Self-dual Palatini action

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 formula_1 and a spin connection formula_2. Much more details and derivations can be found in the article tetradic Palatini action. The spin connection defines a covariant derivative formula_3. The space-time metric is recovered from the tetrad by the formula formula_4 We define the `curvature' by
The Ricci scalar of this curvature is given by formula_6. The Palatini action for general relativity reads
where formula_8. Variation with respect to the spin connection formula_2 implies that the spin connection is determined by the compatibility condition formula_10 and hence becomes the usual covariant derivative formula_11. Hence the connection becomes a function of the tetrads and the curvature formula_12 is replaced by the curvature formula_13 of formula_11. Then formula_15 is the actual Ricci scalar formula_16. 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, formula_18, which is equal to either +1 or −1 depending on whether formula_19 is either an even or odd permutation of formula_20, respectively, and zero if any two indices take the same value. The internal indices of formula_18 are raised with the Minkowski metric formula_22.
Now, given any anti-symmetric tensor formula_23, we define its dual as
The self-dual part of any tensor formula_23 is defined as
with the anti-self-dual part defined as
(the appearance of the imaginary unit formula_28 is related to the Minkowski signature as we will see below).
Tensor decomposition.
Now given any anti-symmetric tensor formula_23, we can decompose it as
where formula_31 and formula_32 are the self-dual and anti-self-dual parts of formula_23 respectively. Define the projector onto (anti-)self-dual part of any tensor as
The meaning of these projectors can be made explicit. Let us concentrate of formula_35,
Then
The Lie bracket.
An important object is the Lie bracket defined by
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):
and
That is the Lie bracket, which defines an algebra, decomposes into two separate independent parts. We write
where formula_42 contains only the self-dual (anti-self-dual) elements of formula_43
The Self-dual Palatini action.
We define the self-dual part, formula_44, of the connection formula_2 as
which can be more compactly written
Define formula_48 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 formula_44. 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 (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. The method of proof follows that given in section II of "The Ashtekar Hamiltonian for General Relativity". We need to establish some results for (anti-)self-dual Lorentzian tensors.
Identities for the totally anti-symmetric tensor.
Since formula_54 has signature formula_55, it follows that
to see this consider,
With this definition one can obtain the following identities,
(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,
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. formula_60, instead there would have been a positive sign. We define formula_61 to be self-dual if and only if
(with Euclidean signature the self-duality condition would have been formula_63). Say formula_61 is self-dual, write it as a real and imaginary part,
Write the self-dual condition in terms of formula_66 and formula_67,
Equating real parts we read off
and so
where formula_23 is the real part of formula_72.
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 formula_75 we obtain
So we have
Consider
where in the first step we have used the anti-symmetry of the Lie bracket to swap formula_79 and formula_80, in the second step we used formula_81 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 formula_85 is a projection it satisfies formula_86, 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 formula_80 can be written as a sum of its self-dual and anti-sef-dual parts, i.e. formula_91. 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 formula_95 and a part that depends only on anti-self-dual Lorentzian tensors and is the anit-self-dual part of formula_96
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, formula_98, is thought of as constructed purely from the connection formula_99, 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 formula_79 is the curvature of the formula_103, the self-dual part of formula_104,
It has been shown that formula_106 is the self-dual part of formula_107
Let formula_108 be the projector onto the three surface and define vector fields
which are orthogonal to formula_110.
Writing
then we can write
where we used formula_113 and formula_114.
So the action can be written
We have formula_116. We now define
An internal tensor formula_61 is self-dual if and only if
and given the curvature formula_120 is self-dual we have
Substituting this into the action (Eq. 12) we have,
where we denoted formula_123. We pick the gauge formula_124 and formula_125 (this means formula_126). Writing formula_127, which in this gauge formula_128. Therefore,
The indices formula_130 range over formula_131 and we denote them with lower case letters in a moment. By the self-duality of formula_132,
where we used
This implies
We replace in the second term in the action formula_136 by formula_137. We need
and
to obtain
The action becomes
where we swapped the dummy variables formula_142 and formula_143 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 formula_145:
The final form of the action we require is
There is a term of the form "formula_148" thus the quantity formula_149 is the conjugate momentum to formula_150. Hence, we can immediately write
Variation of action with respect to the non-dynamical quantities formula_152, that is the time component of the four-connection, the shift function formula_153, and lapse function formula_154 give the constraints
Varying with respect to formula_154 actually gives the last constraint in Eq. 13 divided by formula_159, it has been rescaled to make the constraint polynomial in the fundamental variables. The connection formula_150 can be written
and
where we used
therefore formula_164. 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.