Variational Calculation of the Effective Action

An indication of spontaneous symmetry breaking is found in the two-dimensional $\lambda\phi^4$ model, where attention is paid to the functional form of an effective action. An effective energy, which is an effective action for a static field, is obtained as a functional of the classical field from the ground state of the hamiltonian $H[J]$ interacting with a constant external field. The energy and wavefunction of the ground state are calculated in terms of DLCQ (Discretized Light-Cone Quantization) under antiperiodic boundary conditions. A field configuration that is physically meaningful is found as a solution of the quantum mechanical Euler-Lagrange equation in the $J\to 0$ limit. It is shown that there exists a nonzero field configuration in the broken phase of $Z_2$ symmetry because of a boundary effect.