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Using the cluster expansions for n-point Green functions we derive a closedset of dynamical equations of motion for connected equal-time Green functionsby neglecting all connected functions higher than $4^{th}$ order for the$\lambda \Phi^4$-theory in $1+1$ dimensions. We apply the equations to theinvestigation of spontaneous ground state symmetry breaking, i.e. to theevaluation of the effective potential at temperature $T=0$. Within our momentumspace discretization we obtain a second order phase transition (in agreementwith the Simon-Griffith theorem) and a critical coupling of$\lambda_{crit}/4m^2=2.446$ as compared to a first order phase transition and$\lambda_{crit}/4m^2=2.568$ from the Gaussian effective potential approach.Comment: 25 Revtex pages, 5 figures available via fpt from the directory  ugi-94-11 of ftp@theorie.physik.uni-giessen.de as one postscript file (there  was a bug in our calculations, all numerical results and figures have changed  significantly), ugi-94-1

Connected Green function approach to symmetry breaking in? 1+1 4 -theory