Chapter 23 g=0.25

we can construct the operator $F(\nabla^2\phi(t,\vec x))|_{p=0}$

$$\nabla^2\phi(t,\vec x)=\sum_\mu \phi(x+\mu)-2\phi(x)+\phi(x-\mu)\\ F(\nabla^2\phi(t,\vec x))|_{p=0}=\sum_{\vec x}\sum_\mu \phi(x+\mu)-2\phi(x)+\phi(x-\mu)\\ =\sum_{\vec x}\sum_{\vec\mu} \phi(t+\mu_0, \vec{x}+\vec{\mu})-2\phi(t,\vec{x})+\phi(t-\mu_0, \vec{x}-\vec{\mu})\\ =\tilde{\phi}(t+1)-2\tilde{\phi}(t)+\tilde{\phi}(t-1)$$ and the correlator

$$\langle F(\nabla^2\phi(t))|_{p=0}F(\nabla^2\phi(0))|_{p=0}\rangle\\ =\langle\tilde{\phi}(t)\tilde{\phi}(0) \rangle -2\langle\tilde{\phi}(t+1)\tilde{\phi}(0) \rangle +\langle\tilde{\phi}(t+2)\tilde{\phi}(0) \rangle\\ -2\langle\tilde{\phi}(t-1)\tilde{\phi}(0) \rangle +4\langle\tilde{\phi}(t)\tilde{\phi}(0) \rangle -2\langle\tilde{\phi}(t+1)\tilde{\phi}(0) \rangle\\ +\langle\tilde{\phi}(t-2)\tilde{\phi}(0) \rangle -2\langle\tilde{\phi}(t-1)\tilde{\phi}(0) \rangle +\langle\tilde{\phi}(t)\tilde{\phi}(0) \rangle\\ =6\langle\tilde{\phi}(t)\tilde{\phi}(0) \rangle-4\langle\tilde{\phi}(t+1)\tilde{\phi}(0) \rangle+\langle\tilde{\phi}(t+2)\tilde{\phi}(0) \rangle -4\langle\tilde{\phi}(t-1)\tilde{\phi}(0) \rangle+\langle\tilde{\phi}(t-2)\tilde{\phi}(0) \rangle$$

and construct the GEVP

$$\begin{pmatrix} \langle\tilde{\phi_0}(t)\tilde{\phi_0}(0) \rangle & \langle\tilde{\phi_0}(t)F(\nabla^2\phi_0(0)) \rangle & \langle\tilde{\phi_0}(t)\tilde{\phi_1}(0) \rangle \\ & \langle F(\nabla^2\phi_0(t))F(\nabla^2\phi_0(0)) & \langle F(\nabla^2\phi_0(t)) \tilde{\phi_1}(t) \rangle \rangle \\ & & \langle\tilde{\phi_0}(t)\tilde{\phi_0}(0) \rangle \end{pmatrix}$$