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

Gen Wang<EMAIL_ADDRESS>
(School of Mathematical Sciences, Xiamen University,
Xiamen, 361005, P.R.China.)

###### 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 $G\left(s,a,b\right)=-G\left(s,b,a\right)$
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.

###### Contents

  1. 1 Introduction
    1. 1.1 Dirac’s method for canonical quantization
    2. 1.2 Canonical commutation relation
    3. 1.3 Groenewold’s theorem
  2. 2 Time-evolution of operators and state vectors
    1. 2.1 Three pictures
    2. 2.2 Heisenberg equation of motion
  3. 3 Quantization of Field
    1. 3.1 Canonical quantization of Bose field
    2. 3.2 Canonical quantization of Fermi field
  4. 4 Generalized structural Poisson bracket and geobracket
    1. 4.1 Generalized Covariant Hamilton System
  5. 5 Generalized geometric commutator and geomutator
    1. 5.1 Geomutator
    2. 5.2 Examples for GGC
  6. 6 Quantum covariant Poisson bracket
    1. 6.1 Examples of QCPB
    2. 6.2 Quantum geometric bracket
  7. 7 Quantum covariant Hamiltonian system
    1. 7.1 Covariant dynamics, generalized Heisenberg equation, G-dynamics
    2. 7.2 Imaginary geomenergy
    3. 7.3 Canonical quantum covariant Hamilton equations
    4. 7.4 The QCPB for quantum harmonic oscillator
    5. 7.5 The GSPB and the QCPB
    6. 7.6 Geobracket and quantum geobracket
  8. 8 Covariant quantization of Field
    1. 8.1 Canonical covariant quantization of Bose field
    2. 8.2 Canonical covariant quantization of Fermi field
  9. 9 Geomentum operator and geometric canonical commutation
  10. 10 Conclusions

##  1 Introduction

Our goal in this paper is to propose the following result.

######  Definition 1 (QCPB).

The quantum covariant Poisson bracket in terms of quantum operator
$\hat{f},~{}\hat{g}$ is generally defined by

$\left[\hat{f},\hat{g}\right]={{\left[\hat{f},\hat{g}\right]}_{QPB}}+G\left(s,\hat{f},\hat{g}\right)$

where $G\left(s,\hat{f},\hat{g}\right)$ is called quantum geometric bracket
satisfying $G\left(s,\hat{f},\hat{g}\right)=-G\left(s,\hat{g},\hat{f}\right)$.

Our method of new quantum bracket is inspired by the ideas of the GSPB
111GSPB: Generalized Structural Poisson Bracket; QCPB: Quantum Covariant
Poisson Bracket[22] which can copy the same pattern as Dirac [1, 2] did
before. This GSPB for the generalized covariant Hamilton system is one of the
main things that inspired me to deeply develop a new quantum bracket that is
complete and compatible. The quantum covariant Poisson bracket as a new tool
can mathematically help us better understand and analyze how the quantum
mechanics essentially works.

That is deep reason at the same time realizing that we’re probably will have
much time to reach an incredible field out there and we’re trying to be
optimistic to seek more secrets about quantum world. Quantum geometric bracket
defined by geomutator is such a beautiful thing ever existed in the universe.
New theory has begun to paint a stunning picture of how the quantum secrets
might unfold.

This quantum covariant Poisson bracket takes us on a part of journey to the
truth of quantum mechanics, to discover what the fate of quantum universe may
ultimately be. This is a picture of the covariant quantum mechanics as painted
by quantum geometric bracket—a picture that will surely evolve over time as we
dig for more clues to how our story will unfold. Much of the content of it is
very recent-and hence new puzzle pieces are still waiting to be found in
quantum realm. Indeed, this overhead view of quantum geometric bracket gives a
profound perspective-that the quantum world is closely connected with the
geometric structure.

###  1.1 Dirac’s method for canonical quantization

Dirac’s method for canonical quantization, when it was firstly developed,
quantum physics dealt only with the quantization of the motion of particles.
By conventionally stated, the original form of particle quantum mechanics is
denoted as first quantization, while quantum field theory is formulated in the
language of second quantization.

The following exposition is based on Dirac’s treatise on quantum mechanics for
canonical quantization. According to Dirac’s method [1, 2], Poisson’s bracket
$\left\\{\cdot,\cdot\right\\}_{PB}$ is connected with commutator
$\frac{1}{\sqrt{-1}\hbar}\left[\cdot,\cdot\right]_{QPB}$:

${{\left[\hat{f},\hat{g}\right]}_{QPB}}=\sqrt{-1}\hbar\widehat{{{\left\\{f,g\right\\}}_{PB}}}$

where $\hat{\cdot}:f\mapsto\hat{f}$ is mapping real valued functions
$f\left({{q}_{i}},{{p}_{j}}\right)$ in phase space to real operators $\hat{f}$
in Hilbert space.

Recall that the relation of classical and quantum brackets. Dirac’s book [1,
2] details his popular rule of supplanting Poisson brackets by commutators:

$\\{f,g\\}_{PB}\longmapsto\tfrac{1}{\sqrt{-1}\hbar}[\hat{f},\hat{g}]_{QPB}$
(1)

One might interpret this proposal as saying that we should seek a quantization
map $Q$ mapping a function $f$ on the classical phase space to an operator
${\displaystyle Q_{f}}$ on the quantum Hilbert space such that [1, 2, 8, 9]

${\displaystyle
Q_{\\{f,g\\}_{PB}}={\frac{1}{\sqrt{-1}\hbar}}[Q_{f},Q_{g}]_{QPB}}$

It is now known that there is no reasonable such quantization map satisfying
the above identity exactly for all functions $f$ and $g$.

In quantum mechanics, quantum Poisson brackets
$\left[\hat{f},\hat{g}\right]_{QPB}$ is the commutator of two operators
$\hat{f},~{}\hat{g}$ defined as

$\left[\hat{f},\hat{g}\right]_{QPB}=\hat{f}\hat{g}-\hat{g}\hat{f}$

This relation also holds for classical mechanics, the classical limit of the
above, given the correspondence between Poisson brackets and commutators, [3,
4, 5, 6, 7]

${\displaystyle[\hat{f},\hat{H}]_{QPB}\quad\longleftrightarrow\quad\sqrt{-1}\hbar\\{f,H\\}}_{PB}$

In classical mechanics, for a function $f$ with no explicit time dependence,

${\frac{df}{dt}}={\displaystyle\\{f,H\\}_{PB}}$

This approach also has a more direct similarity to classical physics: by
simply replacing the commutator above by the Poisson bracket, the Heisenberg
equation reduces to an equation in Hamiltonian mechanics. The Dirac bracket is
a generalization of the Poisson bracket developed by Paul Dirac to treat
classical systems with second class constraints in Hamiltonian mechanics, and
to thus allow them to undergo canonical quantization. It is an important part
of Dirac’s development of Hamiltonian mechanics to elegantly handle more
general Lagrangians.

The possible results of a measurement are the eigenvalues of the operator
representing the observable which explains the choice of Hermitian operators,
for which all the eigenvalues are real. The probability distribution of an
observable in a given state can be found by computing the spectral
decomposition of the corresponding operator. Heisenberg’s uncertainty
principle is represented by the statement that the operators corresponding to
certain observables do not commute.

In quantum mechanics in the Heisenberg picture the state vector,
$\left|\psi\right\rangle$ does not change with time, while an observable
$\hat{A}$ is a physical quantity that can be measured which satisfies the
Heisenberg equation of motion [1, 2, 3, 4, 5, 6, 7].

$\frac{d\hat{A}}{dt}={\sqrt{-1}\over\hbar}[\hat{H},\hat{A}]_{QPB}+\frac{\partial\hat{A}}{\partial
t}~{}$

Usually, observables are given by self-adjoint operators that include position
and momentum. In systems governed by classical mechanics, it is a real-valued
function on the set of all possible system states. In quantum physics, it is
an operator, where the property of the quantum state can be determined by some
sequence of operations. For example, these operations might involve submitting
the system to various electromagnetic fields and eventually reading a value.
Dynamical variables $\hat{A}$ such as position, translational linear momentum,
orbital angular momentum, spin, and total angular momentum are each associated
with a Hermitian operator ${\hat{A}}$ that acts on the state of the quantum
system.

A crucial difference between classical quantities and quantum mechanical
observables is that the latter may not be simultaneously measurable, a
property referred to as complementarity. This is mathematically expressed by
non-commutativity of the corresponding operators, to the effect that the
commutator

${\displaystyle\left[\hat{A},\hat{B}\right]_{QPB}=\hat{A}\hat{B}-\hat{B}\hat{A}\neq
0.}$

This inequality expresses a dependence of measurement results on the order in
which measurements of observables $\hat{A}$ and $\hat{B}$ are performed.
Observables corresponding to non-commutative operators are called
incompatible.

Commutator relations may look different than in the Schrödinger picture,
because of the time dependence of operators. For example, consider the
operators $x(t_{1}),x(t_{2}),{{\hat{p}}}^{{\left(cl\right)}}(t_{1})$ and
${{\hat{p}}}^{{\left(cl\right)}}(t_{2})$. The time evolution of those
operators depends on the Hamiltonian of the system. Considering the one-
dimensional harmonic oscillator, [4, 5, 6, 7]

$\hat{H}={\frac{{{\hat{p}}}^{{\left(cl\right)2}}}{2m}}+{\frac{m\omega^{{2}}x^{{2}}}{2}}$
(2)

the evolution of the position and momentum operators is given by:

${d\over
dt}x(t)={\sqrt{-1}\over\hbar}[\hat{H},x(t)]_{QPB}={\frac{{{\hat{p}}}^{{\left(cl\right)}}}{m}}$
(3) ${d\over
dt}{{\hat{p}}}^{{\left(cl\right)}}(t)={\sqrt{-1}\over\hbar}[\hat{H},{{\hat{p}}}^{{\left(cl\right)}}(t)]_{QPB}=-m\omega^{{2}}x$

As for the application of the QPB or the commutation, there are some many
fields including the physics and mathematics, and so on. In particular, the
SUSY, but there are still problems unsolved. This is why we reconsider the QPB
or the commutation.

###  1.2 Canonical commutation relation

The commutator of two operators acting on a Hilbert space is a central concept
in quantum mechanics, since it quantifies how well the two observables
described by these operators can be measured simultaneously. The uncertainty
principle is ultimately a theorem about such commutators, by virtue of the
Robertson-Schrödinger relation. In phase space, equivalent commutators of
function star-products are called Moyal brackets, and are completely
isomorphic to the Hilbert-space commutator structures mentioned.

