Post-Newtonian Hamiltonian description of an atom in a weak gravitational field

Domenico Giulini; Philip K. Schwartz

REVIEW 1 major objections 5 minor 31 references

Post-Newtonian Hamiltonian description of an atom in a weak gravitational field

T0 review · 1 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read An electromagnetically bound atom in a weak gravitational field behaves, to order $c^{-2}$, as a single point particle whose mass includes its internal energy.

desk verdict A careful post-Newtonian Hamiltonian derivation for a composite atom in weak gravity; the central mass-energy claim is real but scoped to the constant-potential approximation, and the paper is transparent about that scope. read the letter →

arxiv 1908.06929 v2 pith:DNDTYV6P submitted 2019-08-19 quant-ph gr-qcphysics.atom-ph

Philip K. Schwartz , Domenico Giulini This is my paper · ORCID

classification quant-phgr-qcphysics.atom-ph MSC 83C1081Q0583C25 PACS 04.25.Nx03.65.-w

keywords post-Newtonianexpansionmass-energyequivalencecompositepointparticlePPNmetricatominterferometryprinciplemultipolarHamiltonianweakgravitationalfield

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper derives, from first principles, the post-Newtonian Hamiltonian for a two-particle atom coupled to both external electromagnetic fields and a weak gravitational field. Its central result is that the center-of-mass part of that Hamiltonian is identical, to order $c^{-2}$, to the Hamiltonian of a single point particle whose mass is the total rest mass plus the internal energy divided by $c^2$. This matters because atom interferometry and quantum optics are approaching sensitivities at which such mass-energy corrections are relevant, and previous treatments often stitched relativistic effects onto non-relativistic models rather than deriving them systematically. If the claim is right, the weak-field equivalence principle survives for this composite quantum system, provided physical distances and momenta are measured with the true spacetime metric.

What carries the argument

The argument is carried by three linked objects: the Eddington-Robertson parametrised post-Newtonian metric, which lets the authors track deviations from general relativity through the parameters $\beta$ and $\gamma$; the multipolar centre-of-mass and relative-coordinate Hamiltonian obtained by canonical quantisation and a Power-Zienau-Woolley transformation; and the final transcription of all distances and momenta into the physical spatial metric $(3)g$. The load-bearing identity is equation (5.8), $H_{C,\mathrm{final}} = H_{\mathrm{point}}(P,R; M + H_{A,\mathrm{final}}/c^2)$, which identifies the composite atom with a single point particle of mass-energy-corrected mass. This identity only emerges after re-expressing terms such as $p_r^2/2\mu$ and the Coulomb interaction in metric-corrected form, showing that the apparent ambiguity between inertial and gravitational mass was an artefact of using the background flat metric rather than the physical one.

What would settle it

Repeat the derivation keeping all terms linear in $r\cdot\nabla\varphi(R)$ through order $c^{-2}$ and check whether the centre-of-mass Hamiltonian still takes the form $H_{\mathrm{point}}(P,R; M + H_{A,\mathrm{final}}/c^2)$; the appearance of any extra coupling between the centre-of-mass momentum $P$ and the internal momentum $p_r$ in that limit would falsify the composite-point-particle claim as stated.

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Extended reading notes

Core claim

Starting from a Poincaré-invariant classical two-particle action, the authors include gravity through the minimal coupling scheme, expand in the Eddington-Robertson parametrised post-Newtonian metric to first order in $c^{-2}$, eliminate the internal electromagnetic field degrees of freedom, canonically quantise, and pass through a Power-Zienau-Woolley transformation to a multipolar Hamiltonian. The final center-of-mass Hamiltonian is $H_{C,\mathrm{final}} = H_{\mathrm{point}}(P,R; M + H_{A,\mathrm{final}}/c^2)$, where $H_{\mathrm{point}}$ is the PPN point-particle Hamiltonian and $H_{A,\mathrm{final}}$ is the internal Hamiltonian. The paper states that the system therefore behaves as a composite point particle whose inertial and gravitational mass is the rest mass of the constituents plus the internal energy divided by $c^2$. A key interpretive step is rewriting the internal kinetic and Coulomb terms using the physical spatial metric; without that step, one would find different inertial and gravitational masses and might wrongly infer a violation of the weak equivalence principle.

Load-bearing premise

The gravitational potential $\varphi$ is assumed constant over the extension of the atom, so all terms involving its gradient $r\cdot\nabla\varphi(R)$ are neglected; if tidal variations across the atom are not negligible, the clean composite-point-particle Hamiltonian (5.8) acquires extra couplings and the simple mass-energy identification is lost.

