Self-dual Palatini action explained

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 Palatini action

See main article: Einstein–Hilbert action, Frame fields in general relativity, spin connection and Tetradic Palatini action. The Palatini action for general relativity has as its independent variables the tetrad

e ^{\alpha} _{I} and a spin connection {\omega ^{IJ} _{\alpha}} . Much more details and derivations can be found in the article tetradic Palatini action. The spin connection defines a covariant derivative D _{\alpha} . The space-time metric is recovered from the tetrad by the formula g _{\alpha} = e ^{I} _{\alpha} e ^{J} _{\beta} η _{IJ} . We define the `curvature' by{\Omega _{\alpha} }^ = \partial_\alpha ^ - \partial_\beta ^ + \omega_\alpha^ ^ - \omega_\beta^ ^ \qquad Eq. 1.

The Ricci scalar of this curvature is given by

e ^{\alpha} _{I} e ^{\beta} _{J} {\Omega _{\alpha} }^. The Palatini action for general relativity readsS = \int d ^{4} x e e ^{\alpha} _{I} e ^{\beta} _{J} {\Omega _{\alpha} }^ [\omega]

where

e = \sqrt{ - g} . Variation with respect to the spin connection {\omega ^{IJ} _{\alpha}} implies that the spin connection is determined by the compatibility condition D _{\alpha} e ^{\beta} _{I} = 0 and hence becomes the usual covariant derivative \nabla _{\alpha} . Hence the connection becomes a function of the tetrads and the curvature {\Omega _{\alpha} }^ is replaced by the curvature {R _{\alpha} }^ of \nabla _{\alpha} . Then e ^{\alpha} _{I} e ^{\beta} _{J} {R _{\alpha} }^ is the actual Ricci scalar R . Variation with respect to the tetrad gives Einsteins equationR _{\alpha} - { 1 \over 2 } g _{\alpha} R = 0 .

Self-dual variables

(Anti-)self-dual parts of a tensor

We will need what is called the totally antisymmetry tensor or Levi-Civita symbol,

\varepsilon _{IJKL} , which is equal to either +1 or −1 depending on whether IJKL is either an even or odd permutation of 0123 , respectively, and zero if any two indices take the same value. The internal indices of \varepsilon _{IJKL} are raised with the Minkowski metric η ^{IJ} .

Now, given any anti-symmetric tensor

T ^{IJ} , we define its dual as*T ^{IJ} = { 1 \over 2 } {\varepsilon _{KL} }^ T^.

The self-dual part of any tensor

T ^{IJ} is defined as{} ^{+T} ^{IJ} : = { 1 \over 2 } \left ( T ^{IJ} - {i \over 2 } {\varepsilon _{KL} }^ T^ \right)

with the anti-self-dual part defined as

{} ^{-T} ^{IJ} : = { 1 \over 2 } \left ( T ^{IJ} + {i \over 2 } {\varepsilon _{KL} }^ T^ \right)

(the appearance of the imaginary unit

i is related to the Minkowski signature as we will see below).

Tensor decomposition

Now given any anti-symmetric tensor

T ^{IJ} , we can decompose it asT ^{IJ} = (1)/(2) \left (T ^{IJ} - (i)/(2) {\varepsilon _{KL} }^ T^ \right) + \frac \left (T^ +\frac ^ T^ \right) =^+T^ +^-T^

where

{} ^{+T} ^{IJ} and {} ^{-T} ^{IJ} are the self-dual and anti-self-dual parts of T ^{IJ} respectively. Define the projector onto (anti-)self-dual part of any tensor asP ^{(\pm)} = { 1 \over 2 } ( 1 \mp i *).

The meaning of these projectors can be made explicit. Let us concentrate of

P ^{+} ,\left (P ^{+} T \right ) ^{IJ} = \left ({ 1 \over 2 } ( 1- i*) T \right ) ^{IJ} = { 1 \over 2 } \left ( {\delta ^{I}} _{K} {\delta ^{J}} _{L} - i { 1 \over 2 } {\varepsilon _{KL} }^ \right) T^ = \left (T^ - ^ T^ \right) = ^+ T^.

Then

{} ^{\pm} T ^{IJ} = \left (P ^{(\pm)} T \right) ^{IJ} .

