Quantized Distances - Quantum Calculus

In the video below I gave a bit of an overview about the “wave calculus project”. At the 5 minute mark , I mention again the fact that we describe a calculus where distances are quantized but where we still have all the symmetries that exist in the continuum. One might think that it is not possible to quantize distances in a Riemannian manifold without breaking symmetries. Our construction does exactly this. Calculus is a theory of differentiation and integration. In our case, we do the calculus in the Hilbert space obtained by taking the closure of differential forms with respect to the inner product $\langle g,f \rangle$ . We define a new exterior derivative $d_h$ which is a bounded operator and has the property that $d_h f(p)$ only depends on $S_h(p)$ , the points in distance $h$ from $p$. Geometries and fields are now just elements in the Hilbert space and Stokes theorem is the simple identity $\langle d_h^* g,f \rangle = \langle g,d_hf \rangle$ . The usual derivative can be regained by taking the limit $d_h/h$ for $h \to 0$ . For small $h$ like the Planck scale, the calculus looks the same from a physical point of view than our calculus. There is no need to build a simplicial complex approximation and break the symmetry. Any discrete frame work automatically breaks symmetries. The wave calculus keeps any symmetries from the classical case intact.

Why could this be important? I wrote about this once here in 2007 under the title three Kepler problems. I propagated there a Kepler test for a theory. It has to work on three scales: on the micro scale and successfully get all the mathematics which gives us “chemistry”, the meso scale and get the physics of our daily life like “mechanics” or “electromagnetism” and the macro scale where it has to be able to handle stars or black holes. This is hardly origional. When I was in graduate schoo, I attended a conference “On three levels” in Leuven. I contributed there even an article about deformations of discrete Laplacians [PDF]. Leuven is important for physics. It is the birthplace of the “Big Bang idea” first conceived by George LeMaitre in 1931. By the way, Lydia Bieri (who had been a postdoc at Harvard) and Harry Nussbaumer (who was my astronomy prof in the first semester at EHTZ and who was later a mentor of my wife Ruth when she was a postdoc in Zuerich), wrote once a nice book about the discovery of the expanding universe.

(*) Side remark about the finite: While finite theories are elegant and nice (and I still pursue this of course myself as this might be our only shelter if ZFC would turn out to be inconsistent), finite discretizations of continuum theories always break continuum symmetries and feel like a numerical scheme. I myself believe that finite theories per se (not intending to model any underlying continuum theory) might be the only thing really to do, if it should turn out that the continuum does not exist (at least in a rigorous strong mathematical frame work). For me, any continuum theory is finite per se, if one thinks with non-standard analysis tools. Any compact set can be described by a finite set, all data we process are finite.

What I want to do here is to have a continuum theory, where “rates of change” only involve a fixed positive distance. This is what happens also in the discrete. When we take the exterior derivative of a (k-1) form f, then df is a k-form which has the property that df(x) only depends on evaluations f(y) of y in distance 1 in the Barycentric refinement complex. We can also in the finite form “wave calculus” versions and it goes essentially the same way. The reason why we can do that in the finite is because we have a geodesic flow on discrete manifolds. This allows us to “push away” the places which influence the value of df(x).

Author: oliverknill