Soft Hair and Subregions in Quantum Gravity

Abstract

In quantum gravity and field theory, large gauge transformations lead to novel degrees of freedom living at the boundary. In the presence of a black hole they are called ‘soft hair’, and it has been suggested that they go some way towards answering the black hole information problem. More generally, they are known as ‘edge modes’, and play a key role in the structure of the quantum state. Another approach to black hole information involves entanglement, and a key theme of this thesis is to explore aspects of the relationship between edge modes and entanglement. We first establish a thermodynamical interpretation of soft hair by deriving generalisations of the laws of black hole mechanics. These laws lead to a natural definition of an entropy density on the black hole horizon, and reveal that soft hairs at neighbouring points are in thermal contact with one another. There are boundary ambiguities in the traditional construction of phase spaces in field theory, and resolving these ambiguities is a step that must be taken before one can fully understand edge modes and soft hair. We provide two possible approaches to such a resolution. The first approach applies to theories without gravity, and involves a direct evaluation of the Poisson structure from a semiclassical path integral. This is then inverted to give the symplectic structure. The second approach applies in holographic theories of quantum gravity. We show how one can recover the symplectic structure in a bulk subregion by measuring an object known as ‘Uhlmann holonomy’ on the boundary, which is a generalisation of Berry phase. The Uhlmann holonomy is actually a direct measure of the entanglement in the quantum state, and so this provides a connection between edge modes and entanglement. In the final part of the thesis we study Uhlmann phase more generally, showing that it may be computed with a holographic path integral in a generic system, so long as the state of the system involves a sufficient degree of entanglement. This suggests that there are deep connections between Uhlmann holonomy, entanglement and holography.