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A supersymmetric Minkowski vacuum can host transient quasi-de Sitter geometries as excited states.

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2026-08-04 03:34 UTC pith:YQRCZJTB

arxiv 2607.20927 v2 pith:YQRCZJTB submitted 2026-07-23 hep-th gr-qc

Coherent states versus Glauber-Sudarshan States: Bootstrapping, Schwinger-Keldysh Contours and Lefschetz Thimbles

Heliudson Bernardo , Tatsuya Daniel , Keshav Dasgupta , Brayden Hull , Yue Katherine Lei , Yiya Selina Li This is my paper
classification hep-th gr-qc
keywords Glauber-Sudarshan statescoherent statesde Sitter as excited stateSchwinger-Keldysh contourLefschetz thimblestrans-seriesWheeler-DeWitt equationbootstrap equation
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper tries to establish that four-dimensional quasi-de Sitter spacetimes need not be vacua or minima of any effective potential: in any diffeomorphism-invariant theory with an exact supersymmetric Minkowski vacuum and nontrivial metric-interaction sectors, they can be realized as Glauber–Sudarshan excited states over that vacuum. Because the Glauber–Sudarshan insertion is a state-defining deformation rather than a varied source, the construction evades the convexity of the 1PI effective action and the Wilsonian no-go logic for metastable de Sitter EFTs. The paper argues that these states differ from ordinary coherent states—they are continuously driven, time-ordered displacement products rather than one-time initial data—and shows how their expectation values are computed consistently on Schwinger–Keldysh contours, organized as trans-series with Lefschetz thimbles, and constrained by a Schwinger–Dyson bootstrap relation. If right, the observed dark energy could be a transient, finite-time expectation value rather than a cosmological constant.

Core claim

The central construction is the Glauber–Sudarshan state |σ⟩ = D(σ)|Ω⟩, where D(σ) = T exp[∫ dt d^3x σ^{μν}(t,x) √−ĝ ĝ_{μν}] and |Ω⟩ is the exact interacting vacuum over a supersymmetric Minkowski minimum. The paper's claim is that for admissible profiles σ_{μν} the semiclassical expectation value ⟨ĝ_{μν}⟩_σ mimics a four-dimensional quasi-de Sitter metric on a finite time interval, with positive energy E_σ = ⟨σ|Ĥ_∂|σ⟩/⟨σ|σ⟩ > 0, while the state remains a physical excited state in the Wheeler–DeWitt sector. Consistency is shown through canonical boundary-Hamiltonian evolution, Schwinger–Keldysh in-in path integrals, trans-series/Lefschetz-thimble decomposition, and a bootstrap equation that f

What carries the argument

The load-bearing object is the Glauber–Sudarshan displacement operator (Eq. 3.26): D(σ) = T exp[∫ dt d^3x σ^{μν}(t,x) √−ĝ ĝ_{μν}], acting on the interacting Minkowski vacuum |Ω⟩. Unlike a coherent-state displacement exp[i∫ σ^{ij} π̂_{ij}] which translates the wavefunctional once, D(σ) is a time-ordered product of infinitesimal 'kicks' that continuously reweights the metric configurations, so the state is driven by its own profile. Consistency is enforced by the admissible class S_phys (Eq. 3.80)—the conditions [Ĥ_⊥, D(σ)]|Ω⟩=0, [Ĥ_i, D(σ)]|Ω⟩=0, and [i∂_t D−[Ĥ_∂, D]]|Ω⟩=0—and by a Schwinger–Dyson bootstrap equation (7.66) that fixes σ_{μν}; the Schwinger–Keldysh contour with trans-series act

Load-bearing premise

The construction requires that a nonzero profile σ_{μν} exists inside the admissible class S_phys—a displacement operator that maps the chosen physical reference state back into the physical Hilbert space, satisfying the WdW and boundary-Hamiltonian intertwining conditions; the paper's own bootstrap section is titled 'Towards solving the bootstrap equations consistently' and no explicit σ is exhibited, so the central claim collapses if no such σ exists or if its expectation v

