### 3.2 Ashtekar variables

To quantize a constrained canonical theory one can use Dirac’s prescription [153] 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, and the solution space has to be equipped with an inner product to
provide the physical Hilbert space. If zero is in the discrete part of the spectrum of a constraint,
as in 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. The main condition for the
physical inner product is that it allows one to quantize real-valued observables. To perform the
first step, we need a Hilbert space of functionals of spatial metrics, as is proposed in
Wheeler–DeWitt quantizations; see, e.g., [310, 173]. Unfortunately, the space of metrics, or, alternatively,
extrinsic curvature tensors, is poorly understood mathematically and not much is known about
suitable inner products. At this point, a new set of variables introduced by Ashtekar [7, 8, 37]
becomes essential. This is a triad formulation, but uses the triad in a densitized form, i.e., it is
multiplied by 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 canonically conjugate to extrinsic curvature coefficients :
where is the gravitational constant. The extrinsic curvature is then replaced by the Ashtekar
connection
with a positive value for , the Barbero–Immirzi parameter [37, 193]. 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
(In a first-order formulation there are additional contributions from torsion in the presence of fermionic
matter; see, e.g., [76] for details.)
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.
Spacetime geometry, however, is more difficult to deduce, since it requires a good knowledge of the
dynamics. In a canonical setting, the dynamics are 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.