LQG and Mathematics

[Macuario]

I am curious about the following question:

Has research in LQG led to any breakthrough in mathematics or any mathematical new result so far?

 

[Niflheim]

I'm not very knowledgeable about LQG, but I do know that things that complex entail a great deal of pure, abstract mathematics just by the nature of you're working with. If there haven't been so far, I would expect theorists working on LQG or string/m theory to eventually make some significant progress. From what I understand, the sheer scope and complexity of the equations make the math such a nightmare that progress will be slow, but will be dramatic when a breakthrough is published.

 

[Macuario]

One thing is to use relatively sophisticated mathematical tools, made available by the mathematicians, and a very different thing is to solve open problems in mathematics in the process of doing research on the theory. ST is expressed in a mathematically sophisticated language and has also solved open problems in mathematics and it has even created new open problems and research areas that the mathematicians had missed, such as mirror symmetry for Calabi-Yau manifolds.

I wonder if there has been any advance in mathematics thanks to LQG. To the best of my knowledge that is not the case, and in fact LQG is regarded by mathematicians as completely uninteresting.

 

[haushofer]

How mathematically well-defined is LQG anyway? Or something like the Wheeler-deWitt equation?

 

[martinbn]

How mathematically well-defined is LQG anyway?

Probably as well as many other things in physics. How well-defined is the path integral? But it surely has been important to mathematicians.

 

[francesca]

I am curious about the following question:

Has research in LQG led to any breakthrough in mathematics or any mathematical new result so far?

- Development of spinnetwork theory (recoupling theory on a graph)
- Developments in twistor theory
- Hamiltonian mechanics of covariant systems (Littlejohn)
- Diff invariant gauge theories on a lattice
- The measure on diff invariant space(ashtekar-Lewandowski measure)
- Generalization of Perelomov coherent states on a lattice (Livine-Speziale CS)
- Twisted geometries as discretization of GR
- Mapping between 3d Chern-Simons theory and 4d simplicial geometry (Han-Haggard-Riello-Kaminski)

...just the first things that come to my mind...

I wonder if there has been any advance in mathematics thanks to LQG. To the best of my knowledge that is not the case, and in fact LQG is regarded by mathematicians as completely uninteresting.

Lack of interest just denote a lack of knowledge. By the way, many researchers doing LQG are based in Mathematical Departments. Because the work of developing LQG deals a lot with some nice mathematics.

Having said all this, however, one must remark that nowadays the declared ambition of strings is mostly to contribute to mathematics or to condensed matter. While the ambition of LQG is to understand what happens to quantum spacetime: physics, not math.

Cheers,
f