⚛️ high-energy theory

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

This paper investigates the construction and dynamics of non-supersymmetric Glauber-Sudarshan transient states in four-dimensional diffeomorphism-invariant theories, demonstrating how they mimic quasi-de Sitter backgrounds and are constrained by a unified bootstrap relation across canonical, path-integral, and Lefschetz-thimble frameworks while exhibiting structural parallels to string theory vertex operators.

Original authors: Heliudson Bernardo, Tatsuya Daniel, Keshav Dasgupta, Brayden Hull, Yue Katherine Lei, Yiya Selina Li

Published 2026-07-24

📖 8 min read🧠 Deep dive

CC BY 4.0

Original authors: Heliudson Bernardo, Tatsuya Daniel, Keshav Dasgupta, Brayden Hull, Yue Katherine Lei, Yiya Selina Li

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Great Cosmic Clock and the Ghost in the Machine

Imagine trying to describe the motion of a river, but you are standing on a boat that is part of the river itself. In the world of quantum gravity—the physics that tries to merge the tiny rules of atoms with the massive rules of gravity—this is exactly the problem. In standard physics, time is like a master clock on the wall that ticks away, telling everything when to move. But in Einstein's theory of gravity, space and time are flexible; they stretch and squeeze. When you try to apply quantum rules to this flexible fabric, the "master clock" disappears. The equations say the total energy of the universe is zero, and nothing seems to change. It's like a movie where the film reel is stuck, and the characters are frozen in place.

This creates a massive headache for cosmologists. We know the universe is expanding, and we see evidence that it once expanded incredibly fast, a phase called "inflation" that looks a lot like a "de Sitter" universe (a fancy name for a space that expands exponentially). But if the fundamental laws of quantum gravity say time doesn't exist and energy is zero, how can we have an expanding universe? The standard answer has been to look for a "vacuum state"—a stable, resting place for the universe that naturally wants to expand. However, recent mathematical arguments suggest that such a stable, expanding resting place might not exist at all. It's as if you're trying to find a flat spot on a trampoline that is constantly bouncing, but the math says the trampoline can never be flat and still bounce. This paper dives into that paradox, asking: if the universe can't be a stable, expanding resting place, what is it?

The Paper's Big Idea: The Universe as a "Drunk" Wave

This paper proposes a radical new way to think about our expanding universe. Instead of looking for a permanent, stable "de Sitter" vacuum (a resting place that expands forever), the authors suggest that our universe is actually a transient excited state. Think of it like this: imagine a calm, flat lake (the "Minkowski" vacuum, which is stable and supersymmetric). If you throw a stone in, you get ripples. If you keep hitting the water with a specific rhythm, you can create a massive, rolling wave that looks like a tsunami for a while. The paper argues that our universe is that rolling wave. It's not a new, stable ocean; it's a temporary, excited splash on top of a calm, supersymmetric sea.

The authors call these temporary, wave-like configurations "Glauber-Sudarshan (GS) states." To understand the difference, imagine two ways to make a wave:

The Coherent State (The Old Way): You push the water once at the edge of the pool and let it ripple out. It's a clean, simple wave that follows the rules of a calm pond.
The Glauber-Sudarshan State (The New Way): You don't just push once; you keep hitting the water with a complex, rhythmic beat while the wave is moving. You are constantly "driving" the wave. This creates a much more complicated, messy, and powerful wave that can look like a giant, expanding ocean for a specific amount of time, even though the water underneath is still calm.

The paper suggests that the "de Sitter" universe we see (the one expanding fast) is actually this second type of wave. It's not a permanent state of the universe; it's a temporary, excited configuration that the universe is "driven" into.

