Quantum state of a nucleating bubble

We consider a field theory consisting of two interacting scalar fields: $\sigma$ and $\Phi$. The scalar field $\sigma$ is assumed to undergo a first-order phase transition via the nucleation of bubbles. We solve the Schrördinger equation for the combined system of a bubble plus the field $\Phi$ with appropriate boundary conditions. This allows us to determine the quantum state of the field $\Phi$ in the background of the nucleating and subsequently expanding bubble. The simplest description of this quantum state is obtained in the picture where $\Phi$ is represented as an infinite set of massive scalar fields in a (2+1)-dimensional de Sitter space. We show that the bubble nucleates with all these fields in de Sitter-invariant quantum states and that the resulting quantum state of the field $\Phi$ is Lorentz invariant.