Initial-value problems in quantum field theory in the large-Napproximation

Abstract

We derive the time-evolution equations appropriate to initial-value problems in λ(${\phi }_{\alpha }$ ${\phi }_{\alpha }$ ${)}^2$ field theory at large N. The Heisenberg equations of motion for this theory are compared to the Schrödinger equation for a wave functional constrained to be in a Gaussian state [the time-dependent Hartree-Fock (TDHF) ansatz]. The TDHF ansatz corresponds to a special choice of initial conditions for the general large-N Heisenberg equations of motion. The renormalization of the theory is discussed in both approaches and a simple method is given to arrive at finite differential equations suitable for numerical integration forward in time. A necessary and sufficient general condition for these equations to be finite is that the initial state contain a finite average number of particles and/or correlated particle pairs per unit volume.