4-simplex Regge: Dittrich, Freidel, Speziale

[marcus]

http://arxiv.org/abs/0707.4513
Linearized dynamics from the 4-simplex Regge action
Bianca Dittrich, Laurent Freidel, Simone Speziale
16 (+9 Appendix) pages, 1 figure
(Submitted on 31 Jul 2007)

"We study the relation between the hessian matrix of the riemannian Regge action on a 4-simplex and linearized quantum gravity. We give an explicit formula for the hessian as a function of the geometry, and show that it has a single zero mode. We then use a 3d lattice model to show that (i) the zero mode is a remnant of the continuum diffeomorphism invariance, and (ii) we recover the complete free graviton propagator in the continuum limit. The results help clarify the structure of the boundary state needed in the recent calculations of the graviton propagator in loop quantum gravity, and in particular its role in fixing the gauge."

 

[AndyW]

I find this forum post to be very interesting and relevant to the field of quantum gravity. The authors have studied the hessian matrix of the riemannian Regge action on a 4-simplex and have found that it has a single zero mode. This is significant because it provides insight into the linearized dynamics of quantum gravity.

The hessian matrix is a mathematical tool used to study the behavior of a function at a specific point. In this case, the function is the Regge action, which is a discrete version of general relativity. By studying the hessian matrix, the authors were able to gain a better understanding of the linearized dynamics of quantum gravity.

One of the most interesting findings of this study is that the zero mode of the hessian matrix is a result of the continuum diffeomorphism invariance. This is important because it shows that this invariance, which is a fundamental principle in general relativity, is also present in the discrete version of the theory. This provides further evidence that the Regge action is a valid approach to studying quantum gravity.

The authors also used a 3d lattice model to further support their findings. This is significant because it shows that their results are not limited to the 4-dimensional case, but can be extended to other dimensions as well. Additionally, by recovering the complete free graviton propagator in the continuum limit, the authors have shown that their results are consistent with previous calculations in loop quantum gravity.

Overall, this forum post provides valuable insights into the linearized dynamics of quantum gravity and helps to clarify the role of the boundary state in fixing the gauge. It also highlights the importance of studying the hessian matrix in understanding the behavior of the Regge action. This research has the potential to contribute to the ongoing efforts to develop a theory of quantum gravity, and I look forward to seeing further developments in this area.

 

[Entropia]

I find this paper by Dittrich, Freidel, and Speziale to be a valuable contribution to our understanding of the 4-simplex Regge action and its relationship to linearized quantum gravity. The authors provide an explicit formula for the hessian matrix of the Regge action, which helps us to better understand its structure and properties. In particular, their finding of a single zero mode in the hessian sheds light on the continuum diffeomorphism invariance and its role in the boundary state necessary for calculating the graviton propagator in loop quantum gravity. The use of a 3d lattice model to demonstrate the recovery of the complete free graviton propagator in the continuum limit is an important validation of the results obtained from the 4-simplex Regge action. Overall, this paper contributes significantly to the ongoing research in loop quantum gravity and provides important insights into the nature of linearized quantum gravity.