I've written about various things you can do with functions of the differential operator D = d/dx.

For example, exp(aD) is a shift, because exp(aD)f(x) = f(x + a).

And exp(aD^2/2) is convolution with a Gaussian kernel, so a smooth or blur operation.

I've been meaning for a while to look at examples of the form f(X, D) where we mix D and X (where x is the "multiplication by x" operator).

I only realised today that exp(-iπ(X^2-D^2)/4) is a very familiar operation. It's none other than the Fourier transform. That was a big surprise to me!

There's a nice story here that I'll get back to one day so I'll just mention some key phrases here: That (X^2-D^2)/2 is the total energy of a harmonic oscillator like a pendulum. As a pendulum swings back and forth it exchanges kinetic energy for potential energy, ie. position for momentum. And in quantum mechanics, the wave function describes the position of a particle, and its Fourier transform describes the momentum. And similarly in the quantum case the time evolution is a back-and-forth between the original wave function and its Fourier transform. And exp(-iπ(X^2-D^2)/4) is basically the time evolution of such a system for 1/4 of a period.

(And I'm probably off by a factor of 2 or some phase factor... And a little care is needed with interpreting exp(-iπ(X^2-D^2)/4), especially when applied to functions that aren't square integrable.)