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.[1][2][3] These proofs were given in terms of spinors. A purely tensorial proof of the new variables in terms of triads was given by Goldberg[4] and in terms of tetrads by Henneaux et al.[5]
The Lie bracket
An important object is the Lie bracket defined by
![{\displaystyle [F,G]^{IJ}:=F^{IK}{G_{K}}^{J}-G^{IK}{F_{K}}^{J},}](//wikimedia.org/api/rest_v1/media/math/render/svg/e153973581572ada321d74dffc6092818e87bcd2)
it appears in the curvature tensor (see the last two terms of Eq. 1), it also defines the algebraic structure. We have the results (proved below):
![{\displaystyle P^{(\pm )}[F,G]^{IJ}=\left[P^{(\pm )}F,G\right]^{IJ}=\left[F,P^{(\pm )}G\right]^{IJ}=\left[P^{(\pm )}F,P^{(\pm )}G\right]^{IJ}\qquad Eq.2}](//wikimedia.org/api/rest_v1/media/math/render/svg/c75bfda351899574a9a10d76a9cbf0e886687699)
and
![{\displaystyle [F,G]=\left[P^{+}F,P^{+}G\right]+\left[P^{-}F,P^{-}G\right].}](//wikimedia.org/api/rest_v1/media/math/render/svg/948bb280c8b5033f52f7e3aacd5f432085af0797)
That is the Lie bracket, which defines an algebra, decomposes into two separate independent parts. We write
![{\displaystyle {\mathfrak {so}}(1,3)_{\mathbb {C} }={\mathfrak {so}}(1,3)_{\mathbb {C} }^{+}+{\mathfrak {so}}(1,3)_{\mathbb {C} }^{-}}](//wikimedia.org/api/rest_v1/media/math/render/svg/61c61424745e0e557a11e71b96765da6df10f065)
where
contains only the self-dual (anti-self-dual) elements of ![{\displaystyle {\mathfrak {so}}(1,3)_{\mathbb {C} }.}](//wikimedia.org/api/rest_v1/media/math/render/svg/1f79438af772e22ee2c523c500c01d740925c197)
We define the self-dual part,
, of the connection
as
![{\displaystyle {A_{\alpha }}^{IJ}={1 \over 2}\left({\omega _{\alpha }}^{IJ}-{i \over 2}{\varepsilon _{KL}}^{IJ}{\omega _{\alpha }}^{KL}\right),}](//wikimedia.org/api/rest_v1/media/math/render/svg/91973f06a46618675fb48213d48e26775e04ad23)
which can be more compactly written
![{\displaystyle {A_{\alpha }}^{IJ}=\left(P^{+}\omega _{\alpha }\right)^{IJ}.}](//wikimedia.org/api/rest_v1/media/math/render/svg/6de8a9978898fa0e2e7b10f4f4e367c37ce6232f)
Define
as the curvature of the self-dual connection
![{\displaystyle {F_{\alpha \beta }}^{IJ}=\partial _{\alpha }{A_{\beta }}^{IJ}-\partial _{\beta }{A_{\alpha }}^{IJ}+{A_{\alpha }}^{IK}{A_{\beta K}}^{J}-{A_{\beta }}^{IK}{A_{\alpha K}}^{J}.}](//wikimedia.org/api/rest_v1/media/math/render/svg/c00bb626fdaf240f5fe83c73ab62bb6d7f14c2e1)
Using Eq. 2 it is easy to see that the curvature of the self-dual connection is the self-dual part of the curvature of the connection,
![{\displaystyle {\begin{aligned}{F_{\alpha \beta }}^{IJ}&=\partial _{\alpha }\left(P^{+}\omega _{\beta }\right)^{IJ}-\partial _{\beta }\left(P^{+}\omega _{\alpha }\right)^{IJ}+\left[P^{+}\omega _{\alpha },P^{+}\omega _{\beta }\right]^{IJ}\\&=\left(P^{+}2\partial _{[\alpha }\omega _{\beta ]}\right)^{IJ}+\left(P^{+}[\omega _{\alpha },\omega _{\beta }]\right)^{IJ}\\&=\left(P^{+}\Omega _{\alpha \beta }\right)^{IJ}\end{aligned}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/0e8634deadc21e6fe97e8170aaf225362940a54a)
The self-dual action is
![{\displaystyle S=\int d^{4}x\;e\;e_{I}^{\alpha }e_{J}^{\beta }\;{F_{\alpha \beta }}^{IJ}.}](//wikimedia.org/api/rest_v1/media/math/render/svg/d1430ec3f2dc25b28d6a885dd9c8eb07cd430a48)
As the connection is complex we are dealing with complex general relativity and appropriate conditions must be specified to recover the real theory. One can repeat the same calculations done for the Palatini action but now with respect to the self-dual connection
. Varying the tetrad field, one obtains a self-dual analog of Einstein's equation:
![{\displaystyle {}^{+}R_{\alpha \beta }-{1 \over 2}g_{\alpha \beta }{}^{+}R=0.}](//wikimedia.org/api/rest_v1/media/math/render/svg/06256beeedbd1ce323ba9d3c1f052dad6fcd59e1)
That the curvature of the self-dual connection is the self-dual part of the curvature of the connection helps to simplify the 3+1 formalism (details of the decomposition into the 3+1 formalism are to be given below). The resulting Hamiltonian formalism resembles that of a Yang-Mills gauge theory (this does not happen with the 3+1 Palatini formalism which basically collapses down to the usual ADM formalism).
