# Are all physically realistic Hamiltonians local?

My understanding of modern physics is that physicists think that, fundamentally, physical laws are local. For system A to interact with system B, they either need to be very close to each other or some signal needs to propagate between them.

A Hamiltonian $$H$$ on a Hilbert space $$\mathcal{H}_1\otimes\dots\otimes\mathcal{H}_n$$ is $$k$$-local if it can be written as a sum of tensor products $$H_1\otimes\dots \otimes H_n$$, where at most $$k$$ of the single-system Hamiltonians $$H_i$$ are not the identity. Physically, a single tensor product term like this seems to represent an actual physical interaction.

Does the first sense of locality imply some degree of locality in the second sense? The converse is not true (I could write down some $$2$$-local Hamiltonian with interaction terms between systems at a huge spatial distance), but it seems like the first concept ought to limit the number of simultaneous interactions in a single term in a Hamiltonian. Does that then imply that all Hamiltonians must be $$k$$-local for some $$k$$?

All the physically-motivated Hamiltonians I've seen are 2-local. But I've seen theoretical constructions that aren't like this. I can write down an arbitrary Hamiltonian $$H=X\otimes X\otimes \dots X$$ (acting on $$n$$ 2-level quantum systems) with any number $$n$$ of $$X$$ Paulis tensored together, and this represents some simultaneous interaction of those $$n$$ quantum systems.

• To clarify: Are you asking if locality in spacetime implies some degree of $k$-locality? If locality in spacetime and Poincaré symmetry are both required, then $k$-locality may be related to renormalizability (or more generally to universality). Are you familiar with those concepts? Nov 18 '21 at 14:21
• If spacetime locality implies a bound on $k$-locality, that would answer my question (although any arguments against spacetime locality, or a different bound on $k$-locality, would also answer it). I am not familiar with either renormalizability or universality. Nov 18 '21 at 14:25
• I don't know if I'm fully understanding the question, but surely this will depend on what the different "parties" -- Hilbert spaces -- represent? If the spaces represent different degrees of freedom of a single particle, then the "nonlocality" of the Hamiltonian might not correspond to spatial nonlocality. On the other hand, if the spaces are being used to represent spatially separated degrees of freedom, then sure, $k$-local Hamiltonian describe interactions between that many physical systems. I'm not aware of any "natural" physical law that isn't describable by pairwise interactions though
– glS
Nov 20 '21 at 17:44