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arxiv: 2604.05901 · v2 · submitted 2026-04-07 · ✦ hep-th · gr-qc

Geodesics from Quantum Field Theory: A Case Study in AdS

Vaibhav Burman , Chethan Krishnan , Livesh Parajuli This is my paper

Pith reviewed 2026-05-10 19:57 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords geodesicsquantum field theoryAdS3stress tensorwave packetssemiclassical limitholographycenter of mass
0
0 comments X p. Extension

The pith

The stress tensor expectation value of a localized quantum state traces a geodesic in any curved spacetime.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that geodesic motion can be extracted directly from a quantum field theory state in curved backgrounds. A covariant center-of-mass trajectory is defined from the expectation value of the stress tensor, and the conservation law alone implies that this trajectory obeys the geodesic equation when the state is localized in the monopole approximation. This construction is tested in global AdS3 by building explicit normalizable wave packets of a free scalar field that reproduce radial, circular, and elliptical timelike and null geodesics. The work also links the radial localization in the bulk to the distribution of the state over global descendants in the dual CFT.

Core claim

We define a covariant center-of-mass trajectory from the expectation value of the stress tensor operator and show, using only ∇_μ ⟨T^{μν}⟩=0, that it obeys the geodesic equation in the monopole approximation in a general spacetime. We construct position operators from the Klein-Gordon inner product and compute their expectation values in generic single-particle wave packet states, then demonstrate analytically and numerically that both prescriptions reproduce the expected radial, circular, and elliptical-like timelike and null geodesics in empty AdS3, including a controlled ultra-relativistic crossover from timelike to null behavior.

What carries the argument

The covariant center-of-mass trajectory derived from the expectation value of the stress tensor operator, which satisfies the geodesic equation via conservation in the sufficiently localized limit.

If this is right

  • This construction supplies a QFT-in-curved-spacetime generalization of the Mathisson-Papapetrou-Dixon framework.
  • Explicit wave packets in global AdS3 confirm that both the stress-tensor and position-operator prescriptions reproduce the expected geodesics.
  • The wave-packet trajectories exhibit a controlled crossover from timelike to null geodesic behavior in an ultra-relativistic regime.
  • Bulk radial localization data is captured by the state's distribution over global descendants of the dual primary on the CFT side.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method could be applied to other backgrounds beyond AdS to extract effective classical trajectories from quantum states.
  • It indicates that the semiclassical limit recovers geodesic motion in general relativity from stress-tensor conservation with minimal additional input.
  • In holographic settings this supplies a direct link between the organization of CFT states and bulk geodesic motion.

Load-bearing premise

The wave packets must remain localized enough that the monopole approximation holds and the center-of-mass trajectory interpretation does not break down.

What would settle it

An explicit calculation of the center-of-mass trajectory from the stress tensor expectation value for a localized wave packet in AdS3 that deviates from the known geodesic path would falsify the claim.

Figures

Figures reproduced from arXiv: 2604.05901 by Chethan Krishnan, Livesh Parajuli, Vaibhav Burman.

