critical electric field that spontaneously generates real pairs

The Schwinger pair production can be understood as follows:

Suppose you have a constant electric field E in some region of space, pointing in the x direction. This is created for example by a large capacitor. Inside this capacitor, the energy of an electron-positron pair separated by some distance $\Delta x$ is $V(\Delta x)= 2m- q E \Delta x$, where $m$ is the mass of the electron (and positron) and $q$ is its charge. The first term is the rest energy of two massive particles, and the second is their potential energy in the presence of the electric field. You can see that for large enough separation the total energy is negative - it becomes energetically favourable to have a pair present instead of an empty capacitor.

This process can be thought of as tunnelling: the configuration of empty space and an electron-positron pair are separated by a barrier $V(\Delta x)$. You therefore have an exponentially small probability to create pairs at the classical turning point $V(\Delta x)=0$, or $\Delta x = \frac{2m} {qE}$. As always in tunnelling process, energy is conserved - it is zero before and after the tunnelling in this case.

Once the particles are created, they accelerate away from each other, and eventually end up neutralizing in part the capacitor, in other words reducing the electric field. Note that there is no critical electric field - pair creation occurs for arbitrarily small electric field, though the probability is exponentially suppressed, roughly as $\exp(- \pi m^2/ q E)$. The derivation of this formula by Schwinger (and corrections to all orders) is a real joy to see, I'd recommend at least for theorists to take a look at the original paper. This may well be the first use of instanton methods in quantum mechanics, though I'm far from being an expert on the history.

Also:

There is no real connection of this to renormalization.

Variants of this calculation are useful in cosmology and QFT in curved spacetime, for example for particle production by time varying backgrounds.

As far as I know, the effect has never been observed due to difficulties creating large enough electric field. I may be wrong on that as well though.