Demystifier said:
Let me put some critical remarks on your attempt to "define" locality.
On the one hand, you require a compatibility with the spacetime manifold. This suggests that space and time should be treated on an equal footing.
The point is: what it suggests is not important. After all, the entire importance to "locality" is its compatibility with relativity, and the central idea to relativity is that things should be defined over the spacetime manifold, and not over a specific preferred foliation of it. The most general requirement is then that all that is important in the theory should be defined as elements, sections, whatever over a fibre bundle of which the base space is the spacetime manifold. This is the same in classical (relativistic) physics: fields are sections over the (co)tangent bundle of spacetime, or of "tensorisations" of such.
Now, and that's where I have to give in a small part to "non-locality", in quantum theory (in the unitary part of quantum theory at least), all "interesting" operators are "indexed" over spacetime (that's another way of saying that they are members of a fibre bundle with base space spacetime, and in many cases, are a section), but they ACT upon a totally different space, which is Hilbert space. However, this hilbert space itself doesn't matter that much, in that we can always "rotate" the "initial" vector to any vector we like with a unitary transformation. So the "initial" vector doesn't even matter, we can make it what we want, by a right choice of unitary transformation. For instance, in QFT, we have the habit of just sandwiching any operator expression between a bra and a ket of the "vacuum". This is just a generic state, which doesn't depend on spacetime, in the same way as the number pi doesn't depend on spacetime.
On the other hand, you note that (in the Heisenberg picture) nonlocality is not *dynamical*, where "dynamics" clearly refers to quantities that depend on time.
No, dynamics refers to a timelike regularity, but doesn't need to introduce a specific foliation. GIVEN a foliation, it becomes an "evolution in time", but dynamics can be a spacetime concept. It actually means: "interaction" or "coupling" (between subsystems, such as the electron field, the EM field, ...).
However, such a dynamical notion of locality clearly does not respect the requirement that space and time should be treated on an equal footing.
In relativity, there is no *equal* footing between space and time, just a similar treatment. But there is a fundamental difference between "spacelike" and "timelike" all together (given by the causal structure, or the signature of the metric).
Further, even if an initial product state may lead to entanglement later, this is certainly not the most general situation. QM allows different cases as well.
True, but one might think that the quantum case where one starts with "non-entangled" systems is probably largely sufficient to reproduce all observable phenomena. After all, that's how things are done in the lab! We use a specific interaction to have "entangled systems", starting from non-entangled product states.
It is an interesting question whether it is necessary to consider initial states which are not product states, or whether this is essentially impossible to find out. I tend to think the latter.
In addition, the words "initial" and "later" again give a special role to time, which may ruin your first requirement above again.
No, there is a precise sense in relativity to causal relationships. Not all pairs of events can be classified as "earlier" and "later", but those who can, are unambiguous.
To conclude, no matter how you reinterpret QM to make it local as much as possible, it seems there will always remain *something* nonlocal about it. This, indeed, is the point of my paper.
True. The point is: in how much is that important ? The whole discussion about "locality" (as I understand it) is essentially: relativity or not ? Out of relativity (and the requirement of things to be defined over a spacetime manifold, independent of a specific foliation) came the idea of "locality", but maybe the concept of locality got a life of itself, and is somewhat more restrictive than what is strictly needed for relativity. QM seems to hide in the little margin between both.