The canonical commutation relation [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13,
14, 15] is the fundamental relation between canonical conjugate quantities.
For example, a set of equal-time commutation relations are introduced as

$\left[\hat{{{x}_{i}}},{{\hat{{{p}_{j}}}}^{\left(cl\right)}}\right]_{QPB}=\sqrt{-1}\hbar{{\delta}_{ij}}$
(4)

between the position operator $\hat{{{x}_{i}}}$ and momentum operator
${{\hat{{{p}_{j}}}}^{\left(cl\right)}}$ in the $\hat{{{x}_{i}}}$ direction of
a point particle in three dimension, where
$\left[\hat{{{x}_{i}}},{{\hat{{{p}_{j}}}}^{\left(cl\right)}}\right]_{QPB}=\hat{{{x}_{i}}}{{\hat{{{p}_{j}}}}^{\left(cl\right)}}-{{\hat{{{p}_{j}}}}^{\left(cl\right)}}\hat{{{x}_{i}}}$
is the commutator of $\hat{{{x}_{i}}}$ and
${{\hat{{{p}_{j}}}}^{\left(cl\right)}}$, $\hbar$ is the reduced Planck’s
constant, where $\delta_{ij}$ is the Kronecker delta. This observation led
Dirac [1, 2] to propose that the quantum counterparts ${\hat{f}},{\hat{g}}$ of
classical observables $f,g$ satisfy [1, 2, 3, 4]

$[\hat{f},\hat{g}]_{QPB}=\sqrt{-1}\hbar\widehat{\\{f,g\\}}_{PB}$ (5)

###  1.3 Groenewold’s theorem

In 1946, Hip Groenewold [8] demonstrated that a general systematic
correspondence between quantum commutators and Poisson brackets could not hold
consistently. However, he did appreciate that such a systematic correspondence
does, in fact, exist between the quantum commutator and a deformation of the
Poisson bracket, the Moyal bracket, and, in general, quantum operators and
classical observables and distributions in phase space. He thus finally
elucidated the correspondence mechanism, the Wigner-Weyl transform, that
underlies an alternate equivalent mathematical representation of quantum
mechanics known as deformation quantization.

Let us accept the following ground rules for the map $Q$ [8]. Firstly, $Q$
should send the constant function 1 to the identity operator. Secondly, $Q$
should take $x$ and $p$ to the usual position and momentum operators $x$ and
$\hat{p}$. Thirdly, $Q$ should take a polynomial in $x$ and $p$ to a
polynomial in $x$ and $\hat{p}$, that is, a finite linear combinations of
products of $x$ and $\hat{p}$, which may be taken in any desired order. The
Groenewold’s theorem says that there is no map satisfying the above ground
rules and also the bracket condition [8]

${\displaystyle
Q_{\\{f,g\\}_{PB}}={\frac{1}{\sqrt{-1}\hbar}}[Q_{f},Q_{g}]_{QPB}}$ (6)

for all polynomials $f$ and $g$.

Actually, the nonexistence of such a map occurs already by the time we reach
polynomials of degree four. Note that the Poisson bracket of two polynomials
of degree four has degree six, so it does not exactly make sense to require a
map on polynomials of degree four to respect the bracket condition. The
Groenewold’s theorem is given as follows:

######  Theorem 1 (Groenewold’s theorem).

[8] There is no quantization map $Q$ on polynomials of degree less than or
equal to four that satisfies

${\displaystyle\quad
Q_{\\{f,g\\}_{PB}}={\frac{1}{\sqrt{-1}\hbar}}[Q_{f},Q_{g}]_{QPB}}$

whenever $f$ and $g$ have degree less than or equal to three.

Note that in this case, ${\displaystyle\\{f,g\\}_{PB}}$ has degree less than
or equal to four.

Axioms for quantization have been stated that if $Q$ represents the
quantization map that acts on functions $f$ in classical phase space, then the
following properties are usually considered desirable [15]:

  1. 1.

$Q_{x}\psi=x\psi$ and $Q_{p}\psi=-\sqrt{-1}\hbar\partial_{x}\psi$ (elementary
position/momentum operators).

  2. 2.

$f\longmapsto Q_{f}$ is a linear map.

  3. 3.

$[Q_{f},Q_{g}]_{QPB}=\sqrt{-1}\hbar Q_{\\{f,g\\}_{PB}}~{}~{}$ (Poisson
bracket)

  4. 4.

$Q_{g\circ f}=g(Q_{f})$ (von Neumann rule).

##  2 Time-evolution of operators and state vectors

In quantum mechanics, there are two parameters for describing the quantum
situation shown as operators $\hat{F}$ and state vectors
$\left|\psi\right\rangle$, the form operators $\hat{F}$ correspond to the
classical mechanical quantities, the latter $\left|\psi\right\rangle$ is for
the quantum state.

The operator corresponding to any mechanical quantity $F$ is expressed as
$\hat{F}$ in quantum mechanics, so there is $F\to\hat{F}$, namely it’s shown
as [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]

$F=F\left(r,p\right)\to\hat{F}=\hat{F}\left(r,\hat{P}\right)=\hat{F}\left(r,-\sqrt{-1}\hbar\nabla\right)$

###  2.1 Three pictures

For the Schrödinger picture, time-evolution of operators and state vectors
respectively satisfy the conditions
$\frac{\partial\left|\psi\right\rangle}{\partial t}\neq
0,\frac{\partial\hat{F}}{\partial t}=0$. We have seen that in the Schrödinger
scheme the dynamical variables of the system remain fixed during a period of
undisturbed motion. However, this is not the only way in which to represent
the time evolution of the system. For the Heisenberg picture, they satisfy the
conditions $\frac{\partial\left|\psi\right\rangle}{\partial
t}=0,\frac{\partial\hat{F}}{\partial t}\neq 0$. For the interaction picture,
the conditions are shown as $\frac{\partial\left|\psi\right\rangle}{\partial
t}\neq 0,\frac{\partial\hat{F}}{\partial t}\neq 0$. As we can see that the
interaction picture is an intermediate representation between the Schrödinger
picture and the Heisenberg picture. Whereas in the other two pictures either
the state vector $\left|\psi\left(t\right)\right\rangle$ or the operators
$\hat{F}$ carry time dependence, in the interaction picture both carry part of
the time dependence of observables [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13,
14, 15].

The Schrödinger picture is a formulation of quantum mechanics in which the
state vectors $\left|\psi\left(t\right)\right\rangle$ evolve in time, but the
operators $\hat{F}$ are constant with respect to time

$\frac{\partial\left|\psi\right\rangle}{\partial t}\neq
0,~{}~{}~{}\frac{\partial\hat{F}}{\partial t}=0$

This differs from the Heisenberg picture which keeps the states constant while
the observables $\hat{F}$ evolve in time. The operators $\hat{F}$ incorporate
a dependency on time, but the state vectors
$\left|\psi\left(t\right)\right\rangle$ are time-independent.

$\frac{\partial\left|\psi\right\rangle}{\partial
t}=0,~{}~{}~{}\frac{\partial\hat{F}}{\partial t}\neq 0$

namely the state vectors $\left|\psi\left(t\right)\right\rangle$ do not change
with time $\frac{\partial\left|\psi\right\rangle}{\partial t}=0$, while
observables $\hat{A}$ satisfy the Heisenberg equation

${\frac{d}{dt}}\hat{A}(t)={\frac{\sqrt{-1}}{\hbar}}[\hat{H},\hat{A}(t)]_{QPB}+\left({\frac{\partial\hat{A}}{\partial
t}}\right)_{\hat{H}}$

where $\hat{H}$ is the Hamiltonian and $\left[\cdot,~{}\cdot\right]_{QPB}$
denotes the commutator of two operators, in this case, two operators are
$\hat{H}$ and $\hat{A}$, respectively.

The interaction picture in which both the states and the observables evolve in
time.

$\frac{\partial\left|\psi\right\rangle}{\partial t}\neq
0,~{}~{}~{}\frac{\partial\hat{F}}{\partial t}\neq 0$

The Schrödinger and Heisenberg pictures are related as active and passive
transformations and commutation relations between operators are preserved in
the passage between the two pictures.

Evolution | Schrödinger | Heisenberg
---|---|---
Ket state | $|\psi(t)\rangle=U(t,t_{0})|\psi(t_{0})\rangle$ | $\frac{\partial\left|\psi\right\rangle}{\partial t}=0$
Observable | $\frac{\partial\hat{A}}{\partial t}=0$ | ${\frac{d}{dt}}\hat{A}(t)={\frac{\sqrt{-1}}{\hbar}}[\hat{H},\hat{A}(t)]_{QPB}$
Table 1: evolutions between Schrödinger and Heisenberg pictures

The Heisenberg picture does not distinguish time from space, so it is better
suited to relativistic theories than the Schrödinger equation. Moreover, the
similarity to classical physics is more manifest: the Hamiltonian equations of
motion for classical mechanics are recovered by replacing the commutator above
by the Poisson bracket. By the Stone-von Neumann theorem, the Heisenberg
picture and the Schrödinger picture must be unitarily equivalent.

###  2.2 Heisenberg equation of motion

In the formalism of quantum mechanics, the state of a system at a given time
is described by a complex wave function, also referred to as state vector in a
complex vector space.

