Description of all conformally invariant differential operators, acting on scalar functions

TL;DR

This paper presents an algorithm to systematically construct all conformally invariant differential operators acting on scalar functions in Minkowski space, revealing their structure and explicit forms for second order operators.

Contribution

It introduces a novel algorithm based on jet bundle formalism to generate all conformally invariant scalar differential operators, including explicit second order examples.

Findings

All second order conformally invariant operators are identified and explicitly realized.

An algorithm for constructing invariant operators of arbitrary order is developed.

The method is illustrated with an example involving the modular group.

Abstract

We give an algorithm to write down all conformally invariant differential operators acting between scalar functions on Minkowski space. All operators of order k are nonlinear and are functions on a finite family of functionally independent invariant operators of order up to k. The independent differential operators of second order are three and we give an explicit realization of them. The applied technique is based on the jet bundle formalism, algebraization of the the differential operators, group action and dimensional reduction. As an illustration of this method we consider the simpler case of differential operators between analytic functions invariant under the modular group. We give a power series generating explicitly all the functionally independent invariant operators of an arbitrary order.