We review the construction of a stationary Gaussian process $X(t)$ starting from the near-critical continuum scaling limit $Φ^h$ of the Ising magnetization and its relation to the mass spectrum of the relativistic quantum field theory associated to $Φ^h$. Then for the near-critical Ising model on $a \mathbb{Z}^2$ with external field $a^{15/8} h$, we study the renormalized magnetization along a vertical line (with horizontal coordinate approximately $t$) and prove that the limit as $a\downarrow 0$ is the same Gaussian process $X(t)$. We also explore the possible extension of this approach to dimensions $d > 2$.