  Wormhole solutions to the equations of general relativity have some
spectacular local and global properties. As these unusual features are not
explicitly forbidden by known physics, wormholes are considered in various
astrophysical and cosmological scenarios. The paradigmatic traversable wormhole
models are described by static spherically-symmetric Ellis--Morris--Thorne and
Simpson--Visser metrics. We show that no dynamical solution of the
semiclassical Einstein equations can have these metrics as their static limit.
On the other hand, possible static limits of the dynamical solutions are not
traversable. Moreover, they lead to violation of a quantum energy inequality
that bounds violations of the null energy condition by quantum fields. This
conclusion does not depend on specific properties of fields that may be
proposed for wormhole construction. As a result, spherically-symmetric
wormholes cannot exist in semiclassical gravity.
