Polynomial Sequences of Binomial Type and Path Integrals

TL;DR

This paper connects polynomial sequences of binomial type with path integrals in phase space, establishing a link between combinatorics and quantum field theory, and proposing an algorithm for quantum computations.

Contribution

It expresses polynomial sequences as path integrals, bridging combinatorics and quantum physics, and introduces a quantum computation algorithm based on this formulation.

Findings

Path integral representation of polynomial sequences of binomial type

Connection between umbral calculus and quantum field theory

Algorithm for parallel quantum computations

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'' $\Space{N}{} \times {[-\pi,\pi]}$. The Hamiltonian is $h(\phi)=\sum_{n=0}^\infty p_n'(0)/n! e^{in\phi}$ and it produces a Schr\"odinger 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, Schr\"odinger equation, propagator, wave function, cumulants, quantum computation.