[math-fun] discrete version of exp(d/dz)
It's straightforward to show that exp(d/dz)f(z) = f(z+1). If we think of a D-dimensional column vector as a function from the set {0, 1, ..., D-1} to the complex numbers, then the circulant matrix |0 1 0 ... 0 0| |0 0 1 ... 0 0| |. . . . . .| |. . . . . .| |. . . . . .| |0 0 0 ... 1 0| |0 0 0 ... 0 1| |1 0 0 ... 0 0| applied to the column vector f computes f((z+1) mod D), where 0 <= z < D. The eigenvectors of this matrix are "plane waves" and form the columns of the discrete fourier transform in D dimensions. I'm interested in finding a DxD matrix "d/dz" such that exp("d/dz") is the matrix above; it'll have {0, 1, ..., D-1} as eigenvalues (or perhaps {-(D-1)/2, ..., 0, ..., (D-1)/2}). I can solve for "d/dz" numerically, but I've been bitten before by the branch cut when taking matrix logs and looking for patterns. Is there a general formula for "d/dz" in dimension D? -- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike http://reperiendi.wordpress.com
On Sun, Dec 27, 2009 at 2:57 PM, Mike Stay <metaweta@gmail.com> wrote:
Is there a general formula for "d/dz" in dimension D?
Or "i d/dz", if that's easier... -- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike http://reperiendi.wordpress.com
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 ------------------------------------------------- www.correo.unam.mx UNAMonos Comunicándonos
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
On Sun, Dec 27, 2009 at 6:40 PM, <mcintosh@servidor.unam.mx> wrote:
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.
That's good to know, thanks. I was trying to follow the sum-over-paths derivation of QM in the first chapter of Zee's "QFT in a nutshell" using spin states instead of position states and using the Hadamard transform to get a conjugate basis for the sum over paths. Has someone done quantum mechanics on discrete spacetime using the q-derivative? Sounds like something Louis Kaufmann would do. -- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike http://reperiendi.wordpress.com
participants (3)
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Eugene Salamin -
mcintosh@servidor.unam.mx -
Mike Stay