Nonminimally coupled scalar field and Ashtekar variables

The nonminimal coupling of a scalar field is considered in the framework of Ashtekar's new variables formulation of gravity. A first-order action functional for this system is derived in which the field variables are a tetrad field, and an $\mathrm{SL}\mathrm{}(2,C)\mathrm{}$ connection, together with the scalar field. The tetrad field and the $\mathrm{SL}\mathrm{}(2,C)\mathrm{}$ connection are related to the Ashtekar variables for the vacuum case by a conformal transformation. A canonical analysis shows that for this coupling the equations of Ashtekar's formulation of canonical gravity are nonpolynomial in the scalar field.