What would settle it

Exhibit an explicit nonzero σ_{μν} satisfying Eq. (3.80) for a concrete action with a supersymmetric Minkowski minimum and compute the one-point function ⟨ĝ_{μν}⟩_σ: if it does not approximate a four-dimensional quasi-de Sitter metric over a finite interval, the central claim fails. Equivalently, prove that the only solution of [Ĥ_⊥, D(σ)]|Ω⟩=0=[Ĥ_i, D(σ)]|Ω⟩ and [i∂_t D−[Ĥ_∂, D]]|Ω⟩=0 is σ=0, or find a state whose expectation value violates the real-metric and positive-energy conditions.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Quasi-de Sitter spacetimes can be realized as excited states over an exact Minkowski vacuum, so no new de Sitter vacuum or metastable potential minimum is required.
  • Since the GS insertion is not a Legendre source, its expectation values are not constrained by 1PI convexity; a positive-energy quasi-dS metric can coexist with a strictly convex effective action.
  • The dark energy scale is not tied to a potential minimum but to the GS profile, and its lifetime is finite, set by the duration of the excitation.
  • Observables must be computed in the in-in (Schwinger–Keldysh) formalism, not in-out amplitudes; the paper derives the doubled-contour representation with branch symmetry enforcing real expectation values.
  • The bootstrap equation turns the consistency of the state into an eigenvalue-like condition, so the possible quasi-dS profiles are fixed points of a map, allowing multiple basins of attraction.

Where Pith is reading between the lines

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

  • If the claim holds, a natural testable extension is to construct an explicit admissible σ for a concrete (e.g., higher-curvature or supergravity) model and compute ⟨ĝ_{μν}⟩_σ directly; the paper does not exhibit such a profile, so the existence of S_phys is the immediate open question.
  • The bootstrap fixed-point picture suggests a gravitational analogue of 'states as attractors': different initial GS data could flow to different quasi-dS profiles, which might explain multiverse-like branches without modifying the vacuum structure.
  • One could examine whether the GS construction provides a UV completion of the open-system/influence-functional description of cosmology: the nonlocal, time-dependent effective action obtained by integrating out modes in accelerating backgrounds (Section 2.2) may be the same as the GS expectation-value dynamics.
  • A sharper consequence: if a GS state can mimic dS for finite time, then cosmic acceleration could be a transient phenomenon with memory of its preparation; this is falsifiable in principle by looking for signatures of finite lifetime in the primordial spectrum.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 8 minor

Summary. The paper develops a formal framework for constructing transient excited states—called Glauber–Sudarshan (GS) states—over a supersymmetric Minkowski vacuum in a four-dimensional diffeomorphism-invariant theory. The GS state is defined as a time-ordered exponential of a prescribed metric operator profile σ_μν acting on the interacting vacuum, in contrast to ordinary coherent states defined by a single-slice momentum displacement. The central claim is that such GS states are generically non-supersymmetric, carry positive boundary energy, and can have semiclassical expectation values that mimic four-dimensional quasi-de Sitter geometries over finite time intervals, without being vacua or extrema of the 1PI effective action. The paper analyzes the canonical structure (ADM constraints, boundary Hamiltonian), the Schwinger–Keldysh in-in path-integral representation of expectation values, the trans-series/Lefschetz-thimble organization of the full gauge-fixed action, a bootstrap equation constraining admissible GS profiles, and the relation between Wheeler–DeWitt constraints, boundary time, and the GS Hilbert-space interpretation. It also draws a structural analogy between bulk GS insertions and world-sheet vertex operators.

Significance. If the central claim were established, the framework would offer a way to realize quasi-de Sitter configurations as excited states in a theory with an exactly supersymmetric Minkowski vacuum, explicitly avoiding the 1PI convexity obstruction and the Wilsonian no-go logic for metastable de Sitter vacua. The paper contains several valuable formal results: the careful distinction between coherent states and GS states in the canonical formalism, the correction of the naive short-time identity (3.33) to the order-by-order expression (3.52), the proof of the reference-time identity (3.42), the explicit Schwinger–Keldysh representation (4.18) with shifted boundary data, and the reality argument for expectation values in §4.4. These manipulations appear internally consistent and constitute a useful toolkit for studying state-dependent semiclassical geometries. The main gap is that the existence of an admissible GS profile realizing the claimed quasi-de Sitter expectation values is not demonstrated; the paper provides a bootstrap formalism for finding such profiles but stops short of solving it. This is a load-bearing missing step rather than a presentation issue.