Editorial extensions

If this is right

Atom-interferometry phase calculations can legitimately approximate the atom as a point particle with mass $M + H_{A,\mathrm{final}}/c^2$ up to order $c^{-2}$, lending support to heuristic treatments already used in experimental proposals.
The internal energy levels of an atom in a gravitational field acquire metric-corrected kinetic and Coulomb terms, so internal spectra depend on the local gravitational environment through the physical spatial metric.
Because the parameters $\beta$ and $\gamma$ appear explicitly in the final Hamiltonian, the same derivation can be used to design quantum experiments that distinguish general relativity from PPN test theories.
Once physical metric quantities are used, the apparent difference between inertial and gravitational mass of the composite system disappears, clarifying how weak equivalence principle statements should be formulated for structured quantum objects.
The gravitationally corrected atom-light interaction terms, including metric factors in the Röntgen term and in the electromagnetic field energy, follow from the same systematic derivation rather than from ad hoc additions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A consequence the paper leaves implicit is that the dropped tidal terms $r\cdot\nabla\varphi(R)$ would couple internal and centre-of-mass motion; in strong gravity gradients the clean point-particle form would fail, possibly producing internal-state-dependent accelerations that are testable with matter-wave interferometry.
The same derivation method should extend to molecules or trapped ions, in which case the composite mass would include rotational and vibrational energy, suggesting molecular interferometry as a further test of mass-energy equivalence.
Because the paper deliberately keeps $\beta$ and $\gamma$ free, the resulting Hamiltonian is a ready-made template for quantum tests of general relativity against PPN test theories, provided the constant-potential approximation is satisfied experimentally.
The critical discussion implies that formulations of the quantum equivalence principle based on proper time or worldlines are state-dependent; the geometry-based coupling used here offers a state-independent alternative, but establishing that requires further conceptual work beyond this calculation.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

First: this is a genuinely useful paper. It takes Sonnleitner and Barnett's machinery, adds an Eddington-Robertson PPN background, and derives the first-order post-Newtonian Hamiltonian for a two-particle atom coupled to both gravity and light. The main results (5.5)-(5.8) are new in their completeness: Marzlin and Lämmerzahl did the electric dipole part, but the gravitational corrections to the electromagnetic field, the modified Coulomb and field-energy prefactors, and the atom-light terms in (5.9) are new. The claim that the centre of mass moves like a point particle of mass M plus H_A/c^2 agrees with the independent calculation by Zych, Rudnicki, and Pikovski, which is a good cross-check. The paper deserves credit for spelling out its assumptions: minimal coupling, c^-2 truncation, neglect of r·∇φ, operator ordering, and dropped backreaction terms. They also correct sign errors in the Sonnleitner-Barnett appendix, which is careful housekeeping.

Soft spots: the stress-test note is right but not fatal. The composite point-particle form (5.8) only holds when the gravitational potential φ is constant over the atom's extension. Keeping the r·∇φ terms adds centre-of-mass/internal couplings that spoil the clean point-particle form. The authors say this explicitly in the text, but the abstract and parts of the discussion do not foreground the qualification, so a casual reader will overstate the result. That is a presentation issue, not a calculation error.

Second, the central result is derived from minimal coupling, which already encodes the equivalence principle for the constituent particles. So the conclusion that the composite behaves mass-energy equivalently is not independent confirmation of the equivalence principle; it is a derivation of what minimal coupling implies for a composite in the PN limit. The authors are aware of this and discuss it in Section 6. It is fine, but it tempers the 'controlled derivation' framing.

Third, the new atom-light couplings in (5.9) are expressed in coordinate components; rewriting them in tetrad components introduces operator ordering ambiguities in the Röntgen term, which the authors handle pragmatically. Minor.

Net: the central claim is sound within its stated scope. The paper is a solid reference for atom interferometry and quantum tests of general relativity. It deserves peer review and publication. I would cite it if I were working on relativistic corrections in quantum systems, and I might bring it to a reading group that cares about Hamiltonian reduction subtleties.

Referee Report

1 major / 5 minor

Summary. The paper derives a post-Newtonian Hamiltonian for an electromagnetically bound two-particle system (an 'atom') in a weak parametrized post-Newtonian gravitational background, extending the calculation of Sonnleitner and Barnett to include gravity. Starting from the PPN metric, the authors apply minimal coupling to the two-particle Darwin Lagrangian, include the gravitational modification of the internal and external electromagnetic fields, perform canonical quantization and a PZW transformation, and express the result in centre-of-mass and relative coordinates. The central result, Eqs. (5.5)-(5.8), is that the centre-of-mass Hamiltonian takes the form of a point particle with mass M + H_A/c^2, where H_A is the internal Hamiltonian, provided the gravitational potential is constant over the atom's size. The paper also discusses the interpretation in terms of the physical metric and comments on formulations of the equivalence principle in quantum mechanics.

