Estimates, renormalized currents, and field equations for the Yukawa₂ field theory

We obtain mathematically rigorous field equations for the Yukawa quantum field theory in two-dimensional space-time. The renormalized boson current j = -λ:ψψ: + δm²φ{symbol} and the fermion current J = λψφ{symbol} are defined as bilinear forms continuous in space and time. As bilinear forms the boson field is shown to be twice continuously differentiable and the fermion field once continuously differentiable. The identities (□ + m²)φ{symbol} = j and [γ₀( ∂ ∂t) + γ₁( ∂ ∂x) + M]ψ = J are shown to hold. As distributions the terms in these equations define operators independent of all cutoffs. The results depend on detailed estimates of the renormalization cancellations in the theory. These estimates have other applications, one of which is a proof of the quadratic estimate dominating the fractional energy operator by the Hamiltonian.