Linear simplicity constraint in Loop Quantum Gravity

1. Jul 21, 2015

Georges Simenon

1. The problem statement, all variables and given/known data
Hi,
I am studying covariant LQG from the recent book by Rovelli & Vidotto, and i am struggling with the linear simplicity constraint. My problems are not with its proof, that i understand, but rather with the physical interpretation in terms of boost generators.
I will try to make my question as much self-consistent as possible. I refer to ch. 3 of the Book.
They start from the Holst action
$$S=\int B\wedge F\quad\text{where}\quad B=\star(e\wedge e)+\frac{1}{\gamma}e\wedge e$$
Then, in sec. 3.3.1, they define the two 2-forms
$$K^I=n_J B^{IJ}|_{\Sigma}\quad\text{and}\quad L^I=n_J(\star B^{IJ})|_{\Sigma}$$
restricted on a spatial hypersurface $\Sigma$ with unit timelike normal $n_I$.
Using these definitions, they prove that the following equation
$$\vec{K}=\gamma\vec{L}\qquad\text{linear simplicity constraint}$$
holds in the "time gauge" $n_Ie^I|_{\Sigma}$.
In sec. 3.4.3, they interpret the components of the vector $\vec{K}$ as the canonical generators of Lorentz boosts. It is this interpretation that i don't understand, and i'm trying to give sense to it.
The same claim you can find in the Zakopane lectures by C. Rovelli. In the notation used there, he identifies
$$(\star(e\wedge e)+\frac{1}{\gamma}e\wedge e)|_{\Sigma}$$
with the $SL(2,\mathbb{C})$ generator.
2. Relevant equations

3. The attempt at a solution
If i am asked to write the Lorentz generators in tetrad notation, i write
$$G^{IJ}=e^I\wedge e^J$$
up to an overall proportionality factor.
This makes sense. Consider, for definiteness, the 2-form
$$e^0\wedge e^1 = d\xi^0\otimes d\xi^1-d\xi^1\otimes d\xi^0$$
It generates Lorentz boosts along the local inertial axis 1. Note: the $\xi$'s are local inertial coordinates.

Last edited: Jul 21, 2015
2. Jul 26, 2015

Greg Bernhardt

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

3. Aug 11, 2015

sgd37

Have you checked if

$$(\star(e\wedge e)+\frac{1}{\gamma}e\wedge e)|_{\Sigma}$$

obeys the so(1,3) algebra commutation relations