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

TL;DR

This paper argues that in higher-dimensional gravitational scattering, the S-matrix cannot be strictly unitary in Fock space due to black hole formation, suggesting a need for a broader framework for quantum gravity.

Contribution

It demonstrates the non-unitarity of the gravitational S-matrix in higher dimensions and explores algebraic and matrix model approaches to reconcile unitarity.

Findings

Finite energy states are normalizable in Fock space for d>4

Black hole physics implies orthogonality of states at infinite energy

S-matrix cannot be strictly unitary in Fock space in high dimensions

Abstract

We argue that for finite energy windows, the final states in gravitational scattering in dimension $d > 4$ are normalizable coherent states in Fock space. However, as the center of the energy window goes to infinity, black hole physics predicts that these states become orthogonal to every state with a finite number of particles. Given that the spectral measure in energy is determined by Poincare invariance, the S-matrix cannot be a unitary operator in Fock space, despite having finite matrix elements in Fock space, and satisfying perturbative unitarity, to all orders in string perturbation theory. We identify regimes in the BFSS matrix model\cite{bfss} and the definition of the S-matrix as the limit of CFT correlators\cite{polchsuss}, which point to the same conclusion. We review a scattering theory based on the quantum mechanics of a finite number of fermionic oscillators, whose algebra formally converges to the Super-Poincare covariant Awada-Gibbons-Shaw\cite{ags} algebra, and argue that a certain class of limiting states on that algebra satisfy all the properties required by physical unitarity in the algebraic formulation of quantum mechanics. The only missing ingredient for a consistent theory is a proof that the S matrix amplitudes themselves are Poincare invariant. We provide suggestive arguments, but no real proof, that this is so.