The results of calculations done here can be found in chapter 3 of notes Ashtekar Variables in Classical Relativity.[6] The method of proof follows that given in section II of The Ashtekar Hamiltonian for General Relativity.[7] We need to establish some results for (anti-)self-dual Lorentzian tensors.
Identities for the totally anti-symmetric tensor
Since
has signature
, it follows that
![{\displaystyle \varepsilon ^{IJKL}=-\varepsilon _{IJKL}.}](//wikimedia.org/api/rest_v1/media/math/render/svg/d24770eb86f1f15f27932a9d87422791d20294ab)
to see this consider,
![{\displaystyle \varepsilon ^{0123}=\eta ^{0I}\eta ^{1J}\eta ^{2K}\eta ^{3L}\varepsilon _{IJKL}=(-1)(1)(1)(1)\varepsilon _{0123}=-\varepsilon _{0123}.}](//wikimedia.org/api/rest_v1/media/math/render/svg/e38afd84e2ee9cefae8466b47b7d7e83200338bf)
With this definition one can obtain the following identities,
![{\displaystyle {\begin{aligned}\varepsilon ^{IJKO}\varepsilon _{LMNO}&=-6\delta _{[L}^{I}\delta _{M}^{J}\delta _{N]}^{K}&&{\text{Eq. 3}}\\\varepsilon ^{IJMN}\varepsilon _{KLMN}&=-4\delta _{[K}^{I}\delta _{L]}^{J}=-2\left(\delta _{K}^{I}\delta _{L}^{J}-\delta _{L}^{I}\delta _{K}^{J}\right)&&{\text{Eq. 4}}\end{aligned}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/67d8a0e0aaa46fed9c71214eb7bf39c9df58094f)
(the square brackets denote anti-symmetrizing over the indices).
Definition of self-dual tensor
It follows from Eq. 4 that the square of the duality operator is minus the identity,
![{\displaystyle **T^{IJ}={1 \over 4}{\varepsilon _{KL}}^{IJ}{\varepsilon _{MN}}^{KL}T^{MN}=-T^{IJ}}](//wikimedia.org/api/rest_v1/media/math/render/svg/f6859bcad70255e148ab06070226593517e66b46)
The minus sign here is due to the minus sign in Eq. 4, which is in turn due to the Minkowski signature. Had we used Euclidean signature, i.e.
, instead there would have been a positive sign. We define
to be self-dual if and only if
![{\displaystyle *S^{IJ}=iS^{IJ}.}](//wikimedia.org/api/rest_v1/media/math/render/svg/ab481633becdc10935dab897c2a4024724b8a57a)
(with Euclidean signature the self-duality condition would have been
). Say
is self-dual, write it as a real and imaginary part,
![{\displaystyle S^{IJ}={1 \over 2}T^{IJ}+{\frac {i}{2}}U^{IJ}.}](//wikimedia.org/api/rest_v1/media/math/render/svg/4215a72c1db797c5a335ba5ed5533329a5e02125)
Write the self-dual condition in terms of
and
,
![{\displaystyle *\left(T^{IJ}+iU^{IJ}\right)={1 \over 2}{\varepsilon _{KL}}^{IJ}\left(T^{KL}+iU^{KL}\right)=i\left(T^{IJ}+iU^{IJ}\right).}](//wikimedia.org/api/rest_v1/media/math/render/svg/c566fef00232fa816f6a2ea2e327e30043d3483a)
Equating real parts we read off
![{\displaystyle U^{IJ}=-{1 \over 2}{\varepsilon _{KL}}^{IJ}T^{KL}}](//wikimedia.org/api/rest_v1/media/math/render/svg/e00b9f1360fdbd560e033cd67ac38ba44ee3007c)
and so
![{\displaystyle S^{IJ}={1 \over 2}\left(T^{IJ}-{i \over 2}{\varepsilon _{KL}}^{IJ}T^{KL}\right)}](//wikimedia.org/api/rest_v1/media/math/render/svg/84b024c6f90dc7bbd630079b0064cdaa1983bb70)
where
is the real part of
.
