The QR formula exp(-t D)= QR actually gives the Lax deformation of the Dirac matrix. This gives an explicit time t map about the symmetry of a finite geometry.
[Update: the paper arxiv.org/abs/... of this note. The source code on the ArXiv gives also access to mathematica code. It is now also pasted into the quantum calculus blog www.quantumcal...]
This is useful for example because we can give now motions within the symmetry that are local. We can also get explicit expressions for deformed matrices where the entries are algebraic. If we think about the deformation with a point of view of Arnold who saw geodesics in a symmetry space as fundamental (like the Euler equations as geodesics in the diffeomorphism group) , we do not want that the geometry far away from a point immediately influences the behavior at the point. That would violate causality. If we look at the Lax deformation, a nonlinear ODE, then like the linear wave equation which is used to measure distance has in the discrete the property that the motion in general is influenced by parts very far away, in the wave equation case simply because the eigen-functions depend globally on the entire space. (I had found in the discrete a discrete time version of the wave equation that honors causality). The video at the beginning and end was done on Friday near the community boating place in Boston.
21 сен 2024