Why the Old Idea Doesn't Work

The authors spend a lot of time explaining why we can't just say, "Okay, let's find a stable de Sitter vacuum." They argue that if you try to treat the expanding universe as a stable, resting place (a vacuum), you run into three big mathematical walls:

The Shape Problem: The math that describes the "energy landscape" of the universe (called the 1PI effective action) is shaped like a bowl. It can have a bottom (a minimum), but it can't have a "hilltop" that stays still. A stable expanding universe would need to sit on a hilltop, which the math says is impossible.
The Time Problem: In an expanding universe, the definition of "fast" and "slow" keeps changing. What looks like a high-energy particle today might look like a low-energy particle tomorrow. This makes it impossible to build a standard "Wilsonian" theory (a way of simplifying physics by ignoring tiny details) because the details you ignore today might become important tomorrow.
The Tower Problem: String theory (the theory behind this paper) predicts an infinite tower of heavy particles. If the universe expands too fast, these heavy particles start to get light and pop into existence, breaking the rules of the simplified theory.

Because of these walls, the paper argues that the idea of a "metastable de Sitter vacuum" (a temporary but stable resting place) is likely a dead end. The universe isn't resting; it's being driven.

How the Paper Solves the Puzzle

So, if the universe isn't resting, how does it move? The authors use a tool called the Schwinger-Keldysh contour. Imagine you are watching a movie, but instead of just watching it play forward, you watch it forward and then immediately backward, and you compare the two. This "in-in" formalism allows physicists to calculate what happens to a system that is changing over time, rather than just calculating how it jumps from one stable state to another.

Using this method, the authors show that you can construct these "Glauber-Sudarshan" states. These states are built on top of a supersymmetric Minkowski vacuum (the calm lake). By applying a specific "operator" (a mathematical push) over a period of time, you can create a state where the average shape of the universe looks exactly like a de Sitter space (the expanding ocean) for a finite time.

The paper introduces a concept called "bootstrapping." Imagine you are trying to balance a broom on your hand. You don't just hold it still; you constantly move your hand to keep it upright. The paper suggests the universe is like that broom. The "drive" that keeps the universe expanding doesn't come from an external force; it comes from within the system itself. The authors derive a "bootstrap equation" that acts like a self-check. The universe says, "If I look like this, then I must be driven like this." If the math checks out, the universe can sustain this expanding wave-like state.

The "Ghost" in the Machine and String Theory

One of the most creative parts of the paper is a comparison to string theory. In string theory, particles are created by "vertex operators"—mathematical tools that poke the string worldsheet to create a new particle. The authors show that their "Glauber-Sudarshan" states are structurally very similar to these vertex operators.

However, there's a twist. In string theory, the "poke" happens on a 2D surface (the string). In this paper, the "poke" happens across the entire 4D history of the universe. It's not just a single moment; it's a continuous deformation of the universe's timeline. The paper argues that just as a string needs to be "conformal" (balanced) to exist, this cosmic wave needs to satisfy a "bootstrap" condition to exist. If the math doesn't balance, the wave collapses.

What This Means for Us

The paper doesn't claim to have solved the mystery of dark energy or proven that the universe is a simulation. Instead, it offers a new, consistent way to think about the universe's expansion without breaking the fundamental laws of quantum gravity.

It rules out the idea that the universe is sitting in a stable, expanding "vacuum" state.
It suggests that the expanding universe is a temporary, excited state—a "Glauber-Sudarshan" wave—riding on top of a calm, supersymmetric background.
It proposes that this state is maintained by a self-consistent "bootstrap" mechanism, where the universe's own internal rules keep the wave going.

The authors are careful to say this is a theoretical framework. They haven't measured this in a lab (because we can't build a universe in a lab), but they have shown that the math works. They suggest that if we look at the universe not as a static object, but as a dynamic, driven wave, we can explain why it expands without violating the strict rules of quantum gravity. It's a shift from asking "Where is the universe resting?" to "How is the universe dancing?" and finding that the dance steps are mathematically consistent, even if they are temporary.

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

1. The Problem: Time, Vacua, and de Sitter Space in Quantum Gravity
The paper addresses a fundamental tension in quantum gravity and string theory: the conflict between the vanishing bulk Hamiltonian of diffeomorphism-invariant theories and the existence of genuine time evolution in semiclassical regimes. In canonical gravity, the bulk Hamiltonian is a sum of constraints ( $H_{\text{bulk}} \approx 0$ ), leading to the Wheeler-DeWitt equation and a "frozen" formalism where physical states do not evolve in an intrinsic time. However, semiclassical observers in asymptotically flat or warped Minkowski backgrounds experience time evolution generated by boundary Hamiltonians.

