Has LQG Research Yielded Any Mathematical Breakthroughs?

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Discussion Overview

The discussion centers on whether research in Loop Quantum Gravity (LQG) has led to any significant breakthroughs in mathematics or new mathematical results. Participants explore the relationship between LQG and mathematical development, considering both theoretical implications and the nature of the mathematics involved.

Discussion Character

  • Debate/contested
  • Exploratory
  • Technical explanation

Main Points Raised

  • One participant expresses curiosity about any mathematical breakthroughs resulting from LQG research.
  • Another participant suggests that while LQG may not have yielded significant mathematical results yet, the complexity of the theory implies that future breakthroughs could be dramatic.
  • A different viewpoint highlights the distinction between using sophisticated mathematical tools and actually solving open mathematical problems, noting that string theory has contributed to mathematics in ways that LQG has not.
  • Questions are raised regarding the mathematical well-definedness of LQG and related concepts, such as the Wheeler-deWitt equation.
  • Several mathematical developments related to LQG are listed, including spinnetwork theory, twistor theory, and various gauge theories, although the participant questions the overall impact of LQG on mathematics.
  • One participant argues that the perceived lack of interest in LQG from mathematicians may stem from a lack of knowledge about the mathematics involved, emphasizing that many LQG researchers are based in mathematical departments.
  • It is noted that the current ambitions of string theory are more aligned with contributing to mathematics and condensed matter, while LQG focuses on understanding quantum spacetime.

Areas of Agreement / Disagreement

Participants express differing views on the mathematical contributions of LQG, with some asserting that it has not produced significant results, while others suggest that it may still lead to breakthroughs in the future. The discussion remains unresolved regarding the overall impact of LQG on mathematics.

Contextual Notes

Participants mention various mathematical concepts and theories related to LQG, but there is uncertainty regarding the well-definedness of LQG and its mathematical implications. The discussion reflects a range of opinions on the relationship between theoretical physics and mathematics.

Macuario
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I am curious about the following question:

Has research in LQG led to any breakthrough in mathematics or any mathematical new result so far?
 
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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.
 
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.
 
How mathematically well-defined is LQG anyway? Or something like the Wheeler-deWitt equation?
 
haushofer said:
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.
 
Macuario said:
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...

Macuario said:
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
 
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