Quantum Popcorn: the shape of a phase transition bubble

Abstract

A quantum tunneling event in field theory is usually depicted as the materialization of a spherical bubble of true vacuum. The bubble appears at rest, and its profile is the time-symmetric slice of Coleman's bounce, the Euclidean solution that controls the decay rate. However, the bubble is spherical only in the leading semiclassical approximation: at the next order the configuration is a Gaussian ensemble of fluctuations around the bounce. We derive the wave functional of the field on the nucleation slice from the half-bounce path integral, and use it to compute the root-mean-square quadrupole moment of a bubble at nucleation. The relative deformation of the bubble wall radius $R$ is $δR/R\sim(\hbar/S_b)^{1/2}$, with $S_b$ the bounce action. Thus $δR/R\sim0.1$ for a typical cosmological phase transition with $S_b/\hbar\sim100$.

What shape does a true-vacuum bubble have immediately after quantum nucleation, beyond the spherical semiclassical bounce? This paper computes Gaussian field fluctuations on the nucleation slice and their contribution to the energy-density quadrupole in a real scalar field theory. A Euclidean half-bounce path integral defines the nucleation wave functional, with its quadratic kernel given by a Dirichlet-to-Neumann map. The equal-time field covariance is obtained from the bounce fluctuation Green’s function restricted to that slice. Angular and radial spectral decompositions yield the quadrupole variance and separate a wall-displacement contribution, available in closed form for a general profile, from internal excitations. For a thin wall of tension $\sigma$ and radius $R$, each quadrupole displacement mode satisfies $\langle|\delta R_{2M}|^2\rangle/R^2=(3\pi^3/64)\hbar/S_b$, giving approximately $0.12$ for $S_b/\hbar=100$. This reproduces the existing world-volume Hartle–Hawking result. Internal-wall corrections scale as $(\ell_w/R)^3$ for a wall with one internal scale. For Fubini-like profiles with $mR\ll1$, fluctuations along the extended tail instead produce a logarithmic enhancement, $\mathcal Q^2=(2\pi R^2/\lambda)[\ln(1/mR)+O(1)]$. Numerical quartic and sextic examples test these limits. The analysis assumes Gaussian fluctuations around a monotonic minimum-action bounce and treats quantum, not thermal, nucleation. Translation modes are fixed, and the escape direction requires a turning-slice constraint; the quadrupole sector avoids these complications. Pointwise wall roughness requires resolution-dependent smoothing, unlike the quadrupole. Subsequent evolution and gravitational-wave emission are not calculated.