The G-dynamics of the QCPB theory

In this paper, we further study the G-dynamics newly emerged in the covariant dynamics defined by the quantum covariant Poisson bracket (QCPB) theory. We propose three new operators based on the G-dynamics, we find that the non-Hermitian operators are inevitable to appear in the framework of the QCPB. Then we seek the precise formulations of the G-dynamics to calculate the practical quantum problems, the G-dynamics of three types in terms of different Hamiltonian are discovered. The geometrinetic energy operator for quantum harmonic oscillator as an application is further calculated.