Combinatorial Interpretation of the Schrödinger Equation
Channel: Math 4 Wisdom
Date: 2023-02-01

I sketch a combinatorial interpretation of the Schroedinger equation:
A) Orthogonal polynomials encode "particle clocks" which relate pairs of reference frames in terms of steps α and β.
B) Integration of a monomial x^n against a weight function w(x) works as a "space-time wrapper" and is given by a sum over a permutation of n.
C) The Hamiltonian acts as a quantum perturbation which adds and removes a compartment and likewise removes and adds a compartment.

The steps α and β are specialized in the fivefold classification of orthogonal Sheffer polynomials.  This defines five zones in scattering processes.

Andrius Kulikauskas (Math 4 Wisdom)(that's me!) presented this work-in-progress on January 9, 2023, to physicist Thomas Gajdosik (Vilnius University) and mathematician John Harland (Palomar College).  Here are the notes which I discuss in the video:
https://www.math4wisdom.com/wiki/Exposition/20230109ResearchProgramFivesome

Timeline:
0:30 John Harland's search for an agency machine, which we discuss later.
6:40 My approach is informed by my PhD in algebraic combinatorics, the fivefold classification of orthogonal Sheffer polynomials, and a fivefold cognitive framework for decision making ("Every effect has had its cause; but not every cause has had its effect; so there is a critical point for deciding")
12:48 Replying to John... My prejudice towards the discrete rather than the continuum.
14:45 Three combinatorial objectives: Orthogonal polynomials, space-time wrapper, Hamiltonian.
17:55 Meixner's and Sheffer's papers relate the recurrence relation and the differential equation.
19:05 Josef Meixner's biography before, during, after the Nazi era.
22:55 Recurrence relation encodes one-step "l" and two-step "k" causality. Combinatorics by Dongsu Kim and Jiang Zeng.
30:20 Relating "l, k" causal tree and "α, β" particle clock interpretations for calculating u'(t) and A'(t)
35:45 Physical interpretation of fivefold classification as zones in scattering theory. Arisal, persistence, collapse of wave function.
38:45 Interpreting a step as "what does not happen". Thus distinguishing the observer and the observed. Relating this to John's agency machine.
42:00 John asks for an example of the consequence for physical calculation. Bound states such as with strong force. Spontaneous collapse theory. Wave function based on subsystem.
47:15 The action of the Hamiltonian on a Hermite polynomial / function.
57:30 Raising and lowering operators interpreted as creating and removing compartments. 
1:00:45 Quantum perturbation. Commutator. Doing the least possible and seeing the effect, which is energy. Potential energy as what is needed to cancel away higher order kinematics.
1:03:00 A(t)=e^γv(t)  is exponential just like e^xu(t). Compared with phase factor i(px-Et)
1:07:15 Compare combinatorics of  "l, k" causal trees and "α, β" particle clocks.
1:08:15 Conclusion and discussion.
1:08:45 A(t)=e^γv(t)  is exponential just like e^xu(t).
1:10:45 Agency machine and statistics of what does not happen.
1:11:45 Minkowski space?
1:12:20 Restating as quantum field theory. Combinatorics of Hermite polynomials matches Wick's theorem. Extending Feynman diagrams by interpreting kinematics of edges.
1:13:30 Interpreting specializations of space-time wrappers.  Continuum arises from the discrete!
1:16:30 Raising and lowering operators.
1:17:00 Combinatorics of the moments.
1:17:45 Ways this model rethinks and extends traditional quantum mechanics.
1:21:50 Meixner's beautiful operator t(D) is the lowering operator.
1:24:30 Self-interacting terms. Sidestepping renormalization issues. Infinite series t(D) acting on finite polynomial is "as big as it needs to be".
1:26:56 Combinatorics of the linearization of Hermite polynomials.
1:31:10 Concluding remarks

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