Spin and Statistics

The spin-statistics theorem of quantum field theory says that particles with half-odd-integer spin (like the electron) must be fermions, while particles with integer spin (like the photon) must be bosons. Fermions obey Fermi-Dirac statistics, and hence obey the Pauli exclusion principle. Bosons obey Bose-Einstein statistics.

The difference in statistics stems from the properties of the exchange operator. This is a unitary operator, say $P_{\rm ex}$, which represents the effect of exchanging two identical fermions, or two identical bosons. For the fermion case, one has at a critical point in the calculations

$$P_{\rm ex}v = -v$$

and for the boson case,

$$P_{\rm ex}v = v$$

The minus sign for fermions ultimately derives from the double covering of $SO(3)$ via $SU(2)$. Spinors also get into the act. Since I don't fully understand the story myself, this seems like a good place to stop.

© 2001 Michael Weiss

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For more on the spin-statistics theorem, try this.