between two independent events ought to have a dynamical meaning, $diverging$ when a singularity is approached [@DMP]. This fact allows a possible solution of the horizon problem [@DMP], and will play a role in our discussion in the last Section.

Our first task is now to formulate and analyze gauge theories on the model of Quantum Spacetime just described. On ordinary classical spacetime, the gauge group of electromagnetism is the group of (regular) functions from Minkowski spacetime $\bR^4$ to $U(1)$, which can be regarded as (a subgroup of) the group of unitaries of the algebra $C_b(\bR^4) = M(C_0(\bR^4))$. Going to Quantum Spacetime, this should be naturally replaced by $\cG = \cU(M(\cE))$, the unitaries of the multipliers of the Quantum Spacetime algebra $\cE$. It is therefore a rather interesting possibility that the gauge group of electromagnetism could also act nontrivially on a real scalar field $\varphi(q)$ on QST, as $$\label{eq:gaugephi}
\varphi(q) \to U\varphi(q) U^*, \qquad U \in \cG.$$ Of course on commutative spacetime the above action is instead trivial, because $U$ and $\varphi$ commute.

In order to find a Lagrangian invariant under the above action, we should introduce a covariant derivative $D_\mu$, i.e., a derivation on $\cE$ such that, under the 