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五切

I am studying the manifestation of the fivesome in the fivefold classification of Sheffer polynomials of A-type 0?.

Spacetime

• Time requires a clock in the immediate vicinity. What does that mean?

Three-dimensional space

• One distinguishable dimension (in which motion occurs)
• Two indistinguishable dimensions (bosonic?)

An infinite cylinder (as in homotopy theory) is the basis for space and the reason why space is three-dimensional. Space is three-dimensional because the boundary separates outside and inside.

• The radial direction depicts outside (and also inside).
• The vertical direction depicts inside (or outside - in parallel).
• The tangent direction (of the circle) depicts the boudnary.

Time is one-dimensional because the present unites the past and the future.

• Empty space (Hermite polynomials) has particle created and then annihilated.
• Nonempty space has particle annihilated and then created (thereby moved).
• Retaining identity in space and time - a principle grounded in the combinatorics of orthogonal polynomials - thus there exists identity as a long term entanglement
• Wolfram's computational equivalence - simple program can be equivalent to the most complicated programs - and add to this the notion of degenerate programs which are less powerful. And an algebra of degeneracy which provides the desired content.
• There is a symmetry of walking away and walking back, {$\alpha$} and {$\beta$}.
• Empty space has unconnected coordinate system for each particle.
• Nonempty space (the other four possibilities) is given by pairs of coordinate systems and they represent entanglements.
• For orthogonal polynomials in quantum field theory, when particles go to another coordinate system and then return, it is because the original coordinate system is understood to be in a broader framework with presumptions about a second coordinate system.

Hermite polynomials

• Gaussian beam and Hermite-Gaussian modes relate to the Gaussian distribution and the Hermite polynomials.

Laguerre polynomials and Laguerre functions

• Laguerre function terms can be matched to the number of injective functions from subsets of [n] to [n].
• Number of terms in Laguerre polynomial.
• 1/n! on the outside of Laguerre polynomials means that the order (on one side) doesn't matter externally although it matters internally.
• In the combinatorics of Sheffer polynomials, {$\alpha$} describes the steps taken in our coordinate system away from our origin to the origin of another coordinate system, and {$\beta$} describes the steps taken from the origin of that coordinate system back to our origin. Thus the Sheffer polynomials ground the creation of space-time.

In the classification of Sheffer polynomials, we have the choices:

• {$\alpha = 0$}, {$\alpha \neq 0$}
• {$\beta = 0$}, {$\beta = \alpha$}, {$\beta \neq \alpha$}, {$\beta = \overline{\alpha}$}

So there are eight possibilities, but some of them are related.

• Wick's theorem? organizes creation and annihilation operators in very much the same way as the Hermite polynomials organize Zeng saplings. This suggests that momentum (the movement of a particle) can be thought of as the annihilation of a particle in one location and the creation of a particle in a different location. I am learning about Feynman diagrams to see how they might relate to Zeng saplings.

Adjoint probabilities

• Adjoint probabilities {$p$} and {$p-1$} are like adjoint primes. One is odd and one is even (the complement of odd) but they are in opposite directions.

读物

Dennis Stanton's paper on orthogonal polynomials and group representations

• Describes the discrete space upon which Hahn polynomials act (and thus Meixner polynomials).
• Orthogonality Theorem. There are three kinds of orthogonal relations possible for group representations that the orthogonal polynomials can express themselves with regard to. The orthogonality of the vector space basis; summing discretely over the basis with regard to the group action; integrating over the group.

Dennis Stanton. An Introduction to Group Representations and Orthogonal Polynomials.

• Weight function has us integrate over the group that the orthogonal polynomials are representing.
• The orthogonal polynomials are expressing the orthogonality of the inner product space.