Consider a dynamical variable $\hat{f}$ corresponding to a fixed linear
operator in the Schrödinger picture. Heisenberg equation of motion can be
written [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]

$\frac{d\hat{f}}{dt}=\frac{1}{\sqrt{-1}\hbar}{{\left[\hat{f},\hat{H}\right]}_{QPB}}$
(7)

where $\hat{H}$ is the Hamiltonian and $\left[\cdot,~{}\cdot\right]_{QPB}$
denotes the commutator of two operators. For the time-independent operators
$\hat{A}$, we have $\partial\hat{A}/\partial t=0$, therefore, the Heisenberg
equation above reduces to:

$\sqrt{-1}\hbar{d\hat{A}\over
dt}=[\hat{A},\hat{H}]_{QPB}=\hat{A}\hat{H}-\hat{H}\hat{A}$

Above equation shows how the dynamical variables of the system evolve in the
Heisenberg picture. It is denoted the Heisenberg equation of motion. Note that
the time-varying dynamical variables in the Heisenberg picture are usually
called Heisenberg dynamical variables to distinguish them from Schrödinger
dynamical variables, which do not evolve in time. Canonical Hamilton equations
of motion is represented by [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,
15]

$\displaystyle\frac{d}{dt}{{\hat{q}}_{\alpha}}={\dot{{\widehat{q}}_{\alpha}}}=\frac{1}{\sqrt{-1}\hbar}{{\left[{{\hat{q}}_{\alpha}},\hat{H}\right]}_{QPB}}$
$\displaystyle\frac{d}{dt}{{\hat{p}}_{\alpha}}={\dot{{\widehat{p}}_{\alpha}}}=\frac{1}{\sqrt{-1}\hbar}{{\left[{{\hat{p}}_{\alpha}},\hat{H}\right]}_{QPB}}$

As we know, the Poisson bracket and Lie bracket have ingenious similarity, the
formal is for classic mechanics, while the latter is used for the quantum
mechanics. In quantum mechanics that a mechanical quantity $f$ is described by
the corresponding operator $\hat{f}$, canonical quantization rule is commonly
used as

$\left\\{f,H\right\\}_{PB}\to\frac{1}{\sqrt{-1}\hbar}\left[\hat{f},\hat{H}\right]_{QPB}\equiv\frac{1}{\sqrt{-1}\hbar}\left(\hat{f}\hat{H}-\hat{H}\hat{f}\right)$

therefore, the condition for the conserved quantity is
$\left[\hat{f},\hat{H}\right]_{QPB}=0$, it’s also known as commutation.
Essentially, this is the Heisenberg picture for an observable $\widehat{f}$ or
operator. The Heisenberg picture is a formulation of quantum mechanics, where
the operators incorporate a dependency on time, but the state vectors are
time-independent, an arbitrary fixed basis rigidly underlying the theory. It
stands in contrast to the Schrödinger picture in which the operators are
constant, instead, and the states evolve in time. The two pictures only differ
by a basis change with respect to time-dependency, which corresponds to the
difference between active and passive transformations. The Heisenberg picture
is the formulation of matrix mechanics in an arbitrary basis, in which the
Hamiltonian is not necessarily diagonal. It further serves to define a third
picture, the interaction picture. In the Heisenberg picture of quantum
mechanics the state vectors do not change with time, while observables
$\hat{f}$ satisfy (7),

Operationally, we can interpret the relationship of dynamical variables
between classical mechanics and quantum mechanics by means of a tabulation.

c and q | Heisenberg’s form of QM | PH form of classical mechanics
---|---|---
Commutators | $\left[\overset{\wedge}{\mathop{{{q}_{i}}}}\,,\overset{\wedge}{\mathop{{{p}_{j}}}}\,\right]_{QPB}=\sqrt{-1}\hbar{{\delta}_{ij}}$ | $\left\\{{{q}_{i}},{{p}_{j}}\right\\}_{PB}={{\delta}_{ij}}$
Hamiltonian | $H\left(\overset{\wedge}{\mathop{q}}\,,\overset{\wedge}{\mathop{p}}\,\right)$ | $H\left(q,p\right)$
Evolution | $\overset{\centerdot}{\mathop{\overset{\wedge}{\mathop{q}}\,}}\,=\frac{1}{\sqrt{-1}\hbar}\left[\overset{\wedge}{\mathop{q}}\,,\overset{\wedge}{\mathop{H}}\,\right]_{QPB},\overset{\centerdot}{\mathop{\overset{\wedge}{\mathop{p}}\,}}\,=\frac{1}{\sqrt{-1}\hbar}\left[\overset{\wedge}{\mathop{p}}\,,\overset{\wedge}{\mathop{H}}\,\right]_{QPB}$ | $\overset{\centerdot}{\mathop{q}}\,=\left\\{q,H\right\\}_{PB},~{}\overset{\centerdot}{\mathop{p}}\,=\left\\{p,H\right\\}_{PB}$
Table 2: The CHS corresponds to the QM, Where
$\left\\{F,H\right\\}_{PB}\equiv\left(\frac{\partial
F}{\partial{{q}_{\alpha}}}\frac{\partial
H}{\partial{{p}_{\alpha}}}-\frac{\partial
H}{\partial{{q}_{\alpha}}}\frac{\partial F}{\partial{{p}_{\alpha}}}\right)$.
PH is an abbreviation of Poission-Hamilton.

This paper will follow the step of Dirac’s method to complete the some rules
of quantum mechanics. The goal of this paper is to define a completely new
bracket which is called quantum covariant Poisson bracket (QCPB) to analyze
the incomplete or missing parts of quantum mechanics.

##  3 Quantization of Field

As we already know, quantization procedure is to generally convert classical
fields into operators acting on quantum states of the field theory. The first
method to be developed for quantization of field theories was canonical
quantization. While this is extremely easy to implement on sufficiently simple
theories, there are many situations where other methods of quantization yield
more efficient procedures for computing quantum amplitudes. Canonical
quantization of a field theory is analogous to the construction of quantum
mechanics from classical mechanics. The classical field is treated as a
dynamical variable called the canonical coordinate, and its time-derivative is
the canonical momentum. One introduces a commutation relation between these
which is exactly the same as the commutation relation between a particle’s
position and momentum in quantum mechanics. This procedure can be applied to
the quantization of any field theory: whether of fermions or bosons, and with
any internal symmetry.

The canonical quantization is a standard procedure for quantizing a classical
theory, while attempting to preserve the formal structure, such as symmetries.
The canonical arises from the Hamiltonian approach to classical mechanics,
where a system’s dynamics is generated via canonical Poisson brackets, a
structure which is only partially preserved in canonical quantization. In
field theory which is to describe the motion of field of field quantities
${{\varphi}^{\alpha}}\left(x\right)\left(x=\left(t,\overrightarrow{x}\right),~{}\alpha=1,2,\cdots
n\right)$ perhaps are multiple components such as the field quantities
${{A}_{\mu}}\left(x\right)$ of electromagnetic field have four components,
here $\alpha$ represents the tensors, spinors, isospin or unitary spin, ect.

###  3.1 Canonical quantization of Bose field

The quantum of a Bose field, the spin is integer or zero, obeying the Bose -
Einstein statistical method, known as bosons, the canonical coordinate
${{\varphi}_{i}}$ and the canonical conjugate momentum ${{p}_{i}}$ of the
field are regarded as the operator of Hilbert space, and the corresponding
quantum commutation relation in discrete field form is [16, 17]

$\left[{{\varphi}_{i}},{{p}_{j}}\right]_{QPB}=\sqrt{-1}\hbar{{\delta}_{ij}}$
$\left[{{\varphi}_{i}},~{}~{}{{\varphi}_{j}}\right]_{QPB}=\left[{{p}_{i}},{{p}_{j}}\right]_{QPB}=0$

Canonical equations of motion for corresponding quantum fields are shown as

$\overset{\centerdot}{\mathop{\varphi}}\,=\frac{\sqrt{-1}}{\hbar}\left[\hat{H},\varphi\right]_{QPB},~{}~{}\overset{\centerdot}{\mathop{\pi}}\,=\frac{\sqrt{-1}}{\hbar}\left[\hat{H},\pi\right]_{QPB}$

The quantization of the field depends entirely on the commutators.

###  3.2 Canonical quantization of Fermi field

A fermion is a particle that follows Fermi-Dirac statistics. These particles
obey the Pauli exclusion principle. Fermions include all quarks and leptons,
as well as all composite particles made of an odd number of these, such as all
baryons and many atoms and nuclei. Fermions differ from bosons, which obey
Bose-Einstein statistics.

Suppose that $\Psi\left(t,x\right)$ is the field function of the Fermi field,
and the canonical coordinates and canonical conjugate momenta of the field are
$\Psi\left(t,x\right)$, $\pi\left(t,x\right)$ respectively, the canonical
motion equation of quantum field is [16, 17]

$\displaystyle\overset{\centerdot}{\mathop{\Psi}}\,\left(t,x\right)=\frac{\sqrt{-1}}{\hbar}\left[\hat{H}\left(t\right),\Psi\left(t,x\right)\right]_{QPB}$
$\displaystyle\overset{\centerdot}{\mathop{\pi}}\,\left(t,x\right)=\frac{\sqrt{-1}}{\hbar}\left[\hat{H}\left(t\right),\pi\left(t,x\right)\right]_{QPB}$

##  4 Generalized structural Poisson bracket and geobracket

In this section, we will briefly review the entire theoretical framework of
generalized covariant Hamiltonian system defined by the generalized structural
Poisson bracket totally based on the paper [22] as a revision of generalized
Poisson bracket [see [22] for more details].

To begin with the generalized Poisson bracket [18, 19, 20] which is defined as
the bilinear operation

${{\left\\{f,g\right\\}}_{GPB}}={{\nabla}^{T}}fJ\nabla
g={{J}_{ij}}\frac{\partial f}{\partial{{x}_{i}}}\frac{\partial
g}{\partial{{x}_{j}}}$

where structural matrix $J$ satisfies antisymmetric
${{J}_{ij}}={{\left\\{{{x}_{i}},{{x}_{j}}\right\\}}_{GPB}}=-{{J}_{ji}}$.

Generalized Hamiltonian system (GHS) is defined as [18, 22]

${\dot{x}}=\frac{dx}{dt}=J\left(x\right)\nabla
H\left(x\right),~{}~{}~{}x\in{{\mathbb{R}}^{m}}$

where $J\left(x\right)$ is structure matrix, $\nabla H\left(x\right)$ is the
gradient of the function Hamilton. Equivalently, Hamiltonian equations can be
further written as:

$\left\\{{{x}_{i}},H\right\\}_{GPB}={{{J}_{ij}}\frac{\partial
H}{\partial{{x}_{j}}}}$

Let $M$ be a smooth manifold and let $s$ be a smooth real structural function
on $M$ which is completely determined by the structure of manifold $M$.
Without loss of generality, we give a GSPB222GPB: Generalized Poisson bracket;
GSPB: Generalized structural Poisson bracket; GCHS: Generalized Covariant
Hamilton System; TGHS: thorough generalized Hamiltonian system; PB: Poisson
bracket; QCPB: Quantum Covariant Poisson Bracket; GGC : generalized geometric
commutator,  in a abstract covariant form,

######  Definition 2.

[22] The GSPB of two functions $f,g\in{{C}^{\infty}}\left(M,\mathbb{R}\right)$
on $M$ is defined as

$\left\\{f,g\right\\}={{\left\\{f,g\right\\}}_{GPB}}+G\left(s,f,g\right)$

where $G\left(s,f,g\right)=-G\left(s,g,f\right)$ is called geometric bracket
or geobracket.

