⚛️ general relativity

On the Unitarity of the Gravitational S-Matrix in High Dimension

The paper argues that the gravitational S-matrix in dimensions greater than four fails to be a unitary operator in Fock space due to the orthogonality of high-energy black hole states, yet suggests that physical unitarity may be recovered through an algebraic formulation based on fermionic oscillators, pending a rigorous proof of Poincaré invariance for the S-matrix amplitudes.

Original authors: T. Banks

Published 2026-02-17

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: T. Banks

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 Big Question: Is Gravity a Perfect Game of Billiards?

Imagine you are watching a game of billiards. You hit the cue ball, it smashes into the others, and they scatter. In a perfect, mathematical world, if you knew the exact position and speed of every ball at the start, you could predict exactly where they would end up. In physics, we call this unitarity. It means information is never lost; the "before" and "after" states are perfectly connected.

For a long time, physicists believed that gravity works just like this billiard game. They thought that if you smashed two particles together, the result would always be a predictable number of other particles flying out. This is what we call the Fock space view: a universe made of distinct, countable particles (like 1, 2, or 3 gravitons).

Tom Banks' paper argues that this view is wrong for gravity, especially in higher dimensions (like 5, 6, or 11 dimensions).

He suggests that when you smash particles together at extremely high energies, the result isn't a neat pile of particles. Instead, the universe gets "fuzzy" and messy in a way that breaks the perfect billiard game rules.

The Problem: The "Infinite Fog"

Let's use an analogy to understand why.

Imagine you are trying to hear a whisper (a single particle) in a quiet room. Easy. Now, imagine you are in a stadium during a rock concert. If you try to whisper, your voice is drowned out by the noise.

In gravity, when you smash particles together with huge energy, they don't just bounce off each other. They create a massive amount of "soft" gravitational radiation. Think of this as a giant, invisible fog of low-energy ripples spreading out everywhere.

The Low Energy Case: If you smash particles gently, the fog is thin. You can still count the main particles (the billiard balls) easily.
The High Energy Case: As you increase the energy, the fog gets thicker and thicker.
- In 4 dimensions (our universe), this fog is messy, but we have tricks to handle it.
- In higher dimensions (5+), the fog becomes overwhelming.

Banks argues that at ultra-high energies, the "fog" becomes so dense that the final state of the collision looks like a coherent state.

Analogy: Imagine a single billiard ball (a particle) vs. a giant, rolling wave of water (a coherent state).
The wave contains infinite tiny droplets. If you try to count the droplets, you can't. If you try to find the original billiard ball inside the wave, it's gone.

The Conclusion: As energy goes to infinity, the final state becomes so "foggy" that it has zero overlap with any state that has a finite number of particles. It's like trying to find a specific grain of sand in a tsunami. The math says the probability of finding a "normal" particle state is zero.

The Black Hole Twist

The paper also brings in black holes.

If you smash particles hard enough, they might form a black hole.
Black holes eventually evaporate (Hawking radiation), releasing a thermal cloud of particles.
Banks argues that this thermal cloud is even "fuzzier" than the wave mentioned above. It is so disordered that it is mathematically impossible to describe it as a collection of a few distinct particles.

The Result: The "S-matrix" (the machine that calculates the "before" and "after" of a collision) cannot be a perfect, unitary machine inside the standard "particle counting" room (Fock space). The rules of the game change when the energy gets too high.

The Solution: A New Kind of Room (Algebraic Scattering)

If the old room (Fock space) doesn't work, where do we go?

Banks proposes moving to a different kind of room called Algebraic Quantum Scattering.

The Old Room: You count particles (1, 2, 3...).
The New Room: You don't count particles. Instead, you look at constraints and patterns.

The Analogy of the "Frozen Q-Bits":
Imagine a giant, flexible net (the causal diamond) that fills space.

Normally, the net is vibrating randomly (the vacuum).
When a particle passes through, it doesn't just "add a ball" to the net. Instead, it freezes a specific pattern of knots in the net.
The "particle" is actually just a specific arrangement of frozen knots relative to the rest of the vibrating net.

Banks suggests that if we describe gravity this way—using a finite number of "q-bits" (quantum bits) that get frozen and unfrozen as they move through time—we can save the concept of unitarity.

