Pair production in a strong electric field with back-reaction

We present a summary of the present status of efforts to solve the problem in which pairs are produced in a strong electric field, are accelerated by it, and then react back on it through the counter-field produced by their current. This picture has been used by Bialas and Czyz and others as a model for effects that may possibly arise in the study of the quark-gluon plasma. We here give a didactic review of recent developments in this back-reaction problem. We first present a simple version of the theory of pair tunneling from a fixed electric field, and then sketch how this has been applied to the quark-gluon plasma. Then we turn to a field formulation of the problem for charged bosons, which leads to the need to carry out a renormalization program, outlined again in simple terms. Numerical results for this program are presented for one spatial dimension, the corresponding physical behaviour of the system is discussed, and the implications for three spatial dimensions are considered. We exhibit a phenomenological transport equation embodying physics that is essentially identical to that of the field formulation, thus helping to tie the model of Bialas and Czyz for the quark-gluon plasma to a field-theory formulation. Last, we note the status of extensions to the problem with three space dimensions; the fermion case; the formulation in terms of boost-invariant variables (as desirable for the quark-gluon plasma); and transport equations derived in a fundamental and consistent fashion. 5 figs., 13 refs. (author)