Geodesics from Quantum Field Theory: A Case Study in AdS

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 $\nablaμ\langle T^{μν}\rangle=0$, 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.

• The paper shows that quantum stress-tensor centroids yield trajectories that satisfy classical geodesic equations under a monopole approximation.
• It employs an operator construction from the Klein-Gordon mode completeness to extract macroscopic positions from smooth, localized wave packets.
• The study quantifies the semiclassical regime breakdown by analyzing variance corrections as packet widths approach the inverse energy scale.

Emergence of Geodesic Motion from Quantum Field Theory in AdS$_3$

Introduction and Conceptual Framework

The paper "Geodesics from Quantum Field Theory: A Case Study in AdS" (2604.05901) presents a comprehensive investigation into how classical geodesic motion emerges from quantum field theoretic constructions in curved spacetimes, focusing primarily on global AdS$_3$. The study addresses the long-standing intuition that sufficiently localized one-particle quantum states should, in semiclassical regimes, propagate along classical geodesics. This intuition, frequently invoked in holographic and semiclassical gravity settings, is here subjected to rigorous scrutiny via two distinct, operator-based frameworks.

The first approach is covariant and employs the energy-weighted centroid, constructed from the expectation value of the quantum stress tensor, generalizing the Mathisson-Papapetrou-Dixon formalism within the context of QFT in curved spacetimes. The second is operatorial, utilizing the completeness of the Klein-Gordon mode functions to construct explicit radial and angular position operators, whose expectation values in smooth, normalizable states track the macroscopic trajectory. Both methodologies are implemented in AdS$_3$, with particular emphasis on the analytic and numerical construction of wave packets possessing tunable initial data (energy, angular momentum, position, and width).

Position, Localization, and the Limitations of Relativistic Quantum Theory

A central theoretical challenge arises from the subtleties of localization in relativistic quantum field theory. It is well known, following Newton-Wigner and Hegerfeldt, that sharply localized position eigenstates are neither physical nor stable under relativistic evolution due to instantaneous spreading and tension with spectral conditions on the Hamiltonian. The approach taken here circumvents these issues by focusing exclusively on smooth, normalizable wave packets and eschewing position eigenstates as physical states. The localization data is instead extracted through expectation values, with attention given to the regime of validity as determined by the ratio of the packet width to intrinsic energy scales.

Formalism I: Stress-Tensor Centroid and Emergent Geodesic Motion

The first formalism defines a "center-of-mass" trajectory by considering the centroid with respect to the expectation value of $T^{tt}$ over a spatial slice, weighted by the ADM lapse. The main technical result establishes that, under the "monopole" (i.e., sufficiently localized) approximation, the resulting trajectory satisfies the geodesic equation up to variance corrections explicitly controlled by the packet width. This result is robust and holds for generic static backgrounds, requiring only local conservation of the stress tensor. For sharply peaked, well-localized states, higher multipole contributions are suppressed and the correspondence with classical geodesics is quantitatively justified.

Figure 1: The packet profile in the beginning. ($t=0.0$)

Formalism II: AdS Position Operators from Mode Completeness

The second formalism constructs explicit position operators using the Sturm-Liouville completeness of the AdS$_3$ scalar field modes. The operator $\hat{\rho}$ acts on the positive-frequency one-particle Hilbert space, with eigenmodes labeled by the quantum numbers $(n,m)$ corresponding to radial and angular excitations. Crucially, the expectation values over smooth one-particle wave packets (with no recourse to position eigenstates) are interpreted as macroscopic locations, permitting a direct comparison with classical motion.

Figure 2: The massless packet profile in the beginning. ($t=0.0$)

Both frameworks give rise to identical centroid dynamics in the optimal localization regime; their deviation in non-ideal regimes provides insight into the breakdown of semiclassical physics.

Explicit Construction and Evolution of Wave Packets

A key technical achievement is the explicit construction and evolution of normalizable Gaussian-like wave packets with tunable mean positions $(\rho_0, \phi_0)$, widths $_3$0, and proxy parameters for radial and angular momentum $_3$1. The time evolution is resolved analytically and numerically via Fourier expansion in the AdS Klein-Gordon eigenbasis, enabling high-precision comparison between quantum expectation values and classical geodesic motion.