Orthogonal polynomials

• Redefine notion of eigenfunction to be the orthogonal polynomial. Weight function (space time) is for orthogonality. It is the space-time measure and there are 5 different kinds. And they refer to probability, they impose it.
• Mesuma K. Atakishiyeva, Natig M. Atakishiyev, Kurt Bernardo Wolf. Kravchuk polynomials and irreducible representations of the rotation group SO(3) related to quantum harmonic oscillator and to SU(2). Mentions that Charlier and Meixner are related to SU(1,1).
• G.Staples. Kravchuk Polynomials and Induced/Reduced Operators on Clifford Algebras
  • Kravchuk polynomials can be interpreted as matrix elements for representations of SU(2). T. Koornwinder, Krawtchouk polynomials, a unification of two different group theoretic interpretations, SIAM J. Math. Anal., 13 (1982), 1011–1023.
  • In quantum theory, Kravchuk matrices interpreted as operators give rise to two new interpretations in the context of both classical and quantum random walks. The latter interpretation underlies the basis of quantum computing. P. Feinsilver, J. Kocik, Krawtchouk matrices from classical and quantum random walks, Contemporary Mathematics, 287 (2001), 83–96.
  • In the context of the classical symmetric random walk, Kravchuk polynomials are elementary symmetric functions in variables taking values ±1. P. Feinsilver, R. Schott, Krawtchouk polynomials and finite probability theory, In: H. Heyer (ed.) Probability Measures on Groups X (Olberwolfach 1990), pp. 129–135, Plenum, New York, 1991.
• Doha, Ahmed. Recurrences and explicit formulae for the expansion and connection coefficients in series of classical discrete orthogonal polynomials Has recurrence formulas for Kravchuk, Meixner, Charlier polynomials.

Dongsu Kim and Jiang Zeng are the authors of the tree paper. "A Combinatorial Formula for the Linearization Coefficients of General Sheffer Polynomials"., 2001.

• Dongsu Kim, Department of Mathematics, KAIST, Taejon 305-701, Korea, E-mail: dskim AT math.kaist.ac.kr
• Jiang Zeng, Institut Girard Desargues, Université Claude Bernard (Lyon I), 69622 Villeurbanne cedex, France, E-mail: zeng AT desargues.univ-lyon1.fr
• Citations of their paper
• [[https://physics.stackexchange.com/questions/304508/why-are-all-my-separable-solutions-orthogonal-polynomials-in-l2 | Why are all my separable solutions orthogonal polynomials in {$L^2$}?

Orthogonal polynomials

• Entangled knowledge {$\alpha$}, {$\overline{\alpha}$} are opposite answers to a shared question.
• {$\alpha$}, {$\beta$} are disentangled knowledge
• Entangled {$\alpha$}, {$\overline{\alpha}$} same subsystem {$\Rightarrow$} disentangled {$\alpha$}, {$\beta$} different subsystems. Is the direction correct?

Coordinate systems - gap between self (where you are - local) and space (where you were - global).

• {$\alpha$}, {$\beta$} nonentangled (after collapse) the global system is distinct
• {$\alpha$}, {$\overline{\alpha}$} entangled - global system is intricately related to the local system
• {$\alpha$}, {$\beta$} expresses the deviation from the bound state
• {$\alpha$}, {$\overline{\alpha}$} ęxpands upon the bound state as given by the imaginary component

Think of orthogonal polynomial classification roots alpha, beta as probabilities. Then the total probability is alpha x beta.

