Noncommutative integral?

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In summary, the conversation discusses the term "noncommutative physics" and its use in eliminating divergences in QFT. The concept is related to non-commutative geometry, a theory created by Alain Connes. It is suggested to refer to Wikipedia and ask about it in the "beyond the standard model" forum. Additionally, the individual asking the question is advised to specify their level of mathematical knowledge.
  • #1
in many papers they use the term of noncommutative physics or something similar and they recall that the use of a noncommutative integral would get rid of divergences in QFT but what is exactly a noncommutative integral and how can you calculate it ?
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  • #2
Do you mean non-commutative geometry? It's a new theory by a guy named Alain Connes that very few of us have studied, or know much about at all. If that's what you meant, you should probably start with Wikipedia, and then ask about it again, in the "beyond the standard model" forum. (Or you could just click the report button and ask the moderators to move this thread there).

It would also be a good idea to tell us how much math you know.
  • #3

A noncommutative integral is a mathematical concept that arises in the field of noncommutative geometry. In this framework, the traditional commutative properties of multiplication are relaxed, allowing for noncommutative spaces to be studied. In physics, this concept has been applied to quantum field theory (QFT) in an attempt to address the issue of divergences.

In traditional QFT, the integral of a function is a commutative operation, meaning that the order in which the variables are multiplied does not affect the result. However, in noncommutative QFT, the integral is a noncommutative operation, meaning that the order in which the variables are multiplied does affect the result. This leads to a different mathematical structure and can potentially eliminate divergences that arise in traditional QFT calculations.

The calculation of a noncommutative integral can be performed using various mathematical techniques, such as spectral triples, noncommutative measure theory, or heat kernel methods. These methods involve understanding the properties of noncommutative spaces and the corresponding noncommutative functions.

It is important to note that the use of noncommutative integrals in QFT is still an active area of research and has not yet been fully developed. While it shows promise in addressing divergences, there are still many challenges and open questions that need to be addressed before it can be fully incorporated into QFT calculations.

1. What is a noncommutative integral?

A noncommutative integral is a mathematical concept that extends the idea of traditional integration to noncommutative algebraic structures. It involves integrating functions over noncommutative domains, where the order of operations matters.

2. How does a noncommutative integral differ from a traditional integral?

In a traditional integral, the order of operations does not affect the final result. However, in a noncommutative integral, the order of operations is crucial and can greatly impact the value of the integral. Additionally, the properties of noncommutative spaces are different from those of commutative spaces, leading to unique properties of noncommutative integrals.

3. What are some applications of noncommutative integrals?

Noncommutative integrals have applications in various fields such as quantum mechanics, representation theory, and noncommutative geometry. They have also been used in the study of noncommutative probability and noncommutative dynamical systems.

4. Are there different types of noncommutative integrals?

Yes, there are different types of noncommutative integrals, such as the Dixmier trace, the Connes trace, and the Wodzicki residue. Each type has its own unique properties and applications.

5. What are some challenges in studying noncommutative integrals?

One of the main challenges in studying noncommutative integrals is the lack of familiar geometric and physical intuition. Noncommutative spaces do not have the same intuitive properties as commutative spaces, making it challenging to visualize and understand noncommutative integrals. Additionally, there are still many open questions and unsolved problems in the field, making it a subject of ongoing research and exploration.

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