  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.
