An integral-free representation of the Dyson series using divided differences

TL;DR

This paper introduces a novel integral-free representation of the Dyson series using divided differences, simplifying quantum time-evolution calculations and enabling direct time-dependent perturbation analysis in the Schrödinger picture.

Contribution

The paper presents a new formalism that replaces the Dyson series with a sum of divided differences, eliminating the need for time ordering and integrals, thus streamlining quantum evolution computations.

Findings

Simplifies Dyson series calculations

Enables direct time-dependent perturbation analysis

Reduces computational complexity of quantum evolution

Abstract

The Dyson series is an infinite sum of multi-dimensional time-ordered integrals, which serves as a formal representation of the quantum time-evolution operator in the interaction-picture. Using the mathematical tool of divided differences, we introduce an alternative representation for the series that is entirely free from both time ordering and integrals. In this new formalism, the Dyson expansion is given as a sum of efficiently-computable divided differences of the exponential function, considerably simplifying the calculation of the Dyson expansion terms, while also allowing for time-dependent perturbation calculations to be performed directly in the Schr{\"o}dinger-picture. We showcase the utility of this novel representation by studying a number of use cases. We also discuss several immediate applications.