The double slit experiment is a real-life manifestation of the Huygens principle. As is well-known, this principle depends on whether the number of dimensions is ever or odd; as Evans1 puts it,
Observe, in contrast to formula $(31)$, that to compute $u(x,t)$ for even $n$ we need information on $u=g$, $u_t=h$ on all of $B(x,t)$, and not just on $\partial B(x,t)$.
Comparing $(31)$ and $(38)$ we observe that if $n$ is odd and $n\ge3$, the data $g$ and $h$ at a given point $x\in\mathbb R^n$ affect the solution $u$ only on the boundary $\{(y,t)|t>0,|x-y|=t\}$ of the cone $C=\{(y,t)|t>0,|x-y|<t\}$. On the other hand, if $n$ is even the data $g$ and $h$ affect $u$ within all of $C$. In other words, a "disturbance" originating at $x$ propagates along a sharp wavefront in odd dimensions, but in even dimensions continues to have effects even after the leading edge of the wavefront passes. This is Huygens' principle.
Thus, my question: qualitatively speaking, how does the double slit experiment look in, say, $4+1$ dimensions? Is there any major difference as compared to the same experiment in our $3+1$ world?
- See page 80. Here, we are solving the wave equation $u_{tt}-\Delta u=0$ in $\mathbb R^n\times(0,\infty)$, with $u(x,0)=g(x)$ and $u_t(x,0)=h(x)$. Moreover, $B(x,r)$ is the closed ball with centre $x$ and radius $r$.
