The energy of the vacuum is given by $$\sum_k \frac{1}{2}\hbar\omega_k.$$ However the frequency $\omega_k$ depends on the wavevector $k$ and some constants like the speed of light $c$, which in turn depends on the permittivity and permeability of free space. I have read about vacuum polarization ...
You are using the terms wrongly. When we apply such a strong electric field, say, until the vacuum polarisation becomes strong enough to create particle-antiparticle pairs, that is called the Schwinger limit, and we never call that "change the energy of the vacuum", because adding an electric field is not changing the energy of the vacuum. It is adding a huge background electric field, which contains energy, on top of the vacuum. At least in standard physics, it is absolutely necessary to fix the energy of the vacuum to zero.
@naturallyInconsistent Ok, I am trying to wrap my head round this. I think I might have some serious misconceptions about this. So when the Schwinger limit is reached and electron-positron pairs are created the vacuum energy still stays the same?
@naturallyInconsistent I have another question about qft. How come there are different vacuum contributions for different fields? Shouldn't there be one vacuum?
$c$ is a constant relating units of space to units of time. It does not depend on anything. Even if you were to change the speed of light, $c$ would remain the same.
Yes. But it is easier to simply think of it as this: We set the energy of the vacuum to always be zero, by definition. We never touch upon it after this. When you add an electric field everywhere, that is something else, not the vacuum. You would have added an extra, new, thing to the vacuum, and that obviously can change the whole system's energy.
Each quantum field adds its own contribution to what it means to be a vacuum. That is just what it means for the vacuum to be vacuum---a lack of things, except the "empty" quantum fields themselves
@naturallyInconsistent and what about phenomena linked to the Euler-Heisenberg Lagrangian, which describes some non-linear dynamics of EM fields such as vacuum birefringence. I read that in these the the permittivity and permeability is effectively changed, by strong em fields.
Euler-Heisenberg Lagrangian comes from QFT. You should take care to study it properly, and not get lost in the jungle. It is extremely difficult and nobody has yet to specify a mathematically rigorous path to learning it; deviation from the explored path is likely to be fatal. What I said is still the least confusing path forward. If you use the terms in old ways or something like that, you can easily get confused.
This energy expression is infinite, so what would "lower" mean? Making it finite? One would have to redefine it somehow so that it sums up to a finite number, e.g. by introducing appropriate weights for the terms, which would push contribution of those terms to zero fast enough.
Discussion on question by eeqesri: Is…