There are some comments I'd like to make about this, and I'll collect them in an answer, although I don' know the required reference.

I remember (it's a long time) reading this in a University textbook on Partial Differential Equations of Prof. V. Iftimie, written in Romanian, and published probably before 1990. I don't have the book, so I can't if there were any references cited to support that point.

Someone who met Feynman told me that once, Feynman asked a mathematical physicist, in relation to his new book on calculus, what does he mean by function which is derivable only twice. The author of the book gave Feynman as example a function which was defined piece-wisely by two analytic functions, so that the function is continuous at $0$, but only the first two derivatives exist there. Feynman replied "that's not a function!". My guess is that Feynman considered analytic functions to be more "real", from physical viewpoint, than the artificial construction which was given as an example.

"a function (let's say scalar) cannot be analytic because otherwise it would violate causality"

Since we can define analytic functions on any differentiable manifold, nothing can stop us to define as many analytic functions as we want on the Minkowski spacetime. That's why I think that the question is about analytic physical fields.

Physical fields are supposed to be solutions of PDE. The PDE can have non-analytic solutions (provided that the initial conditions are not given by analytic functions), which are weak or generalized solutions (for example distributions). It would be interesting if there are examples of natural solutions to the PDE in physics, which are that wild, that they would not be analytic at least when restricted to some open set. My guess is that for any such field there is an open set on which its restriction is analytic. And the problem you raised will affect them too, because one can find in any open set two points which are separated by a spacelike interval.

Even if the physical fields in the universe are analytic, I think that we cannot actually use an analytic function in practice to send messages violating causality, because we don't have the possibility to control its value and the values of all its partial derivatives at $(t_0,x_0)$ within an approximation to predict what happens at $(t,x)$. (If the function is a quantum field we can't know the value and the derivatives in principle.)

OK, we don't need to control everything in detail. It is enough to establish the convention that if we make the function to have values in an interval or not, that's a binary digit, so that we send binary signals. The thing is that we can't even do this, because there are always two analytic functions which respects the constraints we impose to send the message, and one is positive and the other negative at the destination point.

What a non-analytic function can do, an analytic function can do too, in what precision we want. So we can't distinguish them by experiment, even if the experiment involves signals violating relativistic causality.

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