Dualisation of free fields

Abstract

We consider the general free field theory such that system of equations of motion includes a subsystem with a special property. If the subsystem is considered by itself, it would be a topological field theory having no local degrees of freedom. Various well-known field theories admit such a subsystem, including Einstein’s gravity. As the subsystem is a topological theory, the general solution is a pure gauge, modulo global degrees of freedom. Gauge symmetry of the subsystem of linear theory can be explicitly found, providing the general solution to the subsystem. Substituting this solution into entire system, we arrive at an equivalent field theory where the role of the field variables is played by the gauge parameters of the topological subsystem. These fields can be viewed as the potentials for the original ones. A general procedure is proposed for constructing a “parent action” which includes the fields of both dual formulations. At the level of this action, one can switch between dual formulations by imposing appropriate gauge fixing conditions. This general dualisation procedure is exemplified by massless and massive spin two fields. For the massless case, the hook type tensor serves as a potential for the metric, while for the massive case it is the fourth rank tensor with window Young diagram.