Taming the Leibniz Rule on the Lattice

Understand

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.