Geodesics from Quantum Field Theory: A Case Study in AdS

TL;DR

This paper develops and tests two methods to derive geodesic trajectories from quantum field theory in AdS, confirming their consistency with classical geodesics and exploring the quantum-classical transition.

Contribution

It introduces a covariant center-of-mass approach and a position operator method to extract geodesics from QFT in curved spacetime, specifically in AdS$_3$.

Findings

Both methods reproduce expected timelike and null geodesics.

Wave packets exhibit a crossover from timelike to null geodesic behavior in the ultra-relativistic regime.

Bulk localization relates to the distribution over global descendants in the dual CFT.

Abstract

Localized one-particle states of a quantum field theory--whether in flat space or on a curved background--are expected to exhibit geodesic motion in an appropriate semiclassical regime. This expectation is often invoked heuristically: in this work we develop two precise implementations and test them in detail in global AdS$_3$. First, we define a covariant ''center-of-mass'' trajectory from the expectation value of the stress tensor operator and show, using only $\nabla_\mu\langle T^{\mu\nu}\rangle=0$, that it obeys the geodesic equation in the monopole (sufficiently localized) approximation in a general spacetime. This provides a QFT-in-curved-spacetime generalization of the Mathisson-Papapetrou-Dixon framework in classical general relativity. Second, we construct position operators from the Klein--Gordon inner product and mode completeness, and compute their expectation values in generic single-particle wave packet states. We then build explicit normalizable wave packets of a free scalar field in empty AdS$_3$ with tunable energy and angular momentum, and demonstrate analytically and numerically that both prescriptions reproduce the expected radial, circular, and elliptical-like timelike and null geodesics. Our discussion also isolates a natural ultra-relativistic regime in which the wave packet trajectory exhibits a controlled crossover from timelike to null geodesic behavior. We identify precise limits where the localized geodesic interpretation of the wave packet breaks down. On the CFT side, we show that bulk localization--specifically the radial data--is captured by how the state is distributed over global descendants of the dual primary.