Dynamical Connections - Quantum Calculus

In the evening of the 13th of September after giving the talk below, it dawned me after doing lots more experiments that the connection dynamical connection Laplacian and the dynamical connection Green function that were written down on the right hand side of the black board work together if one takes the transpose in one. Here is the setup written down there: G is a finite abstract simplicial complex, a finite set of non-empty sets closed under the operation of taking finite non-empty subsets. The core of a simplex x is the set of all simplices contained in x, the star of a simplex x is the set of all simplices containing x. Every open set is a finite union of stars, every closed set is a finite union of cores. The complex G defines a nice finite topology, the Alexandrov topology. It never really became popular which I believe is due to human bias: Hausdorff convinced Hopf and Alexandrov to focus in their book on Hausdorff spaces.

[P.S. Human bias is a very strong force. There are some “influencer mathematicians” (also today) who decide what is fashion and cool independent of the real value. The rest of the mathematicians follow like sheep; which makes sense as you want to work on something that is fancy and allows you to attract some interest. Eventually you of course want to have a job which allows you to do what you like. I myself did that: I ran onto the “fractal” and “chaos” train when it was fancy (once you get older this becomes irrelevant. You can afford to not care any more what the rest of the community thinks.) It is also personal relations that dictate the outcome of mathematical research and path the way for the future. Heinz Hopf and Pavel Alexandrov were good friends and so needed to compromise on this project. There is a didactic component: non-Hausdorff spaces are less intuitive. We see from experience that any two points can be separated by open sets. Two vertices x,y in a finite abstract simplicial complex that are part of a larger simplex z can not be separated because any neighborhood of x or y both contain z. Remarkable is that Alexandrov followed the advise despite the fact that Alexandrov worked on finite topologies that are non-Hausdorff. But Alexandrov was a nice person and a good friend of Hopf and these two authors had to agree on such fundamental matters. In retrospect it had been a wise decision to focus on Hausdorff spaces. The first truly successful book on topology might not have become so popular if the authors would have allowed for more generality. Too much generality can often be poison for a wider adaption. ]

In the finite, there is no choice. Hausdorff topologies on finite spaces can not be used as it implies the topology is discrete (the topology is the set of all sets, completely unfit as even connnectivity fails). A finite Hausdorff space is completely disconnected as all sets are both open and closed (clopen). I defined the connection matrix $L(x,y) = \chi( C(x) \cap C(y))$ and the Green function matrix $g(x,y) = w(x) w(y) \chi(U(x) \cap U(y))$ , where $w(x) = (-1)^{{\rm dim}(x)}$ and $\chi(A)=\sum_{x \in A} w(x)$ . What happens is remarkable: $g=L^{-1}$ and $\sum_{x,y \in G} g(x,y) = \chi(G)$ and ${\rm det}(L)=\prod_{x \in G} w(x)$ . This was later generalized vastly, where w(x) is a rather arbitrary ring valued function, even division algebra valued non-commutative rings like the quaternions. On Friday, I discovered first experimentally an other generalization which did not quite fit yet while talking. On Saturday night, I saw the correct statement: let $T: G \to G$ is a simplicial map, meaning a continuous map (=order preserving map on the natural poset structure given by the inclusion relation) and coming from a permutation on the vertex set $V={\rm dim}^{-1}(0)$ . Now define the dynamical connection matrix $L(x,y) = \chi(C(x) \cap C(T(y))$ and the dynamical Green function matrix $g(x,y) = w(x) w(y) \chi(U(x) \cap U(T(y))$ . I noticed that both are unimodular but they did not always fit . (They were not inverses of each other). Sometimes they did and what happened is that if they were symmetric, they fit. Late at night on Saturday, I just tried to see whether $g^T=L^{-1}$ and indeed it does! This should easily go over to energized complexes. The most elegant write-up about this is in this paper where already non-symmetric connection matrices occur and where the Green star identity already was g*L = 1 where * is the complex conjugate transpose. Well, it only took a day this time to realize this. The Green-star formula needed many months to hatch as I had focused on finding a formula that used closed sets while U(x) is open. Anyway, it is nice to see that the connection calculus works well with symmetries. An invertible simplicial map can always be thought of as a symmetry of the geometry. The group Aut(G) is of course always a finite group. The most symmetric space is $G=K_n$ , where the automorphism group is $S_{n}$ the symmetric group of n elements. Note that the simplicial complex G has $2^{n}$ elements. For most complexes, the automorphism group is trivial. This is one reason also why I like to extend the Lefschetz fixed point theorem to more general maps, maps from the topology of G to itself. These are then open maps and any choice of inverse of such a map is then continuous. In the talk below I talk a bit more about the map $T: C_n \to C_n, x \to 2x {\rm mod} \; n$ which is not a simplicial map but which defines a map on the topology $\mathcal O$ of the one dimensional complex $G=C_n$ which has 2n elements, the vertices and edges. The algebraic map refers to the algebraic group structure on $C_n$ given by Abelian group $Z_n$ . The fixed point theorem works nicely in that case. If the Koopman operator is correctly defined for a general open map, the fixed point theorem would generalize greatly and allow to model arbitrary continuous maps on topological spaces using finite mathematics alone. The sum of the indices of fixed points is the super trace of that Koopman operator and after applying the heat flow (the only thing which needs to work is McKean-Singer symmetry), one has the super trace on cohomology which is the Lefschetz number.

