Massless Quantum Fields

5.11 Summary

Massless spacetime vector fields (“gauge fields”) {A ^k} ³_k=0 are acted on by the 4-dimensional Minkowski representation of SO ₀(1, 3), like the spacetime translations. They realize, together with the field strengths {F _jk} in the real 6-dimensional adjoint representation, the two fundamental representations of the Lorentz group. Duality pairing for a quantum theory requires a scalar field (“gauge fixing” field) S to complete four (4 = 3 + 1) dual pairs (F _a0,A ^a) ³_a=1 and (S, A ⁰).

The translation representations acting on the four components of the gauge field are in the indefinite unitary group U(1, 3) ⊃ U(1, 1) ×U(2) as supgroup of the indefinite metric Lorentz group SO ₀(1, 3). The Minkowski metric shows up in the indefinite signature (1, 3) metric for the gauge field inner product space. A projection to a probability interpretable vector subspace with the two particle degrees of freedom for left and right circularly polarized photons requires the transition to translation eigenvectors that are determined by a trivial action of the nilpotent part of the dynamics. To define a nilquadratic projection (Becchi-Rouet-Stora transformation) in the quantum algebra, the Bose type gauge fields (A ⁰, S) have to be paired with Lorentz scalar fields (β, γ) of Fermi type (Fadeev-Popov fields). They have no particle degrees of freedom. Translation eigenvectors have trivial Becchi-Rouet-Stora charge; they are “gauge invariant.”