This question is more about trying to feel the waters in our current abilities to compute (or roughly estimate) the refraction index of vacuum, specifically when high numbers of electromagnetic quanta of a given frequency $\omega$ occupy a spherical symmetric incoming wavefront modes $$\frac{e^{i k r}}{r^2}$$

I'm interested in intensities above the Schwinger limit. Do exist analytical or lattice QED estimates?

Why this is interesting?

Usually i try to make my questions as self-contained as possible, but i believe it might be interesting to others why i'm interested in nonlinear vacuum refraction indices, so here it goes:

Let's review what classical theory says about our ability to create micro black holes with electromagnetic radiation. Our sun produces about $10^{26}$ watts of power, so in principle in the future we could harness all that power. The question trying to be answered here is: is the energy output from Sol enough for a sufficiently technically advanced humanity to create micro black holes?

Let's suppose we focus a spherically symmetric beam of light in a single focal point with the purpose of condense enough energy to create a black hole, well, the Schwarzchild radius of a given flow of energy is

$$ R = \frac{ G E }{c^4}$$

substituing constants,

$$ R = 10^{-45} E $$

Now, since this energy propagates electromagnetically as radiation, it needs to stay long enough inside the critical radius so that the black hole forms. If the (radial) refractive index inside a small enough region is $n$, then the light will stay inside the radius a time $T$

$$ R = \frac{cT}{n}$$

equating both terms we are left with an expression for the power that needs to be delivered in a spherical region in order to create a black hole

$$ \frac{cT}{n} = 10^{-45} E $$

$$ \frac{10^{53}}{n} = \frac{E}{T} = P $$

So, assuming a refractive index of vacuum of 1, it means that we require $10^{53}$ watts focused in a spherical region in order to create a black hole. This is $10^{27}$ more power than what would be available for a humanity that managed to create a Dyson shell around our sun!

But if the refractive index could be managed to be above $10^{30}$ before reaching the focus region, then that future humanity would have a chance to create such micro black holes

Even a less generous increase of the refractive index could be useful, if they could store the energy of the sun for a few years and then zapping it into an extremely brief $10^{-20}$ wide pulse, in a similar fashion as to how the National Ignition Facility achieves 500 TeraWatt pulses (by chirp-pulse compression)

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    $\begingroup$ I've just recently begun to read some work by Holger Gies on the application of the worldline formalism to the quantum vacuum probed by strong fields. An intro presentation is here and there are some numbers here P.S. so my idea of using a magnifying glass on a sunny day won't work then? $\endgroup$ – twistor59 Nov 27 '12 at 7:33
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    $\begingroup$ There is another problem: The photons could scatter off each-other and bounce back on the way in. $\endgroup$ – Kevin Kostlan Oct 1 '13 at 22:27

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