[Paper Review] Polynomial Sequences of Binomial Type and Path Integrals

Vladimir V. Kisil|arXiv (Cornell University)|Aug 9, 1998

Algebraic and Geometric Analysis15 references4 citations

TL;DR

This paper establishes a novel connection between enumerative combinatorics and quantum field theory by representing polynomial sequences of binomial type as path integrals in a phase space composed of discrete time and angular momentum variables. Using a Hamiltonian derived from the derivatives of the polynomial at zero, it formulates a Schrödinger-type equation for the sequence, thereby linking umbral calculus to quantum mechanics and enabling a framework for parallel quantum computation.

ABSTRACT

Polynomial sequences $p_n(x)$ of binomial type are a principal tool in the umbral calculus of enumerative combinatorics. We express $p_n(x)$ as a \emph{path integral} in the ``phase space'' $\Space{N}{} imes {[-π,π]}$. The Hamiltonian is $h(ϕ)=\sum_{n=0}^\infty p_n'(0)/n! e^{inϕ}$ and it produces a Schrödinger type equation for $p_n(x)$. This establishes a bridge between enumerative combinatorics and quantum field theory. It also provides an algorithm for parallel quantum computations. Keywords: Feynman path integral, umbral calculus, polynomial sequence of binomial type, token, Schrödinger equation, propagator, wave function, cumulants, quantum computation.

Motivation & Objective

• To bridge enumerative combinatorics and quantum field theory through a novel representation of polynomial sequences.
• To formulate polynomial sequences of binomial type as solutions to a Schrödinger-type equation via path integrals.
• To develop a computational framework for parallel quantum computation using combinatorial structures.
• To generalize the umbral calculus by embedding it in a quantum mechanical phase space formalism.
• To provide a mathematical foundation for interpreting cumulants and generating functions in terms of quantum propagators.

Proposed method

• Represent polynomial sequences of binomial type as path integrals over the phase space $\mathbb{N} \times [-\pi, \pi]$.
• Define a Hamiltonian $h(\phi) = \sum_{n=0}^\infty \frac{p_n'(0)}{n!} e^{in\phi}$ using the derivatives of the polynomial at zero.
• Derive a Schrödinger-type equation for the sequence $p_n(x)$ using this Hamiltonian.
• Construct the propagator and wave function in the path integral formulation to generate the sequence.
• Utilize the Fourier representation of the Hamiltonian to connect combinatorial generating functions with quantum amplitudes.
• Apply the formalism to cumulants and token-based models to model quantum computation.

Experimental results

Research questions

• RQ1 Can polynomial sequences of binomial type be represented as path integrals in a quantum mechanical phase space?
• RQ2 How can the umbral calculus be embedded into a quantum field theory framework using Hamiltonian dynamics?
• RQ3 What is the role of the generating function's derivatives in defining a quantum Hamiltonian?
• RQ4 Can this formulation support a model of parallel quantum computation based on combinatorial structures?
• RQ5 How do cumulants and generating functions emerge from the propagator in this path integral formulation?

Key findings

• Polynomial sequences of binomial type are shown to satisfy a Schrödinger-type equation derived from a phase-space path integral.
• The Hamiltonian is explicitly constructed as a Fourier series in $\phi$, with coefficients given by the derivatives $p_n'(0)/n!$.
• The path integral formulation provides a new algorithmic framework for quantum computation based on combinatorial sequences.
• The wave function in this model corresponds to the generating function of the sequence, with time evolution governed by the Hamiltonian.
• The formalism naturally incorporates cumulants as quantum expectation values in the path integral framework.
• The method establishes a direct correspondence between umbral calculus and quantum field theory, particularly in the context of propagators and time evolution.