On the quantum statistical relation and the covariant phase space Hamiltonian

Jos\'e Luis V. Cerdeira; Tom\'as Ort\'in

REVIEW

On the quantum statistical relation and the covariant phase space Hamiltonian

José Luis V. Cerdeira, Tomás Ortín

High Energy Physics - Theory · Submitted 9 Oct 2026

Pith's verdict: Solid extension of Harlow-Wu to gauge theories that gives a clean closed-form Hamiltonian and connects to the Euclidean path integral; the main results are real, the limitations are honestly stated, and the stress-test concern is largely mitigated by the paper's own caveats.

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arXiv ID2610.13377 (version 2)

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CitationJos\'e Luis V. Cerdeira, Tom\'as Ort\'in. "On the quantum statistical relation and the covariant phase space Hamiltonian." arXiv:2610.13377 (2026). https://pith.science/paper/PBHYLCXW

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The paper delivers what it promises: a closed-form expression for the covariant phase space Hamiltonian in diff-invariant theories with gauge symmetries, generalizing Harlow and Wu. The central formula (3.12b) is new and the derivation is methodical. The connection to the Euclidean path integral and the quantum statistical relation is exected but worked out explicitly, and the examples (GR, Einstein-Maxwell, 5D supergravity with Chern-Simons) show the machinery works in nontrivial cases. The limitations are honestly acknowledged: the construction asumes the existence of boundary terms ℓ, D, R satifying the consistency and gauge-invariance conditions; these are demonstrated for the examples but not proved for arbitrary theories. That is a genuine soft spot, but it is proportional to the scope of the paper — they are giving a recipe, not a theorem for all possible Lagrangians. The stress-test concern about δ_ξ φ not respecting Dirichlet boundary conditions for general diffeomorphisms is partly answered by the paper's own dislaimer that only at the boundary the Hamiltonian acts as a generator (comment 5 after Eq. (3.13)). For non-Killig diffeomorphisms the consistency condition is used at the boundary, where the boundary conditions apply to the variation δφ, not specifically to δ_ξ φ. The paper does not claim the Hamiltonian is a generator away from the boundary for arbitrary ξ, so the concern is less damaging than it first seems. Still, the step from (3.10) to (3.12b) via (2.26) would benefit from a brief remark that δ_ξ φ is allowed because δ_ξ φ on the boundary is not required to vanish, but the condition (2.26) is applied to the variation δ_ξ φ for which the boundary conditions (Dirichlet) may not be satisfied; the paper uses (2.26) for the particular variation δ_ξ φ, which is not a Dirichlet-rompatible variation unless ξ is a boundary-preserving vector field. The examples only use Killing vectors, so this gap does not affect the main applications, but it weakens the claim of generality for arbitrary diffeomorphisms. This is a minor but honest issue. Overall the paper is careful, well-referenced, and contributes a useful general formula. Who is this for? Researchers working in black-hole thermodynamics, covariant phase space, or semiclassical gravity. A serious referee can handle the few fine points. I would recommend acceptance after minor revision addressing the boundary-condition issue for generic ξ. I would cite this in my own work on black-hole charges.

Referee report

Major objections (1)

[§3, Eq. (3.10)–(3.12b)] The derivation of the closed-form Hamiltonian uses the consistency condition (2.26) for the varition δ_ξ φ on the boundary. The consistency condition is required to hold only for variations δφ that respect the Dirichlet boundary conditions (typically fixing the fields on ∂M). For a general diffeomorphism vector ξ, the variation δ_ξ φ = –£_ξ φ + δ_{Λ_ξ} φ does not in general vanish on the boundary and may violate the Dirichlet conditions. The paper does not restrict ξ to preserve the boundary (as in Harlow–Wu), nor does it argue that (2.26) remains valid for such δ_ξ φ. Hence the step from Eq. (3.10) to (3.12b) lacks justification for arbitrary ξ. This gap does not affect the subsequent black‑hole applications (where only Killing vectors are considered), but it weakens the claim that H[ξ] exists for an arbitrary diffeomorphism in an arbitrary diff‑invariant theory with Dirichlet conditions. The authors should either restrict the claim to ξ that preserve the boundary conditions, or provide a proof that δ_ξ φ respects the required boundary conditions for the boundary terms ℓ and D chosen.

