From answers to a previous question, a finite degree polynomial in the distribution $i\Delta^+(x)$, with Fourier transform $2\pi\delta(k^2-m^2)\theta(k_0)$, is a distribution, even though a product of distributions is not in general well-defined. So far I've not been able to find, or be myself sure enough, whether $\exp{\!\left(i\Delta^+(x)\right)}$ is a distribution. It seems that the exponential is the boundary value of the function $\exp{\!\left(\frac{1}{z^2-m^2}\right)}$, but it's not clear to me yet whether that's enough. There seems to be nothing on this in Streater & Wightman (which I own, but don't yet understand well enough to apply the methods), and I don't have heavy lifting books like Reed & Simon, etc.

More specifically, what I currently think I'm really interested in is whether a bounded function of a distribution, such as $\tan{\!{}^{-1}\left(\exp{\left(i\Delta(x)\right)}\right)}$, is a distribution.

  • $\begingroup$ I'm thinking this would be more appropriate for Math Overflow. $\endgroup$ Feb 17, 2011 at 14:48
  • $\begingroup$ Thanks for that, Keenan. I thought of going to Math Overflow, but I don't currently have an account. $\endgroup$ Feb 17, 2011 at 16:37
  • $\begingroup$ I've toyed with offering a bounty on this, but two days later it seems that this is not the right place to start from. I take the lack of attempts to answer the question to be a reflection of that (I'm checking off Roy's offering, though it's clearly not an answer, as an indirect way to kill the question). Math Overflow now seems clearly not appropriate for the question as asked here. In any case, I know precisely how I'll regularize in the construction I'm developing, but it goes against a few prejudices, so it needs a lot of explanation, which is never good in a paper. $\endgroup$ Feb 19, 2011 at 13:34

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Since your original post on this topic I have encountered "Causal Perturbation Theory" which does distribution-like calculations using a modification of distributions based on Scaled Test Functions. This paper does lots of interaction and S matrix calculations. Perhaps you are entirely familiar with this, or it is not suitable but here is the link I have been studying:


  • $\begingroup$ Interesting, relatively nice Mathematics, and I think I hadn't seen it before. arxiv.org/abs/0901.2252, which I had seen before, cites the paper you're looking at. In these terms, $\exp{(i\Delta^+(x))}$ has infinite scaling degree, so I think it's outside. I feel quite disaffected with multinomial expansions in terms of multiplication and derivation operators, since convergence of polynomial expansions to analytic functions is almost always restricted to a (commonly very) limited domain. Thanks, Roy. $\endgroup$ Feb 18, 2011 at 17:26

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