Generating Compatibility Conditions in Mathematical Physics

TL;DR

This paper explores the homological nature of compatibility conditions in differential operators within mathematical physics, providing new theoretical insights and motivating examples for computational applications.

Contribution

It establishes a homological framework linking operator exactness, symbol sequence exactness, and formal integrability, with explicit examples for testing computer algebra methods.

Findings

Homological link between operator exactness and formal integrability

Explicit examples of compatibility conditions in physics

Potential applications in computer algebra systems

Abstract

The search for generating compatibility conditions (CC) for a given operator is a very recent problem met in General Relativity in order to study the Killing operator for various standard useful metrics (Minkowski, Schwarschild and Kerr). In this paper, we prove that the link existing between the lack of formal exactness of an operator sequence on the jet level, the lack of formal exactness of its corresponding symbol sequence and the lack of formal integrability (FI) of the initial operator is of a purely homological nature as it is based on the long exact connecting sequence provided by the so-called snake lemma. It is therefore quite difficult to grasp it in general and even more difficult to use it on explicit examples. It does not seem that any one of the results presented in this paper is known as most of the other authors who studied the above problem of computing the total number of generating CC are confusing this number with a kind of differential transcendence degree, also called degree of generality by A. Einstein in his 1930 letters to E. Cartan. The motivating examples that we provide are among the rare ones known in the literature and could be used as testing examples for future applications of computer algebra.