Field-strength formulation of gauge theories: Transformation of the functional integral

We present a formulation of gauge field theories in which the gauge potentials A_μ(x) are eliminated in a simple way in terms of the field strengths F_μν(x). Our results are closely related to, but are much simpler than, Halpern's dual variable formulation of gauge theories in the axial gauge. We work in the coordinate gauge x^μA_μ(x)=0, and show both analytically and geometrically that the potential A_μ can be determined uniquely from the field strengths F_μν for a suitable class of F's, A→A[F]. We show, furthermore, that a tensor F_μν(x) is a coordinate-gauge field tensor if and only if it satisfies the restricted set of Bianchi identities ε^μνσλx_σD^ρ[F]*F_ρλ=0, D=∂+[A[F], ·]. These results permit us to transform the functional integral for the vacuum-to-vacuum amplitude $Z[J]$ for the gauge theory to a form in which the potentials are completely eliminated in terms of the field strengths. When the Bianchi constraints are eliminated using a set of Lagrange multiplier fields λ_σ(x), the F's can be integrated out completely to obtain a form of the theory which appears to be useful for strong coupling.