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Imagine we have two parallel wires with a potential difference of $V$ volts that form the opposite sides of a square of size $\lambda$.

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Any virtual electron-positron pairs that form between the wires pick up an acceleration $a$ given by

$$a\sim\frac{eV}{m_e\lambda}.\tag{1}$$

According to the Unruh effect the accelerating electron and positron are immersed in a heat bath with temperature $T$ given by

$$T \sim \frac{ha}{ck_B}.\tag{2}$$

For simplicity let us assume that we just have photons of wavelength $\lambda$ such that

$$k_BT\sim\frac{hc}{\lambda}.\tag{3}$$

Combining Equations $(1\hbox{-}3)$ we find that the voltage $V$ must be given by

$$eV\sim m_ec^2.\tag{4}$$

An inertial observer sees the electron-positron pairs radiating at temperature $T$. According to the Stefan-Boltzmann law the radiation flux $j$ through the square is given by

$$j=\sigma T^4\tag{5}$$

where

$$\sigma\sim \frac{k_B^4}{h^3c^2}.\tag{6}$$

If we assume the radiation consists of photons with wavelength $\lambda$ as described by Equation $(3)$ then the number of photons per second, $N$, through the square is given by

$$N \sim \frac{j\lambda^2}{k_BT}\sim\frac{\sigma T^3 \lambda^2}{k_B}.\tag{7}$$

Combining Equations $(3)$,$(6)$ and $(7)$ we find that the number of photons going through the square per second, $N$, is given by

$$N\sim\frac{c}{\lambda}\sim \nu\tag{8}$$

where $\nu$ is the frequency of the radiation of wavelength $\lambda$.

It seems to me that these real photons are generated from the vacuum itself. However as I have not used any quantum field theory in this argument my conclusions are almost certainly completely false!

Reference post: Does an electric field cause the vacuum to emit photons?

  • related https://en.wikipedia.org/wiki/Schwinger_effect – shai horowitz Jun 07 '22 at 19:16
  • I'm not sure your conclusions are correct, but the phrase "According to the Unruh effect" involves a whole lot of quantum field theory in curved spacetime, which you simply didn't write explicitly. – Níckolas Alves Jun 08 '22 at 12:42
  • If your electric field is strong enough to create real electron-positron pairs, then, as @shaihorowitz points out, the Schwinger effect causes the production of pairs of real electron-positron, which in turn could emit real photons according to OP"s argument. If the electric field is not enough, then it is not clear that the reasoning holds (the electron and positron are off-shell) – SolubleFish Jun 08 '22 at 21:34
  • Also, I don't know if it is really safe to talk about the "acceleration" picked up by virtual particles – SolubleFish Jun 08 '22 at 21:35

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