Quantum Narrative as Moments, Poles, Distributions of Orthogonal Sheffer Polynomials
Channel: Math 4 Wisdom
Date: 2024-01-09

Orthogonal Sheffer polynomials are central to key solutions of Schroedinger's equations.  There are five different kinds, and we can think of them as modeling a quantum narrative across five zones of an interaction potential, whereby a wave function arises and collapses.  We can try to understand this from the combinatorics of the polynomal coefficients, the moments, the distributions and also the complex geometry of the poles of the generating functions of the moments.

Andrius Kulikauskas, host of Math 4 Wisdom, gave this presentation on December 21, 2023, to John Harland, leader of the Math 4 Wisdom Physics Study Group.  We invite you to join us!  Sign up for our Math 4 Wisdom email group https://www.freelists.org/list/math4wisdom

1:10 Five types of Sheffer polynomials: Meixner, Charlier, Laguerre, Hermite, Meixner-Pollaczek
1:35 Constraint A(t)e^{xu(t)} satisfied by Sheffer polynomials.
2:20 Polynomial coefficients encode combinatorial objects.
2:30 Moments encode ordered set partitions, set partitions, permutations, perfect matchings, alternating permutations.
3:35 Looking for combinatorics in the weight functions that give the distributions. Deriving the distributions from the moments. John taught me how to do this using the Fourier transform.
https://www.youtube.com/watch?v=AYIdcZHenV0
5:35 Jiang Zeng has a beautiful formula for the moments as weights on permutations.
6:45 The quantum narrative in terms of steps alpha and beta. Five zones: asymptotically free, arisal of wave function, bound state, collapse of wave function, entanglement.
8:35 Particle clocks.
9:15 Adjoint string of forgetful functors on moments and free constructions on distributions?
10:45 Calculating the inverse Fourier transform of the moments. Complex integration with the residue theorem.
11:00 Hermite case. Gaussian integral.
11:40 Laguerre case. A single pole. Residue theorem.
13:45 Charlier case is discrete.
15:30 Meixner case is discrete. I have not done the Meixner-Pollaczek case.
16:00 The combinatorics of the moments gives meaning to the distributions.
17:05 The complex geometry of poles.
17:55 John says in discrete cases we can do Fourier transform over discrete groups. Andrius: Laplace transform, z-transform. Infinitely many poles?
20:20 Quantum narrative starts with discreteness. The mean moves with step alpha minus beta. But then when beta equals alpha, in the degenerate case, the mean does not move and we get continuity.
21:20 Information of the combinatorial world is given in the complex geometry of poles. Quantum narrative tells how space time wrappers change.
22:55 When the wave function collapses, the pole goes away.
23:20 Laguerre polynomial case has a pole because of the Heaviside function. (Erratum!  my slides are wrong, the pole is above the x-axis at i/2pi).
24:55 Closing the loop from moments to distributions. Many technical details. 
26:30 Erratum! This slide is wrong, the first two weights should have minus signs: - alpha, - beta.
26:55 Relation to relativistic causality. The combinatorics of Hermite polynomials is that of Wick's theorem. It may all lift up from the Schroedinger equation to quantum field theory. The importance of quantum field theory, Feynman diagrams.
28:55 John is studying the Dirac equation. Representations of the Lorentz transformation have a spinor aspect - discrete aspect, in addition to the continuous aspect. Rotations.
30:45 Invitation to join our physics study group. Concluding prayer for peace in the world.
32:05 Thank you for support through Patreon.

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