Figure 1
Figure 1. Figure 1: Parameters: M = 25, m0 = −20, ρ0 = 0.5, ϕ0 = π/3, σ1 = 0.09, σ2 = 0.05, nmax = mmax = 75 The plot of ρ¯(t) versus ϕ¯(t) for the above case is presented in Appendix F.1, along with additional examples. 7.2.2 Massive Case: Radial Infall For purely radial infall, we choose the profile defined in (7.3): f(ρ, ϕ) = Nρ e − (ρ−ρ0) 2 4σ2 1 Nϕ e − (ϕ−ϕ0)2 4σ2 2 . (7.8) 23 [PITH_FULL_IMAGE:figures/full_fig_p024_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Parameters: ρ0 = 1.2, ϕ0 = π/3, σ1 = 0.09, σ2 = 0.05, nmax = mmax = 200, M = 40, m0 = 0 The plots of ρ¯(t) vs ϕ¯(t) corresponding to the above case can be found in Appendix F.2. Also see Appendix F.3 for the ρ¯(t) vs t plots for this and other examples. 24 [PITH_FULL_IMAGE:figures/full_fig_p025_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Parameters: ρ0 = 0.5, ϕ0 = π/3, σ1 = 0.09, σ2 = 0.05, nmax = mmax = 300, M = 0, m0 = −40 . The packet reaches the boundary at t = π 2 . The plots of ρ¯(t) vs ϕ¯(t) corresponding to the above case can be found in Appendix F.4. 7.2.4 Null Case: De-localized Wave Packet Using the same choice of wave packet as used for radially infalling massive case (see Eq. (7.3)) results in a highly delocalized evolution fo… view at source ↗
Figure 4
Figure 4. Figure 4: Time evolution of the delocalized wave packet ( [PITH_FULL_IMAGE:figures/full_fig_p028_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Parameters: ρ0 = 0.5, ϕ0 = π/3, σ1 = 0.09, σ2 = 0.05, nmax = mmax = 75, M = 25, m0 = −20, n0 = 0. The plot of ⟨ρ(t)⟩ vs ⟨ϕ(t)⟩ corresponding to this can be found in Appendix G.1 along with other examples. 7.3.2 Massive Case: Radial Infall Radial infall we choose the profile defined in (7.3): f(ρ, ϕ) = Nρ e − (ρ−ρ0) 2 4σ2 1 Nϕ e − (ϕ−ϕ0) 2 4σ2 2 . (7.14) Substituting this profile into the expectation value … view at source ↗
Figure 6
Figure 6. Figure 6: Parameters: ρ0 = 1.2, ϕ0 = π/3, σ1 = 0.09, σ2 = 0.05, nmax = mmax = 200, M = 40, m0 = 0, n0 = 0. The wave packet evolves through the center of the geometry to the antipodal point at ϕ = 4π/3, which is reached at coordinate time t = π. The plots of ⟨ρ(t)⟩ vs ⟨ϕ(t)⟩ corresponding to the above case can be found in Appendix G.2. See Appendix G.3 for the plots of ⟨ρ⟩(t) vs t. 7.3.3 Null Case: Non-radially Infal… view at source ↗
Figure 7
Figure 7. Figure 7: Parameters: ρ0 = 0.5, ϕ0 = π/3, σ1 = 0.09, σ2 = 0.05, nmax = mmax = 300, M = 0, m0 = −40, n0 = 0. The plot of ⟨ρ(t)⟩ vs ⟨ϕ(t)⟩ corresponding to this can be found in Appendix G.4. 7.3.4 Null Case: De-localized Wave Packet We now consider the specific parameter choice m0 = 0 and n0 = 0 in the massless (null) limit, which yields a strictly real wave packet profile. One consequence of reality9 is that d⟨ρˆ⟩ dt… view at source ↗
Figure 8
Figure 8. Figure 8: Parameters: ρ0 = 1.2, ϕ0 = π/3, σ1 = 0.09, σ2 = 0.05, nmax = mmax = 200, M = 0, m0 = 0, n0 = 0 While a rigorous physical justification for this phenomenon is provided in subsequent sections, it fundamentally arises from the existence of a characteristic minimum length scale 34 [PITH_FULL_IMAGE:figures/full_fig_p035_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Energy density distribution. Parameters: [PITH_FULL_IMAGE:figures/full_fig_p040_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Polar plot mapping the trajectory. Parameters: [PITH_FULL_IMAGE:figures/full_fig_p041_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Radial infall delocalization. Parameters: [PITH_FULL_IMAGE:figures/full_fig_p042_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: ρ vs t plot for massive radial infall showing significant deviation from the classical geodesic. While the plots above reflect the stress tensor approach distribution, the behavior remains consistent for the position operator as well. Ultimately, these results confirm that wave packet dynamics in this spacetime are governed by the interplay between spatial width and the characteristic length scale 1/E. It… view at source ↗