Minor objections (5)

[§2.2, Eq. (2.26)] The footnote 11 mentions that the consistency condition need not be imposed over timelike regions; it would be helpful to clarify whether this affects the use of (2.26) for spatial boundaries in the Hamiltonian derivation.
[§3, Eq. (3.16) and discussion] The expression for dH[ξ] on the boundary is given as –i_ξ (L+dℓ). It would be useful to state explicitly that this is an on-shell identity (the derivation uses E_φ=0 earlier).
[§3.1.3, text after Eq. (3.65)] The sentence 'the second term in that line is the electric charge of this theory times the momentum map' could be misinterpreted; consider rewording to clarify which term is referred to.
[§5, Eq. (5.7)] The step from IE = β ∫_∞ H[k] – β ∫_H H[k] to the QSR assumes that the Hamiltonian integrals compute M–ΩJ+Φ_∞Q and TS+Φ_H Q respectively. A brief justification that the Dirichlet boundary conditions indeed yield these combinations (rather than, e.g., the bare Komar charge) would enhance readability.
[Appenix C, Eq. (C.1f)] The transformation of Q_NW under the ambiguities is given as Q_NW + X(φ,δ_ξ φ) + Y[Λ_ξ] + dU[ξ]. It may be worth noting that the combination –Q_NW + i_ξ ℓ is then invariant, as implied by the subsequent claim.

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The paper

The paper claims

A closed-form expression for the covariant phase space Hamiltonian in any diffeomorphism-invariant gauge theory, using boundary terms that ensure a well-posed variational principle and exact gauge invariance.

The reading

This paper presents a closed-form expression for the covariant phase space Hamiltonian that generates diffeomorphisms in any field theory with gauge symmetries, provided the action is supplemented with boundary terms that make the variational problem well-posed and the action exactly gauge invariant. The Hamiltonian is given by $H[\xi] = -Q_{\rm NW}[\xi] + i_\xi \ell + D[\phi, \delta_\xi \phi] - R[\Lambda_\xi]$, where the terms are constructed from the Noether–Wald charge, boundary term $\ell$, and auxiliary forms $D$ and $R$ that encode the consistency and gauge-invariance conditions. Using this, the on-shell Euclidean action of a stationary black hole can be written as an integral of the Hamiltonian over a $(d-2)$-dimensional boundary, reproducing the quantum statistical relation that connects free energy, temperature, entropy, and charges. The paper works out explicit examples: General Relativity, Einstein–Maxwell theory, and minimal five-dimensional supergravity with a Chern–Simons term.

What carries the argument

The central object is the covariant phase space Hamiltonian $H[\xi]$ in Eq. (3.12b), which generates diffeomorphisms. Its $\delta$-integrability relies on two conditions on the boundary: the consistency condition (2.26) that ensures the presymplectic potential satisfies $\Theta(\phi,\delta\phi)+\delta\ell \stackrel{\partial M}{=} dD(\phi,\delta\phi)$, and the exact gauge invariance condition (2.28) $\delta_\Lambda \ell - \Xi[\Lambda] = dR[\Lambda]$. The Noether–Wald charge $Q_{\rm NW}[\xi]$ encodes the diffeomorphism symmetry, and for theories with Chern–Simons terms the paper uses transgression forms to achieve exact gauge invariance.

Extended reading notes

Core claim

The paper proves that the covariant phase space Hamiltonian can be $\delta$-integrated to a closed form for any diffeomorphism-invariant field theory with gauge symmetries, provided the action is supplemented with boundary terms $\ell$ and $D(\phi,\delta\phi)$ satisfying the consistency condition $\Theta(\phi,\delta\phi)+\delta\ell \stackrel{\partial M}{=} dD(\phi,\delta\phi)$ and the exact gauge invariance condition $\delta_\Lambda \ell - \Xi[\Lambda] = dR[\Lambda]$. The Hamiltonian $H[\xi] = -Q_{\rm NW}[\xi] + i_\xi \ell + D[\phi, \delta_\xi \phi] - R[\Lambda_\xi]$ generates diffeomorphisms, and its integral over a $(d-2)$-dimensional boundary gives the on-shell Euclidean action. For stationary black holes, the Hamiltonian evaluated at infinity yields the combination of conserved charges $M - \Omega J + \Phi_\infty Q$, and at the horizon yields $TS + \Phi_H Q$, leading directly to the quantum statistical relation $I_E = \beta(M - \Omega J - \mu Q) - S$ without gauge-fixing.