For me personally, this is so far the biggest mathematical surprise in my entire life: it is the fact that the connection matrix L of a simplicial complex always has an integer inverse which is given by a concrete formula, what I called the green star formula. It is the explicit formula for the entries of the Green function in terms of stars. It had been a tough fight because back then in 2016, I did not think in term of topology as I had looked for formulas involving the Euler characteristic of sub-simplicial complexes. The connection matrix L has as the entries L(x,y) the Euler characteristic of the intersection of the cores C(x),C(y) of x and y. Its inverse, the Green function g(x,y) is the Euler characteristic of the intersection of the stars U(x),U(y) of x and y. The nomenclature for the name Green function comes from the fact that in the continuum, the inverse of a Laplacian is called the Green function. It is one of the most important objects in mathematical physics because we want to solve Poisson type equations L u = b. In electro-statics for example, b is the charge density and u the electric field. In gravity, b is the mass density and u the gravitational field. What Gauss did by replacing ordinary differential equations like the Newton equation by partial differential equations is remarkable, similarly as Maxwell’s description of laws in terms of partial differential equations – the Maxwell equations. In both cases, it allowed to formulate electromagnetic or gravitational fields to be formulated on any Riemannian manifold or more generally on any geometric space with a Laplacian. As for the connection Laplacian, I see it more in line with the Dirac matrix D because both L and D have also negative eigenvalues. In the case of the Dirac matrix, the number of positive and negative eigenvalues is the same, in the case of the connection matrix, the number of positive eigenvalues minus the number of negative eigenvalues is the Euler characteristic. I dubbed this with the slogan “one can hear the Euler characteristic of a simplicial complex”.

What is really nice from the perspective of intuition is that g(x,y) can be interpreted as the potential energy between the x and y. The reason is that the sum over all these potential energies g(x,y) is the Euler characteristic of the complex G which is the sum of the energy values w(x) when x runs over G. The determinant of L and so the determinant of g is equal to the Fermi characteristic, the product of the energy values w(x), when x ranges over the finite set G. I called this the energy theorem. Maybe it should be called the potential energy theorem for simplicial complexes. The fact that a basic Laplacian is invertible and that the spectral gap remains even in van Hove limits of infinite simplicial complexes is truly remarkable. All the difficulties with infinities in the continuum are washed away. In the continuum three dimensional space, the Hodge Laplacian has nasty 1/r singularities. Already in dimension 2 when we look at the Laplacian on the 2-dimensional plane, we have a logarithmic singularity. Again unpleasant. In 4 dimensions and higher the singularities become even nastier because stable planetary motion become impossible by Bertrand’s theorem. If we look at partial differential equations like the wave equation Lu=0, where L is the d’Alembert operator in space time, then this is a nice linear equation which we learn to solve using harmonic analysis. But hell breaks lose, if we perturb it in a non-linear way like Lu = c sin(u), in a sine-Gordon type manner. Non-linear wave equations need lots of mathematical skill [I once took a short “Nachdiplom Vorlesung” run by Michael Struwe at ETHZ about nonlinear wave equations. It is a topic that is heavy in functional analysis, calculus of variations. Nonlinear partial differential equations need discipline and a huge analytic background. I myself prefer simpler mathemnatics.]

