  A description of the space of G-connections using the tangent groupoid is
given. As the tangent groupoid parameter is away from zero, the G-connections
act as convolution operators on a Hilbert space. The gauge action is examined
in the tangent groupoid description of the G-connections. Tetrads are
formulated as Dirac type operators. The connection variables and tetrad
variables in Ashtekar's gravity are presented as operators on a Hilbert space.
