### 3.2 Ashtekar variables

To quantize a constrained canonical theory one can use Dirac’s prescription [105] and first represent the
classical Poisson algebra of a suitable complete set of basic variables on phase space as an operator algebra
on a Hilbert space, called kinematical. This ignores the constraints, which can be written as operators on
the same Hilbert space. At the quantum level the constraints are then solved by determining their kernel,
to be equipped with an inner product so as to define the physical Hilbert space. If zero is in
the discrete part of the spectrum of a constraint, as e.g., for the Gauss constraint when the
structure group is compact, the kernel is a subspace of the kinematical Hilbert space to which the
kinematical inner product can be restricted. If, on the other hand, zero lies in the continuous
part of the spectrum, there are no normalizable eigenstates and one has to construct a new
physical Hilbert space from distributions. This is the case for the diffeomorphism and Hamiltonian
constraints.
To perform the first step we need a Hilbert space of functionals of spatial metrics. Unfortunately,
the space of metrics, or alternatively extrinsic curvature tensors, is mathematically poorly understood and
not much is known about suitable inner products. At this point, a new set of variables introduced by
Ashtekar [7, 8, 30] becomes essential. This is a triad formulation, but uses the triad in a densitized form
(i.e., it is multiplied with an additional factor of a Jacobian under coordinate transformations). The
densitized triad is then related to the triad by but has the same
properties concerning gauge rotations and its orientation (note the absolute value which is often
omitted). The densitized triad is conjugate to extrinsic curvature coefficients :

with the gravitational constant . Extrinsic curvature is then replaced by the Ashtekar connection
with a positive value for , the Barbero-Immirzi parameter [30, 133]. Classically, this number can be
changed by a canonical transformation of the fields, but it will play a more important and fundamental role
upon quantization. The Ashtekar connection is defined in such a way that it is conjugate to the triad,
and obtains its transformation properties as a connection from the spin connection
Spatial geometry is then obtained directly from the densitized triad, which is related to the spatial
metric by

There is more freedom in a triad since it can be rotated without changing the metric. The theory is
independent of such rotations provided the Gauss constraint

is satisfied. Independence from any spatial coordinate system or background is implemented by the
diffeomorphism constraint (modulo Gauss constraint)
with the curvature of the Ashtekar connection. In this setting, one can then discuss spatial geometry
and its quantization.
Space-time geometry, however, is more complicated to deduce since it requires a good knowledge of
the dynamics. In a canonical setting, dynamics is implemented by the Hamiltonian constraint

where extrinsic curvature components have to be understood as functions of the Ashtekar connection and
the densitized triad through the spin connection.