Important lengthy calculation
The proof of Eq. 2 in straightforward. We start by deriving an initial result. All the other important formula easily follow from it. From the definition of the Lie bracket and with the use of the basic identity Eq. 3 we have
![{\displaystyle {\begin{aligned}*[F,*G]^{IJ}&={\frac {1}{2}}{\varepsilon _{MN}}^{IJ}\left(F^{MK}{(*G)_{K}}^{N}-(*G)^{MK}{F_{K}}^{N}\right)\\&={\frac {1}{2}}{\varepsilon _{MN}}^{IJ}\left(F^{MK}{\frac {1}{2}}{\varepsilon _{OPK}}^{N}G^{OP}-{\frac {1}{2}}{\varepsilon _{OP}}^{MK}G^{OP}{F_{K}}^{N}\right)\\&={1 \over 4}\left({\varepsilon _{MN}}^{IJ}{\varepsilon _{OP}}^{KN}+{\varepsilon _{NM}}^{IJ}{\varepsilon _{OP}}^{NK}\right){F^{M}}_{K}G^{OP}\\&={1 \over 2}{\varepsilon _{MN}}^{IJ}{\varepsilon _{OP}}^{KN}{F^{M}}_{K}G^{OP}\\&={1 \over 2}\varepsilon ^{MIJN}\varepsilon _{OPKN}{F_{M}}^{K}G^{OP}\\&=-{\frac {1}{2}}\varepsilon ^{KIJN}\varepsilon _{OPMN}{F^{M}}_{K}G^{OP}\\&={\frac {1}{2}}\left(\delta _{O}^{K}\delta _{P}^{I}\delta _{M}^{J}+\delta _{M}^{K}\delta _{O}^{I}\delta _{P}^{J}+\delta _{P}^{K}\delta _{M}^{I}\delta _{O}^{J}-\delta _{P}^{K}\delta _{O}^{I}\delta _{M}^{J}-\delta _{M}^{K}\delta _{P}^{I}\delta _{O}^{J}-\delta _{O}^{K}\delta _{M}^{I}\delta _{P}^{J}\right){F^{M}}_{K}G^{OP}\\&={\frac {1}{2}}\left({F^{J}}_{K}G^{KI}+{F^{K}}_{K}G^{IJ}+{F^{I}}_{K}G^{JK}-{F^{J}}_{K}G^{IK}-{F^{K}}_{K}G^{JI}-{F^{I}}_{K}G^{KJ}\right)\\&=-F^{IK}{G_{K}}^{J}+G^{IK}{F_{K}}^{J}\\&=-[F,G]^{IJ}\end{aligned}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/69eb5578d1e2d1eb958160cacf9c33b37014b0af)
That gives the formula
![{\displaystyle *[F,*G]^{IJ}=-[F,G]^{IJ}\qquad Eq.5.}](//wikimedia.org/api/rest_v1/media/math/render/svg/082d631e19b51d8bd19a6c18ccd0420053d973c2)
Derivation of important results
Now using Eq.5 in conjunction with
we obtain
![{\displaystyle *(-[F,G]^{IJ})=*(*[F,*G]^{IJ})=**[F,*G]^{IJ}=-[F,*G]^{IJ}.}](//wikimedia.org/api/rest_v1/media/math/render/svg/2fb10ff7c2c67fb2aa880795806f9fd9ee38fe76)
So we have
![{\displaystyle *[F,G]^{IJ}=[F,*G]^{IJ}\qquad Eq.6.}](//wikimedia.org/api/rest_v1/media/math/render/svg/b7bebcf3559edff094074927faf19293265f3890)
Consider
![{\displaystyle *[F,G]^{IJ}=-*[G,F]^{IJ}=-[G,*F]^{IJ}=[*F,G]^{IJ}.}](//wikimedia.org/api/rest_v1/media/math/render/svg/3d01f988d0983918465f56578daa39ee29a5824b)
where in the first step we have used the anti-symmetry of the Lie bracket to swap
and
, in the second step we used
and in the last step we used the anti-symmetry of the Lie bracket again. So we have
![{\displaystyle *[F,G]^{IJ}=[*F,G]^{IJ}\qquad Eq.7.}](//wikimedia.org/api/rest_v1/media/math/render/svg/48a10fccb9846dd9b9a02efa84c8fba2407b339a)
Then
![{\displaystyle {\begin{aligned}\left(P^{(\pm )}[F,G]\right)^{IJ}&={1 \over 2}\left([F,G]^{IJ}\mp i*[F,G]^{IJ}\right)\\&={1 \over 2}\left([F,G]^{IJ}+[F,\mp i*G]^{IJ}\right)\\&=\left[F,P^{(\pm )}G\right]^{IJ}&&{\text{Eq. 8}}\end{aligned}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/bcf4e11728dbe83d2cf2e81bf68b8622a3b83acd)
where we used Eq. 6 going from the first line to the second line. Similarly we have
![{\displaystyle \left(P^{(\pm )}[F,G]\right)^{IJ}=[P^{(\pm )}F,G]^{IJ}\qquad Eq.9}](//wikimedia.org/api/rest_v1/media/math/render/svg/84a944cacbcbb0688189ec4c8acd2d614e5c068b)
by using Eq 7. Now as
