Quartic interaction Hamiltonians and symmetry breaking

A Q-number to C-number transformation has been developed for infinite systems which both simplifies a quartic interaction Hamiltonian and defines the symmetry breaking. The development employs the Lie algebraic properties of the second quantized operators. For those quartic interaction Hamiltonians that allow a Q-number to C-number transform: (1) a reduced bilinear Hamiltonian; (2) a linear equation of motion; (3) source-sink behavior and symmetry breaking: (4) micro-canonical to canonical or grand canonical ensemble transition; and (5) the macroscopic potential for the system are obtained