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

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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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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.