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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:

  1. 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 $\partial \Omega$).
  2. 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?

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But why is it surprising that only 1-forms and 2-forms turn up in physics? After all, 0-forms and 3-forms are essentially functions, and the world is 3-dimensional. So there are no other forms that are possible. Unless you switch to relativity, where the Hodge-Laplacian becomes a wave operator. I'm not sure that 3-forms arise naturally there, but I think it's likely that they do. –  Deane Yang Oct 7 '12 at 22:50
@Yang, actually I excluded the 0- and 3-forms because they are just the usual scalar Laplacians with different boundary conditions. There are way too many examples of them that I don't need to try hard to find plenty of motivation. –  Lizao Li Oct 8 '12 at 22:47

4 Answers 4

up vote 2 down vote accepted

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:

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Very nice. Thanks. –  Lizao Li Oct 8 '12 at 22:51
Hmmm, isn't it the other way around? He uses supesymmetric quantum mechanics to prove Morse inequalities, as far as I understand! And he does not consider exactly the Hodge Laplacian, but a distorted version of it (now called Witten Laplacian). But I agree that indeed, the Witten Laplacian on the full algebra of forms is used. A bit more precisely: the Witten Laplacian is a Schroedinger type operator for every $p=0,\dots,n$ and he uses semiclassical spectral approximation (through Harmonic oscillators) of these Schroedinger operators to get the Morse inequalities. –  Hans Oct 10 '12 at 0:05
@Hans I suppose you are right, but my point was simply that this paper discusses the connection between Morse theory and supersymmetric physics. Of course, you are correct to point out that this paper makes use of a "modified" Laplacian, but I figured the interested reader would find that out after following the link. Anyway, your answer is great, (better than mine!) so +1. –  Kevin Oct 10 '12 at 4:00
I like your answer too, just wanted to clarify some potential ambiguity! –  Hans Oct 15 '12 at 0:26

About the quantitative analysis of metastability in reversible diffusion processes via Witten deformation, there are several papers written by Helffer, Klein and Nier.

without boundary:

with boundary:

Have fun!

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I would suggest you to look at the paper link text

It is written in physics language, so maybe I try to explain:

It is not exactly the Hodge Laplacian which is used, but the Witten Laplacian. The latter
was introduced by Witten in the paper mentioned by Kevin Kordek to study Morse inequalities/complexes on manifolds. It depends on a function $f$ on the manifold (a Morse function in Witten' application) and a small parameter $h>0$ (semiclassical parameter in Witten's application) and is given by

$\Delta_{f,h} = (d_{f,h} + d^*_{f,h})^2$

with $d_{f,h} = h \ e^{-f/h} \ d \ e^{f/h}$ and $d^*_{f,h} = h \ e^{f/h} \ d^* \ e^{-f/h}$

(don't trust me too much for the signs, the point is: the Witten Laplacian is like the Hodge Laplacian, but with the differential distorted by exponential weights. It equals the Hodge laplacian when $f$ is a constant.)

The restriction of the Witten Laplacian to functions ($p=0$) is unitarily equivalent to the generator of a "metastable" stochastic diffusion process (keyword: ground state transformation, it is formula (3) in the paper) moving in the energy landscape given by $f$ (imagien $f$ having several minima). This process follows essentially the (negative) gradient flow of $f$ but a small (with intensity $h$) Brownian noise disturbes the motion and leads to tunneling from one minimum of $f$ to another.

The point of the paper is that metastable properties of this process are much better understood when considering not only the Witten Laplacian (i.e. the generator) for $p=0$ but for every $p=0,dots,n$. For example eigenforms for $p=1$ are related to the metastable transition paths of the process (i.e. the paths along which it tunnels).

There are no boundary conditions here, but of course you can think of similar things by considering a bounded open set of $\mathbb R^n$ and putting boundary conditions.

P.S.: Notation in the mentioned paper: $T$ is my $h$, $E$ is my $f$, $H_{FP}$ is the adjoint of the generator of the diffusion, $H^h$ is the Witten Laplacian (here $h$ stands for hermitian, not to be confused with my $h$). My $p$ is the fermion number. My $d_f,d^*_f$ are denoted by $Q^h, \bar Q^h$.

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Thank you very much. Sounds very interesting. Your mathematical description definitely helped a lot for me to understand it. –  Lizao Li Oct 11 '12 at 20:17

If you allowed 0-forms, then you could talk a lot about superconductivity and the Meissner Effect in superconductors. But we can still say a little bit, using the connection $A$ (i.e. vector potential in Maxwell's theory):

The vector potential is a connection $A$ on a unitary line bundle $\mathcal{L}$. To describe superconductivity, you formulate the Landau-Ginzburg model using a section $s$ of $\mathcal{L}^2$ (this is because superconductivity is formed from Cooper pairs, i.e. two electrons and hence the squared line bundle). Superconductivity will occur when $|s|=a>0$ (some constant) and $d_As=0$ (this ultimately gives the Meissner effect, as $0=d^2_As\;\Rightarrow\; F_A=0$).

In the minimum energy state (described above) we can analyze the solutions of the Landau-Ginzburg equations, by linearizing them (in $A$) around $A=0$. Turns out $(d^*d+4a^2)A=0$ and $d^*A=0$ are produced, and this is where your Laplacian-application arises. Dependency on a single coordinate $x$ of $\mathbb{R}^3$ (local description) will have solutions $e^{-x/\lambda}$. Here $\lambda=\frac{1}{2a}$ is the famous London penetration depth which quantitatively tells you how far the magnetic field penetrates the bulk material before being screened out. In other words, $A$ and $F_A$ vanish exponentially for $x>>\lambda$. This pretty much then defines the ``boundary layer'' of a superconductor.

[[Credit goes to learning about this in Witten's awesome paper (well, 'awesome' is vacuous because they're all awesome), From Superconductors and 4-Manifolds to Weak Interactions.]]

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Indeed many Gauge theories can be formulated as Laplace or Dirac operators on principal-bundle-valued differential forms. Here I am thinking about more down-to-earth scalar valued ones as just motivations for talks and so on... Thanks all the same. –  Lizao Li Oct 8 '12 at 22:57

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