Can the Huygens principle be justified rigorously mathematically?

Question

Can the Huygens principle be justified rigorously mathematically? To what extent and what purpose is it still relevant today? Is it made obsolete by the Maxwell equations? If not, do the Maxwell equations justify it?

Answer 1

Consider the expression for Fresnel diffraction. The electric field is given by: $$E(\vec{r})\propto\intop_{Aperture}{}E(x',y',0)\frac{e^{ikr}}{r}dx'dy'\ $$ The integrand is composed of two parts; the electric field at the aperture $$E(x',y',0)$$ and something that looks like a spherical wave due to a point charge: $$\frac{e^{ikr}}{r}$$ The summation is over the entire aperture.

Essentially, this is the superposition of spherical waves, emitted by point charges that are placed along the entire aperture. This is exactly Huygens Principle.

Clearly, Maxwell's equations justify Huygens principle, since the Fresnel diffraction formula is a direct consequence of the wave equation, which is an immediate consequence of Maxwell's equations.

While in this linear case ('linear' being a keyword) Huygens principle is justified, this doesn't mean that it is necessarily always true.

Answer 2

Huygens principle is not exactly true. This is discussed in Sommerfelds 'Optics'. The problem is for example for an absorbing screen with holes the Greens function used for Huygens principle does not fulfill the boundary condition. One would have to solve Maxwells equations exactly. Unfortunately very few exact solutions are known, one given as well by Sonmerfeld. This solution shows that the deviation from Huygens principle is only of the order of a few wavelength from the boundary. For a modern account on the exact solution you can also refer to Thirrings introduction to mathemazical physics.

Answer 3

The Huygens principle can be rigorously formulated as a statement about the singularities and support properties of the Green operators of the wave equation.To be precise, a forward solution to the zero initial Cauchy data inhomogeneous problem is a smooth solution $\phi\in C^{\infty} (\mathbb{R}^4)$ such that:

a) $-\partial_{tt}\phi+\Delta\phi=f$

b) $\phi(0,x)=0$

c)$ \partial_t\phi(0,x)=0$

for $f\in C_0^{\infty}(\mathbb{R}^4) $ (smooth sources of compact support).

That such a solution $\phi$ exists, is unique and depends continuously with respect the source is equivalent to showing well-posedness. For details you can see this.

In fact, one can define an integral kernel $G(t,x;s,y)$ such that

$\phi(s,y)=\int_{\mathbb{R}^{4}} G(t,x;s,y)f(t,x)$

$G(t,x;s,y)$ corresponds to the advanced Green propagator. The other two are the Green operators associated with the backward inhomogenous problem (retarded Green function) and to the homogeneous initial value problem (causal propagator).

Using techniques of microlocal analysis or by explicitly finding $G(t,x;s,y)$ one can show that $G(x,t;s,y)$ vanishes except at the past light cone the point $(x,t)$ i.e $G(x,t;s,y)=0$ if there isn't a past directed null geodesic between $(x,t)$ and $(s,y)$. Notice, this implies that the singular support of $G$ is the past light cone.

You can see the explicit formula here. Notice that the formula given there is just part of the full formula for the causal propagator and therefore the singular support is different from the one described above. A precise formulation is Equation 4.5.4 in Friedlander

Also, one can see from the formula that the introduction of a mass term doesn't allow the Huygens principle to hold.

The statement that the support of $G$ is only the past light cone of $(x,t)$ can be understood as a rigorous formulation of the Huygens principle.

In a general curved spacetime, i.e. when we are solving $\square_g \phi=f$ with zero initial data on a Cauchy hypersurface $\Sigma$ on a globally hyperbolic spacetime $(\mathbb{R}\times\Sigma, g)$. The support of $G$ is the whole causal past. However, one can still show that the singular support of $G$ is only the past light cone.

Therefore in general the Huygens principle does not hold. One needs specific conditions on the geometry of $g$. It is sufficent that $g$ is flat or a plane wave spacetime. Moreover, the principle depends on the dimensionality of the sapcetime. For example if $n$ is odd the principle is invalid. You can find precise proofs in Friedlander.

Regarding its connection to Maxwell equations. I would like to comment on the following.

I have stated the principle for scalar wave equations. However, one can generalize the approach to tensorial wave equations. In the same way, under suitable gauge choices, Maxwell equations can be formulated as a tensorial wave equation.

The principle would state that we can read the value of the electromagnetic field at a point $p$ in flat even spacetimes by knowing only the information of the electromagnetic field in the past light cone of the point $p$.

The failure of the principle for curved or odd flat spacetimes means that the difference between the value of the solution and the value of an approximation that considers only information on the light cones is not zero. Moreover, the singular behaviour of $G$ at the light cone implies that this difference is smooth function in $(x,t)$. We can restate this as the statement that the singular behaviour of the electromagnetic field (the places where it is not smooth) travels at the speed of light. The electromagnetic field can stop being smooth as consequence of considering that the spacetime metric or the charged distributions are smooth.

These differences can be used to characterise the inhomogeneities of a medium.