{"id":"79b6d446-5bc7-40d5-af7e-89b971c17f0e","arxiv_id":"2411.09748","paper_version":3,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A canonical analysis of unimodular Plebański gravity shows that the Hamiltonian density is constrained only to be constant, promoting the cosmological constant to an integration constant.","lead":"The paper derives the Hamiltonian version of unimodular gravity using Plebański's spin-connection variables, with either a fixed background volume form or a dynamical field. The main result is that the Hamiltonian density is not required to vanish but only to be a constant, so the cosmological constant becomes an integration constant instead of a fixed input.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified","rationale":"The reader's ACCEPT verdict with moderate confidence is appropriate. The central claim—that in preferred-volume unimodular Plebański gravity the Hamiltonian constraint is replaced by the condition that the Hamiltonian density be spatially constant—is well-supported by the derivation of Eq. (36) and the consistency algebra. The weakest assumption identified by the reader, invertibility of the densitized triad, is explicitly stated and restricts the analysis to the non-degenerate sector; it is a limitation rather than a flaw, and the paper does not overclaim applicability to degenerate configurations. The secondary constraint derivation and the closure of the constraint algebra are not shown in full detail, but the steps are standard and internally consistent: the Poisson bracket of the diffeomorphism constraint with the non-constraint Hamiltonian yields K_a, and the preservation of K_a follows from the Hamiltonian constraint algebra. The parametrised Henneaux–Teitelboim analysis is also consistent with known results. No internal contradiction, hidden assumption beyond the stated one, or unsupported leap was found that would change the verdict.","tokens_in":941,"tokens_out":612,"duration_ms":315043,"concrete_test":"Independently compute the Poisson bracket {D(U), ∫ d^3x ~N0 H} in Ashtekar variables to confirm that it equals ∫ d^3x ~N0 U^a ∂_a H, and verify that the bracket {K(T), H(0)} vanishes weakly using the bracket {H(x), H(y)} proportional to the diffeomorphism constraint; if either calculation yields a non-vanishing remainder, the constraint algebra in Sec. IV.A would need revision.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I find no load-bearing flaw in the central claim. The derivation of the secondary constraint K_a = ∂_a H in Eq. (36) follows from the consistency requirement {D(U), H(0)} ≈ 0: since H is a spatial scalar, the bracket is ∫ d^3x ~N0 U^a ∂_a H, and the nowhere-vanishing background density ~N0 implies ∂_a H ≈ 0. The preservation of K_a follows from the standard Hamiltonian constraint algebra, which makes {H, H(0)} weakly zero (proportional to the diffeomorphism constraint). The invertibility assumption on ~E^a_i is explicitly stated after Eq. (7) and confines the analysis to the non-degenerate sector; this is a domain restriction, not an inconsistency, and the central claim holds on that sector. The parametrised-theory analysis is structurally identical to the known Henneaux–Teitelboim result. I therefore see no reason to alter the ACCEPT verdict.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops the canonical (Hamiltonian) analysis of two unimodular versions of chiral Plebanski gravity: a preferred-volume theory with a fixed background volume form, and a parametrised Henneaux-Teitelboim-type theory with a dynamical 3-form T. In the preferred-volume theory the standard Hamiltonian constraint is absent, and demanding consistency with the diffeomorphism constraint produces a secondary constraint K_a = ∂_a H, so that the Hamiltonian density is forced to be spatially constant and the cosmological constant is promoted to an integration constant. In the parametrised theory the trace of the Plebanski matrix becomes a dynamical field λ with conjugate momentum τ~, and a new constraint ∂_a λ ≈ 0 ensures λ is constant. The paper also analyzes reality conditions, the reconstructed metric, and the notion of volume time.","tokens_in":14346,"tokens_out":11993,"duration_ms":112483,"significance":"If the analysis is correct, the paper makes a worthwhile contribution by extending the unimodular-gravity mechanism to the connection-based Plebanski formalism that underlies loop quantum gravity, and by doing so in a careful and explicit way. The derivations are self-contained and parameter-free in the sense that the constraints are obtained from the stated action principles rather than from any fitted input. The discussion of reality conditions and of the four possible metric sectors is particularly useful, and the identification of the volume time in the parametrised formulation is a nice geometric result. The preferred-volume analysis appears to be new, while the parametrised analysis correctly complements and extends earlier