Figure 13
Figure 13. Figure 13: Operator formalism with parameters: ρ0 = 0.50, ϕ0 = π, σ1 = 0.09, σ2 = 0.06, nmax = mmax = 80, M = 30, m0 = −30, n0 = 0, C = −0.49347, D = −0.83922 [PITH_FULL_IMAGE:figures/full_fig_p047_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Center of mass formalism with parameters: [PITH_FULL_IMAGE:figures/full_fig_p047_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Parameters: ρ0 = 0.5, ϕ0 = π/3, σ1 = 0.09, σ2 = 0.05. In plots (a)–(d), the nu￾merically determined wave packet coordinates (¯ρ(t), ϕ¯(t)) (blue dots) are compared directly against the analytical solution of the geodesic equation (red dashed curve). As illustrated in the figures above, varying the mass M produces a range of ellipse-like trajectories, including circular orbits. Fix M, Vary m0 [PITH_FULL_I… view at source ↗
Figure 16
Figure 16. Figure 16: Parameters: ρ0 = 0.5, ϕ0 = π/3, σ1 = 0.09, σ2 = 0.05. In plots (a)–(d), the nu￾merically determined wave packet coordinates (¯ρ(t), ϕ¯(t)) (blue dots) are compared directly against the analytical solution of the geodesic equation (red dashed curve). We clearly see similar behavior as seen in the previous case with m0 = constant. F.2 Massive Case: Radial Infall [PITH_FULL_IMAGE:figures/full_fig_p071_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Parameters: ρ0 = 1.2, ϕ0 = π/3, σ1 = 0.09, σ2 = 0.05. F.3 ρ¯(t) vs t: Massive Case To further characterize the dynamics of these trajectories, we now examine the time evolution of the radial coordinate’s centroid ρ¯(t) as a function of the coordinate time t for each of the scenarios discussed above in [PITH_FULL_IMAGE:figures/full_fig_p072_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Parameters: ρ0 = 1.2, ϕ0 = π/3, σ1 = 0.09, σ2 = 0.05 As observed in these plots, the trajectory of the wave packet deviates slightly from the exact geodesic as it approaches the spatial origin (ρ = 0). Notably, this deviation is systematically suppressed as the scalar mass M is increased. The origin of this discrepancy is fundamentally a coordinate artifact: we have chosen the ρ coordinate to define our o… view at source ↗
Figure 19
Figure 19. Figure 19: and 20 illustrate the resulting 2D trajectories in the (ρ, ϕ) plane for various parameter configurations in the null regime [PITH_FULL_IMAGE:figures/full_fig_p074_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Parameters: ρ0 = 1.2, ϕ0 = π/3, σ1 = 0.09, σ2 = 0.05, nmax = mmax = 300,M = 0, m0 = 0, n0 = 200 73 [PITH_FULL_IMAGE:figures/full_fig_p074_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Parameters: ρ0 = 0.45, ϕ0 = π/3, σ1 = 0.09, σ2 = 0.05, nmax = mmax = 200, M = 120, m0 = −30, n0 = 0, E ≈ 154 [PITH_FULL_IMAGE:figures/full_fig_p075_21.png] view at source ↗
Figure 24
Figure 24. Figure 24: Parameters: ρ0 = 1.10, ϕ0 = π/3, σ1 = 0.09, σ2 = 0.05, nmax = mmax = 250, M = 6.5, m0 = −30, n0 = 0, E ≈ 38 [PITH_FULL_IMAGE:figures/full_fig_p076_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: Parameters: ρ0 = 1.20, ϕ0 = π/3, σ1 = 0.09, σ2 = 0.05, nmax = mmax = 350, M = 2.75, m0 = −30, n0 = 0, E ≈ 34.5 75 [PITH_FULL_IMAGE:figures/full_fig_p076_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: below display 2D plots of ⟨ρˆ⟩ and ⟨ϕˆ⟩ for fixed angular momentum m0 = −20 and mass M = 25, 45, 60, 80 and compares them with the corresponding geodesic solutions for various parameter choices. (a) m0 = −20 , M = 25 , E ≈ 53, nmax = mmax = 75 (b) m0 = −20 , M = 45, E ≈ 69, nmax = mmax = 75 (c) m0 = −20, M = 60, E ≈ 83.5, nmax = mmax = 75 (d) m0 = −20, M = 80, E ≈ 103.8, nmax = mmax = 75 [PITH_FULL_IMAGE… view at source ↗
Figure 27
