Canonical Variables for the Interacting Gravitational and Dirac Fields

The problem of reducing the Lagrangian for the interacting gravitational and Dirac fields to canonical form is discussed, using the vierbein formalism. The arbitrary gauge variables corresponding to local Lorentz transformations of the vierbein are removed by imposing Schwinger's ``time-guage'' condition, and a further condition that the spatial part of the vierbein be symmetric. It is shown that in this guage the Lagrangian can be expressed in a canonical form involving essentially the same gravitational-field variables as in the absence of matter, and that the generators of spatial translations and rotations have the expected form.