Holographic renormalization as a canonical transformation

TL;DR

This paper generalizes holographic renormalization, showing it can be applied as a canonical transformation to make variational problems well-defined in various Hamiltonian systems beyond AdS spaces.

Contribution

It introduces a systematic, intrinsic procedure for well-defining variational problems via asymptotic solutions and boundary terms, extending holographic renormalization beyond AdS holography.

Findings

The method applies to non-AdS gravity, point particles, and strings.

Constructs the space of asymptotic solutions with a well-defined symplectic form.

Identifies boundary terms solving the Hamilton-Jacobi equation to ensure well-posed variational problems.

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.