It is remarkable to see that the GSPB representation admits a dynamical
geometric bracket formula instead of a generalized Poisson bracket structure.
In this abstract representation of the generalized structural Poisson bracket,
we see that geobracket equals

$G\left(s,f,g\right)=\left\\{f,g\right\\}-{{\left\\{f,g\right\\}}_{GPB}},~{}~{}f,g\in{{C}^{\infty}}\left(M,\mathbb{R}\right)$

which always satisfies the covariant condition $G\left(s,f,g\right)\neq 0$.
Figuratively, structure function $s$ is like a stage or the background for the
movement of the objects.

In the following discussions, we mainly discuss an accurate expression of the
geobracket, maybe it’s one of situation in the general covariant form.

######  Theorem 2 (GSPB).

[22] The geobracket is expressed as

$\displaystyle
G\left(s,f,g\right)=f{{\left\\{s,g\right\\}}_{GPB}}-g{{\left\\{s,f\right\\}}_{GPB}}$

for two functions $f,g\in{{C}^{\infty}}\left(M,\mathbb{R}\right)$ on $M$.

Intuitively, one can understand this defined formulation by noting that the
second part, geobracket, can be treated as the second part of the geodesic
equation, and in so doing will make it clear. The GSPB depends smoothly on
both ${{\left\\{f,g\right\\}}_{GPB}}$ and the geobracket
$G\left(s,f,g\right)$. Obviously, the geobracket $G\left(s,f,g\right)$ is
necessary for a complete Hamiltonian syatem, as a result of the geobracket
$G\left(s,f,g\right)$, it can be generally used to depict nonlinear system in
a non-Euclidean space, and for general manifolds. As GSPB defined, the GCHS is
completely assured and determined by the GSPB. Therefore, the GCHS can be
distinctly obtained as follows.

###  4.1 Generalized Covariant Hamilton System

######  Theorem 3 (GCHS).

[22] The GCHS of two functions $f,H\in{{C}^{\infty}}\left(M,\mathbb{R}\right)$
on $M$ is defined as

$\frac{\mathcal{D}f}{dt}=\left\\{f,H\right\\}={{\left\\{f,H\right\\}}_{GPB}}+G\left(s,f,H\right)$

for ${{\mathbb{R}}^{r}}$, the geobracket is

$\displaystyle
G\left(s,f,H\right)=f{{\left\\{s,H\right\\}}_{GPB}}-H{{\left\\{s,f\right\\}}_{GPB}}$

in terms of $f$ and Hamiltonian $H$, $s$ is structural function on $M$.

In fact, general covariance of the GCHS holds naturally. The GCHS is totally
dependent on the GSPB and accordingly defined. Notice that the GCHS is
covariant, it means that the GCHS is complete, and its form is invariant under
the coordinate transformation, just like the covariance of the geodesic
equation. So, the motion is completely determined by the structure of the
manifold $M$. This is also like the idea of general relativity where particles
move on geodesics and the bending is caused by the gravity.

######  Theorem 4 (TGHS,S-dynamics, GCHS).

[22] The TGHS, S-dynamics, GCHS can be respectively formulated as

TGHS:

$\frac{df}{dt}={\dot{f}}={{\left\\{f,H\right\\}}_{GPB}}-H{{\left\\{s,f\right\\}}_{GPB}}$.

S-dynamics:

$\frac{ds}{dt}=w={{\left\\{s,H\right\\}}_{GPB}}=\left\\{1,H\right\\}$.

GCHS:

$\frac{\mathcal{D}f}{dt}=\left\\{f,H\right\\}={{\left\\{f,H\right\\}}_{GPB}}+G\left(s,f,H\right)$.

By defining the GSPB which can basically solve and describe an entire
Hamiltonian system, even nonlinear system. Thus, in order to state this point.
To start with, we recall the basic notions of the GSPB. We explain that the
GCHS corresponds to non-Euclidean space which corresponds to curved coordinate
systems in curved space-time. The geobracket $G\left(s,f,H\right)$ is the part
that has been neglected for a long time which is rightly the correction term
for the general covariance which certainly means nonlinear system. It is
precisely because of the appearance of the second term $G\left(s,f,H\right)$
that the GCHS can describe and explain the generalized nonlinear system, the
nonlinear Hamiltonian system.

In the following, the covariant equilibrium equation of the GCHS as a special
case is naturally generated as follows.

######  Corollary 1.

The covariant equilibrium equation of the GCHS is
$\frac{\mathcal{D}f}{dt}=\left\\{f,H\right\\}=0$, i.e,

${{\left\\{f,H\right\\}}_{GPB}}+G\left(s,f,H\right)=0$

holds, then $f$ is called covariant conserved quantity.

The critical points of the GCHS are precisely corresponding to the geodesics-
like. In an appropriate sense, zeros of the GCHS are thus regarded as function
$f$ through geodesics-like.

######  Theorem 5.

[22] There exists two different canonical Hamilton equations on manifolds
respectively given by

Canonical thorough Hamilton equations

$\frac{d{{x}_{k}}}{dt}={\dot{x}_{k}}={{J}_{kj}}{{D}_{j}}H,~{}~{}\frac{d{{p}_{k}}}{dt}={\dot{p}_{k}}=-{{D}_{k}}H$

canonical covariant Hamilton equations

$\frac{\mathcal{D}{{x}_{k}}}{dt}=\left\\{{{x}_{k}},H\right\\},~{}~{}\frac{\mathcal{D}{{p}_{k}}}{dt}=\left\\{{{p}_{k}},H\right\\}$

where
$\left\\{\cdot,~{}\cdot\right\\}={{\left\\{\cdot,~{}\cdot\right\\}}_{GPB}}+G\left(s,\cdot,~{}\cdot\right)$
is generalized structure Poisson bracket,
${{D}_{i}}={{\partial}_{i}}+{{\partial}_{i}}s$ is generalized derivative
operators.

With the help of the structural function, the GSPB333Notes: GCC: Geometric
canonical commutation; CCHE:canonical covariant Hamilton equations; GHE:
generalized Heisenberg equations; CTHE: Canonical thorough Hamilton
equations.. is well defined for covariant generalized mechanics.

The covariant generalized mechanics
---
GSPB:
$\left\\{f,g\right\\}={{\left\\{f,~{}g\right\\}}_{GPB}}+G\left(s,f,g\right)$,
$G\left(s,f,g\right)=f{{\left\\{s,g\right\\}}_{GPB}}-g{{\left\\{s,f\right\\}}_{GPB}}$
CCHE:
$\left\\{{{x}_{j}},~{}{{p}_{k}}\right\\}={{\delta}_{jk}}+{{x}_{j}}{{\partial}_{k}}s+{{p}_{k}}{{J}_{jq}}{{\partial}_{q}}s$
Hamilton function: $H\left(x,~{}p\right)$
GCHS:
$\frac{\mathcal{D}{{x}_{k}}}{dt}=\left\\{{{x}_{k}},H\right\\}={\dot{x}_{k}}+{{x}_{k}}w,~{}~{}\frac{\mathcal{D}{{p}_{k}}}{dt}=\left\\{{{p}_{k}},H\right\\}=-{{D}_{k}}H+{{p}_{k}}w$
CTHE: ${\dot{x}_{k}}={{J}_{kj}}{{D}_{j}}H,~{}~{}{\dot{p}_{k}}=-{{D}_{k}}H$

where $\left\\{F,G\right\\}_{GPB}={{\nabla}^{T}}FJ\nabla
G={{J}_{ij}}\frac{\partial F}{\partial{{x}_{i}}}\frac{\partial
G}{\partial{{x}_{j}}}$ is generalized Poisson bracket.

Essentially, each different bracket corresponds to a Hamiltonian mechanical
system, as a consequence, HS with CPB, GHS and GPB, GCHS and GSPB. In [22], we
have introduced an geometric approach to solve Hamiltonian mechanical problems
based on the generalized structural Poisson bracket formulation. As a matter
of fact, there always exists a structural function $s$ such that geometric
bracket $G\left(s,f,g\right)$ holds for the GSPB and the GCHS on manifolds.

##  5 Generalized geometric commutator and geomutator

In this section, we shall deduce certain results that how generalized
geometric commutator is defined generally.

Let $a$ and $b$ be any two operators, their commutator is formally defined by
[12, 13, 14, 21]

${\displaystyle[a,b]_{cr}=ab-ba.}$

Note that the operator for commutator can be any mathematical form to be
appeared for the calculation, such as a function, vector, differential
operator, partial differential operator, even a number in a number field, and
so on, it can be arbitrarily chosen according to our needs.

In linear algebra, if two endomorphisms of a space are represented by
commuting matrices in terms of one basis, then they are so represented in
terms of every basis. By using the commutator as a Lie bracket, every
associative algebra can be turned into a Lie algebra.

The commutator has the following properties: [12, 13, 14, 21]

${\displaystyle[a+b,c]_{cr}=[a,c]_{cr}+[b,c]_{cr}}$
${\displaystyle[a,a]_{cr}=0}$ ${\displaystyle[a,b]_{cr}=-[b,a]_{cr}}$
${\displaystyle[a,[b,c]_{cr}]_{cr}+[b,[c,a]_{cr}]_{cr}+[c,[a,b]_{cr}]_{cr}=0}$

The third relation is called anticommutativity, while the fourth is the Jacobi
identity.

Furthermore, let’s give some more specific examples for a better understanding
above definition of the commutator.

  1. 1.

The commutator is

$\displaystyle\left(f,g\right)\mapsto{{\left[\hat{f},\hat{g}\right]}_{cr}}=\hat{f}\hat{g}-\hat{g}\hat{f}$

for the operators $\hat{f},\hat{g}$.

  2. 2.

Let’s consider the Lie bracket [19, 20]

$\displaystyle{{\left[\cdot,\cdot\right]}_{cr}}:{{\mathfrak{X}}^{r}}\left(M\right)\times{{\mathfrak{X}}^{r}}\left(M\right)\to{{\mathfrak{X}}^{r}}\left(M\right)$
$\displaystyle\begin{matrix}{}&{}&{}\\\
\end{matrix}\left(X,Y\right)\mapsto{{\left[X,Y\right]}_{cr}}=XY-
YX\in{{\mathfrak{X}}^{r}}\left(M\right)$

where ${{\mathfrak{X}}^{r}}\left(M\right)$ means vector space.

