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# Generalized geometric commutator theory and quantum geometric bracket and its uses

arXiv: Quantum Physics
Jan 2020

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 $G\left(s, a,b \right)=-G\left(s, b,a \right)$ 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. We find that geomutator $G\left(s, a,b \right)$ of any manifold which can be automatically chosen by $G\left(s, a,b \right)=a{{\left[ s,b \right]}_{cr}}-b{{\left[ s,a \right]}_{cr}}$. Then we use the generalized geometric commutator to define quantum covariant Poisson bracket that is closely related to the quantum geometric bracket defined by geomutator as a generalization of quantum Poisson bracket all associated with the structural function generated by the manifolds to study the quantum mechanics. We find that the covariant dynamics appears along with 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 naturally induced. As an application of quantum covariant Poisson bracket, we reconsider the canonical commutation relation and the quantization of field to be more complete and covariant, as a consequence, we obtain some useful results.

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