Numerical study of truncated Green's-function equations

Abstract

We develop a Green's-function approximation scheme for the $\ensuremath{\lambda}{\ensuremath{\varphi}}^{2N}$ quantum field theory as an alternative to the usual perturbation expansion in powers of $\ensuremath{\lambda}$. The approximation scheme consists of introducing a source term into the Lagrangian, generating a sequence of coupled Green's function equations by repeated differentiation with respect to this source, and forcing the sequence to close by discarding the dependence on higher Green's functions. The validity of this procedure is then checked by using it to calculate the two-point Green's function in $\ensuremath{\lambda}{\ensuremath{\varphi}}^{4}$, $\ensuremath{\lambda}{\ensuremath{\varphi}}^{6}$, $\ensuremath{\lambda}{\ensuremath{\varphi}}^{8}$ theories in one-dimensional space-time. We compare our predictions of the locations of the low-lying poles with previously published tabulations of the eigenvalues of the equivalent quantum-mechanical anharmonic oscillators. The agreement is impressive.