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simplification only occurs if the fermions are massless in the original model. If we had included a mass for the fermions, we would have obtained a sine-Gordon poten- tial for the scalar field, which is an interacting scalar field theory, and a quantum treatment of the bosonized model would become necessary. 

To analyze the discharge of a capacitor, we would like to set up an initial electric field that is localized within a finite region of space and then examine its evolu- tion due to the spontaneous production of fermion pairs. This leads to a physical difficulty because the capacitor plates are necessarily external to the system. This phys- ical difficulty arises even in classical electromagnetism, where Maxwell’s equations are solved but the boundary conditions are provided externally. The difficulty can be avoided in gravitational systems, for example, dur- ing gravitational collapse, because the system naturally tends to evolve toward a black hole. On the other hand, if the gravitational problem is set up to include an eternal black hole as an “external” agent, a similar issue arises. There is no analog of gravitational collapse in the elec- tromagnetic situation since similar electric charges repel. If we were to set up a configuration of φ field correspond- ing to separated localized positive and negative charges, the charge distributions would simply spread out due to mutual repulsion and then annihilate. Hence an external capacitor plate is necessary to set up the problem. 

Here we will set up the capacitor problem in two ways. In the first scheme, we fix the boundary condition that there is a charge + Q at x = − L/ 2 and another charge − Q at x = + L/ 2. These external charges are taken to be fixed and are non-dynamical. The Schwinger process then creates pairs and dissipates the electric field within the capacitor but cannot “evaporate” the charge on the plates. In this case, we find that the capacitor charges are screened due to the Schwinger process and the electric field decays exponentially with distance from a capacitor plate. Our second strategy to set up the capacitor plates is to introduce an external potential (two “bags”) into which we can insert the charges + Q and − Q as config- urations of the φ field itself. These charges are now dy- namical because the field φ is dynamical. The Schwinger process causes evaporation of the charge from the bags. In both physical realizations, the approach to the asymp- totic state is not monotonic but oscillatory. In the first case, the final static state still contains the ± Q charges but the charges are screened by opposite charges. In the second case, the final state is not static. Instead it con- tains bound states of positive and negative charge densi- ties that oscillate without dissipation within the bags. 

In the following two sections we describe the discharge of a capacitor in the two physical setups, first with “ex- ternal charges” and second with “external potential”. We can solve the first setup analytically, allowing us to ob- tain explicit expressions for the late time behavior of the current, electric field, and energy decay law. The so- lution of the second setup has only been obtained nu- merically. The results of both methods are summarized 

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in Sec. IV and show that the capacitor discharge is os- cillatory, the root-mean-square current is proportional to the root-mean-square electric field (Ohm’s law), and that the electrical conductivity of the vacuum is equal to g = $_{e/}$√$_{π}$$_{.}$ 

## II. SETUP I: EXTERNAL CHARGES 

In this section we treat the capacitor as made up of two external charges ± Q placed at x = ∓ L/ 2 respec- tively (see Fig. 1). The electric field due to these charges satisfies Maxwell’s equations and is a non-zero constant in the region between the plates. The value of the elec- tric field in the region − L/ 2 < x < + L/ 2 is E = Q . For | x | > L/ 2, the electric field vanishes. This electric field configuration, together with φ = 0 and ˙ φ ≡ ∂$_{t}$φ = 0, corresponding to no particles, are the initial conditions whose evolution we will consider. 

The equations of motion follow from (2). Including the external charges on the capacitor plates, we get 

with 

and u ν = (1 , 0). 

Maxwell’s equations (6) can be integrated immediately to get 

where 

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