Lattice-Space Quantization of a Nonlinear Field Theory

A method for the approximate diagonalization of certain types of quantum field Hamiltonians is developed which is not limited to weakly nonlinear systems. It consists in omitting the gradient terms in zero order, and diagonalizing the resulting Hamiltonian by replacing the field defined in a continuum space by a field defined in a lattice space. This unperturbed system is equivalent to a countably infinite number of uncoupled nonlinear oscillators, which are then coupled together when the gradient terms are included as a perturbation. The method is applied to the quantization of the classical nonlinear meson theory that was introduced in an earlier paper to provide a qualitative explanation of the saturation of nuclear forces, according to which a positive ${\phi }^4$ term is added to the field Hamiltonian. Although the quantized theory is manifestly noncovariant, it is found that a single-particle solution exists that has an approximately relativistic relation between energy, momentum, and rest mass. It turns out to be essential that the lattice constant be kept finite, as all computed physical quantities become meaningless in the continuum limit (in which the lattice constant approaches zero). It is shown that these particles obey Einstein-Bose statistics, and that they scatter from each other. Nucleons are introduced as classical sources for the meson field, and calculations are made on the nucleon isobaric state, interaction of mesons with nucleons and heavy nuclei, and nucleon-nucleon interaction. Most of the results of the earlier classical theory have close counterparts in the present quantized theory. The possibility of extending the method to the quantization of both meson and nucleon fields when they are strongly coupled together is discussed briefly.