What happens with non-linear perturbations like the sin-Gordon equation or much simpler models like the Frenkel-Kantorova model or even more simpler models like the Chirikov Standard map Lu = c sin(u) with Lu(n)=n(n+1)-2u(n)+u(n-1) is that the perturbation from c=0 to non-zero c is hard. Already in the simplest case, the Standard map, one needs Kolmogorov-Arnold-Moser theory to tackle stability. In higher dimensions this is even not enough mathematically as orbits can leak through the vague attractor of Kologorov and stability for complicated systems like our Solar system only is an “engineering stability” a practical stability. The Nekoroshev leaking which almost certainly happens, is just too slow to matter. Almost certainly, there is no stability in a mathematical sense. Moser’s iconic question “Is the Solar system stable?” is almost certainly “no” in a mathematical sense but since it would take billions or even trillions of years to matter, other factors matter much more. Like in many other cases of statistical mechanics the mathematical questions are different than the engineering question. For example, mathematically, entropy remains constant as basic evolution laws are invertible, the second law of thermodynamics is an illusion as it refers to practicalities, as Zermelo already knew. Less mathematically and more rooted in application-based folks (Ilyia Prigogine for example as well as most contemporary physisists work with the postulate that entropy increases which is non-sense for a mathematician who works with the axiomatic assumption that all fundamental laws of physics are invertible. And this axiomatic assumption is based on experiments. [Indeed, nobody has found a truly fundamental process that is not reversible if everything is taken into account. Exceptions are situations like Black hole evaporation where the mathematical foundations are murky and where we are in the realm of speculation, physics but certainly not rigorous mathematical physics. As t’Hooft already pointed out once that we have no idea what happens with the singularity when a black hole has completely evaporated. ]

What happens if we look at the connection calculus perturbation problem L u = c sin(u), even in infinite simplicial complexes G with a bound on the size of unit spheres is that the vacuum is stable. Mathematically stable. And it does not involve the hard implicit function theorem a la Nash-Moser. It only requires the soft implicit function theorem. It means that the solution u=0 which is unique for c=0 remains a unique stable solution to L u = c sin(u). The question of course is whether this is physically relevant. One of the issues with connection calculus that it is something that only manifests in quantized space meaning that we look at finite abstract simplicial complexes. It is different from discretizations of the Hodge story. If you look at the wave equation on a finite abstract simplicial complex which is an ordinary differential equation telling that the acceleration u” of a function u is H u, where H is the Hodge Laplacian, then we are in trouble already for purely causal reasons. Unlike the wave equation in the continuum which has a finite propagation speed, the naive discrete wave equation in the continuum is non-causal. We learn that already in introduction linear algebra courses. We use Fourier theory to solve the wave equation but the eigenfunctions are not localized. You can try this out. Take a discrete world of a cyclic graph with n nodes, then we can solve the wave equation u”(k) =u(k+1)-2u(k)+u(k-1) =L u by diagonalizing the Jacobi matrix L which is the Kirchhoff matrix of the cyclic graph. If u is an eigenfunction to a non-zero eigenvalue l, then u(t)=u(0) cos(c t) + u'(0) sin(c t)/c where c is the square root of l as we all learned in linear algebra. The upshot is that if we discretize space, then we also have to discretize time. This is not necessary if we work with a Laplacian that is invertible. The miracle in the connection Laplacian case we have not only stability but also finite propagation speed of the wave equation. The Green function does not “reach far”. There is causality. For the discrete pace wave (continuous time) equation on a lattice, even a tiny change 10 billion miles away influences the wave propagation instantaneously. If we use the connection Laplacian (or rather its square as the connection Laplacian L behaves more like the Dirac operator D (and only $L^2$ behaves like $Latex D^2$, spooky action does not happen. Spooky action also does not happen if one discretizes time, but then one is in the realm of cellular automata. I worked in the beginning of this year 2025 on a notion of geodesic flow and interacting particles that is a cellular automaton.

Author: oliverknill