the ordinary or commutative fields $\phi_0$ as $$\phi_\theta(x) = \phi_0(x) e^{\frac{1}{2}\overleftarrow{\partial} \wedge P},
\quad \overleftarrow{\partial} \wedge P \equiv
\overleftarrow{\partial}_\mu \theta^{\mu\nu} P_\nu
\label{twistphi}$$

As ${\cal A}_\theta({\mathbb R}^N)$ is a module for ${\cal
  A}_0({\mathbb R}^N)$, we require that the covariant derivative respects this property. In addition, we also require that in quantum theory, the covariant derivative preserves statistics, and also Poincaré and gauge symmetries. The only possibility that satisfies these requirements is $$D_\mu \phi_\theta = ((D_\mu)_0 \phi_0)e^{\frac{1}{2} \overleftarrow{\partial}
  \wedge P} .
\label{twistD}$$ where $(D_\mu)_0 = \partial_\mu + (A_\mu)_0$ and $(A_\mu)_0$ is the commutative gauge field, depending on $\hat{x}^c$ only.

The commutator of two covariant derivatives gives us the curvature: $$[D_\mu , D_\nu]\phi = ([(D_\mu)_0, (D_\nu)_0])e^{\frac{1}{2}
  \overleftarrow{\partial} \wedge P} = [(F_{\mu\nu})_0 \phi_0]e^{\frac{1}{2}
  \overleftarrow{\partial} \wedge P}.$$

The field strength $(F_{\mu\nu})_0$ transforms correctly (i.e. covariantly) under gauge transformations, so we can use it to construct the Hamiltonian of the quantum gauge theory. Pure gauge theories on the GM plane are thus identical to their commutative counterparts.

Below we will outline an approach to scattering theory of twisted fields based on the LSZ formalism. In that approach, (\[twistphi\]) and (\[twistD\]) are true in the fully interacting case as well. Thus these equations are valid with $P_\mu$ 