Significance. If the result holds, it provides a systematic first-principles derivation of the composite point-particle picture for atoms in gravitational fields, including mass-energy corrections, and clarifies the role of the physical metric in defining inertial and gravitational mass. The explicit dependence on the PPN parameters beta and gamma makes the result useful for future quantum tests of gravity. The paper carefully handles operator ordering, truncation, and tetrad components, and it corrects sign errors in the earlier work it extends. The main limitation, that the gravitational potential is assumed constant over the atom's extension, is stated explicitly in the text, which makes the scope of the central claim clear.

major comments (1)

[Section 5.2, Eq. (5.8)] The composite point-particle form is derived under the assumption, stated in Sections 3.3, 4.1, and 5.1, that phi is constant over the extension of the atom. If the terms proportional to r·∇φ(R) in Eqs. (3.9), (3.12), and (3.14) are retained, H_C acquires O(c^-2) couplings between P and internal variables and cannot be written as H_point(P,R; M + H_A/c^2). The manuscript does flag this limitation, but the abstract and the unqualified sentence 'the system behaves as a composite point particle' should explicitly carry the qualifier 'in a homogeneous (constant-phi) field' to prevent over-reading of the headline claim.

minor comments (5)

[Section 4.1, Eqs. (4.4)-(4.13)] The notation for the current density is confusing because j0 is used for both the contravariant and covariant components; the relation between j^0 and j_0 should be stated with distinct symbols. The final result (4.13) is correct, but the derivation is hard to follow.
[Section 2.2] The statement that the calculation includes terms up to O(c^-2) should be reconciled with the presence of phi^2/c^4 terms in the PPN metric (2.1); a sentence explaining that phi/c^2 is the expansion parameter and that the phi^2/c^4 metric terms contribute at order c^-2 in the Hamiltonian would avoid confusion.
[Section 5.2, Eq. (5.8)] Because H_A,final in Eq. (5.6) contains O(c^-2) terms, namely the p^4 and Darwin corrections, their contribution to the centre-of-mass Hamiltonian through M + H_A,final/c^2 would be O(c^-4). An explicit remark that only the leading, O(c^0) part of H_A,final enters H_C at the working order would make the equality precise.
[Eq. (4.27)] The prefactor (1-(gamma+1)phi/c^2) on the Darwin term is written in the first line and then dropped in the second line; a parenthetical explanation that this correction is O(c^-4) and hence omitted at the chosen order would improve clarity.
[Section 5.3, Eq. (5.15)] The definition of the physical dipole moment via d^a_phys = e^a_b d^b should state that e^a_b are the tetrad components of the identity transformation, to avoid confusing coordinate and physical components.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the composite point-particle Hamiltonian (5.8) is obtained by explicit post-Newtonian calculation from stated minimal-coupling and constant-field assumptions, with no fitted parameters and no self-citation chain forcing the result.

full rationale

The paper's central claim, Eq. (5.8), is derived by algebraically rearranging the explicitly computed Hamiltonians (5.5)-(5.6) and comparing them with the single-particle PPN Hamiltonian (5.7). No parameter is fitted to data, and no output quantity is used to define an input. The starting points are stated assumptions: the classical relativistic two-particle Lagrangian, the PPN metric (2.1), the minimal coupling scheme, and canonical quantization. The composite point-particle form is a nontrivial consequence of the calculation rather than an ansatz. The paper's caveat that the interpretation depends on rewriting kinetic and Coulomb terms with the physical spatial metric is transparency about an interpretive step, not a circular reduction. The only self-citation, ref. [19], motivates the symmetric operator ordering p·φp; under the paper's explicit constant-φ approximation, in which terms proportional to r·∇φ(R) are neglected before Eq. (5.3), ordering ambiguities of this type drop out, so that citation is not load-bearing. The constant-φ approximation is a scope condition limiting the result to homogeneous fields, and the paper states it explicitly; this affects generality, not circularity. Therefore no circular step can be exhibited, and the appropriate finding is no significant circularity.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

All assumptions are explicitly stated in the paper. The central calculation depends on the minimal coupling scheme, the PPN metric with symbolic β and γ, order c^-2 truncation, and the approximation that φ is constant over the atom. No numerical fitting is performed. β and γ are inputs that allow specializing to GR (β=γ=1) or test theories.

free parameters (1)

PPN parameters β and γ = Not fitted; GR corresponds to β=γ=1
Introduced in (2.1) to parametrize the PPN metric. They are kept symbolic so the results can be specialized to general relativity or to test theories; the central conclusion (5.8) holds for all values.

