Quantum dynamics of perfect fluids

We study the quantum field theory of zero temperature perfect fluids. Such systems are defined by quantizing a classical field theory of scalar fields $ϕ^I$ that act as Lagrange coordinates on an internal spatial manifold of fluid configurations. Invariance under volume preserving diffeomorphisms acting on these scalars implies that the long-wavelength spectrum contains vortex (transverse modes) with an exact $ω_T=0$ dispersion relation. As a consequence, physically interpreting the results obtained via perturbative quantization of this theory has proven to be challenging. In this paper, we show that correlators evaluated in a class of semi-classical (Gaussian) initial states prepared at $t=0$ are well-defined and accessible via perturbation theory. The width of the initial state effectively acts as an infrared regulator without explicitly breaking diffeomorphism invariance of the classical action. As an application, we compute the stress tensor two-point correlators and show that vortex modes give a non-trivial contribution to the response function, non-local in both space and time.

💡 Research Summary

The paper revisits the quantum field theory of zero‑temperature perfect fluids, whose low‑energy description is a non‑linear sigma model of scalar fields ϕⁱ(x) that serve as Lagrangian coordinates on an internal spatial manifold. The defining feature of a perfect fluid is invariance under both spacetime Poincaré transformations and volume‑preserving diffeomorphisms (SDiff) acting on the ϕⁱ. This symmetry forces the transverse (vortex) Goldstone modes to have an exact dispersion relation ω_T(k)=0, while the longitudinal (phonon) modes obey the usual ω_L=c_s|k|. Because the vortex sector contains an infinite set of zero‑energy excitations, a normalizable vacuum does not exist and standard perturbative quantization runs into severe infrared (IR) divergences. Earlier attempts introduced a small explicit breaking parameter c_T≪1 to give the vortex modes a linear dispersion ω_T≈c_T|k|, but all physical observables (cross sections, loop corrections to the stress‑tensor correlator, etc.) develop power‑law divergences ∝1/c_Tⁿ as c_T→0, signalling a breakdown of the perturbative expansion and a violation of the underlying SDiff symmetry.

The authors propose a different, symmetry‑preserving IR regulator: instead of deforming the action, they prepare the system at a finite initial time t=0 in a semi‑classical Gaussian wave‑functional. The functional Ψ_i