The typical definition of a causal net of observables in quantum theory is to consider, for the case of a (globally hyperbolic) spacetime $M$, the category of open sets $O(M)$ ordered by inclusion, in which case the causally local net of observables is a functor
\begin{equation} \mathfrak{A} : O(M) \to \mathrm{Alg} \end{equation}
from the open sets of the spacetime to some $C^*$-algebra, obeying the axioms of isotony ($O_1 \subseteq O_2 \to \mathfrak{A}(O_1) \subseteq \mathfrak{A}(O_2)$), causality (if $O_1, O_2$ are spacelike separated, then $[\mathfrak{A}(O_1), \mathfrak{A}(O_2)] = 0$), Poincaré covariance (there's some strongly continuous unitary representation of the Poincaré group such that $\mathfrak{A}(gO) = U(g) \mathfrak{A}(O) U^*(g)$), spectrum condition (momentum is contained in the forward light cone) and the existence of a vacuum state.
Most of these do not seem to depend too strongly on the exact nature of the kinematic group, as we could substitute some other group for the Poincaré group, however the causality axiom poses some issue.
If we consider the classical spacetime, given by some Galilean structure on a manifold, the light cone is degenerate in that it has an angle of $\pi$ and the spacelike region is some lower dimensional object, meaning that there is in fact no two open sets on it which are spacelike separated. For any two open sets, there always exists a timelike curve connecting the two.
I know that there are some analogous methods to consider non-relativistic QM where we only consider the spacelike manifold for this purpose, but is there a more general method that would work on both types of kinematic structures?
