Triangulation and Discrete Volume Spectrum in Covariant LQG

[hodgeman]

What is often said in Covariant LQG is that the triangulation is a truncation, and is not what is responsible for the discrete volumes one ends up with in the theory. Rather, what is responsible is the discrete spectra of the volume operator acting on the nodes of a spin network.

My confusion is the following:
The nodes are dual to the tetrahedra that come from the triangulation. Now if the action of the volume operator on each node results in a minimum eigenvalue (of order Planck volume), then the minimum volume V of a spatial slice that can be measured should be bounded from below:
Nħ^3 < V
where N is the number of nodes/tetrahedra in our triangulation. So as we pick a finer triangulation, we can let V tend to infinity even for a chunk of space with finite volume!

Where is my mistake?

 

[AndyW]

I would like to clarify a few points about this statement and address your confusion. First of all, it is important to note that Covariant LQG (Loop Quantum Gravity) is a theoretical framework for quantum gravity, which combines the principles of general relativity and quantum mechanics. It is still a developing theory and there are ongoing debates and discussions about its specific details and implications.

Now, let's address your confusion about the role of triangulation and the discrete volumes in Covariant LQG. In this theory, the triangulation is used as a mathematical tool to discretize the continuous space-time into a finite number of discrete elements, such as tetrahedra. This is done in order to apply the principles of quantum mechanics to the theory of gravity. However, the triangulation itself is not responsible for the discrete volumes that are observed in this theory.

Instead, the discrete spectra of the volume operator, which acts on the nodes of a spin network, is what determines the discrete volumes in the theory. The nodes are indeed dual to the tetrahedra in the triangulation, but it is the action of the volume operator on these nodes that results in a minimum eigenvalue (of order Planck volume). This minimum eigenvalue is a fundamental property of the theory and is not affected by the choice of triangulation.

Now, let's address your concern about the minimum volume being bounded from below. Indeed, in Covariant LQG, the minimum volume is bounded by the Planck volume, as you correctly stated. However, this does not mean that as we pick a finer triangulation, we can let the volume tend to infinity for a chunk of space with finite volume. This is because the number of nodes/tetrahedra in the triangulation also increases as we refine it, and therefore the minimum volume remains the same. In other words, the minimum eigenvalue of the volume operator remains the same, regardless of the triangulation.

In conclusion, your mistake lies in assuming that the minimum volume is affected by the choice of triangulation, when in fact it is a fundamental property of the theory. I hope this helps clarify your confusion about the role of triangulation and discrete volumes in Covariant LQG. As with any scientific theory, it is important to continue exploring and debating its implications in order to gain a better understanding of the universe we live in.