Application of the Variation Method to Field Quantization

Abstract

The object of this paper is the application of the variation method to a simple field theory. A representation is found in which the state functions are emphasized. Variational trial forms are chosen for these, and optimized by making the expectation value of the field Hamiltonian stationary. Only the simplest case of a neutral, spin-zero, boson field with a fourth-power self-coupling term is considered here, but it is hoped that with further elaboration this may in the future lead to a description of the multipion resonances. The variation method has the advantage of avoiding any limitation on the strength of the self-coupling. Explicit results are obtained for the vacuum, single-particle, and two-particle (scattering and bound) states, and comparison is made with the determinantal method. Finally, a criterion for the variational stability of the vacuum state is obtained.