States and Boundary Terms: Subtleties of Lorentzian AdS/CFT

TL;DR

This paper presents a self-contained Lorentzian AdS/CFT prescription built from the Schwinger variational principle, clarifying how boundary terms at initial and final times encode quantum states. It shows that CFT correlators can be computed from bulk path integrals over regions outside horizons and that leading-order one-point functions are controlled by simple boundary data at null infinity, independent of interior bulk details. The framework unifies Euclidean and Lorentzian AdS/CFT via analytic continuation and provides explicit semiclassical machinery, including a scalar-field toy model, to demonstrate how boundary data and advanced/retarded boundary conditions shape bulk-to-boundary propagators. These insights refine the bulk/boundary dictionary, especially in contexts with propagating bulk states and black hole horizons, and lay groundwork for controlled semiclassical calculations in Lorentzian AdS/CFT.

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.