work.","major_comments":[],"minor_comments":[{"comment":"The assumption that the densitised triad matrix E~^a_i is invertible is stated, but the paper should explicitly note that this restricts the entire analysis to the non-degenerate sector, so that the conclusions do not cover degenerate triads; the same assumption underlies the inverse triad (8), the Hamiltonian density (17), the reconstructed metric (27), and the reality conditions (25).","section":"Sec. III, after Eq. (7)"},{"comment":"The derivation of the secondary constraint K_a = ∂_a H is presented in a single sentence; since this is the central new result, the authors should display the intermediate bracket computation, including the treatment of the fixed background density N~0 as non-dynamical and the assumption that N~0 is nowhere vanishing, as befits a fixed volume form.","section":"Sec. IV.A, Eq. (36)"},{"comment":"The wording that 'the constraint (17) is replaced by a version in which Λ is a free integration constant' is slightly imprecise; because Eq. (17) itself contains the parameter Λ, it would be clearer to state that the new constraint K_a ≈ 0 forces the Hamiltonian density H to be spatially constant, and that this constant plays the role of the cosmological constant.","section":"Sec. IV.A, text after Eq. (43)"},{"comment":"The claim that the constraints (50)-(53) satisfy the same first-class Poisson algebra as the non-unimodular theory is plausible but not demonstrated; because the diffeomorphism constraint D'_a contains the additional term -τ~ ∂_a λ, the authors should give at least one explicit bracket, such as {D'(V), H'(N)} or {H'(N1), H'(N2)}, to show that the algebra still closes.","section":"Sec. IV.B, after Eq. (49)"},{"comment":"The appendix derives gauge transformations of Lagrange multipliers for the canonical Plebanski action only, while the text says the same formalism applies to all theories considered; the authors should either provide the analogous transformation rules for the new constraints K and J in the unimodular theories or explicitly state that those results are left to Ref. [24].","section":"Appendix A"},{"comment":"There are minor formatting issues, such as the stray space in 'Pleba´ nski' in the title and text, and the inconsistent spelling 'Plebanski' in Ref. [2]; these should be corrected in a final production pass.","section":"Throughout"}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is technically sound and the central claim is well supported. The self-citations to Refs. [10] and [11] are appropriate and do not constitute circular reasoning; the paper extends the Lagrangian constructions of Ref. [10] to a full canonical analysis. The only reservations are local presentation issues, mainly the compressed derivation of the secondary constraint and the scope restriction to invertible triads. I see no need for additional refereeing after the minor revisions are made."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a competent, honest canonical analysis. The genuinely new piece is Section IV.A on the preferred-volume theory, where there is no Hamiltonian constraint initially and consistency with the diffeomorphism constraint instead forces the Hamiltonian density to be spatially constant, giving the cosmological constant as an integration constant. That is the unimodular mechanism in connection variables. The volume-time construction around Eq. (61) is also a real addition, and the reality-condition discussion is more detailed than in Smolin's shorter treatment.\n\nThe derivations are explicit and the constraint algebra as presented is internally consistent. I have not recomputed every Poisson bracket, but the structural checks line up: Gauss and diffeomorphism constraints are unchanged in the preferred-volume case, the new K_a constraint has the expected first-class behavior, and the parametrised theory reduces to the known Henneaux-Teitelboim result with a canonically conjugate pair (lambda, tau-tilde). The paper also flags its own boundary: invertibility of the densitised triad is assumed after Eq. (7), so the degenerate sector is out of scope by declaration.\n\nSoft spots, in proportion. First, the secondary constraint (36) is stated without the intermediate calculation. The stress-test note fills it in: {D(U), H(0)} is proportional to the integral of N0 U^a partial_a H, and since N0 is nowhere vanishing, partial_a H must vanish weakly. The paper should show those two lines rather than leaving the result as a claim. Second, the bracket {K(T), K(L)} in Eq. (38) is plausible but I would like to see the computation, since it is the one non-obvious algebraic step. Third, the analysis deliberately does not cover degenerate triads; that is a domain restriction, not a flaw. None of this threatens the central claim.\n\nThe citation pattern is appropriate. Self-citations to Refs. [10] and [11] point to the authors' own Lagrangian actions and to Smolin's parametrised analysis, and none of that material is doing load-bearing work that would make the conclusions circular.