States and Boundary Terms: Subtleties of Lorentzian AdS/CFT

TL;DR

This paper refines the Lorentzian AdS/CFT correspondence by analyzing boundary terms and states, demonstrating that correlators can be computed outside black hole horizons and depend only on boundary data at null infinity.

Contribution

It provides a self-contained Lorentzian formulation of AdS/CFT that avoids Euclidean continuation and clarifies the role of boundary terms and states in correlator computations.

Findings

Correlators can be computed using regions outside black hole horizons.

Boundary terms are determined by quantum states and ensure simple boundary conditions.

CFT one-point functions are given by boundary values of classical solutions at null infinity.

Abstract

We complete the project of specifying the Lorentzian AdS/CFT correspondence and its approximation by bulk semi-classical methods begun by earlier authors. At the end, the Lorentzian treatment is self-contained and requires no analytic continuation from the Euclidean. The new features involve a careful study of boundary terms associated with an initial time $t_-$ and a final time $t_+$. These boundary terms are determined by a choice of quantum states. The main results in the semi-classical approximation are 1) The times $t_\pm$ may be finite, and need only label Cauchy surfaces respectively to the past and future of the points at which one wishes to obtain CFT correlators. Subject to this condition on $t_\pm$, we provide a bulk computation of CFT correlators that is manifestly independent of $t_\pm$. 2) As a result of (1), all CFT correlators can be expressed in terms of a path integral over regions of spacetime {\it outside} of any black hole horizons. 3) The details of the boundary terms at $t_\pm$ serve to guarrantee that, at leading order in this approximation, any CFT one-point function is given by a simple boundary value of the classical bulk solution at null infinity, $I$. This work is dedicated to the memory of Bryce S. DeWitt. The remarks in this paper largely study the relation of the AdS/CFT dictionary to the Schwinger variational principle, which the author first learned from DeWitt as a Ph.D. student.