major comments (4)
  1. [§3.3, Eq. (3.80); §8.1.7] The central construction requires the GS profile σ_μν to belong to the admissible class S_phys, defined by the conditions [Ĥ_⊥, D(σ)]|Ω⟩=0, [Ĥ_i, D(σ)]|Ω⟩=0, and [i∂_t D − [Ĥ_∂, D]]|Ω⟩=0. No explicit profile satisfying these constraints is exhibited, and §8.1.7 itself states that the construction works only 'provided that the displacement operator maps the chosen physical reference state back into that space.' Since D(σ) is a nonlocal, time-ordered exponential of renormalized composite operators, the constraint conditions are highly nontrivial. Without either an existence proof or a concrete admissible σ, the paper has not established that GS states are physical states of the constrained Hilbert space, which is a precondition for the central claim.
  2. [Abstract; §1.1; §7.4.4] The paper's headline claim is that GS-state expectation values can 'temporarily mimic four-dimensional quasi-de Sitter geometries over a finite time interval.' However, no computation of ⟨g_μν⟩_σ for any explicit profile is presented. Figure 2 shows toy time-dependent source profiles σ(t), not the resulting metric expectation values or a demonstration that those profiles satisfy Einstein-type equations or the bootstrap equation. Section 7.4.4 is explicitly titled 'Towards solving the bootstrap equations consistently,' and no solution of the bootstrap equation is given. The quasi-de Sitter conclusion therefore rests on an uncompleted calculation. A concrete family of σ_μν producing a quasi-de Sitter ⟨g_μν⟩_σ, even at leading semiclassical order, is needed to support the abstract's claim.
  3. [§7.5.3] The interpretation of quasi-de Sitter configurations as 'bootstrap fixed points' with local stability is asserted but not demonstrated. The section presents the bootstrap map and discusses basins of attraction in general terms, but it does not exhibit a fixed point, compute its local linearization, or show that the corresponding expectation values approximate a de Sitter metric. Without this analysis, the statement that de Sitter-like states are local, possibly transient, bootstrap fixed points is a conjecture rather than a result. This is directly load-bearing for the paper's main physical message.
  4. [§4.3, Eq. (4.21)] The full gauge-fixed action S_tot is assumed to admit a Borel–Écalle-summable trans-series of the form (4.21), with sectors labelled by A_I and S_I. This is a strong assumption, particularly because the paper later appeals to 'consistency across three complementary descriptions' as evidence for the framework. The trans-series structure is essential to the Lefschetz-thimble arguments and to the treatment of nonperturbative sectors, but no derivation or independent check is provided. If this is meant as an axiom, it should be stated as such; if it is meant to follow from the underlying theory, a justification or reference to a concrete calculation is required. As written, this assumption is a free parameter of the formalism and weakens the claimed consistency.
minor comments (8)
  1. [Title/Abstract] The title and abstract inconsistently capitalize 'Glauber-Sudarshan States' and 'Glauber-Sudarshan states'; please standardize.
  2. [§1, paragraph 1] 'Our overreaching goal' should read 'Our overarching goal.'
  3. [§2.3] The headings 'F atal Point I/II/III' have an erroneous space in 'Fatal'; please correct.
  4. [Footnote 12, §3.2] The explanation of using T instead of T for time-ordering is confusing because both symbols look similar in the typeset text; consider naming the time coordinate differently or explicitly distinguishing the two.
  5. [Figure 2] The caption says 'with same (t )' but the subscript appears missing; also the y-axis label 'field' is ambiguous because the profiles σ(t) are sources/deformations, not physical fields. Please clarify.
  6. [§3.3, Eq. (3.76)-(3.79)] The discussion of the driven Schrödinger equation is clear in the text, but Eq. (3.76) could be misinterpreted as adding a non-Hermitian term to the Hamiltonian. A sentence explicitly stating that this equation is only a formal rewriting and is superseded by (3.78)-(3.79) would help avoid confusion.
  7. [§4.5, footnote 23] The Euclidean expression Ψ_σ[h]=Ψ_Ω[h−σ] is derived for coherent states, but the earlier GS wavefunctional (3.58) involves a bulk insertion rather than a boundary shift. The distinction should be flagged explicitly when using this identity in the Euclidean discussion to prevent the reader from applying it to GS states.
  8. [§8.2] The vertex-operator analogy is clearly marked as structural, which is good. However, the comparison would benefit from a compact table listing the precise correspondence (e.g., GS insertion vs. vertex operator, physical state condition vs. BRST invariance, WdW constraint vs. conformal dimension) and the known mismatches.

