A force with no force-carrier
We have treated the plus-or-minus sign as bookkeeping — a rule about how to write things down. But it has muscle. The remarkable fact is that the symmetry of identical particles makes them behave as if a force pulled them together or pushed them apart, even when no actual force passes between them. There is no push, no pull, no invisible spring — only the requirement that their joint description be symmetric or antisymmetric. Yet the energy of the system comes out different depending on which it is, and a difference in energy with position is exactly what we call a force. This phantom is the exchange interaction, and it is one of the most genuinely quantum effects there is, with no classical cousin at all.
Why the sign moves the particles
The mechanism is easier than it sounds. Recall that an antisymmetric description must vanish whenever the two particles sit in the same state — and that includes sitting at the same place. So two fermions described antisymmetrically are, in effect, kept from being found right on top of each other; their cloud thins out where they would overlap. A symmetric description does the opposite: it actually swells where the two particles coincide, making them likelier to be found close together. The particles literally arrange their probability differently depending on the sign — antisymmetric hollows out, symmetric piles up.
Now bring in the fact that like charges repel. Two electrons, both negative, cost more energy the closer they huddle. The symmetric arrangement, which lets them bunch up, runs up a bigger repulsion bill; the antisymmetric arrangement, which keeps them apart, pays less. The energy difference between the two is real, and it depends on how the spins are oriented — because the spin part and the spatial part of an electron pair are locked together to keep the total antisymmetric. Tilt the spins one way and the electrons are forced apart; tilt them the other way and they crowd in. Spin, through the sign rule, ends up steering the electrons' positions and the energy they carry.
Where magnetism comes from
This spin-dependent energy is the secret engine of magnetism. Each electron is itself a tiny magnet, its north-south axis set by its spin. The puzzle of permanent magnets is why, in iron, vast numbers of these tiny magnets choose to point the same way, since that alignment seems to cost energy. The exchange interaction answers it. In the right material, lining the spins up actually lowers the total energy — because the spin arrangement that aligns the little magnets is precisely the one that lets the repelling electrons keep their distance, saving more energy than the alignment costs. The electrons align their spins not out of magnetic friendliness but to dodge each other's electric repulsion cheaply. A bar magnet's pull, then, is exchange symmetry made visible on the kitchen fridge.
The same logic operates inside a single atom. When several electrons share an outer shell, they minimize their mutual repulsion by spreading out into separate orbitals with their spins aligned — a tendency captured by Hund's rule. It is the same bargain as in a magnet, just within one atom: align the spins so the electrons can stay apart and save energy. From the atom to the fridge magnet, one principle is at work.