  Complex frequency modes occur for a scalar field near a rapidly rotating star
{\it with ergoregion but no event horizon}. Such complex frequency modes must
be included in the quantization of the field. As a model for this system, we
have investigated a real scalar field with mass $\mu $ in a one-dimensional
square-well potential. If the depth of the potential is greater than $\mu^2$,
then there exist imaginary frequency modes. It is possible to quantize this
simple system, but the mode operators for imaginary frequencies satisfy unusual
commutation relations and do not admit a Fock-like representation or a ground
state. Similar properties have been discussed already by Fulling for a complex
scalar field interacting with an external electrostatic potential. We are
interested in the field dynamics in the physical case where the initial state
of the quantum field is specified before the complex frequency modes develop.
As a model for this, we investigated a free scalar field whose ``mass" is
normal in the past and becomes ``tachyonic" in the future. A particle detector
in the far future placed in the in-vacuum state shows non-vanishing excitations
related to the imaginary frequency modes as well. Implications of these results
for the question of vacuum stability near rapidly rotating stars and possible
applications to other fields in physics are discussed briefly.