Figure 27. Figure 27: Parameters: ρ0 = 0.5, ϕ0 = π/3, σ1 = 0.09, σ2 = 0.05 As expected, keeping the mass parameter M and other initial conditions fixed while progressively increasing the angular momentum parameter m0 significantly alters the wave packet’s trajectory. Specifically, the orbit transitions from an elliptical like shape, circularizes 77 [PITH_FULL_IMAGE:figures/full_fig_p078_27.png] view at source ↗
Figure 28
Figure 28. Figure 28: Parameters: ρ0 = 1.2, ϕ0 = π/3, σ1 = 0.09, σ2 = 0.05 78 [PITH_FULL_IMAGE:figures/full_fig_p079_28.png] view at source ↗
Figure 29
Figure 29. Figure 29: Parameters: ρ0 = 1.2, ϕ0 = π/3, σ1 = 0.09, σ2 = 0.05 Consistent with the behavior observed for ρ¯(t), a deviation from the exact classical geodesic (of ρ) manifests as the wave packet traverses the spatial origin. This discrepancy originates from the same physical mechanism discussed previously: loosely, when ⟨ρ⟩ is close to zero, the fluctuations are significant. G.4 Null Case: Localized Wave Packet Here… view at source ↗
Figure 30
Figure 30. Figure 30: Parameters: ϕ0 = π/3, σ1 = 0.09, σ2 = 0.05, nmax = mmax = 300 Left: M = 0, m0 = 0, n0 = 200, ρ0 = 1.2, E ≈ 200, Right: M = 0, m0 = −40, n0 = 0, ρ0 = 0.5, E ≈ 83.5 G.5 ⟨ρˆ⟩ vs t: Null Case We now examine the evolution of ⟨ρˆ⟩ with t for a radially infalling massless scalar with large energy E. The solid blue line is the line with slope ±1, which is the classical path [PITH_FULL_IMAGE:figures/full_fig_p081… view at source ↗
Figure 31
Figure 31. Figure 31: Parameters: ρ0 = 1.2, ϕ0 = π/3, σ1 = 0.09, σ2 = 0.05, nmax = mmax = 300,M = 0, m0 = 0, n0 = 200 Now we also see that there is slight offset at near the boundary ρ = π 2 . Operationally this is because the finite spatial extent of the wave packet causes its leading tail to interact with the AdS boundary before its geometric center arrives. This boundary interaction induces reflection, which accounts for th… view at source ↗
Figure 32
Figure 32. Figure 32: Parameters: ρ0 = 0.45, ϕ0 = π/3, σ1 = 0.09, σ2 = 0.05, nmax = mmax = 200, M = 117, m0 = −30, n0 = 0, E ≈ 151 [PITH_FULL_IMAGE:figures/full_fig_p082_32.png] view at source ↗
Figure 34
Figure 34. Figure 34: Parameters: ρ0 = 0.90, ϕ0 = π/3, σ1 = 0.09, σ2 = 0.05, nmax = mmax = 200, M = 16, m0 = −30, n0 = 0, E ≈ 47.8 [PITH_FULL_IMAGE:figures/full_fig_p082_34.png] view at source ↗
Figure 36
Figure 36. Figure 36: Parameters: ρ0 = 1.20, ϕ0 = π/3, σ1 = 0.09, σ2 = 0.05, nmax = mmax = 350, M = 2.5, m0 = −30, n0 = 0, E ≈ 34.3, H Flat Space Geodesics from Wave Packets H.1 Stress Tensor In this section, we shall show that the center of mass follows the geodesic equation in flat space (i.e straight line) for any choice of packet. In flat space (Minkowski metric), equation (A.3) becomes, x¯ σ = R dV xσ ⟨T tt⟩ R dV ⟨T tt⟩ (… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper defines a covariant center-of-mass worldline from the first moment of the expectation value of the stress-energy tensor in a general curved spacetime and shows, using only the covariant conservation law ∇_μ ⟨T^{μν}⟩ = 0, that this trajectory obeys the geodesic equation under the monopole (sufficiently localized) approximation. It further constructs position operators from the Klein-Gordon inner product and mode completeness, then builds explicit normalizable single-particle wave packets for a free scalar in global AdS₃ with tunable energy and angular momentum. Analytical and numerical results demonstrate that both the stress-tensor and position-operator prescriptions reproduce the expected radial, circular, and elliptical-like timelike and null geodesics, including a controlled ultra-relativistic crossover from timelike to null behavior, while also identifying precise limits where localization fails. The work closes with a CFT-side discussion relating bulk radial localization to the distribution over global descendants of the dual primary.

