Description of all conformally invariant differential operators, acting on scalar functions
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2007-12-01
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Abstract
We present an algorithm to obtain all conformally invariant differential operators acting on scalar functions and taking values on scalar functions as well. There are two cases under consideration: Operators acting on functions defined on the Euclidean space R4 and on the Minkowski space. In both cases each conformally invariant operator of order k is nonlinear and is a function of a finite set of functionally independent invariant operators of order up to k. In particular, the second order independent operators are only three and we give their explicit realization. The applied technique is based on the jet bundle formalism, analysis of the group action, and a dimensional reduction. As a simple illustration of this method we consider the case of differential operators between analytic functions on C which are invariant under the action of the modular group. A power series which generates all functionally independent invariant operators is derived. © 2007 American Institute of Physics.