\n\nWho this is for: canonical quantum gravity people, especially those interested in LQG and the problem of time, plus unimodular-gravity people who want the connection-variable version. It deserves a serious referee and, in my view, acceptance after minor revision, with the main request being to expand the derivation of Eq. (36) and one bracket. I would cite it and would bring it to the reading group.","headline":"A straightforward, honest canonical analysis; the preferred-volume part is genuinely new and the central unimodular mechanism survives scrutiny.","tokens_in":14842,"tokens_out":2801,"would_cite":true,"duration_ms":26546,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["83C05","70H45","83C40"],"pacs":["04.20.Fy","04.60.-m"],"model":"deepseek-v4-flash","headline":"In preferred-volume unimodular Plebański gravity, preserving the constraints under time evolution replaces the Hamiltonian constraint with the condition that the Hamiltonian density be spatially constant, turning the cosmological constant…","keywords":["unimodular gravity","Plebański gravity","canonical analysis","Ashtekar variables","cosmological constant","reality conditions","volume-preserving diffeomorphisms"],"falsifier":"Compute the spatial gradient of the Hamiltonian density, $\\partial_a(\\epsilon_{ijk}F^k_{bc}\\tilde{E}^b_i\\tilde{E}^c_j/(2\\det\\tilde{E}))$, on a spatially homogeneous Plebański solution with the fixed background $\\tilde{N}_0$; if it does not vanish identically, the preferred-volume theory would not admit that solution, directly testing the claim that preservation of constraints forces the Hamiltonian density to be constant.","tokens_in":746,"feed_emoji":"🌀","tokens_out":3792,"duration_ms":147007,"temperature":0.7,"pith_summary":"The paper constructs the Hamiltonian formulation of unimodular gravity in the Plebański framework, where the metric is recovered from a set of complex self-dual two-forms and an SO(3) connection. In the fixed-volume version, there is no Hamiltonian constraint initially; requiring the Gauss and diffeomorphism constraints to be preserved under evolution yields the secondary constraint that the Hamiltonian density is spatially constant, with the constant playing the role of the cosmological constant. In the parametrised version, a field $\\lambda$ plays the role of the cosmological constant, is forced to be spacetime constant, and a conjugate field supplies a preferred volume time. The paper argues that both unimodular versions have the same local degrees of freedom as general relativity plus one global degree of freedom, and that extracting Lorentzian solutions requires careful reality conditions.","feed_headline":"Cosmological constant becomes free in unimodular Plebański gravity","feed_subtitle":"Constraint preservation turns the Hamiltonian density into a spatial constant; the parameter Λ drops out of the action.","key_machinery":"The load-bearing object is the densitised triad $\\tilde{E}^a_i = \\tfrac12\\tilde{\\epsilon}^{abc}\\Sigma^i_{bc}$ extracted from the two-forms, together with its inverse $\\tilde{E}^i_a$. All constraints and the reconstructed metric are expressed in terms of these objects, with $\\det\\tilde{E}$ appearing in denominators. The central identity is the Hamiltonian density $\\mathcal{H} = \\frac{\\epsilon_{ijk}F^k_{ab}\\tilde{E}^a_i\\tilde{E}^b_j}{2\\det\\tilde{E}}$, whose spatial gradient becomes the new constraint $K_a$ in the preferred-volume theory; in the parametrised theory the same density is tied to the dynamical field $\\lambda$ through the constraint $H' = \\mathcal{H} - \\lambda \\approx 0$.","core_discovery":"The central discovery is that unimodularity changes the Hamiltonian constraint, not the kinematical variables. In the preferred-volume theory, the lapse is a fixed background density $\\tilde{N}_0$, so the naive Hamiltonian has a genuine non-constraint part. Preservation of the diffeomorphism constraint under this Hamiltonian produces the secondary constraint $K_a = \\partial_a\\left(\\frac{\\epsilon_{ijk}F^k_{ab}\\tilde{E}^a_i\\tilde{E}^b_j}{2\\det\\tilde{E}}\\right)\\approx 0$, which states that the Hamiltonian density $\\mathcal{H}=\\frac{\\epsilon_{ijk}F^k_{ab}\\tilde{E}^a_i\\tilde{E}^b_j}{2\\det\\tilde{E}}$ is constant on each spatial slice. The value of that constant is not fixed by the action and serves as the cosmological constant of the corresponding general-relativity solution. The paper also shows that the parametrised version is equivalent to adding a canonical pair $(\\lambda,\\tilde{\\tau})$ with $\\lambda=\\mathrm{tr}\\,M$ constrained to be constant and $\\tilde{\\tau}$ measuring the spacetime volume between hypersurfaces.","pith_inferences":["The mechanism that replaces a Hamiltonian constraint by a spatial-constancy condition may extend to other connection-based unimodular formulations, not just the Plebański framework, because it relies only on the lapse becoming a fixed background field.","A canonical quantisation