5.11 Summary
Massless spacetime vector fields (“gauge fields”) {A k} 3k=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) 3a=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.”
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Bibliography
C. Becchi, A. Rouet, R. Stora, Ann. of Phys. 98 (1976), 287.
T. Kugo, I. Ojima, Progr. of Theor. Phys. 60 (1978), 1869.
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(2006). Massless Quantum Fields. In: Operational Quantum Theory II. Operational Physics. Springer, New York, NY . https://doi.org/10.1007/0-387-34644-9_6
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