7 IV. CONCLUSIONS AND DISCUSSION We have studied the quantum discharge of a capacitor in massless QED in 1+1 dimensions by bosonizing the model. The bosonized model is non-interacting and can be solved classically. The solution includes all backreac- tion effects. We now summarize the key results. The final state depends on the setup used to describe the capacitor plates. We have chosen two different ways to describe the capacitor plates. In both cases, the fi- nal state is non-trivial. In setup I, the plates keep their original charge but the charges are screened due to the Schwinger process. In setup II, there are no external charges, but there are external potentials that play the role of capacitor plates. Then the final state consists of fermion-antifermion pairs that are bound to the plates. The energy in the final state depends on the coupling constant, g , and equivalently the mass of the scalar field, m . The final state energy decreases with increasing g , while the time for the capacitor to discharge increases with increasing g , as seen in Eq. (44). So stronger cou- pling leads to more complete discharge but the discharge process itself is slower. We suggest that the longer dis- charge time for larger coupling constant is due to the tighter binding of fermion-antifermion pairs that need to be split apart by the electric field. The discharge process is highly oscillatory, as also seen in the semiclassical analysis [2] and the amplitude of os- cillations falls off rather slowly, as t − 1 / $^{2}$. This suggests that the massless QED system is under damped Our results show that the root-mean-square current in the capacitor is directly proportional to the root-mean- square electric field, indicating that Ohm’s law holds on a macroscopic scale. Thus it makes sense to define the electrical conductivity for the massless QED vacuum to be σ$_{E}$ = j x $_{rms}$/E$_{rms}$ and our results indicate the simple relation σ$_{E}$ = g which can also be written in terms of the fermionic charge as σ$_{E}$ = $_{e/}$√$_{π}$$_{.}$ A correspondence is often made between Schwinger pair creation and Hawking radiation, though we have indicated differences between the two processes that prompt us to use caution in drawing a correspondence. If the oscillatory features of the discharge process carry over to black hole evaporation, we may expect black hole mass oscillations during evaporation. Though, in con- trast to the capacitor, the black hole system is unstable in that smaller mass black holes are hotter and evapo- rate faster, while weaker electric fields in the capacitor do not discharge faster. So it would appear that a fluc- tuation that excessively decreases the mass of the black hole, would make it evaporate yet faster in what may be a runaway process. The issues of black hole formation and the final state of black hole evaporation cannot be resolved by this cor- respondence since the capacitor plates have to be intro- duced externally, whereas there are no such externally set conditions in the case of gravitational collapse. Yet it would be extremely interesting if the electromagnetic FIG. 6: Ratio of the envelopes of current (divided by m ) to electric field at x = 0 for m = 1 . 5, 2 and 2.5. The constant flat ratio implies Ohm’s law and the value of 1 implies that the conductivity is g . FIG. 5: Envelopes for the currents (solid curves) and the elec- tric fields (dashed curves) at x = 0 on a log-log plot. Com- parison with the light straight line with slope − 1 / 2 clearly shows the 1 $_{/}$√$_{t}$ fall off. From black to light grey, the curves represent, respectively, the evolution for m = 1 . 5, 2 and 2.5. FIG. 4: Envelopes of the plots of the electric field at the center of the capacitor, E ( t, x = 0) versus time. From black to light grey, the curves represent, respectively, the evolution for m = 1 . 5 , 2 and 2.5. The rapid oscillations between the envelopes are not shown. 0 200 400 600 800 1000 Time GLYPH<LParen1> t GLYPH<RParen1> GLYPH<Minus> 6 GLYPH<Minus> 4 GLYPH<Minus> 2 0 2 4 6 E GLYPH<LParen1> t,x GLYPH<Equal> 0 GLYPH<RParen1> 10 20 50 100 200 500 1000 Time GLYPH<LParen1> t GLYPH<RParen1> 1 1.5 2 3 5 7 10 15 E GLYPH<LParen1> t,x GLYPH<Equal> 0 GLYPH<RParen1> and j x GLYPH<LParen1> t,x GLYPH<Equal> 0 GLYPH<RParen1> 0 200 400 600 800 1000 Time GLYPH<LParen1> t GLYPH<RParen1> 0.97 0.98 0.99 1 1.01 1.02 1.03 j x GLYPH<LParen1> t,x GLYPH<Equal> 0 GLYPH<RParen1>GLYPH<Slash1>GLYPH<LParen1> m E GLYPH<LParen1> t,x GLYPH<Equal> 0 GLYPH<RParen1>GLYPH<RParen1>