In a paper on the significance of the square of the vector potential, Gubarev, Stodolsky, and Zakharov1 gave the following identity for vector fields that vanish sufficiently rapidly at infinity so that surface terms make no contribution:

d3xA(x)2=d3xd3xA(x)A(x)+×A(x)×A(x)4πxx.
(1)

They offered no proof of the identity other than to remark that the relation can be established by transforming to position space the vector identity (k×a)2=k2a2(ka)2 satisfied by a(k), the Fourier...

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