Fermions: a tale of two minus signs

Professor Susskind presents the quantum mechanics of multi-particle systems, and demonstrates that fermions and bosons are distinguished by the two possible solutions to the wave function equation when two particles are swapped.  When two particles are swapped, the boson wave function equation has a phase factor of +1 whereas the fermion equations has a phase factor of -1.  For fermions, this results in a wave function with zero probability for two particles to be in the same state, thus demonstrating the exclusion principle.  On the other hand, bosons prefer to be in the same state.  This is what makes a photon (boson) laser possible, but an electron (fermion) laser impossible.

The spin variable is required to allow two electrons to occupy the same state in an atom.  Electrons are fermions which have half-integer spins.  This implies that a rotation of the angular momentum by 2π will result in a phase change by -1.  This implies that the identity operation for fermions is not a rotation by 2π, but rather a rotation by 4π, and that a rotation by 2π can be offset or canceled by a swap of two particles.  This is the tale of 2 minus signs.