Infinite Derivatives vs Integral Operators. The Moeller-Zwiebach Puzzle

TL;DR

This paper investigates the relationship between integral and infinite-derivative operators, focusing on specific operators in string theory and non-commutative theories, and addresses a paradox highlighted by Moeller and Zwiebach.

Contribution

It clarifies the connection between integral and infinite-derivative operators and resolves the Moeller-Zwiebach paradox in this context.

Findings

Established links between integral and infinite-derivative operators.

Resolved the apparent discrepancy highlighted by Moeller and Zwiebach.

Provided insights into operators in string and non-commutative theories.

Abstract

We study the relationship between integral and infinite-derivative operators. In particular, we examine the operator $p^{\frac12\,\partial_t^2}\,$ that appears in the theory of $p$-adic string fields, as well as the Moyal product that arises in non-commutative theories. We also attempt to clarify the apparent paradox presented by Moeller and Zwiebach, which highlights the discrepancy between them.