LQG and Mathematics: Exploring the Intersection

In summary, according to the speaker, research in LQG has not resulted in any groundbreaking mathematical results to date. Mathematical theory surrounding LQG is complex and difficult, making progress slow, but significant when a breakthrough is published. However, the speaker notes that most researchers working on LQG are based in mathematical departments, indicating that the field is not purely theoretical.
  • #1
Macuario
4
1
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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  • #2
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.
 
  • #3
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.
 
  • #4
How mathematically well-defined is LQG anyway? Or something like the Wheeler-deWitt equation?
 
  • #5
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.
 
  • #6
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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Likes MTd2, marcus and David Horgan

Related to LQG and Mathematics: Exploring the Intersection

1. What is LQG and how does it relate to mathematics?

LQG (Loop Quantum Gravity) is a theoretical framework that aims to reconcile general relativity and quantum mechanics. It is based on the idea that space and time are quantized at a very small scale. LQG is closely related to mathematics because it uses concepts from geometry, topology, and algebra to describe the quantum properties of space-time.

2. What are the main challenges in applying mathematics to LQG?

One of the main challenges is the fact that LQG deals with the quantum properties of space-time, which is a highly complex and abstract concept. As a result, it requires advanced mathematical tools such as differential geometry, functional analysis, and group theory. Additionally, there is still much debate and ongoing research about the best mathematical approach to describe LQG.

3. How does LQG differ from other theories of quantum gravity?

LQG differs from other theories of quantum gravity, such as string theory, in its fundamental principles and mathematical framework. LQG is based on the concept of discrete quantized space-time, while string theory is based on the idea of vibrating strings in a continuous space-time. Additionally, LQG does not require extra dimensions, unlike string theory.

4. What are some potential applications of LQG in mathematics?

One potential application of LQG in mathematics is the development of new mathematical tools and techniques to study and understand the properties of space-time at a quantum level. LQG may also have implications for other areas of mathematics, such as topology and algebraic geometry, as it provides a new perspective on the structure of space-time.

5. What are some current challenges and future directions in the intersection of LQG and mathematics?

Some current challenges include developing a mathematical framework that can fully describe the quantum properties of space-time and finding ways to test and validate the predictions of LQG. In the future, further research and advancements in both LQG and mathematics may lead to a deeper understanding of the nature of space-time and potentially provide insights into other areas of physics, such as the study of black holes and the early universe.

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