  The Euclidean or Bunch-Davies O(4,1) invariant 'vacuum' state of quantum
fields in global de Sitter space is shown to be unstable to small
perturbations, even for a massive free field with no self-interactions. There
are perturbations of this state with arbitrarily small energy density at early
times that is exponentially blueshifted in the contracting phase of 'eternal'
de Sitter space, and becomes large enough to disturb the classical geometry
through the semiclassical Einstein eqs. at later times. In the closely
analogous case of a constant, uniform electric field, a time symmetric state
equivalent to the de Sitter invariant one is constructed, which is also not a
stable vacuum state under perturbations. The role of a quantum anomaly in the
growth of perturbations and symmetry breaking is emphasized in both cases. In
de Sitter space, the same results are obtained either directly from the
renormalized stress tensor of a massive scalar field, or for massless conformal
fields of any spin, more directly from the effective action and stress tensor
associated with the conformal trace anomaly. The anomaly stress tensor shows
that states invariant under the O(4) subgroup of the de Sitter group are also
unstable to perturbations of lower spatial symmetry, implying that both the
O(4,1) isometry group and its O(4) subgroup are broken by quantum fluctuations.
Consequences of this result for cosmology and the problem of vacuum energy are
discussed.