The "S-matrix" becomes a map that tells us how the pattern of frozen knots changes from the start to the finish.
Even though the "fog" of soft radiation makes it look like particles are disappearing, the information is actually preserved in the complex pattern of the knots.

The "Missing Proof"

The paper ends with a bit of a cliffhanger.
Banks has built a beautiful mathematical model (using these "frozen knots" and a specific algebra called the Awada-Gibbons-Shaw algebra) that should work. It looks like it preserves information (unitarity).

However, there is one big hurdle:
He hasn't fully proven that this model respects Poincaré invariance.

Translation: He hasn't proven that his model looks the same to everyone, regardless of how fast they are moving or where they are standing.
Analogy: He built a perfect car engine, but he hasn't proven yet that the car drives straight on a moving train.

Summary in One Sentence

Tom Banks argues that at extreme energies, gravity creates such a massive "fog" of radiation that we can no longer describe the universe as a collection of distinct particles, but we can save the laws of physics by viewing collisions as changes in a complex, frozen pattern of quantum information.

Why This Matters

This paper challenges the standard way physicists think about string theory and quantum gravity. It suggests that the "particle" view is an illusion that breaks down at high energies, and that the true nature of the universe is more like a shifting, fuzzy tapestry of information than a box of Lego bricks.

1. Problem Statement

The paper addresses a fundamental tension in quantum gravity: the conflict between perturbative unitarity (which holds in Fock space for string theory amplitudes) and non-perturbative expectations derived from black hole physics and infrared (IR) behavior in dimensions $d > 4$ .

The Paradox: In perturbative superstring theory, scattering amplitudes appear to satisfy unitarity order-by-order in the expansion. However, non-perturbative arguments involving black hole formation and soft radiation suggest that at high energies, the final states of scattering processes become orthogonal to any state with a finite number of particles in the standard Fock space.
The Core Question: Can the gravitational S-matrix be defined as a unitary operator acting on a separable Hilbert space (specifically the standard Fock space of gravitons)? The author argues that while perturbative unitarity holds, non-perturbative unitarity in Fock space fails because the spectral measure dictated by Poincaré invariance forces the S-matrix to map finite-particle states into states with infinite particle numbers (coherent states or thermal states) as energy $E \to \infty$ .

2. Methodology

Banks employs a multi-pronged approach combining semi-classical gravity, non-perturbative matrix models, and algebraic quantum field theory (AQFT):

Semi-Classical IR Analysis:
- Utilizes Weinberg's soft theorem to calculate the energy radiated into soft gravitons ( $E_{IR}$ ) during high-energy scattering with impact parameter $b$ .
- Analyzes the overlap between these soft radiation states and finite-particle Fock states.
- Examines the regime of black hole formation ( $b < R_s(E)$ ) and subsequent Hawking decay, treating the final state as a thermal density matrix that is pure but statistically indistinguishable from a thermal state.
Non-Perturbative Model Verification:
- BFSS Matrix Model: Reviews how soft gravitons arise from the excitation of off-diagonal matrix modes in the large $N$ limit.
- AdS/CFT: Analyzes the limit of CFT correlators (Witten diagrams) to show that entanglement entropy diverges as the AdS radius goes to infinity, implying a soft graviton divergence in the Minkowski limit.
Algebraic Scattering Theory Construction:
- Proposes a new framework based on the quantum mechanics of a finite number of fermionic oscillators.
- Constructs a Hilbert bundle over the space of time-like geodesics in Minkowski space, using "nested causal diamonds" as the fundamental geometric units.
- Defines scattering states via constraints on "frozen q-bits" (degrees of freedom) within these diamonds.
- Demonstrates that the algebra of operators in the $N \to \infty$ limit converges to the Awada-Gibbons-Shaw (AGS) algebra, a super-Poincaré covariant algebra defined on the null momentum cone.

3. Key Contributions

A. Proof of Non-Unitarity in Fock Space

The paper provides a rigorous argument that for $d \geq 5$ , the S-matrix cannot be a unitary operator in the standard Fock space.

