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Local and memoryless are easily defined in quantized space and time:

Local: What happens from one time step to the next in one "cell" of quantized space is only influenced by the state of neighboring cells.

Memoryless: The state in the next time step is only influenced by the state in the previous time step (not earlier ones).

My understanding is when we talk about the laws of physics being local, what is actually meant is local and memoryless.

In a continuous spacetime, I guess one could define locality (and I assume that's how it is defined) roughly as "the shorter the time difference, the smaller the neighborhood that can influence what happens during this time". And that is indeed how it works since the light speed imposes exactly such a limit.

But how "local" is physics when we get to really small time differences and distances, since we also get things like uncertainty of position into the picture. That seems to impose a limit after which (I guess) physics seems to behave non-local? Does it still behave memoryless? If I reduce time difference further, does the distance also still shrink to whatever can influence what happens? Is "distance" even still well-defined when uncertainty of position becomes significant?

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Local may have different meaning depending on the context. One common use of this term is in describing the equations of physics as local, because a value of a function/field is determined only by the values of this and other functions/fields at the the same point (or infinitesimally close points), as reflected by the fact that the equations are differential equations of finite order.

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  • $\begingroup$ I accepted this now, but I would like a small explanation: Are these differential equations still a good approximation of what happens at very small scales? $\endgroup$
    – kutschkem
    Nov 22, 2020 at 9:48
  • $\begingroup$ @kutschkem one should distinguish mathematically small (i.e. infinitesimal) and physically small. For example, electrodynamics/elasticity/hydrodynamics of continuous media are local theories, but they operate with quantities average over physically small volumes, containing huge number of atoms, i.e. they do not work on microscale. $\endgroup$
    – Roger V.
    Nov 22, 2020 at 11:30

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