  3. 3.

The case of several sets, that is, mixed set

$\displaystyle{{\left[\cdot,\cdot\right]}_{cr}}:{{\mathfrak{X}}^{r}}\left(M\right)\times{{C}^{\infty}}\left(M\right)\to{{C}^{\infty}}\left(M\right)$
$\displaystyle\begin{matrix}{}&{}&{}\\\
\end{matrix}\left(X,f\right)\mapsto{{\left[X,f\right]}_{cr}}$

For a given a function $g$ and
$X={{X}^{i}}\frac{\partial}{\partial{{x}^{i}}}\in{{\mathfrak{X}}^{r}}\left(M\right)$,
then

$\displaystyle{{\left[X,f\right]}_{cr}}g$ $\displaystyle=X\left(fg\right)-fXg$
$\displaystyle=fXg+g{{X}^{i}}\frac{\partial}{\partial{{x}^{i}}}f-fXg$
$\displaystyle=gXf\in{{C}^{\infty}}\left(M\right)$

Then ${{\left[X,f\right]}_{cr}}=Xf\in{{C}^{\infty}}\left(M\right)$.

Let $a$ and $b$ be any elements of any algebra, or any operators, their
generalized geometric commutator is formally defined by the following.

######  Definition 3.

A generalized geometric commutator of arity two $a,b$ is formally given by

$\left[a,b\right]={{\left[a,b\right]}_{cr}}+G\left(s;a,b\right)$

The geomutator is

$G\left(s;a,b\right)=a{{\left[s,b\right]}_{cr}}-b{{\left[s,a\right]}_{cr}}$

satisfying $G\left(s;a,b\right)=-G\left(s;b,a\right)$, where $s$ is a
geometric structure function given by domain.

Note that a generalized geometric commutator can be regarded as a geometric
binary operation, a geometric binary operation may also involve several sets,
that is, mixed set. In fact, the generalized geometric commutator can be seen
as an algorithm or operational rule for possible needs, this is a expression
of highly formalism, namely, generalized geometric commutator is a deformation
and extension of formalization relative to the commutator, we only consider
this formalized structure for considering the application of it, owing to this
point, it can be used for further considerations. As a consequence, it can be
used with no restrictions.

######  Theorem 6.

The generalized geometric commutator has the following properties:

  1. 1.

$\left[a,a\right]=0$

  2. 2.

$\left[a,c\right]=-\left[c,a\right]$

  3. 3.

$\left[a+b,c\right]=\left[a,c\right]+\left[b,c\right]$

  4. 4.

$\left[ab,c\right]=a{{\left[b,c\right]}_{cr}}+{{\left[a,c\right]}_{cr}}b+G\left(s,ab,c\right)$

  5. 5.

$\left[a,\left[b,c\right]\right]+\left[b,\left[c,a\right]\right]+\left[c,\left[a,b\right]\right]=0$

By modelling generalized geometric commutator, we can consider more
mathematical structure and its applications for other fields. More precisely,
based on the formal expression, the geomutator above is accurately written in
the form

$\displaystyle G\left(s,a,b\right)$ $\displaystyle=asb-bsa+bas-abs$
$\displaystyle=\left\langle a:s:b\right\rangle-\left[a,b\right]_{cr}s$

where

$\displaystyle\left\langle a:s:b\right\rangle=asb-bsa$
$\displaystyle\left[a,b\right]_{cr}s=abs-bas$

Accordingly, the generalized geometric commutator of arity two $a,b$ is
formally rewritten by

$\left[a,b\right]={{\left[a,b\right]}_{cr}}+\left\langle
a:s:b\right\rangle-\left[a,b\right]_{cr}s$

######  Definition 4.

The covariant equilibrium equation is given by $\left[a,b\right]=0$, i.e,

${{\left[a,b\right]}_{cr}}+G\left(s,a,b\right)=0$

for arbitrary operators $a,~{}b$.

Note that covariant equilibrium equation is also written as

${{\left[a,b\right]}_{cr}}+\left\langle
a:s:b\right\rangle-\left[a,b\right]_{cr}s=0$

or

${{\left[a,b\right]}_{cr}}=-G\left(s,a,b\right)$

When generalized geometric commutator comes to the quantum operator, quantum
covariant Poisson bracket appears for a complete extension of classical
quantum Poisson bracket. In such case, the arity two $a,b$ will be replaced by
specific quantum operator, in this applicable fields, the
${{\left[a,b\right]}_{cr}}$ emerges in quantum mechanics as quantum Poisson
bracket form ${{\left[\hat{a},\hat{b}\right]}_{QPB}}$.

###  5.1 Geomutator

Some properties of the geomutator are given by

$\displaystyle
G\left(s,a+b,c+d\right)=G\left(s,a,c\right)+G\left(s,b,d\right)+G\left(s,b,c\right)+G\left(s,a,d\right)$
$\displaystyle G\left(s,a+b,c\right)=G\left(s,a,c\right)+G\left(s,b,c\right)$
$\displaystyle G\left(s,a,c+d\right)=G\left(s,a,c\right)+G\left(s,a,d\right)$

for operators $a,b,c,d$. With some particular properties are given by

$\displaystyle G\left(s,a,a\right)=G\left(s,s,s\right)=0$ $\displaystyle
G\left(s,s,a\right)=s{{\left[s,a\right]}_{cr}}$ $\displaystyle
G\left(s,a,s\right)=s{{\left[a,s\right]}_{cr}}$

As generalized geometric commutator stated above, by using the transformation

$a\to{{a}^{\left(s\right)}}=a+sa,~{}~{}b\to{{b}^{\left(s\right)}}=b+sb$

The generalized geometric commutator is transformed to

$\displaystyle\left[a,b\right]$
$\displaystyle={{\left[a,b\right]}_{cr}}+G\left(s,a,b\right)$
$\displaystyle=ab-ba+asb-bsa+bas-abs$
$\displaystyle=a\left(b+sb\right)-b\left(a+sa\right)-{{\left[a,b\right]}_{cr}}s$
$\displaystyle=a{{b}^{\left(s\right)}}-b{{a}^{\left(s\right)}}-{{\left[a,b\right]}_{cr}}s$

or taking the transformation $a\to{{a}^{\left(sg\right)}}=a+sa-as$, where in
general, $sa-as\neq 0$, then

$\displaystyle\left[a,b\right]$
$\displaystyle={{\left[a,b\right]}_{cr}}+G\left(s,a,b\right)$
$\displaystyle=a\left(b-bs+sb\right)-b\left(a+sa-as\right)$
$\displaystyle=a{{b}^{\left(sg\right)}}-b{{a}^{\left(sg\right)}}$

Obviously, we can easily check that generalized geometric commutator is as a
generalization of commutator mainly reflected in the transformation of the
elements.

###  5.2 Examples for GGC

What’s more, given some more examples for understanding above definition of
the generalized geometric commutator.

  1. 1.

The generalized geometric commutator for the operators $\hat{f},\hat{g}$ is

$\displaystyle\left(f,g\right)\mapsto{{\left[\hat{f},\hat{g}\right]}}={{\left[\hat{f},\hat{g}\right]}_{cr}}+G\left(s,\hat{f},\hat{g}\right)$

where geomutator is

$G\left(s,\hat{f},\hat{g}\right)=\hat{f}{{\left[s,\hat{g}\right]}_{cr}}-\hat{g}{{\left[s,\hat{f}\right]}_{cr}}$

  2. 2.

Reconsider the Lie bracket, it should be generalized,

$\displaystyle\left(X,Y\right)\mapsto{{\left[X,Y\right]}}={{\left[X,Y\right]}_{cr}}+G\left(s,X,Y\right)$

where geomutator takes a form

$G\left(s,X,Y\right)=X{{\left[s,Y\right]}_{cr}}-Y{{\left[s,X\right]}_{cr}}$

  3. 3.

The case of several sets, that is, mixed set

$\displaystyle\left(X,f\right)\mapsto{{\left[X,f\right]}}={{\left[X,f\right]}_{cr}}+G\left(s,X,f\right)$

where geomutator is

$G\left(s,X,f\right)=X{{\left[s,f\right]}_{cr}}-f{{\left[s,X\right]}_{cr}}$

For a given a function $g$ and
$X={{X}^{i}}\frac{\partial}{\partial{{x}^{i}}}\in{{\mathfrak{X}}^{r}}\left(M\right)$,
then

$\displaystyle{{\left[X,f\right]}_{cr}}g$ $\displaystyle=X\left(fg\right)-fXg$
$\displaystyle=fXg+g{{X}^{i}}\frac{\partial}{\partial{{x}^{i}}}f-fXg$
$\displaystyle=gXf\in{{C}^{\infty}}\left(M\right)$

Then ${{\left[X,f\right]}_{cr}}=Xf\in{{C}^{\infty}}\left(M\right)$.

$G\left(s,X,f\right)=-f{{\left[s,X\right]}_{cr}}=f{{\left[X,s\right]}_{cr}}=fXs$

Conclusively, the generalized geometric commutator for $X,f$ is calculated as

$\left[X,f\right]={{\left[X,f\right]}_{cr}}+G\left(s,X,f\right)=Xf+fXs$

##  6 Quantum covariant Poisson bracket

In this section, we start with the definition of quantum covariant Poisson
bracket that generalizes the quantum Poisson bracket by using the structure
function $s$ associated with the structure of space. We shall see that quantum
covariant Poisson bracket is defined by the generalized geometric commutator
as definition 3 stated. Once the framework of the quantum covariant Poisson
bracket is established along with quantum geobracket 444Quantum geometric
bracket is abbreviated as quantum geobracket defined by geomutator, then it
can be continued to generally explain the quantum mechanics more precisely, in
the same way as was done for the GSPB obtained by the same procedure. The QCPB
${{\left[\hat{f},\hat{g}\right]}_{QCPB}}\equiv\left[\hat{f},\hat{g}\right]$
will be denoted throughout the paper.

######  Definition 5 (QCPB).

The QCPB is generally defined as

$\left[\hat{f},\hat{g}\right]={{\left[\hat{f},\hat{g}\right]}_{QPB}}+G\left(s,\hat{f},\hat{g}\right)$

in terms of quantum operator $\hat{f},~{}\hat{g}$, where
$G\left(s,\hat{f},\hat{g}\right)=-G\left(s,\hat{g},\hat{f}\right)$ is called
quantum geometric bracket.

