Motivated by the Hamiltonian of the Hydrogen atom, we can look at an anlogue operator for finite geometries and study the spectrum. There is an open conjecture about the trace of this operator.
The quantum line In one dimension, there is a natural compact metric space D on which one has a translation group which features a smallest unit. The group D of dyadic integers is the Pontyagin dual of the Pruefer group and is a natural “one-dimensional quantum space”. The real analogue is the circle, a compact topological group which is the … ….
The Barycentric limit of the density of states of the connection Laplacian has a mass gap.
The tensor product is defined both for geometric objects as well as for morphisms between geometric objects. It appears naturally in connection calculus.
As we have an internal energy for simplicial complexes and more generally for every element in the Grothendieck ring of CW complexes we can run a Hamiltonian system on each geometry. The Hamiltonian is the Helmholtz free energy of a quantum wave.
Energy theorem The energy theorem tells that given a finite abstract simplicial complex G, the connection Laplacian defined by L(x,y)=1 if x and y intersect and L(x,y)=0 else has an inverse g for which the total energy is equal to the Euler characteristic with . The determinant of is the Fermi characteristic . In the spring 2017 linear algebra Mathematica … ….
Over spring break, the Helmholtz paper [PDF] has finished. (Posted now on “On Helmholtz free energy for finite abstract simplicial complexes”.) As I will have little time during the rest of the semester, it got thrown out now. It is an interesting story, relating to one of the greatest scientist, Hermann von Helmholtz (1821-1894). It is probably one of the … ….
Energy U and Entropy S are fundamental functionals on a simplicial complex equipped with a probability measure. Gibbs free energy U-S combines them and should lead to interesting minima.
Entropy is the most important functional in probability theory, Euler characteristic is the most important functional in topology. Similarly as the twins Apollo and Artemis displayed above they are closely related. Introduction This blog mentions some intriguing analogies between entropy and combinatorial notions. One can push the analogy in an other direction and compare random variables with simplicial complexes, Shannon … ….
An experimental observation : the sum over all Green function values is the Euler characteristic. There seems to be a Gauss-Bonnet connection.