# Infinite derivatives, Locality and Lagrangian

Providing the derivative of a single valued function $f(x)$ is like providing its value at two infinitesimally close points.

My question consists of two parts:

1. Can Higher derivatives be thought of as providing the value of the function at many points in space, all of them infinitesimally close to each other?

Thus providing infinite derivatives is like providing value of function at all points in space? Is this correct?

If so,

1. Is this why we say that the Lagrangian of any quantum field theory cannot have an infinite number of derivatives of the field, because that would lead to a violation of locality?

Can someone explain this precisely?

• – AccidentalFourierTransform Oct 17 '17 at 17:28
• I read it. I am sorry but I did not understand it quite properly. Can you explain a bit to me about what the answer to my question will be. Thanks – Tushar Gopalka Oct 17 '17 at 17:43
• Try writing higher order derivatives of a function using finite differences (central form for easy formula) and see what happens. – Victor Palea Oct 22 '17 at 11:54

For analytic functions there is a certain region (neighborhood) in space-time around a given point $x$ in which the function is given by a converging Taylor expansion involving all these derivatives at point $x$. So yes the Lagrangian would involve the field at other points from $x$ and is nonlocal. This is problematic for many physicists as fields at a certain point in space-time would interact with the same field in other points in space time. Nevertheless, people have worked with nonlocal Lagrangians for example for effective field theories in which these nonlocal terms could represent a deeper theory that is still local.