Lattice Space Quantization of Coupled Meson and Nucleon Fields

Abstract

The lattice space quantization method used by Schiff for treating nonlinear mesons is used in quantizing coupled meson and nucleon fields. Neutral pseudo-scalar mesons with pseudoscalar coupling to neutral Dirac nucleons are assumed. Only the fields at the lattice points of a cubic lattice are considered. An appropriate lattice space Hamiltonian is found and momentum (gradient) terms are treated as perturbations. To zero order the lattice points are uncoupled, so the state function can be written as a product of functions describing a single point. In the representation chosen, finding a particular point function entails the solution of a set of coupled differential equations. These equations can be solved in principle with no assumptions as to the magnitude of the coupling constant $g$ but require numerical integration, and have not been solved. Linear combinations of the zero-order solutions are found which diagonalize the perturbation to lowest order and can be interpreted as one-particle momentum eigen-states. The perturbation energy of these states includes a term proportional to ${k}^{2}$ which is interpreted as the kinetic energy. Linear combinations of two-particle momentum eigenstates are found which approximately diagonalize the perturbation and, from these, scattering cross sections are calculated. The lattice constant $l$ appears in the final results and must be regarded as a parameter of the theory. In order that the perturbation approximation be valid, $l$ must be much larger than the meson Compton wavelength.