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In their paper http://journals.aps.org/prd/abstract/10.1103/PhysRevD.42.3500 (Classical and quantal nonrelativistic Chern-Simons theory) Jackiw and Pi introduced an unusual identity involving two-dimensional delta-function, "which may very well stretch mathematical convention": $$(\partial_x\partial_y-\partial_y\partial_x)\arctan{\frac{y}{x}}=2\pi\delta^{(2)}(x,y).$$ Some further justification of this identity they discuss in http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.66.2682

What is the proper mathematical framework for such kind of identities? Unlike the delta-function, it cannot be a distribution theory, because for distributions partial derivatives commute.

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  • $\begingroup$ This is basically just because arctan does not exist as a single valued function. On $\mathbb{R}^2$ take any non-harmonic smooth function $\phi$, consider the one-form $*\mathrm{d}\phi$. It has a non-zero exterior derivative, which we learned from school that this means there does not exist a smooth function $\psi$ such that $\mathrm{d}\psi = *\mathrm{d}\phi$. But if we pretend that such a function $\psi$ exist, it would be a function satisfying $\mathrm{d}\circ \mathrm{d} \psi = \triangle \phi \not\equiv 0$. $\endgroup$ Sep 15, 2014 at 14:20
  • $\begingroup$ If you don't insist on working with the function $\psi$, and instead work with its exterior derivative $\mathrm{d}\psi$, then the "theory" is perfectly captured within the framework of distribution theory. $\endgroup$ Sep 15, 2014 at 14:23
  • $\begingroup$ It is true that the cause of the identity is multi-valuedness of arctan. This is explained in users.physik.fu-berlin.de/~kleinert/b11/psfiles/mvf.pdf (Multivalued Fields: In Condensed Matter, Electromagnetism, and Gravitation, by Hagen Kleinert, pp. 117-120). From the point of physics, Kleinert's argumentation (as well as Jackiv&Pi's) is perfectly valid. I do not know, however, if it qualifies as perfectly mathematically rigorous. $\endgroup$ Sep 16, 2014 at 4:52

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