Generalized geometric commutator theory and quantum geometric bracket and its uses

Abstract

Inspired by the geometric bracket for the generalized covariant Hamilton system, we abstractly define a generalized geometric commutator $$\left[ a,b \right]={{\left[ a,b \right]}_{cr}}+G\left(s,a,b \right)$$ formally equipped with geomutator $G\left(s, a,b \right)=a{{\left[ s,b \right]}_{cr}}-b{{\left[ s,a \right]}_{cr}}$ defined in terms of structural function $s$ related to the structure of spacetime or manifolds itself for revising the classical representation ${{\left[ a,b \right]}_{cr}}=ab-ba$ for any elements $a$ and $b$ of any algebra. Then we use the generalized geometric commutator to define quantum covariant Poisson bracket that is related to the quantum geometric bracket defined by geomutator as a generalization of quantum Poisson bracket. The covariant dynamics includes the generalized Heisenberg equation as a natural extension of Heisenberg equation and G-dynamics based on the quantum geometric bracket, meanwhile, the geometric canonical commutation relation is induced. As an application, we reconsider the canonical commutation relation and the quantization of field to be more complete.