  It was previously shown by one of us that in any static,
non-globally-hyperbolic, spacetime it is always possible to define a sensible
dynamics for a Klein-Gordon scalar field. The prescription proposed for doing
so involved viewing the spatial derivative part, $A$, of the wave operator as
an operator on a certain $L^2$ Hilbert space $\mathcal H$ and then defining a
positive, self-adjoint operator on $\mathcal H$ by taking the Friedrichs
extension (or other positive extension) of $A$. However, this analysis left
open the possibility that there could be other inequivalent prescriptions of a
completely different nature that might also yield satisfactory definitions of
the dynamics of a scalar field. We show here that this is not the case.
Specifically, we show that if the dynamics agrees locally with the dynamics
defined by the wave equation, if it admits a suitable conserved energy, and if
it satisfies certain other specified conditions, then it must correspond to the
dynamics defined by choosing some positive, self-adjoint extension of $A$ on
$\mathcal H$. Thus, subject to our requirements, the previously given
prescription is the only possible way of defining the dynamics of a scalar
field in a static, non-globally-hyperbolic, spacetime. In a subsequent paper,
this result will be applied to the analysis of scalar, electromagnetic, and
gravitational perturbations of anti-de Sitter spacetime. By doing so, we will
determine all possible choices of boundary conditions at infinity in anti-de
Sitter spacetime that give rise to sensible dynamics.
