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I have heard about a generalization of the calculus named 'quantized calculus'. In this calculus the derivative is defined as

$ df= [F,f]=Ff-fF $

Here $ F(g(x))= \frac{i}{\pi}\int_{-\infty}^{\infty}dt \frac{g(t)-g(s)}{t-s}$. In any case if this is the 'quantized' derivative , how can one defined a 'quantized integral'? How can one recover the usual definition of derivative from this $ \frac{d}{dx} $?

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    Hey, just as a side note... You may want to look up things called deravations. Let $R$ be a ring, then $\rho:R\times R\rightarrow R$. $\rho$ is a derivation if: 1. For every $a, b \in R$ we have that $\rho(ab)=a\rho(b)+\rho(a)b$ and $\rho(a+b)=\rho(a)+\rho(b)$2013-12-20

1 Answers 1

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Basically this is related to the following topics, the Wodzicki trace (1984) for differential operators over closed compact space, and then Connes shows with certain condition Wodzicki residue coindes with Dixmier trace. In short, you define the integral in terms of trace, (or say functional trace which is what it's called in physics,) or residue of the trace when the plain trace diverges, (regularized/ renormalized functional, in physics language.)

This finds some applications in noncommutative geometry, when one calculates the Connes-Moscovici local index formula, which express the cyclic homology/ cohomology pairing in terms of the local residue formula.

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    Thanks for following up. Those references would be a welcome edit to your Answer.2016-10-28