# Extended phase space and edge modes **Extended phase space** is a way to understand the dynamics of a gauge theory in a finite region by studying the degrees of freedom on the boundary carefully. Because gauge theories have constraints, partitioning a Cauchy surface into two leads to subtleties arising from the fact that the data on both sides cannot be independently chosen. One way to deal with this is to add degrees of freedom which one calls **edge modes** to the boundary in a way that takes into account of the gauge transformations that act on the boundary. ## Related topics - [[0529 Corner symmetry]] - [[0176 Bulk interpretation of RT]] - [[0393 CLP extended phase space]] - [[0324 Covariant edge mode]] ## Refs - introduced in [[2016#Donnelly, Freidel]] - [[2017#Geiller (Mar)]] - 3d Abelian CS theory and 3d 1st order gravity - a specific corner ambiguity can be fixed by demanding that the covariant Hamiltonian framework be preserve by *finite field-dependent* gauge transformations, which can be done at the expense of introducing new boundary fields - [[Geiller201712]] - 3d gravity in 1st order connection triad variables - [[2017#Speranza]] - [[0006 Higher-derivative gravity|higher derivative gravity]] - BC chosen such that surface translations are symmetries => acquires a central extensions - [[BalasubramanianParrikar2018]] - open string - [[SetareAdami2018]] - extension to Einstein-Maxwell - [[2019#Geiller, Jai-akson]] - [[0089 Chern-Simons theory|CS]] and [[0557 BF theory|BF theories]] - [[2019#Speranza]] - embedding tools - [[2022#Mertens, Simon, Wong]] - [[2023#Klinger, Leigh, Pai]] - generalised geometrical framework that treats internal symmetries and diffeo on equal footing ## Quantum studies - Series papers by [[2020#Freidel, Geiller, Pranzetti (I-IV)]] - [[DonnellyFreidelMoosavianSperanza2020]] - [[2022#Donnelly, Freidel, Moosavian, Speranza]]: matrix quantisation ## Recent refs - [[MathieuTeh2021]][](https://arxiv.org/abs/2102.06799) - EM edge modes - [[Riello2021]][](https://arxiv.org/abs/2104.10182) - Edge modes without edge modes ## Related - [[0556 Edge mode]]