is a projection it satisfies
, as can easily be verified by direct computation:
![{\displaystyle {\begin{aligned}{}(P^{(\pm )})^{2}&={1 \over 4}(1\mp i*)(1\mp i*)\\{}&={1 \over 4}(1-**\mp 2i*)\\{}&={1 \over 4}(2\mp 2i*)\\{}&=P^{(\pm )}\end{aligned}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/d72ebb504c72c15832c69709b0c7b6e55d1ce9c6)
Applying this in conjunction with Eq. 8 and Eq. 9 we obtain
![{\displaystyle {\begin{aligned}{}\left(P^{(\pm )}[F,G]\right)^{IJ}&=\left((P^{(\pm )})^{2}[F,G]\right)^{IJ}\\&=\left(P^{(\pm )}[F,P^{(\pm )}G]\right)^{IJ}\\{}&=[P^{(\pm )}F,P^{(\pm )}G]^{IJ}\qquad Eq.10.\end{aligned}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/2d1e95fcd7698648fd236a9b3020b318bcddf8ac)
From Eq. 10 and Eq. 9 we have
![{\displaystyle \left[P^{(\pm )}F,P^{(\pm )}G\right]^{IJ}=\left[P^{(\pm )}F,G\right]^{IJ}=\left[P^{(\pm )}F,P^{(\pm )}G+P^{(\mp )}G\right]^{IJ}=\left[P^{(\pm )}F,P^{(\pm )}G\right]^{IJ}+\left[P^{(\pm )}F,P^{(\mp )}G\right]^{IJ}}](//wikimedia.org/api/rest_v1/media/math/render/svg/1e9f6b81cb55f6be2cc2ec723fb4e0597fa32081)
where we have used that any
can be written as a sum of its self-dual and anti-sef-dual parts, i.e.
. This implies:
![{\displaystyle {\begin{aligned}{}\left[P^{+}F,P^{-}G\right]^{IJ}&=0\\{}\left[P^{-}F,P^{+}G\right]^{IJ}&=0\end{aligned}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/1193915614ed2953fca1afdfe5ae536880648fc2)
Summary of main results
Altogether we have,
![{\displaystyle \left(P^{(\pm )}[F,G]\right)^{IJ}=\left[P^{(\pm )}F,G\right]^{IJ}=\left[F,P^{(\pm )}G\right]^{IJ}=\left[P^{(\pm )}F,P^{(\pm )}G\right]^{IJ}}](//wikimedia.org/api/rest_v1/media/math/render/svg/d51665d18ab16618c00a158fb99f7e208fe40e31)
which is our main result, already stated above as Eq. 2. We also have that any bracket splits as
![{\displaystyle [F,G]^{IJ}=\left[P^{+}F+P^{-}F,P^{+}G+P^{-}F\right]^{IJ}=\left[P^{+}F,P^{+}G\right]^{IJ}+\left[P^{-}F,P^{-}G\right]^{IJ}.}](//wikimedia.org/api/rest_v1/media/math/render/svg/dcc77df3caeea50e6e050d9251f90ea53b830fe5)
into a part that depends only on self-dual Lorentzian tensors and is itself the self-dual part of
and a part that depends only on anti-self-dual Lorentzian tensors and is the anit-self-dual part of ![{\displaystyle [F,G]^{IJ}.}](//wikimedia.org/api/rest_v1/media/math/render/svg/070199cf923aa28d488336aae7afdfab1e5aec9f)
The proof given here follows that given in lectures by Jorge Pullin[8]
The Palatini action
![{\displaystyle S(e,\omega )=\int d^{4}xee_{I}^{a}e_{J}^{b}{\Omega _{ab}}^{IJ}[\omega ]\qquad Eq.11}](//wikimedia.org/api/rest_v1/media/math/render/svg/b48b849185248a31f44b9dca5499968f9a717143)
where the Ricci tensor,
, is thought of as constructed purely from the connection
, not using the frame field. Variation with respect to the tetrad gives Einstein's equations written in terms of the tetrads, but for a Ricci tensor constructed from the connection that has no a priori relationship with the tetrad. Variation with respect to the connection tells us the connection satisfies the usual compatibility condition
![{\displaystyle D_{b}e_{a}^{I}=0.}](//wikimedia.org/api/rest_v1/media/math/render/svg/13aa8189c95f50c4b6758fa119f668bac2a59082)
This determines the connection in terms of the tetrad and we recover the usual Ricci tensor.
