A No-Go Theorem for the Compatibility between Involutions of the First Order Differentials on a Lattice and the Continuum Limit

TL;DR

This paper proves that certain algebraic and involutive properties of differentials on a lattice cannot simultaneously hold while maintaining a natural continuum limit, highlighting fundamental incompatibilities.

Contribution

It establishes a no-go theorem showing the incompatibility of three key properties of lattice differentials with the continuum limit, advancing understanding of lattice calculus limitations.

Findings

Proves the incompatibility of algebraic dependence and involution properties on a lattice.

Shows that an involution as an antihomomorphism conflicts with the continuum limit.

Demonstrates fundamental limitations in lattice differential calculus.

Abstract

We prove that the following three properties can not match each other on a lattice, that differentials of coordinate functions are algebraically dependent to their involutive conjugates, that the involution on a lattice is an antihomomorphism and that differential calculus has a natural continuum limit.