Fuzzy Euclidean wormholes in anti-de Sitter space

This paper is devoted to an investigation of Euclidean wormholes made by fuzzy instantons. We investigate the Euclidean path integral in anti-de Sitter space. In Einstein gravity, we introduce a scalar field with a potential. Because of the analyticity, there is a contribution of complex-valued instantons, so-called fuzzy instantons. If we have a massless scalar field, then we obtain Euclidean wormholes, where the probabilities become smaller and smaller as the size of the throat becomes larger and larger. If we introduce a non-trivial potential, then in order to obtain a non-zero tunneling rate, we need to tune the shape of the potential. With the $O(4)$ symmetry, after the analytic continuation to the Lorentzian time, the wormhole throat should expand to infinity. However, by adding mass, one may obtain an instant wormhole that should eventually collapse to the event horizon. The existence of Euclidean wormholes is related to the stability or unitarity issues of anti-de Sitter space. We are not conclusive yet, but we carefully comment on these physical problems.