Holographic renormalization as a canonical transformation

TL;DR

The work argues that holographic renormalization is not restricted to AdS/CFT but is a universal mechanism for rendering certain noncompact variational problems well defined. It develops a general recipe: (i) construct a space of asymptotic solutions carrying a nondegenerate symplectic form, and (ii) derive a boundary term from the Hamilton–Jacobi equation that implements a canonical transformation, diagonalizing the asymptotic symplectic map and preserving the symplectic structure. This approach unifies the treatment across point particles, scalars in various backgrounds, gravity with constraints, and string worldsheets, and it naturally yields finite on-shell actions and well-defined path integrals. When a holographic dual exists, the canonical transformation aligns with the AdS/CFT dictionary and produces the correct renormalized observables, while in other contexts it clarifies the appropriate degrees of freedom and boundary data. The results thus provide a broadly applicable framework for well-posed variational problems in diverse Hamiltonian systems and illuminate the foundational role of boundary terms in quantum theories of gravity and strings.

Abstract

The gauge/string dualities have drawn attention to a class of variational problems on a boundary at infinity, which are not well defined unless a certain boundary term is added to the classical action. In the context of supergravity in asymptotically AdS spaces these problems are systematically addressed by the method of holographic renormalization. We argue that this class of a priori ill defined variational problems extends far beyond the realm of holographic dualities. As we show, exactly the same issues arise in gravity in non asymptotically AdS spaces, in point particles with certain unbounded from below potentials, and even fundamental strings in flat or AdS backgrounds. We show that the variational problem in all such cases can be made well defined by the following procedure, which is intrinsic to the system in question and does not rely on the existence of a holographically dual theory: (i) The first step is the construction of the space of the most general asymptotic solutions of the classical equations of motion that inherits a well defined symplectic form from that on phase space. The requirement of a well defined symplectic form is essential and often leads to a necessary repackaging of the degrees of freedom. (ii) Once the space of asymptotic solutions has been constructed in terms of the correct degrees of freedom, then there exists a boundary term that is obtained as a certain solution of the Hamilton-Jacobi equation which simultaneously makes the variational problem well defined and preserves the symplectic form. This procedure is identical to holographic renormalization in the case of asymptotically AdS gravity, but it is applicable to any Hamiltonian system.