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Research program fivesome, Kim Zeng involution

A Combinatorial Alternative to the Wave Function

• A combinatorial formula for the probability density of a particle's state
• An opportunity to rethink the wave function

Born's rule states that the probability density {$p(x)$} of finding a system in a given state {$x$} (for example, a particular position {$x$}) is proportional to {$|\psi(x)|^2$}, the square of the amplitude of the system's wavefunction {$\psi(x)$} at {$x$}. Note that what is implicit and may not be obvious until we actually do calculations is that this wavefunction is understood within the context of a physical system that is understood to have a domain of possible values and typically a Hamiltonian with a specified potential.

I am exploring an alternative combinatorial understanding which in various cases gives the same results but provides a more explicit and elaborate context that can apply to new situations. My hope is that this interpretation may ultimately prove to be more natural, intuitive and informative than the wave function formalism. I am sharing my current, incomplete and tentative understanding so that others could work on this and we could possibly work together.

The formula for calculating the probability density {$p_n(x)$} of finding a system {$(P_n,\alpha,\beta,c)$} in a state {$x=(\alpha - \beta)k$} is given by:

{$p_n((\alpha - \beta)k) = (1-\frac{\alpha}{\beta})^c \binom{c-1+k}{k}(\frac{\alpha}{\beta})^k\binom{c-1+n}{n}(\frac{1}{\alpha\beta})^nP_n((\alpha - \beta)k)^2$}

Note how this simplifies considerably when {$c=1$}:

{$p_n((\alpha - \beta)k) = (1-\frac{\alpha}{\beta})(\frac{\alpha}{\beta})^k(\frac{1}{\alpha\beta})^nP_n((\alpha - \beta)k)^2$}

This formula will seem elaborate but that is because it is making explicit the context for the wave function. It turns out, surprisingly, that by specifying the relationship with the context, the underlying physical concepts can be understood as discrete and combinatorial, which is desirable. Furthermore, I will explain how this formula is shown by the combinatorics to be natural in ways that the mathematics of the wave function is not.

I will start by explaining what I can about this formula. I am still struggling to interpret it physically. I will do my best and I invite your help. Then I will explain further the steps by which I developed my limited understanding and arrived at this formula.

In studying Schroedinger's equation, notably the case of the quantum harmonic oscillator, I noticed the importance of the Hermite polynomials, which are orthogonal polynomials. I became interested to understand how to interpret combinatorially the coefficients of these polynomials. Similarly, in the simple radial model of the hydrogen atom, we see the importance of the Laguerre polynomials. These are both examples of orthogonal Sheffer polynomials, which have a wonderful combinatorics, especially as developed by Dongsu Kim and Jiang Zeng. In my quest to understand the physical significance, I came to think that the orthogonal polynomials are what carry the information of the wave functions, and indeed, the wave function as such is actually a contrivance. The orthogonal polynomials define an inner product with regard to which they are orthogonal. That inner product is a distribution which typically consists of a weight function over some integral or infinite sum. Orthogonality is manifested by integrating two orthogonal polynomials with respect to this distribution. If they are the same polynomial, then we are integrating the square of that polynomial. This is what we see when we are calculating the square of the amplitude of a system's wavefunction. In my understanding, the wave function, or more precisely, an eigenfunction, is a mythical chimera that fuses together the orthogonal polynomial with the square root of the weight function. Physicists then apply an integral - an inner product - to that eigenfunction. They may consider that inner product to be simply part of their mathematical tool kit which they apply at their convenience. Whereas, as an algebraic combinatorialist, I am studying the combinatorics to interpret the math in terms of actions on combinatorial objects, and thus appreciate how to express algebraically those actions most transparently and naturally. My hope is that this can yield insights into how nature functions or what nature has to say. In my understanding, nature is not taking an integral but applying a distribution. Which is to say, it seems natural to me to define the inner product on the orthogonal polynomials and not on the wave functions.

Thus you will see in the formula the orthogonal Sheffer polynomial {$P_n(x)$} and that its value is squared. The coefficients of the polynomial are integers or rational numbers, but in any event, real numbers, which is to say, they do not include imaginary numbers and so we do not need to take a complex conjugate.

The values {$\alpha, \beta, c$} arise in the recurrence relation of the ... the classification...

Consider the Meixner distribution, similar to the Poisson distribution...

I need to present a few ideas that have been suggesting themselves to me as I have worked on this.

• In physics, from nature's point of view, there is never a single frame, but instead there is a pair of frames, as in describing how a particle participates in an interaction. Thus we always have two ways of looking at things, and in particular, the two frames can look at each other in their own units.