# How can we prove that a non-linear equation of motion for a classical scalar field satisfies causality?

Let $$\phi$$ be a real-valued scalar field in $$N$$-dimensional spacetime with coordinates $$(t,\vec x)$$, and consder the equation of motion $$(\partial_t^2-\nabla^2)\phi(t,\vec x)+V'\big(\phi(t,\vec x)\big)=0 \tag{1}$$ where $$\nabla$$ is the gradient with respect to the spatial coordinates $$\vec x$$, and where $$V'$$ is the derivative of a non-negative polynomial such that $$V(\phi)=0$$ if and only if $$\phi=0$$.

Suppose that smooth initial data ($$\phi$$ and $$\partial_t\phi$$) is spedified at $$t=0$$ for all $$\vec x\in R$$, where $$R$$ is some bounded region of space. Physics folklore says that this determines the solution everywhere in the causal completion of $$(t=0,R)$$, which consists of all points $$(t,\vec x)$$ that are outside the light-cones of all points $$(t=0,\vec x\notin R)$$ where the initial data was not specified. This is illustrated below:

If this folklore is true, then how can it be proven?

• I can prove it when the initial data is zero (see the link to Math SE below), and this implies the result for nonzero initial data when equation (1) is linear, but I don't know how to prove it when the initial data is nonzero in the nonlinear case.

• I've searched the online literature for keywords like hyperbolic PDEs, domain of dependence, and characteristics, among others, and I've learned a lot from this, but so far I haven't recognized anything that answers this question.

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 posted as a self-answer), but the original question remains unanswered.

• Have you tried showing this order-by-order in perturbation theory, using the perturbation theory for classical field theory discussed in physics.stackexchange.com/a/27389/173492? (You know it holds for g=0, and then one would show it holds for $\phi_1$ in the notation of the linked answer, etc.) Apr 1 at 6:50
• The result should also heuristically follow from microcausality of $\phi^4$ QFT (e.g. see physics.stackexchange.com/a/626274/173492 to argue microcausality), combined with taking $\hbar \to 0$: in that limit there should be a wavefunctional solution whose field values are tightly centered on the classical field solution, and then by microcausality, a unitary localized to $|\vec{x}|<R$ should not be able to change the wavefunctional evolution outside of the corresponding lightcone. Of course there should be some more direct+rigorous classical argument, without using QFT... Apr 1 at 7:49
• This sounds like it may be related to Cauchy problems and the Cauchy surface. Aug 16 at 18:20