# Non-local and non-linear

## Main Question or Discussion Point

Is there a strict connection between non-locality and non-linearity? Is it true that non-local systems must have non-linear equations? Are non-linear equations the consequence of non-locality?

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Really? Isn't anyone can answer this?

The response of a material can be non-linear -this depends on the (external) field- while non-locality does not necessarily depend on the field. I don't know if this applies to all cases, this is simply my understanding.

Is it true that non-local systems must have non-linear equations?
You need to elaborate a bit here. What kind of "non-linear equations" do you have in mind ?

Are non-linear equations the consequence of non-locality?
The constitutive relations of a chiral medium can be linear and non-local.

Yes, and I expect a general answer. There might be some exceptional systems as you mentioned. But, can we say that in general?

Nevertheless, let's think the motion of a particle as an example. Let's assume we exposed particle to an instantaneous effect (for example that effect can change his position superluminally). Then this system is non-local, and it must have non-linear equations, right?

martinbn
Isn't quantum mechanics linear and non-local?

Yes, quantum mechanics is linear, and I think that's why it cannot explain some phenomena like wave function collapse.

Nevertheless, let's think the motion of a particle as an example. Let's assume we exposed particle to an instantaneous effect (for example that effect can change his position superluminally). Then this system is non-local, and it must have non-linear equations, right?
A non-local eqn will pop out if the particle's position does not depend only on the applied field, say F, but also on its derivatives, its curl for example. This does not necessarily* introduce nonlinearities in the eqns of motion.

* meaning that you don't always get quadratic or higher order terms of F

martinbn
Yes, quantum mechanics is linear, and I think that's why it cannot explain some phenomena like wave function collapse.
You are asking about a strict connection between non-linear and non-local. If quantum mechanics is linear and non-local, then your question has a negative answer, no?

Demystifier
Gold Member
There is no direct relation between nonlinearity and nonlocality.

Yet, if QM were nonlinear (not merely by a collapse postulate, but at a more physical level), then, depending on the interpretation, it could be even more nonlocal than it is in the standard linear form.
See e.g.
http://xxx.lanl.gov/abs/0707.2319
and Refs. [1,2,3] therein.

A non-local eqn will pop out if the particle's position does not depend only on the applied field, say F, but also on its derivatives, its curl for example. This does not necessarily* introduce nonlinearities in the eqns of motion.

* meaning that you don't always get quadratic or higher order terms of F
So such field should affect particle's acceleration also, to have non-linear equations?

You are asking about a strict connection between non-linear and non-local. If quantum mechanics is linear and non-local, then your question has a negative answer, no?
Or quantum mechanics is wrong. Oops, I have to run There is no direct relation between nonlinearity and nonlocality.

Yet, if QM were nonlinear (not merely by a collapse postulate, but at a more physical level), then, depending on the interpretation, it could be even more nonlocal than it is in the standard linear form.
See e.g.
http://xxx.lanl.gov/abs/0707.2319
and Refs. [1,2,3] therein.
So as far as I understand, although no strict connection between both, nonlocality increases from linear to nonlinear.

So such field should affect particle's acceleration also, to have non-linear equations?
No. Why do you say this ?

So as far as I understand, although no strict connection between both, nonlocality increases from linear to nonlinear.
What do you mean by "nonlocality increases from linear to nonlinear" ?

Isn't quantum mechanics linear ?
not settled yet.

Ok, I guess the relationship between these two is far more than I thought. So there is no substantial connection.