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It is accepted that infinite order of derivatives in field theory lead to non-local effects while finite number of them local.

reference within physics stack exchange

Let’s take a lattice with next-next-neighbour interaction, eg tight binding model for example. There are 2 ways to make a continuum field theory out of it.

  1. via coarse graining of lattice, constructing a statistical field theory.

  2. via solving the lattice and then taking the large wavelength limit, where wavenumber becomes very small and converting differences to derivatives, since the spatial site-to-site difference is very small compared to wavelength of waves.

In both the cases, one gets 2nd order derivative when Hamiltonian depends on next-next-neighbour interaction.

Here the physics is manifestly non-local, even though we have reduced the problem mathematically in such a way to hide this fact.

(Higher number of derivatives allow interpolation in bigger intervals for well-behaved functions via a Taylor Series)

So why is this effect considered local and not non-local?

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    $\begingroup$ "It is accepted that infinite order of derivatives in field theory lead to non-local effects while finite number of them local." Reference, please? $\endgroup$
    – kricheli
    Jun 10, 2023 at 8:23
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    $\begingroup$ Also, I think your second sentence might need clarification. A lattice is discrete. So what is your "field theory of a lattice with next-next neighbour interaction" or how, if continuous, does it relate to the lattice? $\endgroup$
    – kricheli
    Jun 10, 2023 at 8:26
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    $\begingroup$ Have edited the question. Thanks. $\endgroup$ Jun 10, 2023 at 10:06

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First, to clarify and to add a "yes, but..." to your statement

It is accepted that infinite order of derivatives in field theory lead to non-local effects while finite number of them local.

$$ \mathcal{L} = \frac{1}{2}\varepsilon_0 E_i(x) E_i(x) $$ is a local Lagrangian density. To be specific: the free-field Lagrangian density of classical electrostatics. A non-local version of this could look like this: $$ \mathcal{L} = \frac{1}{2} E_i(x) \int \varepsilon(x - x') E_i(x') \text{d}x' $$ (where for simplicity we here assume isotropy). The corresponding material law is $$ D_i(x) = \int \varepsilon(x - x') E_i(x') \text{d}x'\,. $$ Now a third version could be the following Lagrangian density: $$ \mathcal{L} = \frac{1}{2}\varepsilon_0 E_i(x) E_i(x) + \ell^2 \frac{1}{2}\varepsilon_0 \partial_jE_i(x) \partial_jE_i(x) $$ with some parameter $\ell$ that has dimensions of length. (And in principle we could use further terms like $\partial_k\partial_jE_i \partial_k\partial_jE_i$ etc. with even higher orders of derivatives.) This is what people usually call a gradient theory, and somehow in between the local and the fully non-local theory, and some people consider this "weakly non-local".

And it can actually be obtained as an approximation to the non-local theory, because in case of a "weak" non-locality (say $\varepsilon$ being a Gaussian with very small standard deviation, or a compactly supported function with small radius of support) and $E_i$ being sufficiently smooth we can perform a moment expansion in the material law (very similar to the well-known multipole expansion): $$ D_i(x) = \int \varepsilon(x - x') E_i(x') \text{d}x' = \int \varepsilon(x') E_i(x - x') \text{d}x' \\ \approx \int \varepsilon(x') \left[E_i(x) - x'_j \partial_jE_i(x) + \frac{1}{2} x'_jx'_k \partial_k \partial_jE_i(x) + ... \right]\text{d}x' \\ = \varepsilon_0 \left(E_i(x) - \ell^2 \Delta E_i(x) + ...\right) $$ which gives you the material law to the third Lagrangian above, the one for the gradient theory. (For brevity I've omitted some steps here, like symmetry arguments for making some terms vanish in the expansion, defining the parameters $\ell$ and $\varepsilon_0$ as moments of $\varepsilon(x')$ etc.)

Second, to clarify the order of derivatives involved. The local Lagrangian above depends on the first derivative of the potential (because $E_i = -\partial_i \varphi$) and on the zeroth derivative of the field. There is no derivative in the material law and the field equation for the potential (Poisson's equation) contains a second-order derivative. In the gradient theory (with only the one additional term) the Lagrangian depends on the first and second derivative of the potential and on the zeroth and first derivative of the field. There is a derivative in the material law and the field equation for the potential is of fourth order. So you have to be a bit more specific as to where you have the second-order derivative you mention.

Third, to answer your question: From the lattice theory you can actually obtain a non-local or weakly non-local theory. It's just that usually only the leading-order term is used so that you obtain the local theory (as an approximation). Because quite often this is all you need.

If you want to read a bit more on this: https://www.sciencedirect.com/science/article/abs/pii/S0997753802012184 https://link.springer.com/chapter/10.1007/978-3-662-30257-6_37 https://rcin.org.pl/ippt/Content/84221/WA727_90438_P.262-Maugin-Nonlocal.pdf

Fourth, some bonus nitpicking on point 2: When talking about the order of derivatives in the Lagrangian you usually take the Lagrangian among a group of equivalent Lagrangians that has the lowest orders of derivatives in it. Example: For the local electrostatic Lagrangian $$ \mathcal{L} = \frac{1}{2}\varepsilon_0 \partial_i \varphi \partial_i \varphi $$ and $$ \mathcal{L} = - \frac{1}{2}\varepsilon_0 \varphi \Delta \varphi $$ are equivalent in the sense that they yield the same Euler-Lagrange equation because they are identical up to a Null-Lagrangian (a term that can be written as a total derivative). You would use the first one, because the second one unnecessarily depends on a higher order of derivative of the potential.

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  • $\begingroup$ Thanks a lot, the references are really nice. $\endgroup$ Jun 11, 2023 at 7:44

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