Circularity Check

1 steps flagged

Quasi-dS claim is parameterized by the arbitrary GS displacement profile; bootstrap selection is left unsolved.

specific steps
  1. self definitional [Section 3.3, around Eq. (3.84)]
    "Since σij(x, t) is an arbitrary time-dependent function, the GS construction allows one to continuously reshape the probability landscape and thereby steer the dominant support of the state through a far broader region of configuration space. Under sufficiently favorable conditions, namely if the map: σij(x,t) 7→ h∗ij(x,t) is sufficiently non-degenerate, the family of all admissible profiles σij(x,t) generates an extremely large class of effective trajectories."

    The GS state is defined by (3.18) as D(σ)|Ω⟩, with σ a prescribed classical profile. Equation (3.84) then defines the induced dominant metric h* as the image of σ under a map that the paper says is non-degenerate for admissible profiles. Hence the set of possible semiclassical metric expectation values is parameterized by the input profile σ itself. The central claim that these states 'admit' quasi-de Sitter configurations is therefore not derived from the dynamics; it is installed by choosing σ to steer ⟨g⟩σ to the desired dS-like metric. The bootstrap section is only titled 'Towards solving the bootstrap equations consistently', and no admissible σ producing dS is exhibited, so the claimed prediction reduces to the free displacement profile rather than to a computed output.

full rationale

The paper does contain substantial independent structure: the canonical decomposition, Schwinger-Keldysh representations, Lefschetz-thimble organization, and SUSY-breaking arguments are developed explicitly rather than merely cited. Self-citations to [13-20] for the GS construction are not load-bearing because the present work re-derives the displacement-operator and path-integral forms. However, the central quasi-dS claim is not independently demonstrated. The state is defined by an arbitrary profile σ, and the paper itself states that σ can steer the dominant metric configuration over an extremely large class of trajectories. Unless the unsolved bootstrap equations select a specific σ that produces a dS-like expectation value, the quasi-dS outcome is put in by the choice of σ. The caveat in §8.1.7 ('provided that the displacement operator maps the chosen physical reference state back into that space') and the heading in §7.4.4 ('Towards solving...') flag the missing existence proof; these are gaps rather than circular reductions, but they reinforce that the main result is conditional on an unexhibited input. Score 4 reflects a partial, not complete, circularity: the construction could become non-circular if a bootstrap solution or explicit admissible σ for quasi-dS were supplied.

Axiom & Free-Parameter Ledger

3 free parameters · 8 axioms · 2 invented entities

No data fitting anywhere; the paper's only 'parameters' are the functional GS data σ_μν and the assumed trans-series sector data, none of which is pinned down by an exhibited solution. The framework as a whole rests on: existence of a SUSY Minkowski reference vacuum, the resurgent trans-series structure of the action, swampland-like Premise P, convexity of the 1PI effective action, diffeomorphism invariance surviving all corrections, and a WKB split of WdW. Most are standard or explicitly-flagged assumptions, but the trans-series structure is load-bearing for sections 4-6 and is assumed rather than derived.

