Piareround's questions about LQG

[marcus]

Piareround had some questions about LQG. I have to go out for the next hour or so. Maybe someone will take a look and respond. Or I will when I get back.

... https://www.physicsforums.com/showthread.php?t=432962"...

... as I was reading your thread I realized that I assumed two things:
First, quantum loop gravity only requires 4 dimensional Minkowski Space
Second, that quantum foam actually plays a role in quantum loop gravity.

So I was hoping you could help me with the following questions:

1. Are there any limits, such as a minimum or maximum, on the number of dimensions that quantum loop gravity, independent of string theory, would require?
2. Is their a relationship between quantum foam and quantum loop gravity? If so, what role exactly does quantum foam play in quantum loop gravity theory? If not, does quantum loop gravity, replace quantum foam with something analogous to quantum foam purpose in solution the Klein-Gordon Equation(*)?

 

[marcus]

First off, we need to be clear about what is meant by "quantum foam".

In LQG there is something called a spinfoam, which is not assumed to exist. It is a tool for calculating the transition amplitudes between initial and final states of geometry.

You know what it means to speak "figuratively"? You could say that if LQG is a good theory of geometry, then intuitively spacetime has a "foamy" character. But that is mathematically meaningless. It is just a mental picture. You can't calculate with it.

So "quantum foam" is not a clearly defined thing and it plays no role in the theory.

However the spinfoam is visually rather like a foam of soapsuds----vertices, edges, surfaces. A bunch of vertices with edges connecting them, and bits of surface (triangles, fourside polygons, fiveside etc...) surrounded by the edges.
But the spinfoam is used to calculate how the initial geometry evolves into the final geometry. The whole theory is about the quantum evolution of one geometry into another---one shape of space bending and flexing and stretching and changing into another. Transition "probabilities"---how likely each evolutionary path is.

The foam is a tool to organize the information needed to do that. One doesn't try to say what nature or spacetime IS, one tries to predict and calculate the odds of observing this or that.

Answer to #2: quantum foam plays no role. what plays a role is spinfoam. (a "colored 2-complex" where the colors are group reps, used for calculation)

Answer to #1: LQG can be formulated at any dimension. However all the work has gone into studying it in D=3 and D=4. The reason to look at D=3 case is as a toy model, easier to handle, to learn stuff which you can then try to reproduce in the REAL case which is D=4.

LQG research is pretty much completely invested in the D=4 case of LQG. Evidence of other dimensionality would basically falsify all the current work. In my humble opinion (I'm just an observer, one of several here.)

I hope other people will respond to Pia's questions and invite disagreement. If you know the story differently please tell it!