The quantum field theory of free Dirac particles (four‐component massive spin‐ 1/2 particles) is ‘‘derived’’ by a Segal quantization procedure. First, details are given on how the spinor space of Dirac is actually a minimal left ideal of the Clifford algebra associated with a Lorentz inner product space (+,−,−,−), and how the homogeneous group actions break the natural two‐component quaternion structure to give familiar four‐component complex spinors. Second, Wigner’s procedure for constructing unitary representations of the Poincaré group is used to construct the appropriately induced infinite‐dimensional representation of the inhomogeneous group starting from the above four‐dimensional nonunitary representation. Third, and finally, Segal’s procedure for quantizing classical Fermion fields is adapted to this infinite‐dimensional Hilbert space to obtain the Clifford algebra of annihilation–creation operators for spin‐ 1/2 particles. The familiar Fock space appears as a minimal left ideal in this second Clifford algebra.
關聯:
Journal of mathematical physics 31(9), pp.2192-2200