The Lie bracket

An important object is the Lie bracket defined by

[F, G] ^{IJ} : = F ^{IK} {G ^{J} _{K}} - G ^{IK} {F ^{J} _{K}} ,

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):

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} Eq. 2

and

[F, G] = \left [P ^{+} F, P ^{+} G \right ] + \left [P ^{-} F, P ^{-} G \right ].

That is the Lie bracket, which defines an algebra, decomposes into two separate independent parts. We write

ak{so}(1,3) _{\Complex} = ak{so}(1,3) ^{+} _{\Complex} + ak{so}(1,3) ^{-} _{\Complex}

where

ak{so}(1,3) ^{\pm} _{\Complex} contains only the self-dual (anti-self-dual) elements of ak{so}(1,3) _{\Complex.}

The Self-dual Palatini action

We define the self-dual part,

{A ^{IJ} _{\alpha}} , of the connection {\omega ^{IJ} _{\alpha}} as{A ^{IJ} _{\alpha}} = { 1 \over 2 } \left ( {\omega ^{IJ} _{\alpha}} - {i \over 2 } {\varepsilon _{KL} }^ ^ \right),

which can be more compactly written

{A ^{IJ} _{\alpha}} = \left (P ^{+} \omega _{\alpha} \right ) ^{IJ} .

Define

{F _{\alpha} }^ as the curvature of the self-dual connection{F _{\alpha} }^ = \partial_\alpha ^ - \partial_\beta ^ + ^ ^ - ^ ^.

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,

\begin{align} {F _{\alpha} }^ &= \partial_\alpha \left (P^+ \omega_\beta \right)^ - \partial_\beta \left (P^+ \omega_\alpha \right)^ + \left [P^+ \omega_\alpha, P^+ \omega_\beta \right ]^ \\&= \left (P^+ 2 \partial_ \right)^ + \left (P^+ [\omega_\alpha, \omega_\beta] \right)^ \\&= \left (P^+ \Omega_ \right)^\end

The self-dual action is

S = \int d ^{4} x e e ^{\alpha} _{I} e ^{\beta} _{J} {F _{\alpha} }^.

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

{A ^{IJ} _{\alpha}} . Varying the tetrad field, one obtains a self-dual analog of Einstein's equation:{} ^{+ R} _{\alpha \beta} - { 1 \over 2 } g _{\alpha} {} ^{+R} = 0 .

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).

Derivation of main results for self-dual variables

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

η _{IJ} has signature ( - , + , + , + ) , it follows that\varepsilon ^{IJKL} = - \varepsilon _{IJKL} .

to see this consider,

\varepsilon ^{0123} = η ^{0I} η ^{1J} η ^{2K} η ^{3L} \varepsilon _{IJKL} = ( -1 ) ( 1 ) ( 1 ) ( 1 ) \varepsilon _{0123} = - \varepsilon _{0123} .

With this definition one can obtain the following identities,

\begin{align} \varepsilon ^{IJKO} \varepsilon _{LMNO} & = -6 \delta ^{I} _{[L} \delta ^{J} _{M} \delta ^{K} _{N]} && Eq.3 \\ \varepsilon ^{IJMN} \varepsilon _{KLMN} & = -4 \delta ^{I} _{[K} \delta ^{J} _{L]} = - 2 \left (\delta ^{I} _{K} \delta ^{J} _{L} - \delta ^{I} _{L} \delta ^{J} _{K} \right ) && Eq.4 \end{align}

(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,

** T ^{IJ} = { 1 \over 4 } {\varepsilon _{KL} }^ ^ T^ = - T^

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 S ^{IJ} to be self-dual if and only if*S ^{IJ} = i S ^{IJ} .

(with Euclidean signature the self-duality condition would have been

*S ^{IJ} = S ^{IJ} ). Say S ^{IJ} is self-dual, write it as a real and imaginary part,S ^{IJ} = { 1 \over 2 } T ^{IJ} + (i)/(2) U ^{IJ} .

Write the self-dual condition in terms of

U and V ,* \left (T ^{IJ} + i U ^{IJ} \right ) = { 1 \over 2 } {\varepsilon _{KL} }^ \left (T^ + i U^ \right) = i \left (T^ + i U^ \right) .

