Formal proofs of operator identities by a single formal computation

TL;DR

This paper develops a formal algebraic framework using noncommutative polynomials and labelled quivers to rigorously prove operator identities, accounting for domains, codomains, and multiple operator versions.

Contribution

It introduces a method combining algebraic ideal membership with quiver compatibility to verify operator identities in linear categories.

Findings

Proves that polynomial ideal membership combined with quiver compatibility suffices for identity verification.

Models different operator versions by assigning the same label to multiple edges in the quiver.

Ensures that formal computations can be realized as actual operator identities in linear categories.

Abstract

A formal computation proving a new operator identity from known ones is, in principle, restricted by domains and codomains of linear operators involved, since not any two operators can be added or composed. Algebraically, identities can be modelled by noncommutative polynomials and such a formal computation proves that the polynomial corresponding to the new identity lies in the ideal generated by the polynomials corresponding to the known identities. In order to prove an operator identity, however, just proving membership of the polynomial in the ideal is not enough, since the ring of noncommutative polynomials ignores domains and codomains. We show that it suffices to additionally verify compatibility of this polynomial and of the generators of the ideal with the labelled quiver that encodes which polynomials can be realized as linear operators. Then, for every consistent representation of such a quiver in a linear category, there exists a computation in the category that proves the corresponding instance of the identity. Moreover, by assigning the same label to several edges of the quiver, the algebraic framework developed allows to model different versions of an operator by the same indeterminate in the noncommutative polynomials.