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. Here we in particular fill in details of the proof of results for self-dual variables not given in text books.
Contents
- The Palatini action
- Anti self dual parts of a tensor
- Tensor decomposition
- The Lie bracket
- The Self dual Palatini action
- Derivation of main results for self dual variables
- Identities for the totally anti symmetric tensor
- Definition of self dual tensor
- Important lengthy calculation
- Derivation of important results
- Summary of main results
- Derivation of Ashtekars Formalism from the Self dual Action
- Reality conditions
- References
The Palatini action
The Palatini action for general relativity has as its independent variables the tetrad
The Ricci scalar of this curvature is given by
where
(Anti-)self-dual parts of a tensor
We will need what is called the totally antisymmetry tensor or Levi-Civita symbol,
Now, given any anti-symmetric tensor
The self-dual part of any tensor
with the anti-self-dual part defined as
(the appearance of the imaginary unit
Tensor decomposition
Now given any anti-symmetric tensor
where
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 (see the last two terms of
and
That is the Lie bracket, which defines an algebra, decomposes into two separate independent parts. We write
where
The Self-dual Palatini action
We define the self-dual part,
which can be more compactly written
Define
Using
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
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
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
The minus sign here is due to the minus sign in
(with Euclidean signature the self-duality condition would have been
Write the self-dual condition in terms of
Equating real parts we read off
and so
where
Important lengthy calculation
The following lengthy calculation is important as all the other important formula can easily be derived from it. From the definition of the Lie bracket and with the use of
That gives the formula
which is the starting point for everything else.
Derivation of important results
First consider
where in the first step we have used the anti-symmetry of the Lie bracket to swap
where we used
Now if we took
Summarising, we have
Then
where we used
where we have used
Starting with
where we have used that any
Summary of main results
Altogether we have,
which is our main result, already stated above as
into a part that depends only on self-dual Lorentzian tensors and is itself the 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,
This determines the connection in terms of the teterad and we recover the usual Ricci tensor.
The self-dual action for general relativity is given above.
where
It has been shown that
Define vector fields
(where
Writing
where we used
So the action can be written
We have
An internal tensor
and given the curvature
Substituting this into the action (EQ
where we denoted
The indices
where we used
We replace in the second term in the action
and
to obtain
The action becomes
where we swapped the dummy variables
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 ``
Variation of action with respect to the non-dynamical quantities
Varying with respect to
and
where we used
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.