Quantum field theory of self-interacting systems

Abstract

Two self-interacting quantum field theories arising from the interaction Lagrangian lambda phi 4 are discussed. Proceeding from exact particular solutions of the field equations which reduce to the positive and negative frequency solutions of the free field theory for lambda =0, the theories are quantized either by requiring (a) the commutator of the positive and negative frequency solutions to be independent of lambda at t=0 or (b) the propagator formed from the operator solutions of the field equations to be invariant with respect to translations. The solutions satisfying (a) are appropriate for scattering problems extending from t=0 to t= infinity while solutions satisfying (b) are valid from t=- infinity to t= infinity . The solutions satisfying (a) become independent of lambda for t=0 but continue to depend on lambda for large times while those satisfying (b) contain lambda for all times. Both types of solutions satisfy the interacting field equations for all times.