Equating real parts we read off

U ^{IJ} = - { 1 \over 2 } {\varepsilon _{KL} }^ T^

and so

S ^{IJ} = { 1 \over 2 } \left (T ^{IJ} - {i \over 2 } {\varepsilon _{KL} }^ T^ \right)

where

T ^{IJ} is the real part of 2 S ^{IJ} .

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

\begin{align} *[F,*G] ^{IJ} & = (1)/(2) {\varepsilon _{MN} }^ \left (F^ ^ -(*G)^ ^ \right) \\&= \frac ^ \left (F^ \frac ^ G^ - \frac ^ G^ ^ \right) \\&= \left (^ ^ + ^ ^ \right) _ G^ \\&= ^ ^ _ G^ \\&= \varepsilon^ \varepsilon_ ^ G^ \\&= -\frac \varepsilon^ \varepsilon_ _ G^ \\&= \frac \left (\delta^K_O \delta^I_P \delta^J_M + \delta^K_M \delta^I_O \delta^J_P + \delta^K_P \delta^I_M \delta^J_O - \delta^K_P \delta^I_O \delta^J_M - \delta^K_M \delta^I_P \delta^J_O - \delta^K_O \delta^I_M \delta^J_P \right) _ G^ \\&= \frac \left (_ G^ + _ G^ + _ G^ - _ G^ - _ G^ - _ G^ \right) \\&= - F^ ^ + G^ ^ \\&= -[F, G]^\end

That gives the formula

*[F,*G] ^{IJ} = - [F, G] ^{IJ} Eq. 5 .

Derivation of important results

Now using Eq.5 in conjunction with

** = - 1 we obtain* ( - [F, G] ^{IJ} ) = *(*[F,*G] ^{IJ} ) = **[F,*G] ^{IJ} = - [F,*G] ^{IJ} .

So we have

* [F, G] ^{IJ} = [F, *G] ^{IJ} Eq. 6 .

Consider

*[F,G] ^{IJ} = - *[G,F] ^{IJ} = - [G,*F] ^{IJ} = [*F, G] ^{IJ} .

where in the first step we have used the anti-symmetry of the Lie bracket to swap

F and G , in the second step we used Eq. 6 and in the last step we used the anti-symmetry of the Lie bracket again. So we have* [F, G] ^{IJ} = [*F, G] ^{IJ} Eq. 7 .

Then

\begin{align} \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} && Eq.8 \end{align}

where we used Eq. 6 going from the first line to the second line. Similarly we have

\left (P ^{(\pm)} [F, G] \right ) ^{IJ} = [P ^{(\pm)} F, G] ^{IJ} Eq. 9

by using Eq 7. Now as

P ^{(\pm)} is a projection it satisfies (P ^{(\pm)} ) ^{2} = P ^{(\pm)} , as can easily be verified by direct computation:\begin{align} {}(P ^{(\pm)} ) ^{2} & = { 1 \over 4 } ( 1 \mp i *) ( 1 \mp i *) \\ {}& = { 1 \over 4 } ( 1 - ** \mp 2 i *) \\ {}& = { 1 \over 4 } ( 2 \mp 2 i *) \\ {}& = P ^{(\pm)} \end{align}

Applying this in conjunction with Eq. 8 and Eq. 9 we obtain

\begin{align} {}\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} Eq. 10 . \end{align}

From Eq. 10 and Eq. 9 we have

\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}

where we have used that any

G can be written as a sum of its self-dual and anti-sef-dual parts, i.e. G = P ^{(\pm)} G + P ^{(\mp)} G . This implies:\begin{align} {}\left [P ^{+} F, P ^{-} G \right ] ^{IJ} & = 0 \\ {}\left [P ^{-} F, P ^{+} G \right ] ^{IJ} & = 0 \end{align}

Summary of main results

Altogether we have,

\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}

which is our main result, already stated above as Eq. 2. We also have that any bracket splits as

[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} .

into a part that depends only on self-dual Lorentzian tensors and is itself the self-dual part of

[F, G] ^{IJ} , and a part that depends only on anti-self-dual Lorentzian tensors and is the anit-self-dual part of [F, G] ^{IJ} .

