Capacitor Discharge and Vacuum Resistance in Massless QED$_{2}$ Yi-Zen Chu 1 and Tanmay Vachaspati 1 , 2 $^{1}$CERCA, Department of Physics, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, OH 44106-7079, $^{2}$Institute for Advanced Study, Princeton, NJ 08540 A charged parallel plate capacitor will create particle-antiparticle pairs by the Schwinger process and discharge over time. We consider the full quantum discharge process in 1+1 dimensions including backreaction, when the electric field interacts with massless charged fermions. We recover oscillatory features in the electric field observed in a semiclassical analysis and find that the amplitude of the oscillations falls off as t − 1 / 2 and that stronger coupling implies slower decay. Remarkably, Ohm’s law applies to the vacuum and we evaluate the quantum electrical conductivity of the vacuum to be $_{e/}$√$_{π}$$_{, where}$ e is the fermionic charge. Similarities and differences with black hole evaporation are mentioned. Following Schwinger’s 1951 paper [1], it is well-known that quantum effects cause electric fields to produce electron-positron pairs. The effect can be interpreted as the tunneling of virtual electron-positron pairs into real particles. One imagines a virtual e $^{+}$e − pair produced in the vacuum which is then torn apart by the background electric field, say within a capacitor, with the positron accelerated in the direction of the electric field, and the electron in the opposite direction. A similar effect has been proposed in de Sitter space where the rapid expan- sion of spacetime “pulls” particles out of the vacuum. Hawking radiation from black holes has also been inter- preted as a Schwinger process, though the universally attractive nature of gravity suggests that there are dif- ferences. For example, in the electromagnetic case, it is clear that once the charges are created, the positive charge accelerates away from the positively charged ca- pacitor plate due to electromagnetic repulsion. In the black hole case, however, there is radiation even though the black hole attracts all outgoing particles. The energy for pair creation in an electric field must eventually come from the energy in the electric field itself. Hence the electric field has to decay due to the Schwinger process, just as Hawking radiation is assumed to cause black holes to evaporate. The problem of electric field de- cay clearly involves calculation of the backreaction of the Schwinger process and this is a hard problem. There have been several attempts to analyze the decay of the elec- tric field by semiclassical methods, replacing quantum operators by their expectation values [2]. The results are interesting. For example, a uniform electric field will not discharge monotonically but will undergo oscillations. If the conclusion can be directly transported to the black hole case, it would imply oscillations of the black hole mass and not monotonic evaporation. A key difference though is that electric charges can be positive or nega- tive, whereas the particles in Hawking radiation can only have positive mass. (See also the recent work [3], where backreaction in the context of scalar QED in 3+1 dimen- sions was taken into account by solving the equations of The bosonized model is particularly easy to solve be- cause it is quadratic in fields and hence there are no in- teractions. It is sufficient to solve it classically. This where, without any loss of generality, we can assume g ≥ 0. The correspondence between the fermionic and bosonic models is given by identifying the currents at the quantum operator level [10] where, ϵ$_{µν}$ is the Levi-Civita tensor in 1+1D with ϵ$_{01}$ = 1 and 1 The fermions interact with the gauge field by the stan- dard minimal coupling, and an electric field leads to fermion pair production. The advantage of 1+1D QED is that it can be bosonized to yield [7–9], We treat the case of massless QED in 1+1 dimensions In this paper we re-visit the problem of capacitor dis- charge due to the Schwinger process, without restrict- ing ourselves to the semiclassical approximation. We can solve the full quantum problem but the price we pay is that we are then only able to treat massless fermions and the exponential suppression of the classic Schwinger process is absent. motion derived from the one loop Euler-Heisenberg effec- tive action.) arXiv:1001.2559v2 [hep-th] 7 Oct 2013 I. INTRODUCTION S 0 = ∫ d $^{2}$x [ ¯ ψγ $^{µ}$( i∂$_{µ}$ + eA$_{µ}$ ) ψ − 1 4 F$_{µν}$ F µν ] (1) S ′ 0 = ∫ d $^{2}$x [ 1 2 ( ∂φ ) 2 + g 2 φϵ $^{µν}$F$_{µν}$ − 1 4 F$_{µν}$ F µν ] (2) g ≡ e √$_{π}$ (3) : ¯ ψγ $^{µ}$ψ : ↔ ϵ $^{µν}$∂$_{ν}$ φ (4) 1 Note added : Previous versions of this paper contained a spuri- ous factor of 2 in the following formula, i.e., g ≡ 2 $_{e/}$√$_{π}$$_{. (We}$ thank Jaume Garriga and Emil Mottola for bringing this to our attention.) We have, accordingly, dropped this factor of 2 in the electrical conductivity of eq. (41), and everywhere in the text referring to it.