The ground state wave functional for the photon theory is given as(adsbygoogle = window.adsbygoogle || []).push({});

[tex] \Psi_0[\tilde{a}] = \eta \exp \left(-\frac{1}{2} \int \frac{d^3k}{(2\pi)^3} \frac{(\vec{k}\times\tilde{a}(\vec{k}))\cdot(\vec{k}\times\tilde{a}(-\vec{k}))}{|\vec{k}|}\right)[/tex](10.81)

where [tex]\tilde{a}[/tex] is given as the Fourier transform of [tex]a[/tex], that is,

[tex] a_i(\vec{x}) = \int \frac{d^3k}{(2\pi)^3}\tilde{a}_i(\vec{k})e^{i\vec{k}\cdot\vec{x}} [/tex](10.67)

Transforming back to [tex]a[/tex], the book now says that (10.81) is equivalent to

[tex] \Psi_0[a] = \eta \exp \left(-\frac{1}{(2\pi)^2} \int d^3x d^3y \frac{(\nabla\times\vec{a}(\vec{x}))\cdot(\nabla\times\vec{a}(\vec{y}))}{|\vec{x}-\vec{y}|^2}\right) [/tex](10.83)

I've had to think about this for a long time, and I'm still not sure I understand it exactly.

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# Fourier transorm problem has me stumped

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