The self-dual action for general relativity is given above.
![{\displaystyle S(e,A)=\int d^{4}xee_{I}^{a}e_{J}^{b}{F_{ab}}^{IJ}[A]}](//wikimedia.org/api/rest_v1/media/math/render/svg/35370f732ce9c5425252db77539aa3f6ab1c30a3)
where
is the curvature of the
, the self-dual part of
,
![{\displaystyle A_{a}^{IJ}={1 \over 2}\left(\omega _{a}^{IJ}-{i \over 2}{\varepsilon ^{IJ}}_{MN}\omega _{a}^{MN}\right).}](//wikimedia.org/api/rest_v1/media/math/render/svg/db9385b2fef8833410db373de512c1ac47f838dd)
It has been shown that
is the self-dual part of ![{\displaystyle \Omega [\omega ].}](//wikimedia.org/api/rest_v1/media/math/render/svg/9fdb3669dd009565ddb2470c35c4b59cf2bd8c2d)
Let
be the projector onto the three surface and define vector fields
![{\displaystyle E_{I}^{a}=q_{b}^{a}e_{I}^{b},}](//wikimedia.org/api/rest_v1/media/math/render/svg/a6d385a9494ed1cd644a8e9dec8b1446a2e74d27)
which are orthogonal to
.
Writing
![{\displaystyle E_{I}^{a}=\left(\delta _{b}^{a}+n_{b}n^{a}\right)e_{I}^{b}}](//wikimedia.org/api/rest_v1/media/math/render/svg/619388b744ee01d87364858938e6cdd64720eadc)
then we can write
![{\displaystyle {\begin{aligned}\int &d^{4}x\left(eE_{I}^{a}E_{J}^{b}{F_{ab}}^{IJ}-2eE_{I}^{a}e_{J}^{d}n_{d}n^{b}{F_{ab}}^{IJ}\right)=\\&=\int d^{4}x\left(e\left(\delta _{c}^{a}+n_{c}n^{a}\right)e_{I}^{c}\left(\delta _{d}^{b}+n_{d}n^{b}\right)e_{J}^{d}{F_{ab}}^{IJ}-2e\left(\delta _{c}^{a}+n_{c}n^{a}\right)e_{I}^{c}e_{J}^{d}n_{d}n^{b}{F_{ab}}^{IJ}\right)\\&=\int d^{4}x\left(ee_{I}^{a}e_{J}^{b}{F_{ab}}^{IJ}+en_{c}n^{a}e_{I}^{c}e_{J}^{b}{F_{ab}}^{IJ}+ee_{I}^{a}n_{d}n^{b}e_{J}^{d}{F_{ab}}^{IJ}+en_{c}n^{a}n_{d}n^{b}E_{I}^{c}E_{J}^{d}{F_{ab}}^{IJ}-2ee_{I}^{a}e_{J}^{d}n_{d}n^{b}{F_{ab}}^{IJ}-2n_{c}n^{a}e_{I}^{c}e_{J}^{d}n_{d}n^{b}{F_{ab}}^{IJ}\right)\\&=\int d^{4}xee_{I}^{a}e_{J}^{b}{F_{ab}}^{IJ}\\&=S(E,A)\end{aligned}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/903396f4ac0630eb7c06ffe7f86f0d4fe1ce3baf)
where we used
and
.
