Properties of the false vacuum as a quantum unstable state

Abstract

We analyze properties of unstable vacuum states from the point of view of the quantum theory. In the literature one can find some suggestions that some of false (unstable) vacuum states may survive up to times when their survival probability has a non-exponential form. At asymptotically late times the survival probability as a function of time $t$ has an inverse power--like form. We show that at this time region the energy of the false vacuum states tends to the energy of the true vacuum state as $1/t^{2}$ for $t \to \infty$. This means that the energy density in the unstable vacuum state should have analogous properties and hence the cosmological constant $\Lambda = \Lambda (t)$ too. The conclusion is that $\Lambda$ in the Universe with the unstable vacuum should have a form of the sum of the "bare" cosmological constant and of the term of a type $1/t^{2}$: $\Lambda(t) \equiv \Lambda_{bare} + d/ t^{2}$ (where $\Lambda_{bare}$ is the cosmological constant for the Universe with the true vacuum).