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This is a bit of a qualitative question.

A rigorous treatment of QFT comes down to making sense of multiplication of distributions, as far as I understand. This is in the aim of constructing and manipulating with the interacting field operators after all.

For this purpose, I am under the impression that the theory of regularity structure is regarded as a powerful tool these days.

However, the approach by means of the Colombeau algebra does not seem as popular, though it seems more general in nature.

Could anyone please explain the reason for this?

I would appreciate any insight.

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For free fields Colombeau algebra is equivalent to normal ordering (Wick ordering) of products of creation and annilation operators. The latter is more easily described, which may explain the reduced popularity of Colombeau algebra.

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    $\begingroup$ In what sense are they equivalent? Normal ordering applies for the free field operators on a Fock space, as far as I know, $\endgroup$
    – Isaac
    Commented Apr 27, 2023 at 12:42
  • $\begingroup$ I may be mistaken, but I thought that normal ordering the $\phi^4$ interaction gives the same result as in Colombeau algebra. $\endgroup$
    – Carlo Beenakker
    Commented Apr 27, 2023 at 15:34
  • $\begingroup$ In which spacetime dimensions? For the$\phi^4$ theory, it is a very crucial factor.. $\endgroup$
    – Isaac
    Commented Apr 27, 2023 at 15:43
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    $\begingroup$ I'm only familiar with $d=2$, when normal ordering removes the singularities. If Colombeau is applicable also for $d=3$ then, indeed, it is stronger than Wick. In any case my "equivalent" statement was too strong, I will change that, thanks for your feedback. $\endgroup$
    – Carlo Beenakker
    Commented Apr 27, 2023 at 16:56

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