The S-matrix and boundary correlators in flat space

TL;DR

This paper develops a framework to extract the S-matrix from boundary correlation functions in Minkowski spacetime, revealing how boundary data encodes bulk scattering and unitarity constraints.

Contribution

It introduces a novel boundary correlator approach to derive the S-matrix in flat space, extending concepts from AdS/CFT to Minkowski spacetime and analyzing singularities related to bulk scattering.

Findings

Boundary correlators encode the S-matrix after smearing.

Massive case: boundary correlators dominated by particle scattering saddle points.

Massless case: singularities in boundary correlators correspond to light-like scattering and recover S-matrix residues.

Abstract

We consider the path integral of a quantum field theory in Minkowski spacetime with fixed boundary values (for the elementary fields) on asymptotic boundaries. We define and study the corresponding boundary correlation functions obtained by taking derivatives of this path integral with respect to the boundary values. The S-matrix of the QFT can be extracted directly from these boundary correlation functions after smearing. We interpret this relation in terms of coherent state quantization and derive the constraints on the path-integral as a function of boundary values that follow from the unitarity of the S-matrix. We then study the locality structure of boundary correlation functions. In the massive case, we find that the boundary correlation functions for generic locations of boundary points are dominated by a saddle point which has the interpretation of particles scattering in a small elevator in the bulk, where the location of the elevator is determined dynamically, and the S-matrix can be recovered after stripping off some dynamically determined but non-local ``renormalization'' factors. In the massless case, we find that while the boundary correlation functions are generically analytic as a function on the whole manifold of locations of boundary points, they have special singularities on a sub-manifold, points on which correspond to light-like scattering in the bulk. We completely characterize this singular scattering sub-manifold, and find that the corresponding residues of the boundary correlations at these singularities are precisely given by S-matrices. This analysis parallels the analysis of bulk-point singularities in AdS/CFT and generalizes it to the case of multi-bulk point singularities.