Infinite derivatives vs integral operators. The Moeller-Zwiebach puzzle

Abstract

We study the relationship between integral and infinite-derivative operators. In particular, we examine the operator $p^{\frac{1}{2}\,\partial_t^2}\,$ that appears in the theory of $p$-adic string fields, as well as the Moyal product that arises in non-commutative theories. We also try to clarify the apparent paradox raised by Moeller and Zwiebach, which highlights the discrepancy between them.