based on these variables would need to impose a softened Hamiltonian constraint, $\\hat{K}_a\\Psi=0$ rather than $\\hat{\\mathcal{H}}\\Psi=0$, which could alter the role of time in loop quantum gravity; the paper does not develop this.","The analysis excludes configurations where $\\det\\tilde{E}=0$, so any quantum state built on degenerate triads falls outside the claimed equivalence; extending the formalism to the degenerate sector would require a separate treatment."],"forward_implications":["In the preferred-volume theory, the cosmological constant is no longer a fixed parameter but an integration constant determined by initial data, exactly as in metric unimodular gravity.","The new constraint $K_a$ generates a restricted set of time reparametrisations, corresponding to volume-preserving diffeomorphisms, so the reduced symmetry is consistent with the fixed background volume form.","In the parametrised theory, $\\lambda$ is forced to be constant in spacetime, and $\\tilde{\\tau}$ provides a preferred volume time, giving a physical interpretation to the extra global degree of freedom.","Reality conditions require $\\tilde{N}_0$ to be purely imaginary for Lorentzian solutions, and an additional condition is needed to exclude purely imaginary spacetime metrics.","The constraint algebra remains first class, with the Gauss constraint from the internal SO(3) symmetry and the diffeomorphism constraint as expected for connection variables."],"supporting_citations":[{"why":"Proposed the two unimodular Plebański actions analyzed in this paper.","marker":"[10]"},{"why":"Earlier canonical analysis of the parametrised theory; the present paper extends it and adds reality conditions.","marker":"[11]"},{"why":"Original Henneaux–Teitelboim parametrised unimodular gravity, the source of the volume-time construction.","marker":"[9]"},{"why":"Identified the densitised triad as the canonical momentum in the self-dual two-form formalism.","marker":"[3]"},{"why":"Introduced Ashtekar connection variables and the polynomial constraint structure that the Plebański canonical theory reproduces.","marker":"[4]"},{"why":"Discussion of unimodular gravity and gauging used for comparison with the metric formulation.","marker":"[7]"}],"fun_headline_variants":["Unimodular Plebański gravity: Λ is a constant","Hamiltonian density becomes constant in unimodular Plebański","Λ emerges as integration constant in Plebański unimodular gravity","Canonical analysis shows Λ constant in unimodular Plebański"],"cache_read_input_tokens":17152,"weakest_assumption_plain":"The matrix $\\tilde{E}^a_i$ must be invertible everywhere on the spatial slice, because the inverse triad, the Hamiltonian density, and the reconstructed metric are all undefined when $\\det\\tilde{E}=0$.","fun_headline_variants_meta":{"raw":{"variants":["Unimodular Plebański gravity: Λ is a constant","Hamiltonian density becomes constant in unimodular Plebański","Λ emerges as integration constant in Plebański unimodular gravity","Canonical analysis shows Λ constant in unimodular Plebański"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000552,"raw_usage":{"total_tokens":2604,"prompt_tokens":889,"completion_tokens":1715,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":505,"completion_tokens_details":{"reasoning_tokens":1640}},"tokens_in":505,"tokens_out":1715,"duration_ms":13819,"temperature":1.0,"reasoning_tokens":1640,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T20:22:23.250291+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the spatial gradient of the Hamiltonian density, $\\partial_a(\\epsilon_{ijk}F^k_{bc}\\tilde{E}^b_i\\tilde{E}^c_j/(2\\det\\tilde{E}))$, on a spatially homogeneous Plebański solution with the fixed background $\\tilde{N}_0$; if it does not vanish identically, the preferred-volume theory would not admit that solution, directly testing the claim that preservation of constraints forces the Hamiltonian density to be constant.","supporting_citations":[{"cited_title":"Unimodular Pleba\\'{n}ski Gravity","cited_arxiv_id":"2309.06333","evidence_quote":"Proposed the two unimodular Plebański actions analyzed in this paper."},{"cited_title":"The cosmological con- stant and general covariance,","cited_arxiv_id":null,"evidence_quote":"Original Henneaux–Teitelboim parametrised unimodular gravity, the source of the volume-time construction."},{"cited_title":"Self- dual 2-forms and gravity,","cited_arxiv_id":null,"evidence_quote":"Identified the densitised triad as the canonical momentum in the self-dual two-form formalism."},{"cited_title":"New Variables for Classical and Quantum Gravity,","cited_arxiv_id":null,"evidence_quote":"Introduced Ashtekar connection variables and the polynomial constraint structure that the Plebański canonical theory reproduces."},{"cited_title":"Gravity with more or less gauging","cited_arxiv_id":"1805.11626","evidence_quote":"Discussion of unimodular gravity and gauging used for comparison with the metric formulation."}],"review_version":1}