22.2 Laplace smearing

We introduce a fictitious smearing time $\tau$ and we solve $$\partial_\tau \phi(\tau,t,\vec x)=\nabla^2 \phi(\tau,t,\vec x) \,\quad\mbox{with}\,\quad \phi(0,t,\vec x)=\phi(t,\vec x)\,.$$ this smearing do not effect the zero momentum component because the laplacian $\nabla^2=\partial_{x_1}^2+\partial_{x_2}^2+\partial_{x_3}^2$ in momentum space is zero $$\partial_\tau \tilde\phi(\tau,t,\vec p)=|\vec{p}|^2 \tilde\phi(\tau,t,\vec p)$$