Fourier transform of a Lorentz invariant generalized function

Question

Consider on $\mathbb{R}^{n+1}$ the indefinite quadratic form defining the Minkowski metric $$B(p)=(p^0)^2-(p^1)^2-\dots-(p^n)^2.$$ Let $\mu$ be a generalized function on $\mathbb{R}^{n+1}$ which is supported on the light cone, i.e. on the set $$\{p|\, B(p)\geq 0 \mbox{ and } p^0\geq 0 \}.$$ Assume also that $\mu$ is Lorentz invariant, i.e. $\mu$ is invariant under the connected component of the group of linear transformations preserving $B$. Let $\tilde \mu (x)$ be the Fourier transform of $\mu$.

Assume that $\frac{\partial}{\partial x_k}\tilde \mu (x)=0$ for any $k$ and any $x$ being space-like, i.e. $B(x)<0$. Is it true that $\mu =0$?

Answer 1

Interesting question. I don't have an answer but the following could be of help. There is a complete description of distributions which are invariant under the restricted Lorentz group. For example, see this article by Rieckers and Güttinger. Morally, it amounts to a reduction to one-dimensional distributions in the variable $B(p)$. A modification of your question which one could look at as a preliminary step is as follows.

Let $V=\{p\ |\ B(p)\ge 0\}$. Are there Lorentz invariant distributions $T$ such that $T$ as well as $\widehat{T}$ have support contained in $V$?

Note that there is a rather vast literature on various versions of a "qualitative uncertainty principle" where you impose conditions on the support of a function as well as its Fourier transform. This new question is of this type. Finally, you might want to see if there is an explicit formula for the Rieckers-Güttinger spectral representation of the Fourier transform in terms of that of the original distribution. If so you might have a one-dimensional reduction of the problem, albeit perhaps with a strange 1d Fourier transform.

Answer 2

I'm changing the names of the coordinates slightly to $(t,x)$. The light cone $K:=\{(t,x)\in\mathbb{R}^{1+n} \mid t\geq 0 \wedge t^2 - \|x\|^2 \geq 0 \}=\{(t,x) \mid t\geq \|x\|\}$ is a closed, convex, cone. Consider the dual cone $K^\vee := \{(s,y) \mid st+\langle y,x\rangle \leq 0\}=\{(s,y) \mid s\leq 0 \wedge s^2-\|y\| \geq 0\}$.

A general form of the Paley-Wiener theorem tells you that any for tempered distribution $u\in\mathcal{S}'(\mathbb{R}^{1+n})$ with $supp(u)\subseteq K$, the Fourier transform $\mathcal{F}u$ extends to a holomorphic function $F: \mathbb{R}^n + i (K^\vee)^\circ \to \mathbb{C}$ that satisfies a certain growth condition. Here "extends" means that if you consider the "slice" functions $F_b:\mathbb{R}^n\to\mathbb{C}, a\mapsto F(a+ib)$, then $\lim_{t\searrow 0} F_{tb} =\mathcal{F}u$ (limit w.r.t. the strong topology on $\mathcal{S}'$) for $b$ in the interior of $K^\vee$.

Note that $F_b=\mathcal{F}(e^{\langle b,x\rangle}u(x))$.

This is not quite the situation of the identity theorem or the edge-of-the-wedge theorem, but it's close enough that I think one can show from this that $F$ must be constant everywhere and therefore zero so that $u=0$.