Spin Statistics Theorem

Systems of identical particles whose state kets are totally symmetric with respect to particle interchange are said to
obey *Bose-Einstein statistics* [14,39]. Moreover, such particles are termed *bosons*. On the other hand,
systems of identical particles whose state kets are totally antisymmetric with respect to particle interchange are said to
obey *Fermi-Dirac statistics* [43,28], and the constituent particles are called *fermions*. In non-relativistic quantum mechanics, the rule that all integer-spin particles are bosons, whereas all half-integer spin
particles are fermions, must be accepted as an empirical fact. However, in relativistic quantum mechanics, it is possible
to formulate reasonably convincing arguments that half-integer-spin particles cannot be bosons, and integer-spin particles cannot be fermions [47,81].
Incidentally, electrons, protons, and neutrons are all fermions.

The *Pauli exclusion principle* [79] is an immediate consequence of the fact that electrons obey Fermi-Dirac statistics. This principle
states that no two electrons in a multi-electron system can possess identical sets of observable eigenvalues. For instance,
in the case of a three-electron system, the state ket is

(9.42) |

[See Equation (9.41).] Note, however, that

(9.43) |

In other words, if two of the electrons in the system possess the same set of observable eigenvalues then the state ket becomes the null ket, which corresponds to the absence of a state.