Significance. If the central derivations hold, the manuscript supplies a parameter-free, first-principles QFT-in-curved-spacetime generalization of the Mathisson-Papapetrou-Dixon framework that recovers geodesic motion directly from stress-tensor conservation. The explicit, normalizable wave-packet constructions in AdS₃, together with both analytic limits and numerical tracking of geodesics, furnish concrete, falsifiable benchmarks for semiclassical regimes and holographic interpretations. The identification of controlled breakdown conditions and the ultra-relativistic crossover further strengthens the result's utility for understanding when the geodesic picture remains valid.

minor comments (2)
  1. The numerical demonstrations of wave-packet trajectories would benefit from an explicit statement of the discretization scheme, convergence criteria, and quantitative error measures used to establish agreement with the analytic geodesics.
  2. A short paragraph clarifying the precise sense in which the monopole truncation error is controlled for the AdS₃ wave packets (e.g., via moments of the stress-tensor support) would make the transition from the general-spacetime derivation to the concrete examples more transparent.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their detailed and positive summary of the manuscript, as well as for the recommendation of minor revision. We appreciate the recognition of the central results on the stress-tensor center-of-mass trajectory and the position-operator construction in AdS3.

Circularity Check

0 steps flagged

No circularity: derivations follow directly from conservation laws and standard QFT constructions

full rationale

The paper's central claim defines the center-of-mass worldline from the first moment of ⟨T^{μν}⟩ and derives the geodesic equation solely from its covariant conservation ∇_μ⟨T^{μν}⟩=0 under the monopole truncation; this is a direct algebraic consequence of the given definition and the Bianchi identity for the stress tensor, with no fitted parameters or self-referential inputs. The second construction uses the standard Klein-Gordon inner product and mode completeness to define position operators whose expectation values are then computed in explicit normalizable wave packets whose dynamics are compared against independently solved geodesic equations in AdS₃. No self-citation chains, ansätze, or uniqueness theorems imported from prior author work appear in the load-bearing steps; the CFT-side statement on global descendants is a direct consequence of the bulk mode expansion rather than a renaming or circular reduction. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard axioms of QFT in curved spacetime including stress tensor conservation and the Klein-Gordon equation for free fields, plus the assumption of localized one-particle states; no free parameters or invented entities are introduced.

axioms (2)
  • standard math Conservation of the stress-energy tensor: ∇_μ T^{μν} = 0
    Invoked directly to derive the geodesic equation for the center-of-mass trajectory from the expectation value.
  • domain assumption Klein-Gordon equation and mode completeness for the scalar field
    Basis for defining position operators via the inner product and constructing the wave packets.

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