Is there a formulation of Huygens' principle using the language of algebraic geometry?

Question

Huygens' principle is a way to describe the propagation of light rays in a manifold $\mathcal{M}$. Suppose we start with an infinitesimal ball of light at position $X$ in a manifold. Let $B(X,t)$ be the set of points to which light rays can arrive at times less than or equal to $t$. Then $\partial B(X,t)=\Sigma(X,t)$ is a closed surface that constitutes the set of points to which light arrive precisely at time $t$. Consider both $\Sigma(X,t_-)$ and $\Sigma(X,t_+)$

Consider also the class of surfaces $\mathcal{A}(t_--t_+)=\Sigma(Y,t_+-t_-)$ for each $Y\in \Sigma(X,t_-)$

Then $\Sigma(X,t_+)$ is the unique closed surface in $\mathcal{M}$ that is tangent to each element in $\mathcal{A}(t_--t_+)$

This is Huygens' theorem.

My question is whether this can be reframed in the language of algebraic geometry. My question is vague because I know very little in algebraic geometry. I guess I am trying to ask if there is a generalization of Huygens' principle to more exotic spaces like supermanifolds, for instance.

Answer 1

While this is not exactly algebraic geometry, Anders Kock has given a description of Huygen's principle in terms of synthetic differential geometry, which does have some connections to algebraic geometry.

Answer 2

An analogue of the Huygens principle (including interference, so more precisely a Huygens-Fresnel principle) in superspace has been formulated by Gomes in arXiv:gr-qc/0602092. The formulation is used to find the analogue of the WKB approximation to the Wheeler DeWitt equation.

Answer 3

From your question, I guess that you start out with a "smooth Riemannian manifold $\mathcal{M}$". Algebraic geometry is, very roughly, the realm of zero loci of polynomial (or perhaps rational) equations, and polynomial (rational) maps between them. A general smooth Riemannian manifold need not belong to that realm, hence algebraic geometry would have very little to say about such an $\mathcal{M}$, or the Huygens' principle on it.

That said, I have some brief observations.

1. The way you describe Huygens' principle corresponds to the notion of taking an envelope of curves or surfaces (the $\mathcal{A}$ family in your case). This is an operation that makes sense in algebraic geometry because the envelope of a polynomial family of algebraic varieties is also a variety, I believe.

2. There is a couple of famous articles bringing together algebraic geometry and the Huygens' principle (or something very closely related to it) for constant coefficient hyperbolic equations (at order 2, such equations basically generalize the wave equation on Euclidean space, while at higher orders they are no longer describable by simple Riemannian geometry):

• Atiyah, Michael F.; Bott, Raoul; Gårding, Lars, Lacunas for hyperbolic differential operators with constant coefficients. I, Acta Math. 124, 109-189 (1970). ZBL0191.11203.
• Atiyah, Michael F.; Bott, Raoul; Gårding, Lars, Lacunas for hyperbolic differential operators with constant coefficients. II, Acta Math. 131, 145-206 (1973). ZBL0266.35045.

3. As was already implied in the previous point, the notion of light ray (or more generally characteristic ray) only sometimes coincides with that of a geodesic in a Riemannian manifold. The notion itself is more closely tied with the finite speed of propagation for a hyperbolic equation. Such equations can be formulated on supermanifolds and for superfields. Roughly speaking, any supermanifold can be seen as a bundle of odd or super directions over a regular manifold (the body, cf. Serre-Swan theorem). Getting one's hands dirty, and thinking as an analyst, a hyperbolic equation in this setting can be represented as a hierarchical system of regular hyperbolic equations on the body, where the super directions appear in a filtered manner at higher orders of the hierarchy. At the base of the hierarchy, you have a regular hyperbolic PDE on the body without any super information entering into it, and it is only this equation that controls the speed of propagation and hence the Huygens' principle. This approach was pioneered by Choquet-Bruhat in the seminal paper

• Choquet-Bruhat, Yvonne, The Cauchy problem in classical supergravity, Lett. Math. Phys. 7, 459-467 (1983). ZBL0529.58039.