Editing Loop quantum gravity (section)

=== Pre-history and Ashtekar new variables ===
{{main|Frame fields in general relativity| Ashtekar variables| Self-dual Palatini action}}
Many of the technical problems in canonical quantum gravity revolve around the constraints. Canonical general relativity was originally formulated in terms of metric variables, but there seemed to be insurmountable mathematical difficulties in promoting the constraints to [[quantum operator]]s because of their highly non-linear dependence on the canonical variables. The equations were much simplified with the introduction of Ashtekar's new variables. Ashtekar variables describe canonical general relativity in terms of a new pair of canonical variables closer to those of gauge theories. The first step consists of using densitized triads <math>\tilde{E}_i^a</math> (a triad <math>E_i^a</math> is simply three orthogonal vector fields labeled by <math>i = 1,2,3</math> and the densitized triad is defined by <math display="inline">\tilde{E}_i^a = \sqrt{\det (q)} E_i^a</math>) to encode information about the spatial metric,
<math display="block">\det(q) q^{ab} = \tilde{E}_i^a \tilde{E}_j^b \delta^{ij}.</math>
(where <math>\delta^{ij}</math> is the flat space metric, and the above equation expresses that <math>q^{ab}</math>, when written in terms of the basis <math>E_i^a</math>, is locally flat). (Formulating general relativity with triads instead of metrics was not new.) The densitized triads are not unique, and in fact one can perform a local in space [[rotation]] with respect to the internal indices <math>i</math>. The canonically conjugate variable is related to the extrinsic curvature by <math display="inline">K_a^i = K_{ab} \tilde{E}^{ai} / \sqrt{\det (q)}</math>. But problems similar to using the metric formulation arise when one tries to quantize the theory. Ashtekar's new insight was to introduce a new configuration variable,
<math display="block">A_a^i = \Gamma_a^i - i K_a^i</math>
that behaves as a complex <math>\operatorname{SU}(2)</math> connection where <math>\Gamma_a^i</math> is related to the so-called [[spin connection]] via <math>\Gamma_a^i = \Gamma_{ajk} \epsilon^{jki}</math>. Here <math>A_a^i</math> is called the chiral spin connection. It defines a covariant derivative <math>\mathcal{D}_a</math>. It turns out that <math>\tilde{E}^a_i</math> is the conjugate momentum of <math>A_a^i</math>, and together these form Ashtekar's new variables.

The expressions for the constraints in Ashtekar variables; Gauss's theorem, the spatial diffeomorphism constraint and the (densitized) Hamiltonian constraint then read:
<math display="block">G^i = \mathcal{D}_a \tilde{E}_i^a = 0</math>
<math display="block">C_a = \tilde{E}_i^b F^i_{ab} - A_a^i (\mathcal{D}_b \tilde{E}_i^b) = V_a - A_a^i G^i = 0,</math>
<math display="block">\tilde{H} = \epsilon_{ijk} \tilde{E}_i^a \tilde{E}_j^b F^k_{ab} = 0</math>
respectively, where <math>F^i_{ab}</math> is the field strength tensor of the connection <math>A_a^i</math> and where <math>V_a</math> is referred to as the vector constraint. The above-mentioned local in space rotational invariance is the original of the <math>\operatorname{SU}(2)</math> gauge invariance here expressed by Gauss's theorem. Note that these constraints are polynomial in the fundamental variables, unlike the constraints in the metric formulation. This dramatic simplification seemed to open up the way to quantizing the constraints. (See the article [[Self-dual Palatini action]] for a derivation of Ashtekar's formalism).

With Ashtekar's new variables, given the configuration variable <math>A^i_a</math>, it is natural to consider wavefunctions <math>\Psi (A^i_a)</math>. This is the connection representation. It is analogous to ordinary quantum mechanics with configuration variable <math>q</math> and wavefunctions <math>\psi (q)</math>. The configuration variable gets promoted to a quantum operator via:
<math display="block">\hat{A}_a^i \Psi (A) = A_a^i \Psi (A),</math>
(analogous to <math>\hat{q} \psi (q) = q \psi (q)</math>) and the triads are (functional) derivatives,
<math display="block">\hat{\tilde{E_i^a}} \Psi (A) = - i {\delta \Psi (A) \over \delta A_a^i}.</math>
(analogous to <math>\hat{p} \psi (q) = -i \hbar d \psi (q) / dq</math>). In passing over to the quantum theory the constraints become operators on a kinematic Hilbert space (the unconstrained <math>\operatorname{SU}(2)</math> Yang–Mills Hilbert space). Note that different ordering of the <math>A</math>'s and <math>\tilde{E}</math>'s when replacing the <math>\tilde{E}</math>'s with derivatives give rise to different operators – the choice made is called the factor ordering and should be chosen via physical reasoning. Formally they read
<math display="block">\hat{G}_j \vert\psi \rangle = 0</math>
<math display="block">\hat{C}_a \vert\psi \rangle = 0</math>
<math display="block">\hat{\tilde{H}} \vert\psi \rangle = 0.</math>

There are still problems in properly defining all these equations and solving them. For example, the Hamiltonian constraint Ashtekar worked with was the densitized version instead of the original Hamiltonian, that is, he worked with <math display="inline">\tilde{H} = \sqrt{\det (q)} H</math>. There were serious difficulties in promoting this quantity to a quantum operator. Moreover, although Ashtekar variables had the virtue of simplifying the Hamiltonian, they are complex. When one quantizes the theory, it is difficult to ensure that one recovers real general relativity as opposed to complex general relativity.