Derivation of Ashtekar's Formalism from the Self-dual Action

The proof given here follows that given in lectures by Jorge Pullin^[8]

The Palatini action

S(e,\omega) = \int d ^{4} x e e ^{a} _{I} e ^{b} _{J} {\Omega _{ab} }^ [\omega] \qquad Eq. 11

where the Ricci tensor,

{\Omega _{ab} }^, is thought of as constructed purely from the connection \omega ^{IJ} _{a} , 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 conditionD _{b} e ^{I} _{a} = 0 .

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.

S(e,A) = \int d ^{4} x e e ^{a} _{I} e ^{b} _{J} {F _{ab} }^ [A]

where

F is the curvature of the A , the self-dual part of \omega ,A ^{IJ} _{a} = { 1 \over 2 } \left (\omega ^{IJ} _{a} - {i \over 2 } {\varepsilon ^{IJ} }_ \omega_a^ \right).

It has been shown that

F[A] is the self-dual part of \Omega [\omega].

Let

q ^{a} _{b} = \delta ^{a} _{b} + n ^{a} n _{b} be the projector onto the three surface and define vector fieldsE ^{a} _{I} = q ^{a} _{b} e ^{b} _{I,}

which are orthogonal to

n ^{a} .

Writing

E ^{a} _{I} = \left (\delta ^{a} _{b} + n _{b} n ^{a} \right ) e ^{b} _{I}

then we can write

\begin{align} \int & d ^{4} x \left (e E ^{a} _{I} E ^{b} _{J} {F _{ab} }^ - 2 e E^a_I e^d_J n_d n^b ^ \right) = \\&= \int d^4 x \left (e \left (\delta_c^a + n_c n^a \right) e^c_I \left (\delta_d^b + n_d n^b \right) e^d_J ^ - 2 e \left (\delta_c^a + n_c n^a \right) e^c_I e^d_J n_d n^b ^ \right) \\&= \int d^4 x \left (e e^a_I e^b_J ^ + e n_c n^a e^c_I e^b_J ^ + e e^a_I n_d n^b e^d_J ^ + e n_c n^a n_d n^b E^c_I E^d_J ^ - 2e e^a_I e^d_J n_d n^b ^ - 2 n_c n^a e^c_I e^d_J n_d n^b ^ \right) \\&= \int d^4 x e e^a_I e^b_J ^ \\&= S(E,A) \end

where we used

{F _{ab} }^ = ^ and n ^{a} n ^{b} F ^{i} _{ab} = 0 .

So the action can be written

S(E,A) = \int d ^{4} x \left (e E ^{a} _{I} E ^{b} _{J} {F _{ab} }^ - 2 e E^a_I e^d_J n_d n^b ^ \right) \qquad Eq. 12

We have

e = N \sqrt{q} . We now define\tilde{E} ^{a} _{I} = \sqrt{q} E ^{a} _{I}

An internal tensor

S ^{IJ} is self-dual if and only if*S ^{IJ} : = { 1 \over 2 } {\varepsilon ^{IJ} }_ S^ = i S^

and given the curvature

{F _{ab} }^ is self-dual we have{F _{ab} }^ = -i _ ^

Substituting this into the action (Eq. 12) we have,

S(E,A) = \int d ^{4} x \left ( - i (1)/(2) \left ( (N)/(\sqrt{q) } \right) \tilde^a_I \tilde^b_J _ ^-2 N n^b \tilde^a_I n_J ^ \right)

where we denoted

n _{J} = e ^{d} _{J} n _{d} . We pick the gauge \tilde{E} ^{a} _{0} = 0 and n ^{I} = \delta ^{I} _{0} (this means n _{I} = η _{IJ} n ^{J} = η _{00} \delta ^{I} _{0} = - \delta ^{I} _{0} ). Writing \varepsilon _{IJKL} n ^{L} = \varepsilon _{IJK} , which in this gauge \varepsilon _{IJK0} = \varepsilon _{IJK} . Therefore,\begin{align} S(E,A) & = \int d ^{4} x \left ( - i { 1 \over 2 } \left ({N \over \sqrt{q}} \right ) \tilde{E} ^{a} _{I} \tilde{E} ^{b} _{J} \left ({\varepsilon ^{IJ} }_ ^ + _ ^ \right) - 2 N n^b \tilde^a_I n_J ^ \right) \\&= \int d^4 x \left (- i \left (\right) \tilde^a_I \tilde^b_J _ ^ + 2 N n^b \tilde^a_I ^ \right)\end

The indices

I,J,M range over 1 , 2 , 3 and we denote them with lower case letters in a moment. By the self-duality of A ^{IJ} _{a} ,A ^{i 0} _{a} = - i { 1 \over 2 } {\varepsilon ^{i0} }_ A_a^ = i _ A_a^= i A_a^i.

where we used

{\varepsilon ^{i0} }_ = -_ = -_ = -_.

