Self-dual Palatini action


, 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 and a spin connection. Much more details and derivations can be found in the article tetradic Palatini action. The spin connection defines a covariant derivative. The space-time metric is recovered from the tetrad by the formula We define the `curvature' by
The Ricci scalar of this curvature is given by. The Palatini action for general relativity reads
where. Variation with respect to the spin connection implies that the spin connection is determined by the compatibility condition and hence becomes the usual covariant derivative. Hence the connection becomes a function of the tetrads and the curvature is replaced by the curvature of. Then is the actual Ricci scalar. Variation with respect to the tetrad gives Einsteins equation

Self-dual variables

(Anti-)self-dual parts of a tensor

We will need what is called the totally antisymmetry tensor or Levi-Civita symbol,, which is equal to either +1 or −1 depending on whether is either an even or odd permutation of, respectively, and zero if any two indices take the same value. The internal indices of are raised with the Minkowski metric.
Now, given any anti-symmetric tensor, we define its dual as
The self-dual part of any tensor is defined as
with the anti-self-dual part defined as
.

Tensor decomposition

Now given any anti-symmetric tensor, we can decompose it as
where and are the self-dual and anti-self-dual parts of respectively. Define the projector onto self-dual part of any tensor as
The meaning of these projectors can be made explicit. Let us concentrate of,
Then

The Lie bracket

An important object is the Lie bracket defined by
it appears in the curvature tensor, it also defines the algebraic structure. We have the results :
and
That is the Lie bracket, which defines an algebra, decomposes into two separate independent parts. We write
where contains only the self-dual elements of

The Self-dual Palatini action

We define the self-dual part,, of the connection as
which can be more compactly written
Define as the curvature of the self-dual connection
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,
The self-dual action is
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:
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. The resulting Hamiltonian formalism resembles that of a Yang-Mills gauge theory.

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. The method of proof follows that given in section II of The Ashtekar Hamiltonian for General Relativity. We need to establish some results for self-dual Lorentzian tensors.

Identities for the totally anti-symmetric tensor

Since has signature, it follows that
to see this consider,
With this definition one can obtain the following identities,
.

Definition of self-dual tensor

It follows from Eq. 4 that the square of the duality operator is minus the identity,
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
. Say is self-dual, write it as a real and imaginary part,
Write the self-dual condition in terms of and,
Equating real parts we read off
and so
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
That gives the formula

Derivation of important results

Now using Eq.5 in conjunction with we obtain
So we have
Consider
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
Then
where we used Eq. 6 going from the first line to the second line. Similarly we have
by using Eq 7. Now as is a projection it satisfies, as can easily be verified by direct computation:
Applying this in conjunction with Eq. 8 and Eq. 9 we obtain
From Eq. 10 and Eq. 9 we have
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:

Summary of main results

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

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

The proof given here follows that given in lectures by Jorge Pullin
The Palatini action
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
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.
where is the curvature of the, the self-dual part of,
It has been shown that is the self-dual part of
Let be the projector onto the three surface and define vector fields
which are orthogonal to.
Writing
then we can write
where we used and.
So the action can be written
We have. We now define
An internal tensor is self-dual if and only if
and given the curvature is self-dual we have
Substituting this into the action we have,
where we denoted. We pick the gauge and . Writing, which in this gauge. Therefore,
The indices range over and we denote them with lower case letters in a moment. By the self-duality of,
where we used
This implies
We replace in the second term in the action by. We need
and
to obtain
The action becomes
where we swapped the dummy variables and in the second term of the first line. Integrating by parts on the second term,
where we have thrown away the boundary term and where we used the formula for the covariant derivative on a vector density :
The final form of the action we require is
There is a term of the form "" thus the quantity is the conjugate momentum to. Hence, we can immediately write
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
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
and
where we used
therefore. So the connection reads
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.