Coherent State Orthogonality: For impact parameters $b \sim R_s(E)$ , the soft theorem predicts the emission of a coherent state of gravitons with wavelength $\lambda \sim b$ . The overlap of this state with any Fock state containing $\leq M$ particles scales as:
$\sum_{n \leq M} |\langle n | \text{coherent}(E) \rangle|^2 \sim e^{-c (E/M_P)^{\frac{d-2}{d-3}}}$
As $E \to \infty$ , this overlap vanishes exponentially.
Black Hole Orthogonality: If black holes form, the final state (even if pure) behaves like a thermal density matrix at the Hawking temperature. Since $T_H \to 0$ for large black holes, the state becomes orthogonal to all finite-particle states.
Conclusion: Since the spectral measure is fixed by Poincaré invariance, the S-matrix must integrate over these energy windows. If the operator maps finite-particle states to orthogonal infinite-particle states, it cannot be unitary on the Fock space, despite having finite matrix elements and satisfying perturbative unitarity.

B. The Algebraic Scattering Framework

Banks proposes a resolution where unitarity is preserved not in Fock space, but in an algebraic formulation:

Finite Dimensional Approximation: The theory is defined on a finite-dimensional Hilbert space representing a causal diamond.
Constraints as Scattering: Scattering is defined by constraints on irreducible representations of fermionic oscillators. "Localized excitations" are defined by freezing a specific subset of q-bits relative to the "empty diamond" (vacuum) state.
Emergent Conservation Laws: Energy and momentum emerge as asymptotic conservation laws associated with the number of constrained q-bits divided by $N$ (the diamond size) in the $N \to \infty$ limit.
AGS Algebra Limit: The algebra of operators formally converges to the AGS algebra, where generators are operator-valued half-measures on the null cone.

C. Reconciliation with String Theory and Matrix Models

The paper clarifies why these results do not contradict existing theories:

String Perturbation Theory: The series is not Borel summable; therefore, perturbative unitarity does not guarantee a unitary non-perturbative completion in Fock space.
BFSS/AdS-CFT: These models naturally incorporate the soft graviton divergences (via off-diagonal matrix modes or CFT entanglement) but do not possess a standard Fock space S-matrix in the strict sense.

4. Results

Infrared Divergence in $d > 4$ : Unlike $d=4$ where IR divergences are logarithmic, in $d > 4$ , the number of soft gravitons grows rapidly with energy, leading to states orthogonal to the finite-particle sector.
Failure of Fock Space Unitarity: The S-matrix is not a unitary operator on the separable Hilbert space of Fock states. The "missing" unitarity is recovered only in a generalized algebraic sense.
Existence of Algebraic Unitarity: A "locally normal state" on the limiting AGS algebra exists where the S-matrix is unitary. This state is a "fuzzy" generalization of a local normal state, defined by the empty diamond density matrix.
Black Hole Formation as Scattering: The paper models black hole formation as a transition where constraints on q-bits lead to a high-entropy equilibrium state, consistent with thermal Hawking decay statistics.

5. Significance and Open Questions

Conceptual Shift: The paper challenges the standard assumption that a unitary S-matrix must act on a Fock space. It suggests that in quantum gravity, the physical Hilbert space is likely non-separable or requires an algebraic formulation (AQFT style) rather than a particle-based Fock space.
Resolution of the Information Paradox: By framing the final state as a pure state in an algebraic framework that is orthogonal to finite-particle states, the paper offers a mechanism for information preservation without requiring a violation of unitarity in the traditional sense.
Remaining Challenge: The author admits that while the algebraic structure supports unitarity, a definitive proof of Poincaré invariance for the S-matrix amplitudes within this specific algebraic model is still missing. The paper provides "suggestive arguments" but no rigorous proof that the constructed S-matrix commutes with Lorentz transformations.

Summary: Tom Banks argues that gravitational scattering in high dimensions inevitably produces states orthogonal to the Fock space due to soft radiation and black hole physics. Consequently, the S-matrix is not unitary in Fock space. He proposes an alternative algebraic scattering theory based on finite-dimensional quantum mechanics and causal diamonds, which recovers unitarity in a generalized "locally normal" state and converges to the AGS algebra, though full Poincaré covariance remains to be proven.