Load-bearing premise

The paper assumes that for any given diffeomorphism-invariant gauge theory, there exist boundary terms ℓ and D(φ, δφ) that satisfy the consistency condition (2.26) and the exact gauge invariance condition (2.28) for the chosen boundary conditions, so that the Hamiltonian can be δ-integrated.

Editorial extensions

If this is right

The closed-form Hamiltonian provides a systematic prescription for computing conserved charges and thermodynamics for any diffeomorphism-invariant gauge theory, including those with Chern–Simons terms.
The Euclidean on-shell action for stationary black holes is expressed as a boundary integral of the Hamiltonian, giving a direct path from the action to the quantum statistical relation without gauge-fixing or ad-hoc normalization.
The construction clarifies the essential role of boundary terms that guarantee well-posed variational problems and exact gauge invariance, connecting classical and quantum properties of the action.
The method applies to a wide class of theories, including those with higher-form gauge fields and non-Abelian symmetries, and can be extended to other boundary conditions (Neumann, mixed).
The gauge invariance of the Hamiltonian charge up to total derivatives ensures that physical charges are well-defined and path-independent on the boundary.

Reading between the lines

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

The closed form may enable the study of non-stationary or dynamical black-hole processes, where the Hamiltonian charge for arbitrary diffeomorphisms could be used to define local or flux-balance laws.
In holographic contexts, the Hamiltonian boundary term directly computes the grand-canonical free energy, suggesting a covariant approach to holographic renormalization for gauge-gravity duals with gauge fields.
The explicit construction for 5D minimal supergravity provides a template for handling Chern–Simons terms in supergravity theories, potentially extending to higher dimensions or more complicated gaugings.
The framework likely extends to higher-derivative gravity theories, the cosmological constant, and scalar fields with gauge symmetries, as the authors note in their conclusions, but the technical complexity of boundary terms in such cases remains a challenge.

Original abstract

We show how the covariant phase space Hamiltonian can be constructed in theories in which some of the fields have gauge freedoms. The addition to the action of boundary terms that guarantee the well-posedness of a variational principle and its exact gauge invariance for the prescribed boundary conditions turn out to be essential in this construction. The same ingredients are essential in the use of the action in the framework of Gibbons and Hawking's Euclidean approach to Quantum Gravity and we show how the on-shell action constructed with a Lagrangian with those properties can be expressed in terms of the covariant phase space Hamiltonian. We compute the Hamiltonian of several theories, one of which has Chern--Simons terms, using our prescription and show that, in black-hole backgrounds, the Hamiltonian gives directly the quantum statistical relation including terms related to non-gravitational charges and chemical potentials, as expected.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-12: 1 request, 3,813 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.10 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.47 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We show how the covariant phase space Hamiltonian can be constructed in theories in which some of the fields have gauge freedoms. The addition to the action of boundary terms that guarantee the well-posedness of a variational principle and its exact gauge invariance for the prescribed boundary conditions turn out to be essential in this construction. The same ingredients are essential in the use of the action in the framework of Gibbons and Hawking's Euclidean approach to Quantum Gravity and we show how the on-shell action constructed with a Lagrangian with those properties can be expressed in terms of the covariant phase space Hamiltonian. We compute the Hamiltonian of several theories, one of which has Chern--Simons terms, using our prescription and show that, in black-hole backgrounds, the Hamiltonian gives directly the quantum statistical relation including terms related to non-gravitational charges and chemical potentials, as expected.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We show how the covariant phase space Hamiltonian can be constructed in theories in which some of the fields have gauge freedoms.	contribution 0.99	0.94	0.49	asserted 1.00
s1	The addition to the action of boundary terms that guarantee the well-posedness of a variational principle and its exact gauge invariance for the prescribed boundary conditions turn out to be essential in this construction.	method 0.62	0.76	0.11	asserted 1.00
s2	The same ingredients are essential in the use of the action in the framework of Gibbons and Hawking's Euclidean approach to Quantum Gravity and we show how the on-shell action constructed with a Lagrangian with those properties can be expressed in terms of the covariant phase space Hamiltonian.	contribution 0.58	0.89	0.19	asserted 1.00
s3	We compute the Hamiltonian of several theories, one of which has Chern--Simons terms, using our prescription and show that, in black-hole backgrounds, the Hamiltonian gives directly the quantum statistical relation including terms related to non-gravitational charges and chemical potentials, as expected.	result 0.96	0.96	0.47	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.63 · evidence from proof or derivation 1.00.

What was examined, and what was not

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arXiv listing abstract v1

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S sections
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Discussion

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