A secondary, related problem is the realization of de Sitter (dS) spacetime. Standard effective field theory (EFT) logic, particularly the convexity of the 1PI effective action and the requirements of Wilsonian renormalization, suggests that metastable de Sitter vacua with positive vacuum energy are inconsistent in quantum gravity. Specifically:

1PI Convexity: The exact 1PI effective action $\Gamma$ is convex; thus, any stationary point must be a global minimum. This forbids local positive-energy minima (dS vacua).
Wilsonian Obstruction: In accelerating backgrounds, the separation of UV and IR scales is time-dependent due to redshift, rendering a time-independent Wilsonian effective action ill-defined.
No-Go Logic: A configuration with $V > 0$ cannot simultaneously satisfy quasi-stationarity, parametric longevity, and scale separation (the "Premise P" of the paper).

The authors argue that these obstacles apply to vacua (extrema of $\Gamma$ ) but not necessarily to excited states. The central question is whether transient quasi-de Sitter geometries can be realized as excited states over supersymmetric Minkowski minima without violating the constraints of quantum gravity.

2. Methodology and Framework
The paper develops a unified framework comparing conventional Coherent States (CS) with Glauber-Sudarshan (GS) states within a diffeomorphism-invariant theory (including gravity, matter, ghosts, and higher-curvature corrections). The methodology employs several complementary formalisms:

Canonical Formalism: The authors analyze the Hamiltonian structure of the Einstein-Hilbert action (plus Gibbons-Hawking-York boundary terms and higher-order corrections). They demonstrate that while the bulk Hamiltonian vanishes on-shell, the physical Hamiltonian is a boundary term ( $H_{\text{phys}} = H_{\partial\Sigma}$ ). This boundary Hamiltonian generates time evolution for fluctuations around a fixed background.
Path Integral Formulations:
- Schwinger-Keldysh (SK) In-In Formalism: Used to compute real-time expectation values $\langle O \rangle_\sigma$ for time-dependent states. This avoids the issues of in-out transition amplitudes which are ill-suited for transient cosmological backgrounds.
- Trans-Series and Lefschetz Thimbles: The total action $S_{\text{tot}}$ is treated as a trans-series object containing perturbative, non-perturbative (instanton), and non-local sectors. The path integral is decomposed into Lefschetz thimbles (steepest descent cycles) associated with complex saddles to organize the semi-classical expansion.
Bootstrap Equations: The authors derive Schwinger-Dyson (SD) equations in the presence of GS insertions. They formulate a "bootstrap" procedure where the GS profile must be self-consistent with the backreacted equations of motion. This leads to a nonlinear map whose fixed points correspond to consistent semiclassical configurations.
Structural Analogy: A detailed comparison is drawn between bulk GS insertions and worldsheet vertex operators in string theory, analyzing their roles in deforming path integrals and satisfying consistency conditions (beta functions vs. bootstrap equations).

3. Key Distinctions: Coherent States vs. Glauber-Sudarshan States
The paper rigorously distinguishes between two types of states:

Coherent States (CS): Defined as $|\alpha\rangle = D_{\text{CS}}(\alpha)|0\rangle$ , where $D_{\text{CS}}$ is a displacement operator acting on a free or perturbative vacuum at a single Cauchy slice. These represent sharp wavepackets evolving under a free or weakly interacting Hamiltonian. They are tied to boundary data and perturbative expansions.
Glauber-Sudarshan States (GS): Defined as $|\sigma\rangle = D_{\text{GS}}(\sigma)|\Omega\rangle$ $∣ σ ⟩ = D_{GS} (σ) ∣Ω ⟩$ , where $|\Omega\rangle$ $∣Ω ⟩$ is the exact interacting vacuum and $D_{\text{GS}}$ $D_{GS}$ is a time-ordered operator insertion over a spacetime history (or SK contour).
- Structure: The GS displacement operator involves a time-ordered exponential of composite operators (e.g., $\int d^4x \sqrt{-g} \sigma_{\mu\nu} g^{\mu\nu}$ ) rather than a simple boundary translation.
- Dynamics: GS states are "driven" configurations. The source $\sigma$ is not an external parameter for a Legendre transform but a state-defining parameter.
- Supersymmetry: GS states can break supersymmetry in their expectation values while being constructed over a supersymmetric Minkowski reference vacuum. This allows for non-supersymmetric quasi-dS profiles without requiring a non-supersymmetric vacuum.

