Infinite energy from an electric field?

[p78653]

Imagine an electric field between two charged plates that is so intense that its energy density is enough to produce real electron-positron pairs.

These electron-positron pairs annihilate to produce photons that radiate away.

Does the electric field between the charged plates regenerate so that the process repeats indefinitely?

 

[hutchphd]

Its an imaginary field. You tell me.........

 

[Vanadium 50]

Can God create a stone so big He can't lift it?

 

[Nugatory]

Does the electric field between the charged plates regenerate so that the process repeats indefinitely?

Sure, as long as we do something like running an electrical generator connected to the plates to regenerate it as energy is radiated away.

Some serious quantum electrodynamics is needed to properly analyze this system, but you might try thinking about what happens to the electric field between the plates if the electrons and positrons are attracted to the plates (positron to the cathode, electron to the anode) instead of each other.

 

[Nugatory]

This thread will remain closed unless someone wants to contribute a proper analysis, as opposed to the hopelessly oversimplified heuristic I provided above.

 

[p78653]

According to the Uncertainty Principle electron-positron pairs can appear for a short time $\Delta t$ in the vacuum provided the uncertainty in their energy $\Delta E$ obeys:
$$\Delta E \times \Delta t \sim \hbar.\tag{1}$$
Let us assume that $\Delta E \sim m_e c^2$ so that the stationary electron-positron pairs are on-shell and therefore real.

Suppose there is an electric field $F$ which accelerates each electron and positron to a velocity $\Delta v$ in time $\Delta t$ so that
$$\Delta v=\frac{F}{m_e}\Delta t.\tag{2}$$
The energy $\Delta E$ of the electron-positron pair is Doppler-shifted by energy $\epsilon$ given by
$$\epsilon =\frac{\Delta v}{c}\Delta E.\tag{3}$$
Substituting Eqn.$(1)$ and Eqn.$(2)$ into Eqn.$(3)$ we find
$$\epsilon \sim \frac{\hbar}{c}\frac{F}{m_e}.\tag{4}$$
Are photons with energy $\epsilon$ radiated out of the vacuum without the electric field $F$ being diminished?

 

[haushofer]

No, because that violates energy conservation, because your hand-waving interpretation of virtual particles as "energy-conservation violating entities".

I.e. you're taking this analogy too seriously.

See also

https://arxiv.org/abs/2007.03030

 

[PeterDonis]

Let us assume that $\Delta E \sim m_e c^2$ so that the stationary electron-positron pairs are on-shell and therefore real.

No, that's not correct. An on shell particle would have that energy indefinitely, not just for a very short time. The particles described by $\Delta E$ (in this hand-waving, heuristic description) are virtual particles, not real particles, because they only have that energy for a short time.

 

[Vanadium 50]

I don't think a single statement you made is correct.

You seem to be trying to learn physics by stitching together various sound bites from popularizations. This does not work. You don't get a consistent picture this way - you get a patchwork mess.

 

[JimWhoKnew]

Are photons with energy $\epsilon$ radiated out of the vacuum without the electric field $F$ being diminished?

I think your previous question on this subject, the one that was blocked for reasons I fail to understand, was stated better.

The (yet hypothetical) pair production by a strong EM field is called Schwinger Effect. It is related to Klein Paradox and Hawking Radiation. Note that the created electron and positron are expected to be accelerated in opposing directions, so they get further apart and less likely to annihilate^*. The energy for the creation and acceleration must come from the EM field, so in a closed isolated system it has to diminish. It does so by a quantum process which is referred to as Backreaction, a complex task to analyze.

^* In section 1.7 of "Introduction to quantum effects in gravity" by Mukhanov & Winitzki, they say that one of the reasons that Hawking's result was so surprising, is because the created pairs are expected to experience the same fall in the background gravitational field, rather than be drawn apart.

 

[PeterDonis]

I think your previous question on this subject, the one that was blocked for reasons I fail to understand

It wasn't blocked, it was closed after the best responses were given that could be given since the question was very vaguely formulated.

The (yet hypothetical) pair production by a strong EM field is called Schwinger Effect.

A "strong EM field" is not a vacuum in the sense that matters for this discussion. "Vacuum" would mean that all quantum fields are in their vacuum states. In between the plates in the Schwinger Effect, the quantum electromagnetic field is not in the vacuum state; it is in a state that contains a lot of energy over the vacuum (ground) state energy. Pair production is then just a transfer of some of that energy to the quantum electron-positron field from the quantum electromagnetic field. You don't need any "virtual particles" or "uncertainty principle" or any other hand-waving for that. So if this is the sort of process the OP had in mind, then @Vanadium 50 in post #4 was perfectly correct that none of the claims made about it in the OP were correct.

It is related to Klein Paradox and Hawking Radiation.

Reference, please?

 

[JimWhoKnew]

It wasn't blocked, it was closed after the best responses were given that could be given since the question was very vaguely formulated.

I found his previous question interesting and wanted to respond (similar things to these in post #5), just to find out it was already locked.

A "strong EM field" is not a vacuum in the sense that matters for this discussion.

The OP says "Suppose there is an electric field $F$ which accelerates each electron and positron to a velocity...".

You don't need any "virtual particles" or "uncertainty principle" or any other hand-waving for that.

Of course. That's why I said his previous question (which I consider the same, despite the differences) was stated better. Had he received a satisfactory answer there, he could have done without this thread (speculative, I admit).

Reference, please?

HR and KP are discussed in Fulling's book "Aspects of..."
SE and KP are discussed by Greiner et al. in "Quantum Electrodynamics" and "QED in strong fields" (although they don't call it SE).
HR and SE are discussed in Mukhanov & Winitzki, mentioned in #5.
More can be found by a Google search.

 

[PeterDonis]

The OP says "Suppose there is an electric field $F$ which accelerates each electron and positron to a velocity...".

And the OP also says:

According to the Uncertainty Principle electron-positron pairs can appear for a short time $\Delta t$ in the vacuum

In other words, the OP is confused about what "vacuum" means and doesn't realize that "suppose there is an electric field" means there isn't a vacuum.

That's why I said his previous question (which I consider the same, despite the differences) was stated better.

If we take the previous question literally and assume the OP understands the physical meaning of what was asked, the answer was simple and was given by @Nugatory in the thread. What you posted in post #5 of this thread would have been relevant in that previous thread, but that's not the same as saying that thread should have been left open. (Or, for that matter, that this one should remain open.)