Radiation from a vacuum bubble: Two-dimensional case

The question is raised whether a bubble of true vacuum, expanding into an infinite space filled with false vacuum, can radiate some of the energy released, by the conversion of false vacuum to true, back into the interior of the bubble in the form of particles. In particular, we study vacuum particle production for the case of a scalar field in one space dimension, obeying the equation $\square \varphi -V(x,t)\mathrm{}\varphi =0\mathrm{}$, where $V(x,t)\mathrm{}$ is taken to be of the form $V(x,t)=F\left({(x^2-t^2)}^{\frac{1}{2}}\right)\theta \mathrm{}\left(t\right)+m^2$, and where $F\left(\xi \right)$ is an attractive, short-range well centered at $\xi =R$. This equation is of the general form which arises from consideration of the linear perturbations of a thin-walled vacuum bubble which has been created at $t=0\mathrm{}$ by tunneling and which moves along the trajectory $x={(R^2+t^2)}^{\frac{1}{2}}$. For general wells $F\left(\xi \right)$ it is found that the fields grow as a coupling-constant-dependent power of time $t^{\nu }$ for large time. The exponent $\nu$ is a (generally nonintegral) angular momentum parameter in a Euclidean bound-state problem. For the case in which $F\left(\xi \right)$ is consistently calculated from the classical "bounce" solution, the leading power ${\nu }_0$ is given by unity. However, this growth is probably caused by an incorrect treatment of the translational mode. For the case of a $\delta$-function well the stress-energy tensor is calculated for the purpose of showing that the O(1,1) invariance of the perturbation alone does not prohibit radiation. The approach is generalizable to the four-dimensional case.