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3.2 Ashtekar variables

To quantize a constrained canonical theory one can use Dirac’s prescription [153Jump To The Next Citation Point] 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 ψ[q] of spatial metrics, as is proposed in Wheeler–DeWitt quantizations; see, e.g., [310173]. 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 [7837Jump To The Next Citation Point] 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 Eai is then related to the triad by | |−1 Eai = |det ebj| eai 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 Ki := Kabeb a i:
i b b i {K a(x),E j(y)} = 8πG δaδjδ(x,y) (1 )
where G is the gravitational constant. The extrinsic curvature is then replaced by the Ashtekar connection
i i i Aa = Γ a + γK a (2 )
with a positive value for γ, the Barbero–Immirzi parameter [37193]. 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,
{Aia(x),Ebj(y)} = 8πγG δbaδijδ(x,y ) (3 )
and obtains its transformation properties as a connection from the spin connection
Γ ia = − εijkebj(∂[aeke] + 12eckela∂[celb]) . (4 )
(In a first-order formulation there are additional contributions from torsion in the presence of fermionic matter; see, e.g., [76Jump To The Next Citation Point] for details.)

Spatial geometry is then obtained directly from the densitized triad, which is related to the spatial metric by

a b ab E i E i = q detq .

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,

1 ∫ 1 ∫ G [Λ] = ------ d3x ΛiDaEai = ------ d3x Λi(∂aEai + εijkAjaEak) ≈ 0, (5 ) 8πγG Σ 8πγG Σ
is satisfied. Independence from any spatial coordinate system or background is implemented by the diffeomorphism constraint (modulo Gauss constraint)
∫ D [N a] = --1--- d3xN aFi Eb ≈ 0 (6 ) 8π γG Σ ab i
with the curvature i F ab 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

∫ 1 3 −1∕2( i a b 2 i j a b) H [N ] = ------- d xN |det E | εijkFabE jEk − 2(1 + γ )K [aK b]Ei E j ≈ 0, (7 ) 16π γG Σ
where extrinsic-curvature components have to be understood as functions of the Ashtekar connection and the densitized triad through the spin connection.
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