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3.2 Ashtekar variables

To quantize a constrained canonical theory one can use Dirac’s prescription [105Jump 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, 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 y[q] 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 [7830Jump 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 with an additional factor of a Jacobian under coordinate transformations). The densitized triad Ea i is then related to the triad by Ea = ||det eb||-1 ea i j i 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 i b Ka := Kabe i:

i b b i {K a(x),E j(y)}= 8pGd adjd(x,y) (1)
with the gravitational constant G. Extrinsic curvature is then replaced by the Ashtekar connection
Aia = Gia + gKia (2)
with a positive value for g, the Barbero-Immirzi parameter [30133]. 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,
i b b i {Aa(x),E j(y)}= 8pgGd adjd(x,y) (3)
and obtains its transformation properties as a connection from the spin connection
Gia = - eijkebj(@[aeke] + 1eckela@[celb]). (4) 2

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 integral 1 integral G[/\] = ------ d3x /\iDaEai = ------ d3x /\i(@aEai + eijkAjaEak) ~~ 0 (5) 8pgG S 8pgG S
is satisfied. Independence from any spatial coordinate system or background is implemented by the diffeomorphism constraint (modulo Gauss constraint)
integral D[N a] =---1-- d3x N aF iEb ~~ 0, (6) 8pgG S ab i
with the curvature i F ab 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

integral 1 3 -1/2( i a b 2 i j a b) H[N ] = ------- d xN |det E | eijkFabE jEk - 2(1 + g )K [aK b]Ei E j ~ ~ 0, (7) 16pgG S
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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