How to tell if a theory is "local"? Suppose I have a collection of $N$ quantum systems, which I would like to think of as lattice sites. If you tell me that these $N$ sites have some particular embeddings $\vec{x}_i$ in $\mathbb{R}^d$, then what I usually think of as a "local" theory is one in which (at the weakest level) every term in the Hamiltonian has a finite radius of support.
But now imagine that I am not aware of how to embed the lattice into some Euclidean space, I only have knowledge of the intersite couplings $J_{ij}$ (let's say the theory is Gaussian for simplicity). Is there now a way to determine whether my theory is local in some dimension $d$? My first instinct is that this is a sort of weighted graph embedding problem, although it's not clear that the choice of weights can be uniquely decided in a physically meaningful way. I was wondering if there is a simpler answer to this problem, and/or if it is addressed anywhere in the literature.
 A: In my opinion locality is a mathematical property. So for me in your example, it is actually $J_{ij}$ that determines the locality and not any embedding into any physical space per se. The reason for this is simple in your free model: $J_{ij}$ models jumps across sites whatever they represent. If the jumps are independent of the sites, i.e. $J_{ij}=J$ or $J_{ij}=i.i.d.$ etc, then your system is all-to-all and thus zero dimensional (c.f. a gas). Now if $J_{ij}$ is something like $J_{ij}=\delta_{j,j+1}+h.c.$ then your theory has a sense of distance as you could calculate some correlation function and see that it takes (quite generally ballistic, see Lieb-Robinson bound) finite time to propagate. The dimension of this theory is now 1D as for all you know this happens along the sequence $i=1,2,...$. But you could promote $i\rightarrow\vec{i}$ and now this could be higher dimensional again depending on your definition of $J$. However, $J_{i,j,k,l}=\delta_{i+1,i,i,i}+h.c.$ might look higher dimensional but is still 1D (where the multiple indicies are shorthand for higher dimensions and not a many body interaction).