Negative probability

• My perspective: negative probability is the opposite probability in the reverse time direction. So the probability p is balanced by the probability p-1 = -(1-p). Something happening in one direction is balanced by something unhappening in the opposite direction.
• My perspective: probability and negative probability take place as steps in opposite sides of a particle clock. If we measure the probability on one side, then we destroy the probability on the other side.
• Richard Feynman. Negative probability.
• X is the actual probability, Y is the measurement error of X, and Z=X+Y is the observed value. Thus X may be negative but it is shielded by Y.
• The negative regions of the distribution are shielded from direct observation by the quantum uncertainty principle: typically, the moments of such a non-positive-semidefinite quasiprobability distribution are highly constrained, and prevent direct measurability of the negative regions of the distribution. Nevertheless these regions contribute negatively and crucially to the expected values of observable quantities computed through such distributions.
• In Convolution quotients of nonnegative definite functions[5] and Algebraic Probability Theory [6] Imre Z. Ruzsa and Gábor J. Székely proved that if a random variable X has a signed or quasi distribution where some of the probabilities are negative then one can always find two random variables, Y and Z, with ordinary (not signed / not quasi) distributions such that X, Y are independent and X + Y = Z in distribution. Thus X can always be interpreted as the "difference" of two ordinary random variables, Z and Y. If Y is interpreted as a measurement error of X and the observed value is Z then the negative regions of the distribution of X are masked / shielded by the error Y.
• The proposed “propensity” concept in Xie et al.[14] turns out to be what Feynman and others referred to as “quasi-probability.” Note that when a quasi-probability is larger than 1, then 1 minus this value gives a negative probability. In the reliable facility location context, the truly physically verifiable observation is the facility disruption states (whose probabilities are ensured to be within the conventional range [0,1]), but there is no direct information on the station disruption states or their corresponding probabilities. Hence the disruption "probabilities" of the stations, interpreted as “probabilities of imagined intermediary states,” could exceed unity, and thus are referred to as quasi-probabilities.
• Wigner quasiprobability distribution is the Wigner–Weyl transform of the density matrix, so the realization of that operator in phase space. In 1949, José Enrique Moyal, who had derived it independently, recognized it as the quantum moment-generating functional,[5] and thus as the basis of an elegant encoding of all quantum expectation values, and hence quantum mechanics, in phase space.
• Phase-space formulation The chief advantage of the phase-space formulation is that it makes quantum mechanics appear as similar to Hamiltonian mechanics as possible by avoiding the operator formalism, thereby "'freeing' the quantization of the 'burden' of the Hilbert space".[5] This formulation is statistical in nature and offers logical connections between quantum mechanics and classical statistical mechanics, enabling a natural comparison between the two (see classical limit).
• The decision point (present, boundary) is the quantum {$x$} which commutes with either ground so that {$\int p_k x p_n$} can be interpreted ambiguously, and so relations with distant, nonproximate levels can be discarded.
• The Sheffer polynomials are perhaps characterized by the operator {$x$} and its action on {$P_n(x)$} whereas the classical orthogonal polynomials are perhaps characterized by the operator {$\frac{\textrm{d}}{\textrm{dx}}$} as with the Rodrigues formula.
• Moments - the numbers give the impact of {$x^k$}.
• Hankel determinant - the numbers describe what momentum is.
• Wolfram's cellular automata can be thought of as acting on the hierarchy of levels, as with raising and lowering operators.
• Krautchouk polynomials are a family that consists of only finitely many polynomials. The higher polynomials are zero. This introduces the notion of finiteness. It may be a sixth possibility that expresses the finite human situation, morality, the sixsome.

Sheffer classification gives 5 possibilities for synchronization. Consider {$\alpha$} and {$beta$} as vectors in the plane. Then we can have:

• Two points at the origin.
• A point at the origin and a vector.
• The same vector twice.
• Two nonzero vectors of different length on a line.
• Complex conjugates in the plane
• The configurations that orthogonal polynomials code explain how phase space can be divided into distinct trees or cycles. The division of phase space into more compartments requires higher energy.
• Orthogonal polynomial growth configurations (Zeng nurseries) can be thought of as a nondeterministic algorithm.
• Consider Aristotle's four causes in terms of the two kinds of causality in decision making.
• Empty space is given by the weight. Differentiating it yields the same (almost) - a difference by a quantum. Differentiating {$x^k$} yields a lower degree, thus clearly different, and similarly with {$P_n(x)$}.
• https://en.m.wikipedia.org/wiki/Sheffer_sequence is what i am interested in.

