The pre-Lie structure of the time-ordered exponential
Kurusch Ebrahimi-Fard, Frederic Patras
TL;DR
This paper explores the algebraic structure underlying time-ordered exponentials using pre-Lie algebras, providing new formulas and insights relevant to physics and mathematics.
Contribution
It introduces a novel approach connecting time-ordering with pre-Lie algebra structures, deriving explicit formulas and generalizations for operator products.
Findings
Recovered noncommutative Bohnenblust-Spitzer formula
Derived explicit formulas for time-ordered exponential products
Established connections between time-ordering and pre-Lie algebra theory
Abstract
The usual time-ordering operation and the corresponding time-ordered exponential play a fundamental role in physics and applied mathematics. In this work we study a new approach to the understanding of time-ordering relying on recent progress made in the context of enveloping algebras of pre-Lie algebras. Various general formulas for pre-Lie and Rota-Baxter algebras are obtained in the process. Among others, we recover the noncommutative analog of the classical Bohnenblust-Spitzer formula, and get explicit formulae for operator products of time-ordered exponentials.
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