So the action can be written
![{\displaystyle S(E,A)=\int d^{4}x\left(eE_{I}^{a}E_{J}^{b}{F_{ab}}^{IJ}-2eE_{I}^{a}e_{J}^{d}n_{d}n^{b}{F_{ab}}^{IJ}\right)\qquad Eq.12}](//wikimedia.org/api/rest_v1/media/math/render/svg/3faa6d6db31c5e75ea7d164a95ad7423ddded87e)
We have
. We now define
![{\displaystyle {\tilde {E}}_{I}^{a}={\sqrt {q}}E_{I}^{a}}](//wikimedia.org/api/rest_v1/media/math/render/svg/7142250dae64138b567442e5c09e4d278907d343)
An internal tensor
is self-dual if and only if
![{\displaystyle *S^{IJ}:={1 \over 2}{\varepsilon ^{IJ}}_{MN}S^{MN}=iS^{IJ}}](//wikimedia.org/api/rest_v1/media/math/render/svg/3ff4c293557b107d15e49ac860b3b394eceb9ba2)
and given the curvature
is self-dual we have
![{\displaystyle {F_{ab}}^{IJ}=-i{1 \over 2}{\varepsilon ^{IJ}}_{MN}{F_{ab}}^{MN}}](//wikimedia.org/api/rest_v1/media/math/render/svg/74bad9108332f737bf3af7fd1b0702f12e5c3d10)
Substituting this into the action (Eq. 12) we have,
![{\displaystyle S(E,A)=\int d^{4}x\left(-i{\frac {1}{2}}\left({\frac {N}{\sqrt {q}}}\right){\tilde {E}}_{I}^{a}{\tilde {E}}_{J}^{b}{\varepsilon ^{IJ}}_{MN}{F_{ab}}^{MN}-2Nn^{b}{\tilde {E}}_{I}^{a}n_{J}{F_{ab}}^{IJ}\right)}](//wikimedia.org/api/rest_v1/media/math/render/svg/c1361f6e0977daabb9c046ddaaa0e6b9ee7219dc)
where we denoted
. We pick the gauge
and
(this means
). Writing
, which in this gauge
. Therefore,
![{\displaystyle {\begin{aligned}S(E,A)&=\int d^{4}x\left(-i{1 \over 2}\left({N \over {\sqrt {q}}}\right){\tilde {E}}_{I}^{a}{\tilde {E}}_{J}^{b}\left({\varepsilon ^{IJ}}_{M0}{F_{ab}}^{M0}+{\varepsilon ^{IJ}}_{0M}{F_{ab}}^{0M}\right)-2Nn^{b}{\tilde {E}}_{I}^{a}n_{J}{F_{ab}}^{IJ}\right)\\&=\int d^{4}x\left(-i\left({N \over {\sqrt {q}}}\right){\tilde {E}}_{I}^{a}{\tilde {E}}_{J}^{b}{\varepsilon ^{IJ}}_{M}{F_{ab}}^{M0}+2Nn^{b}{\tilde {E}}_{I}^{a}{F_{ab}}^{I0}\right)\end{aligned}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/8810c5fa4a92d7ea2c8fd66f6ecfbb31ea48468c)
The indices
range over
and we denote them with lower case letters in a moment. By the self-duality of
,
![{\displaystyle A_{a}^{i0}=-i{1 \over 2}{\varepsilon ^{i0}}_{jk}A_{a}^{jk}=i{1 \over 2}{\varepsilon ^{i}}_{jk}A_{a}^{jk}=iA_{a}^{i}.}](//wikimedia.org/api/rest_v1/media/math/render/svg/7f2ab293cc84f59e4e04c2e9130d4ceee5e10224)
where we used
![{\displaystyle {\varepsilon ^{i0}}_{jk}=-{\varepsilon ^{i}}_{0jk}=-{\varepsilon ^{i}}_{jk0}=-{\varepsilon ^{i}}_{jk}.}](//wikimedia.org/api/rest_v1/media/math/render/svg/b0aa6b37b3c875e3174aa4e5a10677fdee2293fe)
This implies
![{\displaystyle {\begin{aligned}{F_{ab}}^{i0}&=\partial _{a}A_{b}^{i0}-\partial _{b}A_{a}^{i0}+A_{a}^{ik}{A_{bk}}^{0}-A_{b}^{ik}{A_{ak}}^{0}\\&=i\left(\partial _{a}A_{b}^{i}-\partial _{b}A_{a}^{i}+A_{a}^{ik}A_{bk}-A_{b}^{ik}A_{ak}\right)\\&=i\left(\partial _{a}A_{b}^{i}-\partial _{b}A_{a}^{i}+\varepsilon _{ijk}A_{a}^{j}A_{b}^{k}\right)\\&=iF_{ab}^{i}\end{aligned}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/e0c005d4bb928fc3db0331d1787b6229bdd59fa3)