Sturmian functions - harmonic oscillator

• Sturmian words - billiard ball bouncing between horizontal and vertical sides of table
• Sturmian functions https://arxiv.org/pdf/quant-ph/9809062.pdf
• This paper relates Sturmian words, Sturmian functions and Sturmian numbers.
• Relativistic quantum harmonic oscillator
• A.Poszwa. Relativistic Generalizations of the Quantum Harmonic Oscillator
• Spherical harmonics
• R. P. Martı́nez-y-Romero. Relativistic quantum mechanics of a Dirac oscillator 1999. Free article. Refers to spinor spherical harmonics.
• The spinor spherical harmonics are the natural spinor analog of the vector spherical harmonics. While the standard spherical harmonics are a basis for the angular momentum operator, the spinor spherical harmonics are a basis for the total angular momentum operator (angular momentum plus spin). These functions are used in analytical solutions to Dirac equation in a radial potential.
• See: Greiner, Walter (6 December 2012). "9.3 Separation of the Variables for the Dirac Equation with Central Potential (minimally coupled)". Relativistic Quantum Mechanics: Wave Equations. Springer. ISBN 978-3-642-88082-7.
• Relativistic hydrogen atom
• R. P. Martı́nez-y-Romero. Relativistic hydrogen atom revisited American Journal of Physics 68, 1050 (2000). The hydrogen atom is solved using a simple method. We show that this system has an exact solution that can be written in terms of Laguerre polynomials of noninteger index, instead of the hypergeometric series. This point is important because Laguerre polynomials of integer index appear in the solution of the nonrelativistic hydrogen atom, giving students a more unified point of view for this system.
• A. D. Alhaidari, H. Bahlouli, M. E. H. Ismail. The Dirac-Coulomb Problem: a mathematical revisit Alhaidari researches orthogonal polynomials. Describes the relativistic hydrogen atom in terms of Pollaczek polynomials.
• Guillermo Palma, Ulrich Raff. The One Dimensional Hydrogen Atom Revisited
• Spacetime is made of events (the coordinates). But in quantum mechanics, an event is only when a wave function collapses. So time and space combine only when the wave function collapses. So relativity is only important for the actual collapse of the wave function. The wrapper {$e^{-x^2}$} may explain the quadratic nature of gravity. But how?
• Orthogonality measures self-duality. Two pyramids give the perspectives of position and momentum.
• Delayed signals? (Kinks?) Attiyah: Retarded Dirac operator. Attiyah, Moore. A Shifted View of Fundamental Physics.
• Time-delayed Dirac delta, Group delay and phase delay
• Subsystem requires 2 coordinate systems, thus the orthogonal polynomial combinatorics.
• Gravity is the relation between 2 coordinate systems.
• {$\alpha$}, {$\beta$} unentangled, {$\alpha$}, {$\overline{\alpha}$} entangled.
• Qft in terms of discrete time events based on the energy level. No need for continuum.
• Supersymmetry describes question-containers for particles.
• The conceptual foundation from which we can derive Minkowski space, special relativity, quantum field theory.
• When you reach the global quantum (or zero) then you can say if you have been counting up or down.
• Hermite. {$\alpha,\beta =0$} means you won't run into the global quantum {$\gamma$}, which then goes to infinity.
• Dictionary data structure relates combinatorics and computer science.
• Dictionary data structure relates questions and answers, thus global and local.
• Add or delete particles.
• Histoire expresses memory.
• Morality means experiments that link the local and global through independence of the scaling of x. An experiment asserts that scaling and translation are independent.
• The fivesome, Minkowski space don't honor independent experiments as morality honors.

