action of $\cG$, $$D_\mu\varphi(q) \to UD_\mu\varphi(q)U^*.$$ This is accomplished by defining $$\label{eq:covariantderivative}
D_\mu\varphi(q) := \partial_\mu \varphi(q) -ie[A_\mu(q),\varphi(q)],$$ where $e$ is the electron charge (see below for a discussion of this choice) which describes the coupling with the gauge field, $\partial_\mu$ is the derivation on $\cE$ defined by $$\partial_\mu \varphi(q) = \frac{\partial}{\partial a_\mu} \varphi(q+a {\mathbbm{1}})|_{a=0},$$ and $A_\mu$ is the electromagnetic potential on QST, on which $\cG$ is assumed to act as $$\label{eq:gaugeA}
A_\mu(q) \to UA_\mu(q)U^* +ie^{-1}U\partial_\mu U^*,$$ which reduces to the ordinary gauge transformation on commutative spacetime by writing $U = e^{i\Lambda}$. This also explains the choice of $e$ in , as the coupling constant between the electromagnetic potential and the neutral field $\varphi$. In fact, $A_\mu$ will also interact with the electron field $\psi$, which transforms as $\psi(q) \to U\psi(q)$, and therefore the choice $D_\mu\psi(q) = \partial_\mu\psi(q)-ieA_\mu(q)\psi(q)$ for its covariant derivative gives the correct interaction. A potential problem in this respect is represented by the fact that it seems difficult to write the interaction, on Quantum Spacetime, of $A_\mu$ with a field of charge different from $0,\pm e$ (like the quark fields). For a discussion in the framework of formal \*-products and the Seiberg-Witten map see  [@CJSWW].

The fact that  