free parameters (3)
  • GS profile σ_μν(x,t)
    A hand-chosen classical profile defining the state via (3.26)/(3.18). The quasi-dS mimicry claim hinges on its choice; the admissibility conditions (3.80) and bootstrap equation (7.66) that would fix it are not solved in the visible text. Functionally infinite-dimensional; no fitted numerics.
  • Figure 2 toy profiles σ(t) (constant/decaying/oscillatory/pulse/growing)
    Schematic illustrations of 'field vs time' evolution for the GS state; the generating dynamics are not given in the visible text.
  • Trans-series sector data A_I, S_I
    Instanton actions and sector actions entering the assumed trans-series form of S_tot (4.21); introduced as inputs from the conjectured resurgent structure of the theory, not derived in this paper.
axioms (8)
  • domain assumption The underlying theory has a (possibly supersymmetric) Minkowski vacuum with asymptotic time and a well-defined boundary Hamiltonian (ADM/Iyer-Wald).
    Invoked throughout (§1.6, §2, §3.1); the GS construction and the emergence of Schrodinger evolution require this reference branch. Abstract: 'over supersymmetric minima.'
  • ad hoc to paper The full gauge-fixed action S_tot admits a Borel-Ecalle-summable trans-series of the form (4.21).
    Eq. (4.21), footnote 20 (§4.3). Underpins the thimble decomposition and the 'persistence of the Minkowski minimum' argument of §5; assumed with citations to the authors' own and standard resurgence literature.
  • domain assumption Premise P: no quasi-stationary dS configuration with parametric scale separation; positive vacuum energy forces tower descent m_tower ≲ H.
    §2.3. The entire no-go logic motivating 'dS only as excited state' rests on this swampland-like premise, which is assumed, not derived.
  • standard math The exact 1PI effective action is convex (second derivative positive semidefinite via Legendre transform).
    §2.1, Eqs. (2.1)-(2.4). Standard result, though the authors themselves flag its applicability to quantum gravity as nontrivial.
  • domain assumption Diffeomorphism invariance survives all higher-derivative and matter corrections, so lapse and shift remain nondynamical (π_N ≈ 0).
    §1.3-1.4 (Step 3). Needed for the universal constraint form (1.13)/(1.30) and for the 'Hamiltonian is holographic' conclusion.
  • domain assumption WKB/Born-Oppenheimer split of the WdW wavefunctional Ψ ≈ exp(iS_0/ℏ)ψ, yielding t_WKB and the Schrodinger equation.
    §1.6, §3.1, §8.1. The emergence of time and the boundary/bulk matching depend on this semiclassical split.
  • domain assumption Ghost/gauge-fixing boundary charges vanish under standard asymptotic falloffs (c, c̄, b → 0).
    §1.5, footnote 4, Table 1. Needed so H_phys is the gravitational boundary charge only.
  • domain assumption The canonical commutator [ĝ_ij, π̂^kl] = iδ δ^3 with density-weight +1 momenta, and composite-operator renormalization preserves the Trotterized time-ordered exponential (3.26).
    §3.2, Eqs. (3.19)-(3.30), footnote 14. The GS state definition depends on this.
invented entities (2)
  • Glauber-Sudarshan states |σ⟩ = Texp(∫ σ_μν Ĝ^μν) |Ω⟩ no independent evidence
    purpose: Realize quasi-dS spacetimes as transient excited states over a supersymmetric Minkowski vacuum, evading vacuum no-go theorems.
    Introduced to explain quasi-dS; no falsifiable handle outside the framework (no predicted mass, lifetime, or amplitude computed). Admissibility is conditioned on unsolved equations (3.80).
  • Admissible class S_phys of GS profiles no independent evidence
    purpose: Restrict σ so the GS state stays in the physical (WdW-annihilated) Hilbert space and evolves under the boundary Hamiltonian alone.
    Eq. (3.80); defined by [Ĥ_⊥, D]|Ω⟩ = [Ĥ_i, D]|Ω⟩ = 0 and the intertwining condition; no elements of S_phys are exhibited.

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We investigate how, in a highly constrained system such as a four-dimensional diffeomorphism-invariant theory with vanishing bulk Hamiltonian and non-trivial interactions between the metric and additional degrees of freedom, transient excited states--called Glauber-Sudarshan states--can be constructed over supersymmetric minima. These states are generically non-supersymmetric and, although they are not minima of any potential, they admit positive-energy metric configurations that effectively mimic four-dimensional quasi-de Sitter backgrounds. We analyze how such states differ from conventional coherent states by studying their time evolution both in the canonical formalism--via boundary Hamiltonians--and in the in-in path-integral framework through Schwinger-Keldysh contours. We also examine the consistency of the construction across three complementary descriptions: the 1PI effective action, the Wilsonian (or exact renormalization group) effective action, and the Picard-Lefschetz (Lefschetz-thimble) decomposition of the Schwinger-Keldysh path integral, which provides the trans-series organization of the theory. In the presence of gauge-fixing and ghost sectors, we show that the dynamics of these transient configurations are governed by a nontrivial bootstrap relation that simultaneously constrains their behavior near supersymmetric Minkowski vacua and along quasi-de Sitter-like trajectories. Finally, we examine the Wheeler-DeWitt equation, the notion of time in the presence of transient excited states, and the emergence of the bulk Schrodinger equation. We further show that these states admit a natural structural interpretation analogous to the vertex operators that arise in the two-dimensional world-sheet formulation of string theory.

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