We replace in the second term in the action
by
. We need
![{\displaystyle {\mathcal {L}}_{t}A_{b}^{i}=t^{a}\partial _{a}A_{b}^{i}+A_{a}^{i}\partial _{b}t^{a}}](//wikimedia.org/api/rest_v1/media/math/render/svg/84b13b77f3d9007dd2fc6c96a91b7e442037c280)
and
![{\displaystyle {\mathcal {D}}_{b}\left(t^{a}A_{a}^{i}\right)=\partial _{b}\left(t^{a}A_{a}^{i}\right)+\varepsilon _{ijk}A_{b}^{j}\left(t^{a}A_{a}^{k}\right)}](//wikimedia.org/api/rest_v1/media/math/render/svg/2749a6ccf2ab8ffada495231bd9f4e28a1615d01)
to obtain
![{\displaystyle {\mathcal {L}}_{t}A_{b}^{i}-{\mathcal {D}}_{b}\left(t^{a}A_{a}^{i}\right)=t^{a}\left(\partial _{a}A_{b}^{i}-\partial _{b}A_{a}^{i}+\varepsilon _{ijk}A_{a}^{j}A_{b}^{k}\right)=t^{a}F_{ab}^{i}.}](//wikimedia.org/api/rest_v1/media/math/render/svg/afa1126839cf017e3c8f5038e783edbea4a8e123)
The action becomes
![{\displaystyle {\begin{aligned}S&=\int d^{4}x\left(-i\left({N \over {\sqrt {q}}}\right){\tilde {E}}_{I}^{a}{\tilde {E}}_{J}^{b}{\varepsilon ^{IJ}}_{M}{F_{ab}}^{M0}-2\left(t^{a}-N^{a}\right){\tilde {E}}_{I}^{b}{F_{ab}}^{I0}\right)\\&=\int d^{4}x\left(-2i{\tilde {E}}_{i}^{b}{\mathcal {L}}_{t}A_{b}^{i}+2i{\tilde {E}}_{i}^{b}{\mathcal {D}}_{b}\left(t^{a}A_{a}^{i}\right)+2iN^{a}{\tilde {E}}_{i}^{b}F_{ab}^{i}-\left({N \over {\sqrt {q}}}\right)\varepsilon _{ijk}{\tilde {E}}_{i}^{a}{\tilde {E}}_{j}^{b}F_{ab}^{k}\right)\end{aligned}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/52b54d5b3dd33ccc98401bc0105d9f96057dfab3)
where we swapped the dummy variables
and
in the second term of the first line. Integrating by parts on the second term,
![{\displaystyle {\begin{aligned}\int d^{4}x{\tilde {E}}_{i}^{b}{\mathcal {D}}_{b}\left(t^{a}A_{a}^{i}\right)&=\int dtd^{3}x{\tilde {E}}_{i}^{b}\left(\partial _{b}(t^{a}A_{a}^{i})+\varepsilon _{ijk}A_{b}^{j}(t^{a}A_{a}^{k})\right)\\&=-\int dtd^{3}xt^{a}A_{a}^{i}\left(\partial _{b}{\tilde {E}}_{i}^{b}+\varepsilon _{ijk}A_{b}^{j}{\tilde {E}}_{k}^{b}\right)\\&=-\int d^{4}xt^{a}A_{a}^{i}{\mathcal {D}}_{b}{\tilde {E}}_{i}^{b}\end{aligned}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/0b8a21213d6bf2f8fe7dccf99c8d3ece8b1d98b6)
where we have thrown away the boundary term and where we used the formula for the covariant derivative on a vector density
:
![{\displaystyle {\mathcal {D}}_{b}{\tilde {V}}_{i}^{b}=\partial _{b}{\tilde {V}}_{i}^{b}+\varepsilon _{ijk}A_{b}^{j}{\tilde {V}}_{k}^{b}.}](//wikimedia.org/api/rest_v1/media/math/render/svg/09bf7d0ebbcc3ad3a9027f1567f44cfed78077a4)
The final form of the action we require is
![{\displaystyle S=\int d^{4}x\left(-2i{\tilde {E}}_{i}^{b}{\mathcal {L}}_{t}A_{b}^{i}-2i\left(t^{a}A_{a}^{i}\right){\mathcal {D}}_{b}{\tilde {E}}_{i}^{b}+2iN^{a}{\tilde {E}}_{i}^{b}F_{ab}^{i}+\left({N \over {\sqrt {q}}}\right)\varepsilon _{ijk}{\tilde {E}}_{i}^{a}{\tilde {E}}_{j}^{b}F_{ab}^{k}\right)}](//wikimedia.org/api/rest_v1/media/math/render/svg/4154ad2e8d0a7349d1bd88d9d4c15cca6061310a)
