  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.

General Physics