Quantum dynamics of perfect fluids

TL;DR

The work tackles the challenge of quantizing zero-temperature perfect fluids with SDiff invariance, where vortex modes have $\omega_T(\vec{k})=0$ leading to ill-defined vacua and obstructing standard perturbation theory. It introduces a Schwinger-Keldysh in-in framework with semi-classical Gaussian initial states at $t=0$, enabling perturbative calculations that separately treat longitudinal phonons and transverse vortons. The main result is a explicit one-loop computation of the retarded stress-tensor correlator $G^R_{\vec p}(t)$, showing nonlocal-in-time and nonlocal-in-space contributions from vortex modes via the TL and LL channels, with explicit dependence on the initial-state spectral index $\Delta$, the transverse speed $\hat{c}_T$, and scale $\mu$; the TT channel does not contribute to the retarded response. These findings demonstrate that physically meaningful hydrodynamic quantum observables can be extracted in SDiff-invariant perfect fluids without ad hoc infrared regulators, at least for short times and small momenta, and they open avenues for extending to other observables and to parity-violating or non-Gaussian initial states. The results also hint at connections with non-perturbative vacua and vorton descriptions, providing a bridge between effective field theory methods and alternative formulations of quantum hydrodynamics.

Abstract

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 exact $ω_T=0$ dispersion relation. As a result, physically interpreting the perturbative quantization of this theory by standard methods has proven to be challenging. In this paper, we show that correlators evaluated in the 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. As an application, we compute 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.

Figures (2)

• Figure 1: Leading order Feynman diagram contributions to the momentum space Wightman function $\langle T_{ij}(x_1) T_{kl}(x_2)\rangle=\langle T^-_{ij}(x_1) T^+_{kl}(x_2)\rangle$ from (a) $TT$ (b) mixed $TL$, (c) and $LL$ two-phonon ("superfluid") internal propagators.
• Figure 2: $TL$ contribution to the response function, given by Eq. \ref{['eq:LTresult']}, ploted for range of values of the spectral parameter $\Delta$.