It is zero if and only if $\hat{f}$ and $\hat{g}$ covariant commute, i,e.
$\left[\hat{f},\hat{g}\right]=0$. It is remarkable to see that the QCPB
representation admits a dynamical geometric bracket formula on the manifold.
Note that structural function $s$ represents the background property of
spacetime.

Note that the QCPB can be also treated as the generalized geometric
commutator, somehow, it has the following properties:

  1. 1.

$\left[\hat{f}+\hat{h},\hat{g}\right]=\left[\hat{f},\hat{g}\right]+\left[\hat{h},\hat{g}\right]$.

  2. 2.

$\left[\hat{f},\hat{g}\right]=-\left[\hat{g},\hat{f}\right],\left[\hat{f},\hat{f}\right]=0$.

  3. 3.

$\left[\left[\hat{f},\hat{g}\right],\hat{h}\right]+\left[\left[\hat{g},\hat{h}\right],\hat{f}\right]+\left[\left[\hat{h},\hat{f}\right],\hat{g}\right]=0$.

  4. 4.

$\left[\hat{f},\hat{h}\hat{g}\right]=\hat{h}{{\left[\hat{f},\hat{g}\right]}_{QPB}}+{{\left[\hat{f},\hat{h}\right]}_{QPB}}\hat{g}+G\left(s,\hat{f},\hat{h}\hat{g}\right)$.

Relation 2 is called anticommutativity, while 3 is the generalized Jacobi
identity, identity 4 can be interpreted as a generalized Leibniz rule. Note
that the generalized Jacobi identity has a complicated form

$\displaystyle\left[\left[\hat{f},\hat{g}\right],\hat{h}\right]$
$\displaystyle=\left[{{\left[\hat{f},\hat{g}\right]}_{QPB}}+G\left(s,\hat{f},\hat{g}\right),\hat{h}\right]=\left[{{\left[\hat{f},\hat{g}\right]}_{QPB}},\hat{h}\right]+\left[G\left(s,\hat{f},\hat{g}\right),\hat{h}\right]$
$\displaystyle={{\left[\left[\hat{f},\hat{g}\right],\hat{h}\right]}_{QPB}}+G\left(s,\left[\hat{f},\hat{g}\right],\hat{h}\right)$
$\displaystyle={{\left[{{\left[\hat{f},\hat{g}\right]}_{QPB}}+G\left(s,\hat{f},\hat{g}\right),\hat{h}\right]}_{QPB}}+G\left(s,{{\left[\hat{f},\hat{g}\right]}_{QPB}}+G\left(s,\hat{f},\hat{g}\right),\hat{h}\right)$
$\displaystyle={{\left[{{\left[\hat{f},\hat{g}\right]}_{QPB}},\hat{h}\right]}_{QPB}}+G\left(s,{{\left[\hat{f},\hat{g}\right]}_{QPB}},\hat{h}\right)+{{\left[G\left(s,\hat{f},\hat{g}\right),\hat{h}\right]}_{QPB}}+G\left(s,G\left(s,\hat{f},\hat{g}\right),\hat{h}\right)$

Analogously,

$\displaystyle\left[\left[\hat{g},\widehat{h}\right],\hat{f}\right]={{\left[{{\left[\hat{g},\widehat{h}\right]}_{QPB}},\hat{f}\right]}_{QPB}}+G\left(s,{{\left[\hat{g},\widehat{h}\right]}_{QPB}},\hat{f}\right)+{{\left[G\left(s,\hat{g},\widehat{h}\right),\hat{f}\right]}_{QPB}}+G\left(s,G\left(s,\hat{g},\widehat{h}\right),\hat{f}\right)$
$\displaystyle\left[\left[\widehat{h},\hat{f}\right],\hat{g}\right]={{\left[{{\left[\widehat{h},\hat{f}\right]}_{QPB}},\hat{g}\right]}_{QPB}}+G\left(s,{{\left[\widehat{h},\hat{f}\right]}_{QPB}},\hat{g}\right)+{{\left[G\left(s,\widehat{h},\hat{f}\right),\hat{g}\right]}_{QPB}}+G\left(s,G\left(s,\widehat{h},\hat{f}\right),\hat{g}\right)$

As a result, the generalized Jacobi identity is completely rewritten as

$\displaystyle{{N}_{cl}}\left(\hat{f},\hat{g},\widehat{h}\right)$
$\displaystyle=\left[\left[\hat{f},\hat{g}\right],\widehat{h}\right]+\left[\left[\hat{g},\widehat{h}\right],\hat{f}\right]+\left[\left[\widehat{h},\hat{f}\right],\hat{g}\right]$
$\displaystyle={{N}_{cc}}\left(\hat{f},\hat{g},\widehat{h}\right)+{{N}_{ll}}\left(s,\hat{f},\hat{g},\widehat{h}\right)=0$

where the classical Jacobi identity is denoted as

${{N}_{cc}}\left(\hat{f},\hat{g},\widehat{h}\right)={{\left[{{\left[\hat{f},\hat{g}\right]}_{QPB}},\widehat{h}\right]}_{QPB}}+{{\left[{{\left[\hat{g},\widehat{h}\right]}_{QPB}},\hat{f}\right]}_{QPB}}+{{\left[{{\left[\widehat{h},\hat{f}\right]}_{QPB}},\hat{g}\right]}_{QPB}}$

and

$\displaystyle{{N}_{ll}}\left(s,\hat{f},\hat{g},\widehat{h}\right)=G\left(s,{{\left[\hat{f},\hat{g}\right]}_{QPB}},\widehat{h}\right)+{{\left[G\left(s,\hat{f},\hat{g}\right),\widehat{h}\right]}_{QPB}}+G\left(s,G\left(s,\hat{f},\hat{g}\right),\widehat{h}\right)$
$\displaystyle+G\left(s,{{\left[\hat{g},\widehat{h}\right]}_{QPB}},\hat{f}\right)+{{\left[G\left(s,\hat{g},\widehat{h}\right),\hat{f}\right]}_{QPB}}+G\left(s,G\left(s,\hat{g},\widehat{h}\right),\hat{f}\right)$
$\displaystyle+G\left(s,{{\left[\widehat{h},\hat{f}\right]}_{QPB}},\hat{g}\right)+{{\left[G\left(s,\widehat{h},\hat{f}\right),\hat{g}\right]}_{QPB}}+G\left(s,G\left(s,\widehat{h},\hat{f}\right),\hat{g}\right)$

In the following discussions, we mainly propose and discuss an accurate
expression of the quantum geobracket, maybe it’s one of situation in the
general covariant form.

######  Definition 6 (quantum geometric bracket).

The quantum geometric bracket is

$G\left(s,\hat{f},\hat{g}\right)=\hat{f}{{\left[s,\hat{g}\right]}_{QPB}}-\hat{g}{{\left[s,\hat{f}\right]}_{QPB}}$

where $s$ represents the globally condition of space.

As defined above,
$\left[\cdot,~{}\cdot\right]={{\left[\cdot,~{}\cdot\right]}_{QPB}}+G\left(s,\cdot,~{}\cdot\right)$
denotes the generalized geometric commutator (GGC) which is associated with
structure function $s$ of two operators, $G\left(s,~{}\cdot,~{}\cdot\right)$
is amendatory formula to the classical bracket, as a corrected term of the
QCPB to the QPB.

Notice that
${{\left[\hat{f},\hat{g}\right]}_{QPB}}=\hat{f}\hat{g}-\hat{g}\hat{f}$, then
quantum geobracket can be precisely expressed as

$\displaystyle G\left(s,\hat{f},\hat{g}\right)$
$\displaystyle=\hat{f}{{\left[s,\hat{g}\right]}_{QPB}}-\hat{g}{{\left[s,\hat{f}\right]}_{QPB}}$
$\displaystyle=\hat{f}\left(s\hat{g}\right)-\hat{g}\left(s\hat{f}\right)+\hat{g}\left(\hat{f}s\right)-\hat{f}\left(\hat{g}s\right)$
$\displaystyle=-G\left(s,\hat{g},\hat{f}\right)$

that is a antisymmetric bracket dependent on the structural function $s$.
Thus, the QCPB satisfies the antisymmetric property

$\left[\hat{f},\hat{g}\right]={{\left[\hat{f},\hat{g}\right]}_{QPB}}+G\left(s,\hat{f},\hat{g}\right)=-{{\left[\hat{g},\hat{f}\right]}_{QPB}}-G\left(s,\hat{g},\hat{f}\right)=-\left[\hat{g},\hat{f}\right]$

or the quantum geobracket is given by

$G\left(s,\hat{f},\hat{g}\right)=\left\langle\hat{f}:s:\hat{g}\right\rangle-\left[\hat{f},\hat{g}\right]s$

where
$\left\langle\hat{f}:s:\hat{g}\right\rangle=\hat{f}\left(s\hat{g}\right)-\hat{g}\left(s\hat{f}\right)$.

Above all, the QCPB is completely written as

$\displaystyle\left[\hat{f},\hat{g}\right]$
$\displaystyle={{\left[\hat{f},\hat{g}\right]}_{QPB}}+G\left(s,\hat{f},\hat{g}\right)$
$\displaystyle={{\left[\hat{f},\hat{g}\right]}_{QPB}}+\hat{f}{{\left[s,\hat{g}\right]}_{QPB}}-\hat{g}{{\left[s,\hat{f}\right]}_{QPB}}$
$\displaystyle={{\left[\hat{f},\hat{g}\right]}_{QPB}}+\left\langle\hat{f}:s:\hat{g}\right\rangle-\left[\hat{f},\hat{g}\right]s$

######  Definition 7.

The covariant equilibrium equation is given by
$\left[\hat{f},\hat{g}\right]=0$, i.e,

${{\left[\hat{f},\hat{g}\right]}_{QPB}}+G\left(s,\hat{f},\hat{g}\right)=0$

for operators $\hat{f},~{}\hat{g}$.

