductors, and Cooper pairs are examples of Bosons.
If matter were only made of bosons, all the electrons would fall into the same state and be “one body”. If such a universe could manage a Big Bang, matter would immediately begin to condense into black holes; there would be no stars or planets, no light except the tremendously dim glow of Hawking radiation. Black holes would join and fuse until the entire universe was a single black hole. When the big black hole died, the universe would consist only of random thermal vibrations.
