Quantum Vacuum Instability of ``Eternal'' de Sitter Space

The Euclidean or Bunch-Davies 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. 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 $O(4)$ subgroup are broken by quantum fluctuations. Consequences of this result for cosmology and the problem of vacuum energy will be discussed.