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Additional info for Analytic Properties of Feynman Diagrams in Quantum Field Theory
Note, by the way, that since (~by )2 = 0, the state cannot be lled twice. More generally, the anticommutation relations imply that any multiparticle state is antisymmetric under the interchange of two particles: ayp ayq j0i = ;ayqayp j0i. Thus we conclude that if the ladder operators obey anticommutation relations, the corresponding particles obey Fermi-Dirac statistics. We have just shown that in order to insure that the vacuum has only positive-energy excitations, we must quantize the Dirac eld with anticommutation relations under these conditions the particles associated with the Dirac eld obey Fermi-Dirac statistics.
But we haven't proved yet that this \spin" has anything to do with angular momentum. To do this, we must write down the angular momentum operator. 2 by looking for the conserved quantity associated with translational invariance. We can nd the angular momentum operator in a similar way as a consequence of rotational invariance. Under a rotation (or any Lorentz transformation), the Dirac eld transforms (in our original convention) according to (x) ! 0 (x) = 12 ( 1 x): To apply Noether's theorem we must compute the change in the eld at a xed point, that is, = 0 (x) ; (x) = 12 ( 1 x) ; (x): Consider for de niteness an in nitesimal rotation of coordinates by an angle about the z -axis.
Thus we conclude that if the ladder operators obey anticommutation relations, the corresponding particles obey Fermi-Dirac statistics. We have just shown that in order to insure that the vacuum has only positive-energy excitations, we must quantize the Dirac eld with anticommutation relations under these conditions the particles associated with the Dirac eld obey Fermi-Dirac statistics. This conclusion is part of a more gen- 58 Chapter 3 The Dirac Field eral result, rst derived by Pauli*: Lorentz invariance, positive energies, positive norms, and causality together imply that particles of integer spin obey Bose-Einstein statistics, while particles of half-odd-integer spin obey FermiDirac statistics.