There is a term of the form "
" thus the quantity
is the conjugate momentum to
. Hence, we can immediately write
![{\displaystyle \left\{A_{a}^{i}(x),{\tilde {E}}_{j}^{b}(y)\right\}={i \over 2}\delta _{a}^{b}\delta _{j}^{i}\delta ^{3}(x,y).}](//wikimedia.org/api/rest_v1/media/math/render/svg/9d720a0a036f5623039e8becf77cfa841e5cd2e6)
Variation of action with respect to the non-dynamical quantities
, that is the time component of the four-connection, the shift function
, and lapse function
give the constraints
![{\displaystyle {\mathcal {D}}_{a}{\tilde {E}}_{i}^{a}=0,}](//wikimedia.org/api/rest_v1/media/math/render/svg/419502378902dc39211f259937413a99bf87ac24)
![{\displaystyle F_{ab}^{i}{\tilde {E}}_{i}^{b}=0,}](//wikimedia.org/api/rest_v1/media/math/render/svg/a061ac4d0149624b9f9c809eefff72c60f8c8bd4)
![{\displaystyle \varepsilon _{ijk}{\tilde {E}}_{i}^{a}{\tilde {E}}_{j}^{b}F_{ab}^{k}=0\qquad Eq.13.}](//wikimedia.org/api/rest_v1/media/math/render/svg/2544238640ecfeada1119e7195db6e46c08a4631)
Varying with respect to
actually gives the last constraint in Eq. 13 divided by
, it has been rescaled to make the constraint polynomial in the fundamental variables. The connection
can be written
![{\displaystyle A_{a}^{i}={1 \over 2}{\varepsilon ^{i}}_{jk}A_{a}^{jk}={1 \over 2}{\varepsilon ^{i}}_{jk}\left(\omega _{a}^{jk}-i{1 \over 2}\left({\varepsilon ^{jk}}_{m0}\omega _{a}^{m0}+{\varepsilon ^{jk}}_{0m}\omega _{a}^{0m}\right)\right)=\Gamma _{a}^{i}-i\omega _{a}^{0i}}](//wikimedia.org/api/rest_v1/media/math/render/svg/0915c17935d87ff3bc70c1225182e81ecb20dbc5)
and
![{\displaystyle E_{ci}\omega _{a}^{0i}=-q_{a}^{b}E_{ci}\omega _{b}^{i0}=-q_{a}^{b}E_{ci}e^{di}\nabla _{b}e_{d}^{0}=q_{a}^{b}q_{c}^{d}\nabla _{b}n_{d}=K_{ac}}](//wikimedia.org/api/rest_v1/media/math/render/svg/0de69638b223c564a1edd3502951f12a74d52cf4)
where we used
![{\displaystyle e_{d}^{0}=\eta ^{0I}g_{dc}e_{I}^{c}=-g_{dc}e_{0}^{c}=-n_{d},}](//wikimedia.org/api/rest_v1/media/math/render/svg/15648f8ebcf029908d2e63018cf44bffe23471b2)
therefore
. So the connection reads
![{\displaystyle A_{a}^{i}=\Gamma _{a}^{i}-iK_{a}^{i}.}](//wikimedia.org/api/rest_v1/media/math/render/svg/db024c95333442c15e1fe2ab7bddbb1aef3466c3)
This is the so-called chiral spin connection.
Because Ashtekar's variables are complex it results in complex general relativity. To recover the real theory one has to impose what are known as the reality conditions. These require that the densitized triad be real and that the real part of the Ashtekar connection equals the compatible spin connection.
More to be said on this, later.
Ashtekar Variables in Classical General Relativity, Domenico Giulini, Springer Lecture Notes in Physics 434 (1994), 81-112, arXiv:gr-qc/9312032 The Ashtekar Hamiltonian for General Relativity by Ceddric Beny
Knot theory and quantum gravity in loop space: a primer by Jorge Pullin; AIP Conf.Proc.317:141-190,1994, arXiv:hep-th/9301028