  The Ising-Kac model is a variant of the ferromagnetic Ising model in which
each spin variable interacts with all spins in a neighbourhood of radius
$\gamma^{-1}$ for $\gamma \ll 1$ around its base point. We study the Glauber
dynamics for this model on a discrete two-dimensional torus $\mathbb{Z}^2/
(2N+1)\mathbb{Z}^2$, for a system size $N \gg \gamma^{-1}$ and for an inverse
temperature close to the critical value of the mean field model. We show that
the suitably rescaled coarse-grained spin field converges in distribution to
the solution of a non-linear stochastic partial differential equation.
  This equation is the dynamic version of the $\Phi^4_2$ quantum field theory,
which is formally given by a reaction diffusion equation driven by an additive
space-time white noise. It is well-known that in two spatial dimensions, such
equations are distribution valued and a Wick renormalisation has to be
performed in order to define the non-linear term. Formally, this
renormalisation corresponds to adding an infinite mass term to the equation. We
show that this need for renormalisation for the limiting equation is reflected
in the discrete system by a shift of the critical temperature away from its
mean field value.
