A first draft is now available [PDF]. The entire topic is a bit off main stream, but that’s why I’m actually quite proud about it, especially because it is not in a field, I’m most familiar with (partial differential equations). It needs courage to work in a new field and I myself I’m always afread of this but a 2 week winter break was a good opportunity to get into this again after 15 years. Maybe a few reasons, why I think the result is nice (false humility is also a lie. and I really think it is nice):
- There is a wave equation in arbitrary dimensions which satisfies the strong Huygens principle and has nice explicit solution as
which has the same form as the solution
for the standard wave equation. But while the later only in odd dimensions satisfies the Huygens principle, the modified wave equation has sharp wave fronts in all dimensions.
- Having a bounded exterior derivative that is not involving limits and keeps all symmetries intact,was a dream of mine for decades but discretisations always invoke some sort of lattice approximation or triangulation, which all break symmetries. No single physics experiment has ever hinted at a triangulated or lattice structure or granular structure or random structure of space. Still, if we take the hints of quantum mechanics seriously, we must ridicule any calculus on scales smaller than the Planck scale.
- It is nice to see Bessel functions alive. The Mark I (youtube) computer at Harvard had been called “Bessie” because it was mostly computing Bessel functions! The subject of Bessel funcdtions is usually presented in a rather encyclopedic way. Also PDE’s are a heavy subject, filled with messy stuff. Numerical schemes in particular are not pretty, if they want to be stable and accurate.
- The subject hits a classical topic and central for physics. What is the true nature of space, time and matter? It is the old Newton-Huygens dispute: is a particle a wave or a corpuscle? At the moment the school book approach is to be schizophrenic and take the picture that fits each case: particle when we measure, wave if it moves.
- Having a sharp wave front also in 4 dimensions allows to look at a wave front as “space” in a space-time manifold. This is not possible in a traditional calculus, as in 4 dimensions, the classical wave equation does not satisfy the Huygens principle. The wave front picture has some automatic features like “expanding space”. It can model also inflation as near a point with large negative curvature, the expansion of the wave front is fast.
- It is a “deformation of the Dirac operator” theme which I have been fascinated me for a long time: Witten deformation (I learned about in grad school thanks to Konrad Osterwalder), deformation and factorization filled a large part of my thesis was about isospectral deformations of Jacobi matrices and factorization of these matrices
leading to operators with spectra on Julia sets as this is a quadratic map in operators, Dirac matrices (square roots of Laplacians) appeared there. Dirac matrices turned out to be helpful also in discrete geometry, especially the McKean-Singer symmetry which leads to elegant proofs of the Lefschetz fixed point theorem. (The most recent writing about this appears here, which was a contribution to Barry Simon’s 80th birthday. (The paper does not make it into a festschrift. Some referee wanted to have it completely rewritten, which is not an option for me.) And then there is the isospectral Toda type deformation for Dirac matrices. (See “remarks about connection and Dirac matrices” and “Block Jacobi matrices, barycentric limits and manifolds“.)