The operator identity

$G\left(s,\hat{f},\hat{h}\hat{g}\right)=\hat{f}{{\left[s,\hat{h}\hat{g}\right]}_{QPB}}-\hat{h}\hat{g}{{\left[s,\hat{f}\right]}_{QPB}}$

allows the evaluation of the commutator of $\hat{f},\hat{h}$ with $\hat{g}$,

$\displaystyle G\left(s,\hat{f},\hat{h}\hat{g}\right)$
$\displaystyle=\hat{f}{{\left[s,\hat{h}\hat{g}\right]}_{QPB}}-\hat{h}\hat{g}{{\left[s,\hat{f}\right]}_{QPB}}$
$\displaystyle=\widehat{f}{{\left[s,\hat{h}\right]}_{QPB}}\hat{g}+\hat{f}\hat{h}{{\left[s,\hat{g}\right]}_{QPB}}-\hat{h}\hat{g}{{\left[s,\hat{f}\right]}_{QPB}}$

where we have used
${{\left[s,\hat{h}\hat{g}\right]}_{QPB}}={{\left[s,\hat{h}\right]}_{QPB}}\hat{g}+\hat{h}{{\left[s,\hat{g}\right]}_{QPB}}$,
and it implies that

$\displaystyle\left[\hat{f},\hat{h}\hat{g}\right]$
$\displaystyle=\hat{h}\left({{\left[\hat{f},\hat{g}\right]}_{QPB}}-\hat{g}{{\left[s,\hat{f}\right]}_{QPB}}\right)+{{\left[\hat{f},\hat{h}\right]}_{QPB}}\hat{g}$
$\displaystyle\begin{matrix}{}&{}&{}&{}\\\
\end{matrix}+\hat{f}\left({{\left[s,\hat{h}\right]}_{QPB}}\hat{g}+\hat{h}{{\left[s,\hat{g}\right]}_{QPB}}\right)$
$\displaystyle=\hat{h}\left({{\left[\hat{f},\hat{g}\right]}_{QPB}}-\hat{g}{{\left[s,\hat{f}\right]}_{QPB}}\right)+{{\left[\hat{f},\hat{h}\right]}_{QPB}}\hat{g}+\hat{f}{{\left[s,\hat{h}\hat{g}\right]}_{QPB}}$

###  6.1 Examples of QCPB

We can give a example for better understanding of the QCPB, for instance, if
let

$\hat{F}=-\sqrt{-1}{{e}^{\sqrt{-1}x}}\frac{d}{dx},~{}~{}\hat{G}={{e}^{\sqrt{-1}x}}$

be given, then the classical operation leads to the consequence

$\displaystyle[\hat{F},\hat{G}]_{QPB}\psi(x)$
$\displaystyle=(\hat{F}\hat{G}-\hat{G}\hat{F})\psi(x)$
$\displaystyle=\left(-\sqrt{-1}e^{\sqrt{-1}x}\frac{d}{dx}e^{\sqrt{-1}x}-e^{\sqrt{-1}x}\left(-\sqrt{-1}e^{\sqrt{-1}x}\frac{d}{dx}\right)\right)\psi(x)$
$\displaystyle=e^{2\sqrt{-1}x}\psi(x)-\sqrt{-1}e^{2\sqrt{-1}x}\frac{d}{dx}\psi(x)+\sqrt{-1}e^{2\sqrt{-1}x}\frac{d}{dx}\psi(x)$
$\displaystyle=e^{2\sqrt{-1}x}\psi(x)$

Then $[\hat{F},\hat{G}]_{QPB}=e^{2\sqrt{-1}x}$. But by using the QCPB, we have

$\left[\hat{F},\hat{G}\right]={{\left[\hat{F},\hat{G}\right]}_{QPB}}+G\left(s,\hat{F},\hat{G}\right)$

Hence, we only need to evaluate quantum geometric bracket
$G\left(s,\hat{F},\hat{G}\right)$ expressed as

$G\left(s,\hat{F},\hat{G}\right)=\hat{F}{{\left[s,\hat{G}\right]}_{QPB}}-\hat{G}{{\left[s,\hat{F}\right]}_{QPB}}$

More precisely,

$\displaystyle\hat{F}{{\left[s,\hat{G}\right]}_{QPB}}\psi$
$\displaystyle=-\sqrt{-1}{{e}^{\sqrt{-1}x}}\frac{d}{dx}\left(s{{e}^{\sqrt{-1}x}}\psi-{{e}^{\sqrt{-1}x}}s\psi\right)=0$
$\displaystyle\hat{G}{{\left[s,\hat{F}\right]}_{QPB}}\psi=\psi\sqrt{-1}{{e}^{2\sqrt{-1}x}}\frac{ds}{dx}=-\psi{{e}^{\sqrt{-1}x}}\hat{F}s$

As a result of above computation, we get

$\hat{G}{{\left[s,\hat{F}\right]}_{QPB}}=-{{e}^{\sqrt{-1}x}}\hat{F}s$

And the quantum geometric bracket is finally shown as

$\displaystyle G\left(s,\hat{F},\hat{G}\right)\psi$
$\displaystyle=\hat{F}{{\left[s,\hat{G}\right]}_{QPB}}\psi-\hat{G}{{\left[s,\hat{F}\right]}_{QPB}}\psi$
$\displaystyle=-\hat{G}{{\left[s,\hat{F}\right]}_{QPB}}\psi=\psi{{e}^{\sqrt{-1}x}}\hat{F}s$
$\displaystyle=-\psi\sqrt{-1}{{e}^{2\sqrt{-1}x}}\frac{ds}{dx}$

Accordingly, the QCPB is obtained

$\displaystyle\left[\hat{F},\hat{G}\right]$
$\displaystyle={{\left[\hat{F},\hat{G}\right]}_{QPB}}+G\left(s,\hat{F},\hat{G}\right)={{e}^{2\sqrt{-1}x}}+{{e}^{\sqrt{-1}x}}\hat{F}s$
$\displaystyle={{e}^{2\sqrt{-1}x}}\left(1-\sqrt{-1}\frac{ds}{dx}\right)$

We can check that if $\frac{ds}{dx}=0$ is given, then
$\left[\hat{F},\hat{G}\right]$ is degenerated to the classical case
${{\left[\hat{F},\hat{G}\right]}_{QPB}}$, equivalently,
$G\left(s,\hat{F},\hat{G}\right)=0$ disappears.

For another example, let $\hat{F}=\frac{d}{dx},\hat{G}=x$ be given, and wave
function is $\psi\left(x\right)=\sin x$, then the QCPB is given by

$\displaystyle\left[\frac{d}{dx},x\right]\psi\left(x\right)$
$\displaystyle={{\left[\frac{d}{dx},x\right]}_{QPB}}\psi\left(x\right)+G\left(s,\frac{d}{dx},x\right)\psi\left(x\right)$
$\displaystyle={{\left[\frac{d}{dx},x\right]}_{QPB}}\sin
x+G\left(s,\frac{d}{dx},x\right)\sin x$

The QPB and the quantum geometric bracket can be respectively evaluated by

${{\left[\frac{d}{dx},x\right]}_{QPB}}\sin x=\sin x$

and

$\displaystyle G\left(s,\frac{d}{dx},x\right)\sin x$
$\displaystyle=\frac{d}{dx}{{\left[s,x\right]}_{QPB}}\sin
x-x{{\left[s,\frac{d}{dx}\right]}_{QPB}}\sin x$
$\displaystyle=x{{\left[\frac{d}{dx},s\right]}_{QPB}}\sin x=x\sin
x\frac{ds}{dx}$

then it leads to the outcome

$\left[\frac{d}{dx},x\right]\psi\left(x\right)=\left(1+x\frac{ds}{dx}\right)\psi\left(x\right)$

###  6.2 Quantum geometric bracket

As previously stated, it turns out that the quantum geometric bracket is given
by

$G\left(s,\hat{f},\hat{g}\right)=\left[\hat{f},\hat{g}\right]-{{\left[\hat{f},\hat{g}\right]}_{QPB}}$

Actually, it’s a map expressed as

$\hat{f},\hat{g}\to G\left(s,\hat{f},\hat{g}\right)$

with respect to two quantum operators $\hat{f},~{}\hat{g}$. In general, in
order to preserve the antisymmetry of the QPB, the quantum geometric bracket
is always asked to satisfy the antisymmetry, i,e.

$G\left(s,\hat{f},\hat{g}\right)=-G\left(s,\hat{g},\hat{f}\right)$

Thus, let’s consider the properties of quantum geobracket, more precisely,

$G\left(s,\hat{f},\hat{g}\right)=\hat{f}{{\left[s,\hat{g}\right]}_{QPB}}-\widehat{g}{{\left[s,\hat{f}\right]}_{QPB}}$

that fulfils the antisymmetry
$G\left(s,\hat{f},\hat{g}\right)=-G\left(s,\hat{g},\hat{f}\right)$, then some
special properties of the quantum geometric bracket can be represented by the
following

$\displaystyle G\left(s,\hat{f},\hat{f}\right)=G\left(s,s,s\right)=0$
$\displaystyle G\left(s,s,\hat{g}\right)=s{{\left[s,\hat{g}\right]}_{QPB}}$
$\displaystyle G\left(s,\hat{f},s\right)=s{{\left[\hat{f},s\right]}_{QPB}}$

The QCPB has greatly enlarged the research scope of canonical commutation
relation, obviously, it completes and improves the quantum Poisson brackets.
In short, the QCPB is completely written as

$\displaystyle\left[\hat{f},\hat{g}\right]$
$\displaystyle={{\left[\hat{f},\hat{g}\right]}_{QPB}}+\hat{f}{{\left[s,\hat{g}\right]}_{QPB}}-\hat{g}{{\left[s,\hat{f}\right]}_{QPB}}$
$\displaystyle=\hat{f}\hat{g}-\hat{g}\hat{f}+\hat{f}\left(s\hat{g}\right)-\hat{g}\left(s\hat{f}\right)+\hat{g}\left(\hat{f}s\right)-\hat{f}\left(\hat{g}s\right)$

Thus, for a given wave function $\varphi$ onto the QCPB, it gives a
generalized commutation relation

$\displaystyle\left[\hat{f},\hat{g}\right]\varphi$
$\displaystyle={{\left[\hat{f},\hat{g}\right]}_{QPB}}\varphi+\hat{f}{{\left[s,\hat{g}\right]}_{QPB}}\varphi-\hat{g}{{\left[s,\hat{f}\right]}_{QPB}}\varphi$
$\displaystyle=\hat{f}\hat{g}\varphi-\hat{g}\hat{f}\varphi+\hat{f}\left(s\hat{g}\varphi\right)-\hat{g}\left(s\hat{f}\varphi\right)+\hat{g}\hat{f}\left(s\varphi\right)-\hat{f}\hat{g}\left(s\varphi\right)$
$\displaystyle={{\left[\hat{f},\hat{g}\right]}_{QPB}}\varphi+{{\left[\hat{g},\hat{f}\right]}_{QPB}}\left(s\varphi\right)+\hat{f}\left(s\hat{g}\varphi\right)-\hat{g}\left(s\hat{f}\varphi\right)$

As one can see, the QCPB has largely enriched the QPB for a better
understanding of quantum mechanics, it clearly appears more parts such as

$G\left(s,\hat{f},\hat{g}\right)=\hat{f}{{\left[s,\hat{g}\right]}_{QPB}}-\hat{g}{{\left[s,\hat{f}\right]}_{QPB}}=\hat{f}\left(s\hat{g}\right)-\hat{g}\left(s\hat{f}\right)+\hat{g}\left(\hat{f}s\right)-\hat{f}\left(\hat{g}s\right)$

This terms never emerges in the quantum mechanics for general thinking,
therefore, the quantum geobracket $G\left(s,\hat{f},\hat{g}\right)$ is rightly
a correction to remedy the QPB.

