Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Generalized geometric commutator theory and quantum geometric bracket
  and its uses   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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The categories of this paper are: supr-con
11
The categories of this paper are: hep-lat
28
The categories of this paper are: alg-geom
33
The categories of this paper are: plasm-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: funct-an
24
The categories of this paper are: dg-ga
3
The categories of this paper are: physics
5
The categories of this paper are: gr-qc
32
The categories of this paper are: atom-ph
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The categories of this paper are: solv-int
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The categories of this paper are: chem-ph
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The categories of this paper are: stat
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The categories of this paper are: q-fin
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The categories of this paper are: ao-sci
8
The categories of this paper are: hep-ex