Infinite derivatives, Locality and Lagrangian

Question

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,

2. 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?

Thanks in advance.

Answer 1

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.

Answer 2

Note that even in 0+1D, the paths integrated over in the path integral are far from analytic. In fact, almost all of them are not even differentiable. See this answer. It takes a lot of care to make sense of the derivatives for most of these paths. One way people try to do it is by letting the paths themselves have some distributional character. Look up the Fokker-Planck equation, and see this paper. So far the stochastic derivatives need to be defined in a case by case basis.

One reason we do not include arbitrarily high order derivative terms in the Lagrangian is that they tend to be non-renormalizable. This means that in the UV, locality is violated (in a way that makes the theory admit no action principle at all), somewhat like you asked about. I think this is likely connected to the difficulty of defining higher derivatives of distributions, but I'm no expert.

Answer 3

I'm a student, not a professor, but I think that's quite correct: we can always expand a field around any of its points (admitting the necessary conditions) if we dispose of infinite derivatives of that field, so the locality of our theory implies that a Lagrangian can depend just on a finite number of derivatives of the field because it has to depend on the neighbourhood of a single point. That's to say that a Lagrangian (thus the equations of motion!) can not depend upon an arbitrarily far point