Gaussian effective potential on the Hamiltonian lattice.

Abstract

We compare the Gaussian effective potential in (1+1)-dimensional ${\ensuremath{\lambda}}_{0}$${\ensuremath{\varphi}}^{4}$ theory with a result obtained on a Hamiltonian lattice using a basis wave-functional expansion method. The lattice effective potential is obtained by exploiting the Symanzik relation between vacuum energy with a source J and the generating functional of the connected Green's functions W[J]. The vacuum energy eigenvalues are obtained by numerically diagonalizing the Hamiltonian matrix. The effective potential is then obtained using a functional Legendre transformation. We find that the Gaussian effective potential is visually indistinguishable from the lattice effective potential for large ${\ensuremath{\varphi}}_{\mathrm{cl}}$. For interest we also compute the one-loop effective potential and find it disagrees quite dramatically with the Gaussian and lattice effective potentials.