Quoting Dan Asimov <dasimov@earthlink.net>:
Can someone please elaborate? I'm not getting the connection between Mike's question and being in good company with anyone, or with a complex logarithm, or with the equation AB - BA = I.
Since Mike first mentioned that exp(d/dz)(f)(z) = f(z+1) (presumably on R or C), I thought he was looking for a D-dimensional analogue of this formula. No?
Yes! The point is, there isn't any; but that needn't deter someone who doesn't know that. Think of Dirac's delta function (unit matrix), or Heaviside's operator calculus. The formula in question defines a shift operator, which is nicely enough related to Taylor's series. When and where do shift operators apply? Schroedinger was impressed with the fact that p + iq (d/dx + ix) and its conjugate ran you through the harmonic oscillator spectrum, and conjectured that other Hamiltonians had cimilar operators. They almost do, but the whole subject is fairly complicated. Mike's ideas are valid enough for finite matrices (over the reals (or complex)); it is getting a limit which would satisfy a "real" mathematician which is the problem, and he could be reassured to know that multiple valuedness of logarithms isn't the source of the problem. Wintner's paper, as I recall, used the trace argument and the ladder operators had infinite traces, so there was a convergence anomaly; in 1950 this did seem to strike at the roots of quantum mechanics. -hvm ------------------------------------------------- www.correo.unam.mx UNAMonos Comunicándonos