/sci/ - Science & Math

>>9683260
Indeed. Let [math]M[/math] be a Riemannian 4-manifold with a metric [math]g[/math], then the Yang-Mills action for the EM field is [eqn]\mathcal{A} = \frac{1}{4\pi^2}\int_M F^2 = \frac{1}{4\pi^2}\int_{\mathbb{R}^4}dV\sqrt{\operatorname{det}(-g)}F^2[/eqn] where [math]F =dA \in \Omega^2(M) \otimes \mathbb{R}[/math] is the [math]U(1)[/math] field strength. By a conformal expansion [math]g_{\mu\nu} \sim I + \epsilon_{\mu\nu}\partial^{\mu}\partial^\nu[/math] in the metric, a coupling between gravitons [math]\epsilon[/math] and photons [math]A[/math] is possible.
>>9684432
Take a local section of the the Hopf fibration [math]S^3 \rightarrow S^2[/math].

>>9671492
[math]|\phi|_{L^2} = |\mathcal{F}\phi|_{L^2}[/math] and [math]\phi = \frac{\lambda}{2}\mathcal{F}\phi[/math] implies [math]|\phi|_{L^2} = \frac{\lambda}{2}|\mathcal{F}\phi|_{L^2} = \frac{\lambda}{2}|\phi|_{L^2}[/math].
Also to go forward with the Neumann series, if you assuming [math]\phi[/math] is absolutely continuous then [math]\mathcal{F}T = T\mathcal{F} = \mathcal{F}^{-1}[/math] where [math]Tf(x) = f(-x)[/math] is the inversion satisfying [math]T^2 = I[/math], so [math]\mathcal{F}^{2} = \mathcal{F} T^2 \mathcal{F} = \mathcal{F}^{-2}[/math], which implies that [math]\mathcal{F}^m = I[/math] for any [math]m \equiv 0 \mod 4[/math].
At the end you obtain
[eqn]
\phi = \sum_n\left(\frac{\lambda}{2}\right)^2(I
+ \mathcal{F}+ \mathcal{F}^{-1} + \mathcal{F}^2)f.[/eqn]

>>9655361
[math]\int_\mathbb{R}\frac{dp}{2\pi}\frac{e^{ipx}{p}[/math]
>>9655388
Sign is defined piecewise.

>>9576333
Yes.