  Let $\Phi^h(x)$ with $x=(t,y)$ denote the near-critical scaling limit of the
planar Ising magnetization field. We take the limit of $\Phi^h$ as the spatial
coordinate $y$ scales to infinity with $t$ fixed and prove that it is a
stationary Gaussian process $X(t)$ whose covariance function is the Laplace
transform of a mass spectral measure $\rho$ of the relativistic quantum field
theory associated to the Euclidean field $\Phi^h$. Our analysis of the small
distance/time behavior of the covariance functions of $\Phi^h$ and $X(t)$ shows
that $\rho$ is finite but has infinite first moment.
