The Hodge-de Rham Laplacian $L=(d+d^*)^2$, where $d$ is the boundary operator of the de Rham complex, is well-known in the math community. Recently, I tried very hard to search for examples of its use in physics and engineering applications (for giving talks or future teaching). More specifically, I want to find applications of the Hodge-de Rham Laplacian on scalar(real or complex)-valued differential p-forms on $\Omega\in\mathbb{R}^n$ with natural boundary conditions and $p\neq 0,n$, where not only the cohomology but also the solution itself is of interest (thinking about PDEs).
To my surprise, I can only find one such example: in $\Omega\in \mathbb{R}^3$, the 1-form and 2-form Laplacians are the better-known vector Laplacian used in the Maxwell equations.
Note that "natural boundary" condition is very important. It means the boundary condition comes from the need to perform integration by parts. For example in coordinates, one of the homogeneous natural boundary conditions for 2-Laplacian in $\mathbb{R}^3$ is $u\times n =0$ and $(\nabla\times u)\cdot n =0$. But $u=0$ on $\partial \Omega$ is not "natural" here. That's why:
- Stokes or Navier-Stokes does not count, because the vector Laplacian there is really the strain opeartor on div-free fields (the boundary condition is $u=0$ on $\partial \Omega$).
- For the similar reason about the boundary condition, the "vector Laplacian" given by a system of $m$ scalar Laplacians (or its corresponding heat or wave) equation does not qualify.
It is somewhat unbelievable that the Maxwell equations is the only example. Are there any more applications?
Best Answer
Witten wrote a classic paper in which he uses Morse theory to study supersymmetric quantum mechanics. Harmonic forms play a central role. Here's the link: http://intlpress.com/JDG/archive/1982/17-4-661.pdf