# How can we prove that a positive-frequency wave and its time-derivative cannot both have compact support in space?

Consider a complex-valued function of the form $$\newcommand{\bfx}{\mathbf{x}} \newcommand{\bfk}{\mathbf{k}} f(t,\bfx)=\int d^Nk\ g(\bfk)\exp\big(-i\omega(\bfk)t-i\bfk\cdot\bfx\big) \tag{1}$$ where boldface denotes a list of $$N$$ real variables, the dot-product is defined as usual, and $$\omega(\bfk)\equiv \sqrt{\strut{}1+\bfk\cdot\bfk}. \tag{2}$$ (I'm calling this a "wave," but notice the constant term under the square root.) Suppose that $$f(0,\bfx)$$ is nonzero at least for some $$\bfx$$. Can we choose $$g(\bfk)$$ so that $$f(t,\bfx)$$ and $$df(t,\bfx)/dt$$ both have compact support in $$\bfx$$ at $$t=0$$?

The answer must be no, because otherwise I could use the Paley-Wiener theorem to construct a contradiction to the Reeh-Schlieder theorem. But that's a very indirect argument that uses relativistic quantum field theory, which surely isn't necessary for the simple question I'm asking here! How can we prove more directly that no such $$g(\bfk)$$ exists?

• I am trying to understand the same issue but for more general functions, since recently I found the paper Finite Time Differential Equations I am trying to figure out if their findings can be extended to more than one dimensions (unsuccessfully so far). Maybe it could help you. Feb 21, 2022 at 2:52
• @Joako Thank you for the comment! I don't know how to adjust that example to construct a counter-example, though. If we omit the "$1$" term under the square root in my equation (2), then any function of the form $f(kx-ct)$ with compact support at $t=0$ would be a counter-example with $N=1$. It's that pesky "$1$" term under the square root that makes things difficult. Mar 28 at 23:29
• @Joako By the way, the function $E(x,y,z,t)=f(k_xx-ct, k_yy-ct, k_zz-ct)$ shown in that example doesn't satisfy the usual wave equation. Not sure what the author of that answer meant. Mar 28 at 23:29
• @Joako Now let $f(x_1,...,x_N)$ be any smooth function, and suppose $$E(t,x_1,...,x_N)=f(x_1-t,x_2-t,...,x_N-t).$$ Any such function $E(t,x_1,...,x_N)$ satisfies $$\partial_t^2 E = (\partial_1+\partial_2+\cdots+\partial_N)^2 E.$$ If $E$ also satisfies the wave equation, then this immediately implies $$\sum_{j\neq k} \partial_j\partial_k E = 0.$$ By inspection, the example described in the other answer doesn't satisfy this when $N\geq 2$, not even when $t=0$. Mar 29 at 2:29