Interpreting how an observable as an operator that is acting as a shift upon the Hermite polynomial (the eigenfunction)

• Potential - is a term that is a second derivative of space-time
• Observable (energy) is (derivative of space-time)(derivative of information)
• Second derivative of information gives zero (after imposing orthogonality)

Sheffer polynomials - Quantum field theory

• In calculating the moments, the permutations are considered in terms of ascents (raising operators) and descents (lowering operators). Thus the ascents and descents express the workings of the particle's clock. There are not only creation and annihilation operators.
• For Sheffer polynomials, translation {$f$} occurs only in {$A(t)$} (for compactness) and not in {$u(t)x$} (the Lie algebra).
• In a permutation, ascents and descents turn around whether we are counting forwards or backwards.
• Relevant for the particle clock: Principle of least proper time. PBS Space Time. Is ACTION The Most Fundamental Property in Physics?
• Particles don't move, they disappear and reappear. Their clock is given by all the possible clocks according to their zone (what state they are acting on, which may be the null state, or the bound state, or so on.)
• In my theory, based on the Sheffer polynomials, the interaction has more time events, or rather, some of them are internal (kinematic).
• A particle-clock relates two vertices as two coordinate systems and the paths back and forth. This can be thought of as two-point kinematics.
• Have <creation | interaction | annihilation>.
• Make the Hermitian conjugate of the creation open.
• Energy levels? go as {$(m+1)^2=m^2+2m+1$}. So we get nonzero energy because of spin because {$m$} bleeds over. Spin is the "square root" because of nonmeasurability that arises between raising and lowering, creation and annihilation - it is as if a mid-state for that. Halfway between levels. Consider the contributions of creating and annihilation {$\sqrt{m}\sqrt{m+1}$}.
• Fivesome - see picture in Griffiths - (1 and 5) Radiation zone, (2 and 4) Intermediate region V=0 (centrifugal force), (3) Scattering region ({$V\neq 0$}).

Minkowski space-time

• Scaling and translation of orthogonal polynomials are dependent.
• Conflicting requirements that f=0 and that the polynomial be monic. Add a sixth variable (morality) as a coefficient times the monic, and this will be similar to making x = pN, but independently scaled. Morality is this independent scaling, what is small and what is large (just the large by the small)(show good will - small is enough).
• The 5 natural bases for {$L^2$} or the 6 natural bases for the symmetric functions are perhaps related to M-theory.

Kinks

• There are 5 conic sections plus 3 more: the point C (where the cones meet), two lines that cross in an X, and the line segment (which arises in the limiting case as the hyperbola passes over to an ellipse). Given any other point P on the lower cone, P not C, then we get conic figures by rotating the plane that contains C along the axis that is tangent to the cone. We can start with a circle, then it becomes an ellipse ever longer, until it reaches infinity and becomes a parabola, which then intersects the upper cone as a hyperbola, which then becomes a line as the hyperbola gets squeezed and infinitely narrow, but then the line drops down to a line segment from P to C, from where it opens up as an ellipse, which grows in width until it becomes the circle. So the line segment is a moment of ambiguity where it equals the line and yet the limit cases from either side are different. The line segment seems like a string or loop in string theory.
• Nurseries, Zeng saplings
• An observer carves up spacetime into kinks.
• Kinks express the need that a question be answered, that there be resolution, that tension be resolved, which is important in all conceptual languages.
• Kinks express the yearning for justice, that everything be set right, that there be no loose ends.
• Kinks express the obligations met by a pushdown automata, as with balanced parentheses.
• Measurement is a trivial translation of coordinate system from x to x-a and thus relates to the trivial self-entanglement of a tree.
• Physicists know how to copy equations, how to manipulate them, how to solve them, how to set them up, but not how to read them mathematically, conceptually. Good physicists can read the physics and yet not the math.
• There is an alternating view (swapping momentum and position) where a shift in momentum is manifestation of position.
• {$x^n$} is n units of free space. Related to nth energy level.
• Space-time is a back drop of randomness which expresses orthogonality.
• Ehrenfest model ) of diffusion was proposed by Tatiana and Paul Ehrenfest to explain the second law of thermodynamics. The model considers N particles in two containers. Particles independently change container at a rate λ. If X(t) = i is defined to be the number of particles in one container at time t, then it is a birth–death process with transition rates. Related to Kravchuk polynomials.

Kravchuk polynomials are a special case of the Mexiner polynomials of the first kind. They are only a finite sequence.