assumptions (6)

domain assumption Minimal coupling scheme: the Minkowski metric is replaced by the PPN metric, and partial derivatives by Levi-Civita covariant derivatives
Stated in the introduction as the foundational principle for coupling matter to gravity; the derivation of all gravitational corrections is based on it.
domain assumption The PPN metric (2.1) with parameters β, γ describes a weak gravitational field to first post-Newtonian order, with φ/c^2 small
The expansion of the metric defines the approximation scheme, and all results are truncated at order c^-2.
domain assumption The gravitational potential φ is constant over the extension of the atom, so terms proportional to r·∇φ(R) are neglected
Assumed in Section 4.1 and used to drop terms in (3.12), (3.14), and Section 5.1; the paper explicitly states 'we assumed φ to be constant over the extension of the atom'.
domain assumption Canonical quantization, the PZW transformation, and the electric dipole approximation remain valid for the gravitational correction terms
Inherited from [1] and applied in Sections 3.2 and 5.1; the action of the PZW operator on the new terms is stated to reduce to \bar p_i → p_i at this order.
standard math The electromagnetic dynamics is described by the standard minimal-coupled Maxwell action (4.1) and the fixed-particle-sector Darwin Lagrangian from [1]
Takes [1] as starting point and the standard EM action in curved spacetime [20-22] as input.
domain assumption The system is below the pair-production threshold and the Hamiltonian formalism is valid; the no-interaction theorem is avoided by working to order c^-2
Stated in the introduction and conclusion; limits the validity of the Hamiltonian approach.

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Cite this review

Pith. "Pith review of Post-Newtonian Hamiltonian description of an atom in a weak gravitational field." pith.science (2026). https://pith.science/paper/DNDTYV6P

@misc{pith2026190806929,
  author       = {Pith},
  title        = {Pith review of: Post-Newtonian Hamiltonian description of an atom in a weak gravitational field},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DNDTYV6P}},
  note         = {Machine review of arXiv:1908.06929}
}

read the original abstract

We extend the systematic calculation of an approximately relativistic Hamiltonian for centre of mass and internal dynamics of an electromagnetically bound two-particle system by Sonnleitner and Barnett [1] to the case including a weak post-Newtonian gravitational background field, described by the Eddington--Robertson parametrised post-Newtonian metric. Starting from a proper relativistic description of the situation, this approach allows to systematically derive the coupling of the model system to gravity, instead of `guessing' it by means of classical notions of relativistic effects. We embed this technical result into a critical discussion concerning the problem of implementing and interpreting general couplings to the gravitational field and the connected problem of how to properly address the question concerning the validity of the Equivalence Principle in Quantum Mechanics.

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Mass-energy and anomalous friction in quantum optics,

M. Sonnleitner and S. M. Barnett, “Mass-energy and anomalous friction in quantum optics,” Phys. Rev. A98, 042106 (2018)

work page 2018

[2] [2]

General relativistic eﬀects in atom interferometry,

S. Dimopoulos, P. W. Graham, J. M. Hogan, and M. A. Kasevich, “General relativistic eﬀects in atom interferometry,” Phys. Rev. D78, 042003 (2008)

work page 2008

[3] [3]

Quantum interferometric visibility as a witness of general relativistic proper time,

M. Zych, F. Costa, I. Pikovski, and Č. Brukner, “Quantum interferometric visibility as a witness of general relativistic proper time,” Nat. Commun.2, 505 (2011)

work page 2011

[4] [4]

Universal decoherence due to gravitational time dilation,

I. Pikovski, M. Zych, F. Costa, and Č. Brukner, “Universal decoherence due to gravitational time dilation,” Nat. Phys.11, 668–672 (2015)

work page 2015

[5] [5]

Gravitational redshift in quantum-clock interferometry

A. Roura, “Gravitational redshift in quantum-clock interferometry,” (2018), arXiv:1810.06744 [physics.atom-ph]

work page Pith review arXiv 2018

[6] [6]

Proper time in atom interferometers: Diﬀractive versus specular mirrors,

E. Giese, A. Friedrich, F. Di Pumpo, A. Roura, W. P. Schleich, D. M. Greenberger, and E. M. Rasel, “Proper time in atom interferometers: Diﬀractive versus specular mirrors,” Phys. Rev. A99, 013627 (2019)

work page 2019

[7] [7]

Interference of clocks: A quantum twin paradox,

S. Loriani, A. Friedrich, C. Ufrecht, F. Di Pumpo, S. Kleinert, S. Abend, N. Gaaloul, C. Meiners, C. Schubert, D. Tell, E. Wodey, M. Zych, W. Ertmer, A. Roura, D. Schlippert, W. P. Schleich, E. M. Rasel, and E. Giese, “Interference of clocks: A quantum twin paradox,” Sci. Adv.5, eaax8966 (2019)

work page 2019

[8] [8]

Gravitational mass of composite systems,

M. Zych, Ł. Rudnicki, and I. Pikovski, “Gravitational mass of composite systems,” Phys. Rev. D99, 104029 (2019)

work page 2019