3
$\begingroup$

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} $?

  • 0
    I can tell $F$ is an operator, but I have no idea what $[F,f]$ means nor what $H$ is. In any case, this smells like physics...2012-04-24
  • 0
    $ [A,B] $ is the commutator between the 2 operator.. or similar , i think $ f$ is the element of an algebra.2012-04-24
  • 0
    I assume by the letter $H$ you actually meant $F$ then right? Would I be correct in also assuming $f$ is a function and $df$ as an operator?2012-04-24
  • 0
    ah of course i meant $F$ instead of H :) ... and $ df$ should be an operator ..2012-04-24
  • 0
    I don't know anything at all about quantized calculus, but most of the top hits Google shows me for the phrase state that the analogue of the classical integral in this setting is the "Dixmier trace" (whatever that is).2012-04-24
  • 0
    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

0

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.

  • 0
    I applaud your effort to answer a long open question, but this introduces a number of undefined terms. Since a speedy response will not be as beneficial as a more leisurely and expositive one, I would urge the introduction of some definitions or links to definitions that help future Readers.2016-10-13
  • 1
    Thank you much for the comment, hardmath. Indeed, I should have said more about each concepts I mentioned. One can find some preliminary introduction on Dixmier trace on [Wiki](https://en.wikipedia.org/wiki/Dixmier_trace) and Wodzicki residue [here](https://arxiv.org/abs/math/0211361). In regards to the cyclic (co-)homology please refer to Connes' [book](http://www.alainconnes.org/docs/book94bigpdf.pdf) Chapter 3.2016-10-28
  • 0
    Thanks for following up. Those references would be a welcome edit to your Answer.2016-10-28