Both radial ($_3$2) and non-radial ($_3$3) cases are analyzed, with the latter including circular and elliptical-like orbits. The expectation values of $_3$4, $_3$5, and the energy density $_3$6 are shown to track the corresponding geodesics with high fidelity as long as the conditions for semiclassical localization are satisfied.

Figure 3: Polar plot mapping the trajectory. Parameters: $_3$7, $_3$8, $_3$9, and $_3$0.

Delocalization, Breakdown, and Localization Scale

One of the paper’s quantitatively substantiated claims is the breakdown of the geodesic approximation when the packet width falls below a critical length scale set by the energy, specifically $_3$1. In this regime, variance corrections become significant and the centroid or expectation trajectory deviates from the classical geodesic (see Figs. 17 and 22), with massless and low-energy cases manifesting rapid delocalization and even packet splitting. Such deviations can be delayed by increasing the energy or mass (thus reducing $_3$2), but cannot be entirely eliminated due to the underlying relativistic quantum structure.

Figure 4: $_3$3—packet shows onset of delocalization for small widths.

This analysis not only clarifies the range of validity for the geodesic correspondence but also makes precise the sense in which the semiclassical story emerges and, crucially, breaks down.

Exact Operator Relations in AdS: Selection Rules and Quantum-Classical Correspondence

A particularly sharp result relies on AdS-specific spectral properties. The selection rules arising from the Jacobi polynomial structure of the radial modes, together with the equally spaced eigenvalues $_3$4, enable exact operator equations for certain combinations. For arbitrary one-particle states, the expectation values $_3$5 and $_3$6 are shown to satisfy the same second-order differential equations as their classical geodesic counterparts, up to explicit variance corrections. This reflects a deeper symmetry in quantum AdS dynamics and provides a concrete realization of quantum-classical correspondence for specific observables.

Figure 5: Operator formalism with parameters: $_3$7, $_3$8, $_3$9, $T^{tt}$0, $T^{tt}$1, $T^{tt}$2.

Conserved Charges: Wave Packet vs. Geodesic Identification

An explicit discussion is presented on the divergence between the classical conserved charges (energy and angular momentum) associated to a best-fit geodesic, and those directly extracted from wave packet expectation values. Finite width effects induce a systematic shift, with the quantum energy always exceeding the naive classical value due to variance contributions—an effect mirrored in quantum anharmonic oscillator toy models. Thus, direct identification of quantum and classical charges is only approximate, with exact correspondence arising only in the infinitely localized, high-energy (classical) limit.

Holographic and CFT Interpretation

The AdS/CFT connection is made precise by mapping the bulk one-particle Hilbert space to the global conformal module of the dual CFT scalar primary. The bulk-localized Gaussian packet corresponds to a specific superposition of CFT global descendants, while the mean energy and angular momentum map onto quantum numbers in the CFT. The "radial" data in the bulk is thereby translated into the probability distribution over CFT descendant levels. The paper elucidates how the initial position, width, and momentum data of the bulk wave packet may be reconstructed from the expectation values and interference patterns in the $T^{tt}$3 descendant basis.

Practical and Theoretical Implications

This work provides a technically robust framework for connecting quantum field theoretic data to effective classical dynamics in AdS backgrounds. The implications are multifold:

• Holographic bulk localization: The results clarify how CFT data organizes into semiclassical bulk trajectories, forming a foundational component for more ambitious reconstructions (e.g., black hole interior).
• Limits of semiclassical gravity: The analysis of the breakdown of the geodesic regime in terms of packet width, energy, and mass is immediately relevant for studies of quantum gravity, localization, and emergent general relativity.
• Operator construction in curved backgrounds: The explicit realization of position operators adapted to curved backgrounds via mode completeness provides a template for further studies in higher-dimensional AdS and more general spacetimes.

Figure 6: Parameters: $T^{tt}$4—demonstrating quantum-classical correspondence and breakdown at small widths.

Conclusion

This paper establishes a precise operational framework for deriving classical geodesic motion from quantum field theoretic constructs in AdS$T^{tt}$5, combining explicit operator constructions, analytic control via mode decompositions, and detailed numerical evolution. The demonstrated agreement between quantum centroid trajectories and classical geodesics, and the quantified regimes of validity for localization, provide a solid foundation for future developments in quantum gravity, AdS/CFT correspondence, and semiclassical analysis of QFT in curved backgrounds. The exact operator-geodesic correspondence, enabled by AdS spectral properties, may suggest fruitful avenues in symmetry-based approaches and quantum information aspects of holography.