Electromagnetic Unruh/Hawking effect? (Improved argument) [closed]

This is an improved version of the argument in Electromagnetic Unruh effect?

In the quantum vacuum particle pairs, with total energy $E_x$, can come into existence provided they annihilate within a time $t$ according to the uncertainty principle $$E_x\ t \sim \hbar.$$ If we let $t=x/c$ then we have $$E_x \sim \frac{\hbar c}{x}$$ where $x$ is the Compton wavelength of the particle pair.

Let us assume that there is a force field present that immediately gives the particles an acceleration $a$ as soon as they appear out of the vacuum.

Approximately, the extra distance, $\Delta x$, that a particle travels before it is annihilated is $$\Delta x \sim a t^2 \sim \frac{ax^2}{c^2}.$$ Therefore the particle pairs have a new Compton wavelength, $X$, given by $$X \sim x + \Delta x \sim x + \frac{ax^2}{c^2}.$$ Accordingly the energy $E_X$ of the particle pairs, after time $t$, is related to their new Compton wavelength $X$ by \begin{eqnarray} E_X &\sim& \frac{\hbar c}{X}\\ &\sim& \frac{\hbar c}{x(1+ax/c^2)}\\ &\sim& \frac{\hbar c}{x}(1-ax/c^2)\\ &\sim& E_x - \frac{\hbar a}{c}. \end{eqnarray} Thus the particle pair energy $E_X$ needed to satisfy the uncertainty principle after time $t$ is less than the energy $E_x$ that was borrowed from the vacuum in the first place. When the particle pair annihilates the excess energy $\Delta E=\hbar a/c$ produces a photon of electromagnetic radiation with temperature $T$ given by $$T \sim \frac{\hbar a}{c k_B}.$$ Thus we have derived an Unruh radiation-like formula for a vacuum that is being accelerated by a field. If the field is the gravitational field then we have derived the Hawking temperature. By the equivalence principle this is the same as the vacuum temperature observed by an accelerating observer. But this formula should be valid for any force field.

Let us assume that the force field is a static electric field $\vec{E}$ and that the particle pair is an electron-positron pair, each with charge $e$ and mass $m_e$. The classical equation of motion for each particle is then $$e\ \vec{E}=m_e\ \vec{a}.$$ Substituting the magnitudes of the electric field and acceleration into the Unruh formula gives $$T \sim \frac{\hbar}{c k_B}\frac{e|\vec{E}|}{m_e}.$$ If we take the electric field strength $|\vec{E}|=1$ MV/m then the electromagnetic Unruh/Hawking temperature is $$T\approx 10^{-2}\ \hbox{K}.$$ If this temperature could be measured then one could experimentally confirm the general Unruh/Hawking effect.

Is there any merit to this admittedly non-rigorous argument or can the Unruh/Hawking effect only be analyzed using quantum field theory?

closed as unclear what you're asking by ACuriousMind♦Nov 22 '17 at 11:32

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