system
Given the following post, respond as if you're commenting on Reddit:
human
Why does Pauli's Exclusion Principle exist?

I get that it doesn't allow fermions like electrons and quarks to get cramed together past a certain point, but why?
gpt
It’s a fundamental fact of nature that particles of the same type are exactly identical to each other.

This implies that probability densities for multiple particles must be invariant under exchange of any two identical particles.

This implies that the multi-particle state vector can only change by a phase under exchange of any two identical particles.

We define bosons to be particles such that this phase is 1, and fermions to be particles such that this phase is -1.

Then the spin-statistics theorem is what relates these two types of particles to certain values of their spins.

It’s easy to show that for particles where the phase is -1 (fermions), the state vector can only be identically equal to zero if any two of the identical particles are placed in the same state. Therefore there does not exist any valid state vector in the multi-particle Hilbert space where two identical fermions occupy the same state.

That just comes directly from the definition of what a fermion is, and how the state vector has to transform under permutation symmetry (because particles are identical).

Why are “matter particles” (quarks and electrons) fermions rather than bosons? That’s just how it is in our universe.