Ashtekar variables, which were a new canonical formalism of general relativity, raised new hopes for the canonical quantization of general relativity and eventually led to loop quantum gravity. Smolin and others independently discovered that there exists in fact a Lagrangian formulation of the theory by considering the self-dual formulation of the Tetradic Palatini action principle of general relativity. These proofs were given in terms of spinors. A purely tensorial proof of the new variables in terms of triads was given by Goldberg and in terms of tetrads by Henneaux et al.
The Palatini action
The Palatini action for general relativity has as its independent variables the tetrad e I α {\displaystyle e_{I}^{\alpha }} and a spin connection ω α I J {\displaystyle {\omega _{\alpha }}^{IJ}} . Many more details and derivations can be found in the article tetradic Palatini action. The spin connection defines a covariant derivative D α {\displaystyle D_{\alpha }} . The space-time metric is recovered from the tetrad by the formula g α β = e α I e β J η I J . {\displaystyle g_{\alpha \beta }=e_{\alpha }^{I}e_{\beta }^{J}\eta _{IJ}.} We define the curvature as
Ω α β I J = ∂ α ω β I J − ∂ β ω α I J + ω α I K ω β K J − ω β I K ω α K J . {\displaystyle {\Omega _{\alpha \beta }}^{IJ}=\partial _{\alpha }{\omega _{\beta }}^{IJ}-\partial _{\beta }{\omega _{\alpha }}^{IJ}+\omega _{\alpha }^{IK}{\omega _{\beta K}}^{J}-\omega _{\beta }^{IK}{\omega _{\alpha K}}^{J}.}
The Ricci scalar of this curvature is given by e I α e J β Ω α β I J {\displaystyle e_{I}^{\alpha }e_{J}^{\beta }{\Omega _{\alpha \beta }}^{IJ}} . The Palatini action for general relativity reads
S = ∫ d 4 x e e I α e J β Ω α β I J [ ω ] , {\displaystyle S=\int d^{4}x\;e\;e_{I}^{\alpha }e_{J}^{\beta }\;{\Omega _{\alpha \beta }}^{IJ}[\omega ],}
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