Multiple time scales and the φ4 model of quantum field theory

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Abstract
Multiple time scales perturbation theory is applied to the weakly nonlinear φ4 quantum field theory model. The multiple time scales perturbation equations are solved to lowest order, leading to the removal of secular and quasisecular terms from the standard perturbative solution. This removal occurs in a manner similar to that developed in a previous quasisecular perturbative approach which focused on small energy denominators. The multiple time scales approach provides a better rationale for the quasisecular perturbation theory, as well as providing a systematic method which can be extended to higher orders in the coupling constant. It leads to the natural introduction of a first-order renormalized Hamiltonian, which is a well-defined self-adjoint operator on a certain Hilbert space of physical states. This renormalized Hamiltonian is a direct sum of Schrödinger Hamiltonians on N-particle subspaces, which describe the interactions of pairs of particles via a nonlocal potential.