Quantum Statistical Metastability Revisited

We calculate the decay rate for a state prepared in a thermal density matrix centered on a metastable ground state. We find a rate that is intrinsically time dependent, as opposed to the constant rates of previous works. The rate vanishes at early times, rises to a maximum and eventually falls-off to zero as a consequence of unitary time evolution. Finally, we discuss extensions of this calculation to field theories and possible implications for both sphaleron mediated transitions and first order inflationary theories. The analysis of the decay of metastable states has always been a important and interesting topic in physics. Recently, however, this subject has assumed even greater importance due to the discovery that there are field configurations in the standard model (so-called sphalerons) that mediate baryon number violating transitions which are unsuppressed at high temperature. Needless to say, this has important implications for the evolution of baryon number in the early universe. Another reason why there has been a rekindling of interest in the evolution of metastable states in the early universe is due to the development of viable models of inflation (extended inflation) that go back to Guth’s idea of ending the inflationary era via false vacuum decay. In short, there are good reasons for making sure that the decay of metastable states at finite temperature is, in fact, well understood. We will argue in this Letter that some aspects of the calculation of the decay rate of metastable states must be rethought. In particular, we will make the point that different choices of initial state can make for significant changes in the decay rate. We show that under realistic conditions, the decay rate must be time dependent. This is consistent with some recent experimental data which we discuss below. Two of the seminal works on this topic are those by Langer and Affleck. It will be instructive to review these calculations, since our results are quite different from theirs. Langer develops a Fokker-Planck type equation for the probability of finding the system in a given configuration at time t. This probability obeys a continuity equation and the associated current gives the flow of probability in the configuration space. In a one dimensional system, this current, evaluated at the saddle point (which is the top of the barrier), is identified with the rate of activation of the system over the barrier. To compute this current, Langer then constructs a steady state solution to his FokkerPlanck equation. This is tantamount to setting up a steady state situation by continously replenishing the metastable state at a rate equal to the rate at which it is leaking across