Taming the Leibniz Rule on the Lattice

TL;DR

This paper proves a no-go theorem showing the impossibility of constructing a lattice difference operator and product rule that satisfy translation invariance, locality, and Leibniz rule, and proposes a formalism to circumvent this limitation.

Contribution

It establishes a no-go theorem for lattice operators satisfying key properties and introduces a formalism using infinite flavors or matrices to overcome this challenge.

Findings

No lattice difference operator can satisfy all three properties simultaneously.

A formalism with infinite flavors or matrices can bypass the no-go theorem.

Provides a framework for constructing lattice theories with Leibniz rule.

Abstract

We study a product rule and a difference operator equipped with Leibniz rule in a general framework of lattice field theory. It is shown that the difference operator can be determined by the product rule and some initial data through the Leibniz rule. This observation leads to a no-go theorem that it is impossible to construct any difference operator and product rule on a lattice with the properties of (i) translation invariance, (ii) locality and (iii) Leibniz rule. We present a formalism to overcome the difficulty by an infinite flavor extension or a matrix expression of a lattice field theory.