Lectures on Holographic Renormalization

  • Ioannis PapadimitriouEmail author
Conference paper
Part of the Springer Proceedings in Physics book series (SPPHY, volume 176)


We provide a pedagogical introduction to the method of holographic renormalization, in its Hamiltonian incarnation. We begin by reviewing the description of local observables, global symmetries, and ultraviolet divergences in local quantum field theories, in a language that does not require a weak coupling Lagrangian description. In particular, we review the formulation of the Renormalization Group as a Hamiltonian flow, which allows us to present the holographic dictionary in a precise and suggestive language. The method of holographic renormalization is then introduced by first computing the renormalized two-point function of a scalar operator in conformal field theory and comparing with the holographic computation. We then proceed with the general method, formulating the bulk theory in a radial Hamiltonian language and deriving the Hamilton–Jacobi equation. Two methods for solving recursively the Hamilton–Jacobi equation are then presented, based on covariant expansions in eigenfunctions of certain functional operators on the space of field theory couplings. These algorithms constitute the core of the method of holographic renormalization and allow us to obtain the holographic Ward identities and the asymptotic expansions of the bulk fields.


Ward Identity Dilatation Operator Canonical Momentum Holographic Renormalization Holographic Dictionary 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



I would like to thank the International Institute of Physics, Natal, for the hospitality and financial support during the School on Theoretical Frontiers in Black Holes and Cosmology, June 8–19, 2015, where these lectures were delivered.


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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.SISSA and INFN - Sezione di TriesteTriesteItaly

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