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 [a, b] = [a, b] cr + G (s, a, b) formally equipped with geomutator G (s, a, b) = −G (s, b, a) defined in terms of structural function s related to the structure of spacetime or manifolds itself for revising the classical representation [a, b] cr = ab − ba for any elements a and b of any algebra. We find that geomutator G (s, a, b) of any manifold which can be automatically chosen by G (s, a, b) = a[s, b] cr − b[s, a] 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.

The G-dynamics of the QCPB theory

“…As [3] stated, the quantum covariant Poisson bracket (QCPB) is defined by generalized geometric commutator (GGC) while quantum geometric bracket (QGB) is given based on the geomutator. More precisely, Definition 4 (QCPB).…”

“…This section will give the covariant dynamics which contains two different sub-dynamics: the generalized Heisenberg equation and G-dynamics [3]. It tells that the generalized Heisenberg equation has improved the classical Heisenberg equation by considering the quantum geometric bracket.…”

“…Definition 7 (QCHS). [3] The QCHS defined by QCPB in terms of quantum operator f , Ĥ is generally given by f , Ĥ = f , Ĥ…”

“…In this section, we will briefly review the entire theoretical framework of quantum covariant Hamiltonian system defined by the quantum covariant Poisson bracket totally based on the paper [3]. More precisely, the time covariant evolution of any observable f in the covariant dynamics is given by both the generalized Heisenberg equation of motion and G-dynamics.…”

“…In this section, we simply review some basic results given by the paper [3] in QCPB. As quantum mechanics illustrated, the commutator relations may look different than in the Schrödinger picture, because of the time dependence of operators.…”