This implies

\begin{align} {F _{ab} }^ &= \partial_a A_b^ - \partial_b A_a^ + A_a^ ^ - A_b^ ^ \\&= i \left (\partial_a A_b^i - \partial_b A_a^i + A_a^ A_ - A_b^ A_ \right) \\&= i \left (\partial_a A_b^i - \partial_b A_a^i + \varepsilon_ A_a^j A_b^k \right) \\&= i F_^i\end

We replace in the second term in the action

N n ^{b} by t ^{b} - n ^{b} . We needl{L} _{t} A ^{i} _{b} = t ^{a} \partial _{a} A ^{i} _{b} + A ^{i} _{a} \partial _{b} t ^{a}

and

l{D} _{b} \left (t ^{a} A ^{i} _{a} \right ) = \partial _{b} \left (t ^{a} A ^{i} _{a} \right ) + \varepsilon _{ijk} A ^{j} _{b} \left (t ^{a} A ^{k} _{a} \right )

to obtain

l{L} _{t} A ^{i} _{b} - l{D} _{b} \left (t ^{a} A ^{i} _{a} \right ) = t ^{a} \left (\partial _{a} A ^{i} _{b} - \partial _{b} A ^{i} _{a} + \varepsilon _{ijk} A ^{j} _{a} A ^{k} _{b} \right ) = t ^{a} F ^{i.} _{ab}

The action becomes

\begin{align} S & = \int d ^{4} x \left ( - i \left ({N \over \sqrt{q}} \right ) \tilde{E} ^{a} _{I} \tilde{E} ^{b} _{J} {\varepsilon ^{IJ} }_ ^ - 2 \left (t^a - N^a \right) \tilde^b_I ^ \right) \\&= \int d^4 x \left (- 2 i \tilde_i^b \mathcal_t A_b^i + 2 i \tilde_i^b \mathcal_b \left (t^a A_a^i \right) + 2 i N^a \tilde^b_i F_^i - \left (\right) \varepsilon_ \tilde^a_i \tilde^b_j F_^k \right)\end

where we swapped the dummy variables

a and b in the second term of the first line. Integrating by parts on the second term,\begin{align} \int d ^{4} x \tilde{E} ^{b} _{i} l{D} _{b} \left (t ^{a} A ^{i} _{a} \right ) & = \int dt d ^{3} x \tilde{E} ^{b} _{i} \left (\partial _{b} (t ^{a} A ^{i)} _{a} + \varepsilon _{ijk} A ^{j} _{b} (t ^{a} A ^{k)} _{a} \right ) \\ & = - \int dt d ^{3} x t ^{a} A ^{i} _{a} \left (\partial _{b} \tilde{E} ^{b} _{i} + \varepsilon _{ijk} A ^{j} _{b} \tilde{E} ^{b} _{k} \right ) \\ & = - \int d ^{4} x t ^{a} A ^{i} _{a} l{D} _{b} \tilde{E} ^{b \end{align}} _{i}

where we have thrown away the boundary term and where we used the formula for the covariant derivative on a vector density

\tilde{V} ^{b} _{i} :l{D} _{b} \tilde{V} ^{b} _{i} = \partial _{b} \tilde{V} ^{b} _{i} + \varepsilon _{ijk} A ^{j} _{b} \tilde{V} ^{b} _{k} .