4. Key Results and Contributions

Transient de Sitter as an Excited State: The paper demonstrates that quasi-de Sitter geometries can be realized as the expectation values of GS states over supersymmetric Minkowski sectors for a finite time interval ( $t_i < t < t_f$ ). These are not eternal vacua or minima of an effective potential, but transient excited configurations. This circumvents the convexity of the 1PI action and the Wilsonian no-go theorems, as the state is not an extremum of $\Gamma$ .
The Bootstrap Mechanism: A central result is the derivation of a bootstrap equation for GS states. The consistency condition requires that the chosen GS profile $\sigma$ $σ$ and the resulting expectation values $\langle g_{\mu\nu} \rangle_\sigma$ $⟨ g_{μν} ⟩_{σ}$ satisfy a coupled system of equations:
1. An effective Einstein-type equation for the mean fields.
2. A self-consistency equation for the GS insertion itself (derived from the variation of the displacement operator).
  The paper shows that quasi-dS configurations correspond to fixed points (or approximate fixed points) of a bootstrap map. The non-uniqueness of these fixed points reflects the existence of multiple semiclassical branches.
Time Evolution and the Wheeler-DeWitt Equation: The authors reconcile the frozen formalism with time evolution. They show that while the exact wavefunctional satisfies the Wheeler-DeWitt constraint ( $\hat{H}_{\text{bulk}}\Psi = 0$ ), a semiclassical branch (selected by a Minkowski minimum) admits an emergent Schrödinger equation with a WKB time $t_{\text{WKB}}$ . Furthermore, the boundary Hamiltonian generates physical time evolution for the asymptotic clock. The GS construction allows for a consistent description of time-dependent expectation values within this constrained framework.
Worldsheet Analogy: The paper establishes a structural analogy between bulk GS insertions and integrated worldsheet vertex operators. Both act as extended deformations of the path integral weight. However, the consistency conditions differ: worldsheet deformations require vanishing beta functions (conformal invariance), while bulk GS deformations require satisfying the coupled bootstrap equations. The GS insertion is not merely a source but part of a self-consistent state definition.
Trans-Series Organization: The analysis confirms that the full path integral for these states must be organized as a trans-series, incorporating instanton sectors and non-perturbative effects via Lefschetz thimble decomposition. The GS expectation values are naturally trans-series objects.

5. Significance and Claims
The paper claims to provide a coherent framework for describing finite-time cosmological backgrounds (like quasi-de Sitter space) within a theory of quantum gravity that strictly adheres to diffeomorphism invariance and the absence of positive-energy vacua.

Resolution of the dS Problem: It offers a way to realize de Sitter-like physics without postulating a metastable dS vacuum, thereby avoiding the Swampland constraints associated with dS vacua. The dS geometry is an excited state phenomenon, not a vacuum property.
State-Dependent Dynamics: It emphasizes that in quantum gravity, the definition of a state (via GS operators) is as crucial as the dynamics. The "drive" for the geometry comes from the self-consistent bootstrap of the state itself, rather than an external potential.
Methodological Advancement: The work integrates canonical boundary Hamiltonians, SK contours, trans-series, and bootstrap techniques into a single consistent picture, offering a practical framework for analyzing off-shell and transient configurations in string/M-theory compactifications.

The authors remain modest, noting that while the framework is established, explicit solutions to the bootstrap equations in specific string/M-theory compactifications and a more rigorous treatment of the thimble decomposition remain open problems. The primary contribution is the structural proposal that transient quasi-de Sitter geometries are consistent excited states over supersymmetric Minkowski minima, governed by a bootstrap self-consistency condition.