We propose new equations of motion under the theory of the Brownian motion to connect the states of quantum, diffusion, soliton, and periodic localization. The new equations are nothing but the classical equations of motion with two additional terms
and the one of them can be regarded as the the quantum potential. By choosing a parameter space, various important states are obtained. Further, the equations contain other interesting phenomena such as general dynamics of diffusion process, collapse of the soliton, the nonlinear extension of the Schrdinger equation, and the dynamics of phase transition.
We introduce new techniques that can preserve unitarity of the system including ghost particles. Negative norms of the particles can be involved in zero-norm states by constraints of the physical space. These are useful to apply the higher-derivative
propagator for quantum gravity to suppress divergences of vacuum energy and graviton mass correction. The quantum effects are mainly depending on the ghost mass scale. As the scale can be chosen in any order, the observed cosmological constant is realized. Further, applying ghost partners for the standard model particles, quantum gravity with matter fields becomes renormalizable with power counting arguments.
