Let $\phi$ be a classical scalar field in $1+D$-dimensional spacetime with coordinates $(t,\vec x)$, and consder the equation of motion $$ \newcommand{\pl}{\partial} (\pl_t^2-\nabla^2)\phi+m^2\phi+ g\phi^3=0 \tag{1} $$ where $\nabla$ is the gradient with respect to the spatial coordinates $\vec x$. If two solutions $\phi_1$ and $\phi_2$ and their first time-derivatives are equal to each other at $t=0$ for all $|\vec x|>R$, then presumably they must also be equal to each other for all $|\vec x|>R+|t|$ for every $t$. In other words, an "initial" difference between two solutions presumably cannot propagate faster than the speed of light. (I'm using units where the speed of light is $1$.) This is what I mean by causality in the title of the question.

How can we prove that equation (1) has this property?


I originally posted this question on Math Stack Exchange, where a comment gave me a hint that helped me prove a weaker result, which I also posted there. The weaker result considers a single solution that is zero for all $|\vec x|>R$ at $t=0$ and proves that it remains zero for $|\vec x|>R+|t|$ for all $t$. In other words, the (smallest) region where the solution is non-zero cannot spread faster than the speed of light. This weaker result can be used to answer my question in the case $g=0$, because in that case, the difference between two solutions is another solution. However, my original question remains unanswered: when $g\neq 0$, I still don't know how to prove that the (smallest) region where two generic solutions differ cannot spread faster than the speed of light.


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