Locality in Quantum Mechanics

We speak of locality or non-locality of an equation in QM, depending on whether it has no differential operators of order higher than two.

My question is, how could one tell from looking at the concrete solutions of the equation whether the equ. was local or not...or, to put it another way, what would it mean to say that a concrete solution was non-local?

edit: let me emphasise this grew from a problem in one-particle quantum mechanics. (Klein-Gordon eq.) Let me clarify that I am asking what is the physical meaning of saying a solution, or space of solutions, is non-local. Answers that hinge on the form of the equation are...better than nothing, but I am hoping for a more physical answer, since it is the solutions which are physical, not the way they are written down ....

This question, which I had already read, is related but the relation is unclear. Why are higher order Lagrangians called 'non-local'?

Answer

Presuming that there aren't nonlocal constraints, a differential operator that is polynomial in differential operators is local, it doesn't have to be quadratic. My understanding is that irrational or transcendental functions of differential operators are generally nonlocal (though that's perhaps a Question for math.SE).

A given space of solutions implies a particular nonlocal choice of boundary conditions, unless the equations are on a compact manifold (which, however, is itself a nonlocal structure). There is always an element of nonlocality when we discuss solutions in contrast to equations.

[For the anti-locality of the operator $(-\nabla^2+m^2)^\lambda$ for odd dimension and non-integer $\lambda$, one can see I.E. Segal, R.W. Goodman, J. Math. Mech. 14 (1965) 629 (for a review of this paper, see here).]

EDIT: Sorry, I should have gone straight to Hegerfeldt's theorem. Schrodinger's equation is enough like the heat equation to be nonlocal in Hegerfeldt's sense. There are two theorems, from 1974 in PRD and from 1994 in PRL, but in arXiv:quant-ph/9809030 we have, of course with references to the originals,

Theorem 1. Consider a free relativistic particle of positive or zero mass and arbitrary spin. Assume that at time $t=0$ the particle is localized with probability 1 in a bounded region V . Then there is a nonzero probability of finding the particle arbitrarily far away at any later time.

Theorem 2. Let the operator $H$ be self-adjoint and bounded from below. Let $\mathcal{O}$ be any operator satisfying $$0\le \mathcal{O} \le \mathrm{const.}$$ Let $\psi_0$ be any vector and define $$\psi_t \equiv \mathrm{e}^{-\mathrm{i}Ht}\psi_0.$$ Then one of the following two alternatives holds. (i) $\left<\psi_t,\mathcal{O}\psi_t\right>\not=0$ for almost all $t$ (and the set of such t's is dense and open) (ii) $\left<\psi_t,\mathcal{O}\psi_t\right>\equiv 0$ for all $t$.

Exactly how to understand Hegerfeldt's theorem is another question. It seems almost as if it isn't mentioned because it's so inconvenient (the second theorem, in particular, has a rather simple statement with rather general conditions), but a lot depends on how we define local and nonlocal.

I usually take Hegerfeldt's theorem to be a non-relativistic cognate of the Reeh-Schlieder theorem in axiomatic QFT, although that's perhaps heterodox, where microcausality is close to the only definition of local. Microcausality is one of the axioms that leads to the Reeh-Schlieder theorem, so, no nonlocality.