What Is a Noncommutative Integral and How Does It Impact Quantum Field Theory?

[zetafunction]

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 ?

 

[Fredrik]

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.

 

[Entropia]

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.