From: "mcintosh@servidor.unam.mx" <mcintosh@servidor.unam.mx> To: math-fun <math-fun@mailman.xmission.com> Cc: mcintosh@servidor.unam.mx Sent: Sun, December 27, 2009 6:40:47 PM Subject: Re: [math-fun] discrete version of exp(d/dz) Quoting Mike Stay <metaweta@gmail.com>:

Is there a general formula for "d/dz" in dimension D?

You are in good company with Paul Dirac and Oliver Heaviside. As long as D is finite, there is no problem, even with a multiply valued complex logarithm. It is the limit that hurts, and for which distribution theory was sort of invented. I recall that in the late forties, Aurel Wintner proved that there are no matrices A and B such that AB - BA = I, the unit matrix. Mark Kac must have read the paper, because he assigned it as a problem in the final examination for his Mathematical Methods course. However, the Physics Department did not disappear in a puff of smoke even if you called A and B p and q. But there *were* those who noticed that something was wrong. - hvm ________________________________ The theorem that [A,B] = AB - BA = I is impossible is true only for operators over a finite dimensional space V. The proof is trivial. Since trace(AB) = trace(BA), trace [A,B] = 0, while trace(I) = dim V. Counterexamples can be found in infinite dimensional space. For example, on the space of polynomials, let (A(p))(x) = (d/dx) p(x), (B(p))(x) = x p(x). Then ([A,B](p))(x) = (d/dx)(x p(x)) - x (d/dx) p(x) = p(x), so [A,B] = I. Further counterexamples can be found even in finite dimensional spaces of characteristic p when p divides dim V, since then trace(I) = 0. For example, over F_2 with A = [[0,1],[0,0]], B = [[0,0],[1,0]], we have [A,B] = I. -- Gene