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Since the vacuum is thought to be virtual particles, is it possible that there might be virtual micro-black holes with virtual horizons and virtual hawking radiation which form and evaporate at a rate that makes them immeasurable or virtual?

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This is a virtual question and probably has a virtual answer :) – mcodesmart Oct 4 '13 at 17:49
But their high energy density makes their lifetime very much shorter than typical for other virtual particles in the vacuum. – Pieter Geerkens Oct 5 '13 at 3:46
@PieterGeerkens Is the idea that virtual micro-black holes are more short-lived than virtual particles a reason that they wouldn't virtually form and virtually evaporate? – Jonathan Langdale Nov 9 '13 at 21:29
Heisenberg uncertainty still holds. – Pieter Geerkens Nov 9 '13 at 21:50

Yes. The Bekenstein-Hawking entropy is $S=\frac{A}{4}$. If one gives the black hole a tiny entropy, then the energy of the black hole is extremely high. Thus, by the Heisenberg uncertainty principle, i.e., $\Delta E\cdot\Delta t\geq\frac{\hbar}{2}$, the black hole exists for an extremely short time, i.e., if one allows parity symmetry to define an "anti-black hole" (whatever that may mean), then the black hole is virtual. This means that through the uncertainty relation, if one allows spontaneous "black-hole pair production", then the resulting black holes will be virtual if $A$ is infinitesimal.

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That was kind of the idea behind the theory of quantum foam - a scale where quantum uncertainty is so large that it causes gravitational effects (the definition of the Planck length is that the Compton wavelength is identical to the Schwarzschild radius).

I seem to recall that the notion of quantum foam implied some dispersion of light on very large distances and that the measured dispersion of light indicated that it might not be real, though. But that is from memory.

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No. Virtual particles are not real and arise in formulations of quantum field theory. In these constructions, for every unique fundamental particle there is an associated field. Fields in quantum field theory are the Fourier Transforms of annihilation $a(\vec{p})$ and creation $a^\dagger(\vec{p})$ operators that exist at every point in spacetime.

For annihilation operators we have: $$A(x)=\sum_{\vec{p}} \dfrac{1}{2Vp_0}a(\vec{p})\ e^{i\vec{p}x}$$ For creation operators we have:

$$A^\dagger(x)=\sum_{\vec{p}} \dfrac{1}{2Vp_0}a^\dagger(\vec{p})\ e^{-i\vec{p}x}$$

The field $\phi(x)$ is simply the sum of the creation and annihilation operators:

$$\phi(x) = A(x) + A^\dagger(x)$$

Virtual particles are particles where the creation and annihilation operators of the particles are coherent, e.g. they are destroyed in regions outside that governed by the uncertainty principle.

Real particles are ones that do not have coherent creation and annihilation operators. They are viewed as excitations of the underlying field and exist outside the bounds of uncertainty.

At this point it might seem that if would make sense to propose that we could have a "black hole" field. However, this is a confused concept since black holes are classical objects and are predicted in solutions to the Einstein Field Equations which are derived in from the Theory of General Relativity. As such they are macroscopic, composite objects and are not fundamental particles which are used as the basis for quantum field theories (QFT).

Because we still do not have a fully developed theory of quantum gravity, our interpretation of black holes in quantum field theories is not agreed upon. Specifically, black holes do not have a fundamental mass in the same sense as fundamental particles in QFT. As a classical object, a black hole can have any rest mass. This means that there are no mass bounds associated with creation and annihilation fields one would want to associate with a black hole if one were to attempt to treat it as a fundamental particle. One would need to have a field for every possible mass of the black hole. This would leave an infinite number of particle fields which we know is energetically impossible (see Planck's Law for overview of issues with ultraviolet divergences).

So my answer is No. It is not possible in the physical world for us to consider black holes as fundamental particles in the context of QFT.

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They do not have to be fundamental particles in the context of QFT, though. Using the path integral formalism, you could for instance ask what the measure of states of the gravitational fields containing black holes is. (sorry I accidentally pressed enter before finishing the message the first time). – Slereah Jan 12 '14 at 14:26
@Slereah If you are talking about virtual particles you are talking QFT. – user11547 Jan 12 '14 at 14:27

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