Scalar formalism for non-Abelian gauge theory

Abstract

The gauge field theory of an N-dimensional multiplet of spin- 1/2 particles is investigated using the Klein–Gordon-type wave equation {Π ⋅ (1+iσ) ⋅ Π+m2}Φ=0, Πμ≡∂/∂ixμ−eAμ, investigated before by a number of authors, to describe the fermions. Here Φ is a 2×1 Pauli spinor, and σ repesents a Lorentz spin tensor whose components σμν are ordinary 2×2 Pauli spin matrices. Feynman rules for the scalar formalism for non-Abelian gauge theory are derived starting from the conventional field theory of the multiplet and converting it to the new description. The equivalence of the new and the old formalism for arbitrary radiative processes is thereby established. The conversion to the scalar formalism is accomplished in a novel way by working in terms of the path integral representation of the generating functional of the vacuum τ-functions, τ(2,1, ⋅⋅⋅ 3 ⋅⋅⋅)≡〈0−‖T(Ψin(2) Ψ̄in(1) ⋅⋅⋅ Aμ(3)in ⋅⋅⋅ S)‖0−〉, where Ψin is a Heisenberg operator belonging to a 4N×1 Dirac wave function of the multiplet. The Feynman rules obtained generalize earlier results for the Abelian case of quantum electrodynamics.