The final form of the action we require is

S = \int d ^{4} x \left ( - 2 i \tilde{E} ^{b} _{i} l{L} _{t} A ^{i} _{b} - 2 i \left (t ^{a} A ^{i} _{a} \right ) l{D} _{b} \tilde{E} ^{b} _{i} + 2 i N ^{a} \tilde{E} ^{b} _{i} F ^{i} _{ab} + \left ({N \over \sqrt{q}} \right ) \varepsilon _{ijk} \tilde{E} ^{a} _{i} \tilde{E} ^{b} _{j} F ^{k} _{ab} \right )

There is a term of the form "

p • q " thus the quantity \tilde{E} ^{a} _{i} is the conjugate momentum to A ^{i} _{a} . Hence, we can immediately write\left \{ A ^{i} _{a} (x), \tilde{E} ^{b} _{j} (y) \right \} = {i \over 2 } \delta ^{b} _{a} \delta ^{i} _{j} \delta ^{3} (x,y) .

Variation of action with respect to the non-dynamical quantities

(t ^{a} A ^{i)} _{a} , that is the time component of the four-connection, the shift function N ^{b} , and lapse function N give the constraintsl{D} _{a} \tilde{E} ^{a} _{i} = 0 , F ^{i} _{ab} \tilde{E} ^{b} _{i} = 0 , \varepsilon _{ijk} \tilde{E} ^{a} _{i} \tilde{E} ^{b} _{j} F ^{k} _{ab} = 0 Eq. 13 .

Varying with respect to

N actually gives the last constraint in Eq. 13 divided by \sqrt{q} , it has been rescaled to make the constraint polynomial in the fundamental variables. The connection A ^{i} _{a} can be writtenA ^{i} _{a} = { 1 \over 2 } {\varepsilon ^{i} }_ A^_a = _ \left (\omega^_a - i \left (_ \omega^_a + _ \omega^_a \right) \right) = \Gamma_a^i - i \omega^_a

and

E _{ci} \omega ^{0 i} _{a} = - q ^{b} _{a} E _{ci} \omega ^{i 0} _{b} = - q ^{b} _{a} E _{ci} e ^{di} \nabla _{b} e ^{0} _{d} = q ^{b} _{a} q ^{d} _{c} \nabla _{b} n _{d} = K _{ac}

where we used

e ^{0} _{d} = η ^{0I} g _{dc} e ^{c} _{I} = - g _{dc} e ^{c} _{0} = - n _{d,}

therefore

\omega ^{0 i} _{a} = K ^{i} _{a} . So the connection readsA ^{i} _{a} = \Gamma ^{i} _{a} - i K ^{i} _{a} .

This is the so-called chiral spin connection.

Reality conditions

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.

See also

• Lie algebra
• Plebanski action
• BF model

Notes and References

1. Samuel . Joseph . A lagrangian basis for ashtekar's reformulation of canonical gravity . Pramana . Springer Science and Business Media LLC . 28 . 4 . 1987 . 0304-4289 . 10.1007/bf02847105 . L429–L432. 1987Prama..28L.429S . 120704976 .
2. Jacobson . Ted . Smolin . Lee . The left-handed spin connection as a variable for canonical gravity . Physics Letters B . Elsevier BV . 196 . 1 . 1987 . 0370-2693 . 10.1016/0370-2693(87)91672-8 . 39–42. 1987PhLB..196...39J .
3. Jacobson . T . Smolin . L . Covariant action for Ashtekar's form of canonical gravity . Classical and Quantum Gravity . IOP Publishing . 5 . 4 . 1988-04-01 . 0264-9381 . 10.1088/0264-9381/5/4/006 . 583–594. 1988CQGra...5..583J . 250866876 .
4. Goldberg . J. N. . Triad approach to the Hamiltonian of general relativity . Physical Review D . American Physical Society (APS) . 37 . 8 . 1988-04-15 . 0556-2821 . 10.1103/physrevd.37.2116 . 2116–2120. 9958915 . 1988PhRvD..37.2116G .
5. Henneaux . M. . Nelson . J. E. . Schomblond . C. . Derivation of Ashtekar variables from tetrad gravity . Physical Review D . American Physical Society (APS) . 39 . 2 . 1989-01-15 . 0556-2821 . 10.1103/physrevd.39.434 . 434–437. 9959655 . 1989PhRvD..39..434H .
6. Ashtekar Variables in Classical General Relativity, Domenico Giulini, Springer Lecture Notes in Physics 434 (1994), 81-112,
7. The Ashtekar Hamiltonian for General Relativity by Ceddric Beny
8. Knot theory and quantum gravity in loop space: a primer by Jorge Pullin; AIP Conf.Proc.317:141-190,1994, arXiv:hep-th/9301028