To consider the corresponding equilibrium of the QCPB, namley,
$\left[\hat{f},\hat{g}\right]=0$, in other words,

${{\left[\hat{f},\hat{g}\right]}_{QPB}}+G\left(s,\hat{f},\hat{g}\right)=0$

or specifically,

${{\left[\hat{f},\hat{g}\right]}_{QPB}}=\hat{g}{{\left[s,\hat{f}\right]}_{QPB}}-\hat{f}{{\left[s,\hat{g}\right]}_{QPB}}=G\left(s,\hat{g},\hat{f}\right)$

If let $\hat{f}=s$ be given for the QCPB, then it accordingly derives

$\displaystyle\left[s,\hat{g}\right]$
$\displaystyle={{\left[s,\hat{g}\right]}_{QPB}}+G\left(s,s,\widehat{g}\right)={{\left[s,\hat{g}\right]}_{QPB}}+s{{\left[s,\hat{g}\right]}_{QPB}}$
$\displaystyle=\left(1+s\right){{\left[s,\hat{g}\right]}_{QPB}}$

The QCPB has following properties

$\displaystyle\left[\hat{f},\hat{g}\right]=-\left[\hat{g},\hat{f}\right],~{}~{}\left[\hat{f},\hat{f}\right]=0$
$\displaystyle\left[\hat{f},\hat{h}+\hat{g}\right]=\left[\hat{f},\hat{g}\right]+\left[\hat{f},\hat{h}\right]$
$\displaystyle\left[\hat{h}+\hat{g},\hat{f}\right]=\left[\hat{g},\hat{f}\right]+\left[\hat{h},\hat{f}\right]$
$\displaystyle\left[\hat{f},c\hat{g}\right]=c\left[\hat{f},\hat{g}\right]$
$\displaystyle\left[\hat{f},\hat{h}\hat{g}\right]=\hat{h}{{\left[\hat{f},\hat{g}\right]}_{QPB}}+{{\left[\hat{f},\hat{h}\right]}_{QPB}}\hat{g}+G\left(s,\hat{f},\hat{h}\hat{g}\right)$

for any quantum operators $\hat{f},\hat{h},\hat{g}$, where $c$ is a complex
constant.

Let

$\hat{f}={{\hat{f}}_{+}}+\sqrt{-1}{{\hat{f}}_{-}},~{}~{}\hat{g}={{\hat{g}}_{+}}+\sqrt{-1}{{\hat{g}}_{-}}$

be given for generally calculations of the QCPB, where
${{\hat{f}}_{+}},{{\hat{f}}_{-}},{{\hat{g}}_{+}},{{\hat{g}}_{-}}$ are
Hermitian operators.

$\displaystyle\left[\hat{f},\hat{g}\right]$
$\displaystyle=\left[{{{\hat{f}}}_{+}}+\sqrt{-1}{{{\hat{f}}}_{-}},{{{\hat{g}}}_{+}}+\sqrt{-1}{{{\hat{g}}}_{-}}\right]$
$\displaystyle={{\left[{{{\hat{f}}}_{+}}+\sqrt{-1}{{{\hat{f}}}_{-}},{{{\hat{g}}}_{+}}+\sqrt{-1}{{{\hat{g}}}_{-}}\right]}_{QPB}}+G\left(s,{{{\hat{f}}}_{+}}+\sqrt{-1}{{{\hat{f}}}_{-}},{{{\hat{g}}}_{+}}+\sqrt{-1}{{{\hat{g}}}_{-}}\right)$

As a consequence, we only need to compute two parts of the QCPB respectively,
the QPB and the quantum geometric bracket. To begin with the QPB is given by

$\displaystyle{{\left[{{{\hat{f}}}_{+}}+\sqrt{-1}{{{\hat{f}}}_{-}},{{{\hat{g}}}_{+}}+\sqrt{-1}{{{\hat{g}}}_{-}}\right]}_{QPB}}={{\left[{{{\hat{f}}}_{+}},{{{\hat{g}}}_{+}}\right]}_{QPB}}+{{\left[{{{\hat{f}}}_{+}},\sqrt{-1}{{{\hat{g}}}_{-}}\right]}_{QPB}}$
$\displaystyle\begin{matrix}{}&{}&{}\\\
\end{matrix}+{{\left[\sqrt{-1}{{{\hat{f}}}_{-}},{{{\hat{g}}}_{+}}\right]}_{QPB}}+{{\left[\sqrt{-1}{{{\hat{f}}}_{-}},\sqrt{-1}{{{\hat{g}}}_{-}}\right]}_{QPB}}$
$\displaystyle={{\left[{{{\hat{f}}}_{+}},{{{\hat{g}}}_{+}}\right]}_{QPB}}-{{\left[{{{\hat{f}}}_{-}},{{{\hat{g}}}_{-}}\right]}_{QPB}}+\sqrt{-1}\left({{\left[{{{\hat{f}}}_{-}},{{{\hat{g}}}_{+}}\right]}_{QPB}}+{{\left[{{{\hat{f}}}_{+}},{{{\hat{g}}}_{-}}\right]}_{QPB}}\right)$

Subsequently, the quantum geometric bracket follows

$\displaystyle
G\left(s,{{{\hat{f}}}_{+}}+\sqrt{-1}{{{\hat{f}}}_{-}},{{{\hat{g}}}_{+}}+\sqrt{-1}{{{\hat{g}}}_{-}}\right)$
$\displaystyle=\left({{{\hat{f}}}_{+}}+\sqrt{-1}{{{\hat{f}}}_{-}}\right){{\left[s,{{{\hat{g}}}_{+}}+\sqrt{-1}{{{\hat{g}}}_{-}}\right]}_{QPB}}-\left({{{\hat{g}}}_{+}}+\sqrt{-1}{{{\hat{g}}}_{-}}\right){{\left[s,{{{\hat{f}}}_{+}}+\sqrt{-1}{{{\hat{f}}}_{-}}\right]}_{QPB}}$
$\displaystyle={{{\hat{f}}}_{+}}{{\left[s,{{{\hat{g}}}_{+}}+\sqrt{-1}{{{\hat{g}}}_{-}}\right]}_{QPB}}-{{{\hat{g}}}_{+}}{{\left[s,{{{\hat{f}}}_{+}}+\sqrt{-1}{{{\hat{f}}}_{-}}\right]}_{QPB}}$
$\displaystyle\begin{matrix}{}&{}&{}&{}\\\
\end{matrix}+\sqrt{-1}\left\\{{{{\hat{f}}}_{-}}{{\left[s,{{{\hat{g}}}_{+}}+\sqrt{-1}{{{\hat{g}}}_{-}}\right]}_{QPB}}-{{{\hat{g}}}_{-}}{{\left[s,{{{\hat{f}}}_{+}}+\sqrt{-1}{{{\hat{f}}}_{-}}\right]}_{QPB}}\right\\}$
$\displaystyle={{{\hat{f}}}_{+}}{{\left[s,{{{\hat{g}}}_{+}}\right]}_{QPB}}-{{{\hat{f}}}_{-}}{{\left[s,{{{\hat{g}}}_{-}}\right]}_{QPB}}-{{{\hat{g}}}_{+}}{{\left[s,{{{\hat{f}}}_{+}}\right]}_{QPB}}+{{{\hat{g}}}_{-}}{{\left[s,{{{\hat{f}}}_{-}}\right]}_{QPB}}$
$\displaystyle+\sqrt{-1}\left\\{{{{\hat{f}}}_{+}}{{\left[s,{{{\hat{g}}}_{-}}\right]}_{QPB}}+{{{\hat{f}}}_{-}}{{\left[s,{{{\hat{g}}}_{+}}\right]}_{QPB}}-{{{\hat{g}}}_{+}}{{\left[s,{{{\hat{f}}}_{-}}\right]}_{QPB}}-{{{\hat{g}}}_{-}}{{\left[s,{{{\hat{f}}}_{+}}\right]}_{QPB}}\right\\}$

Hence, we obtain the QCPB

$\displaystyle\left[\hat{f},\hat{g}\right]=\left[{{{\hat{f}}}_{+}}+\sqrt{-1}{{{\hat{f}}}_{-}},{{{\hat{g}}}_{+}}+\sqrt{-1}{{{\hat{g}}}_{-}}\right]$
$\displaystyle={{\left[{{{\hat{f}}}_{+}},{{{\hat{g}}}_{+}}\right]}_{QPB}}+{{{\hat{f}}}_{+}}{{\left[s,{{{\hat{g}}}_{+}}\right]}_{QPB}}-{{{\hat{g}}}_{+}}{{\left[s,{{{\hat{f}}}_{+}}\right]}_{QPB}}$
$\displaystyle\begin{matrix}{}&{}&{}&{}\\\
\end{matrix}-{{\left[{{{\hat{f}}}_{-}},{{{\hat{g}}}_{-}}\right]}_{QPB}}-{{{\hat{f}}}_{-}}{{\left[s,{{{\hat{g}}}_{-}}\right]}_{QPB}}+{{{\hat{g}}}_{-}}{{\left[s,{{{\hat{f}}}_{-}}\right]}_{QPB}}$
$\displaystyle+\sqrt{-1}\left\\{{{\left[{{{\hat{f}}}_{-}},{{{\hat{g}}}_{+}}\right]}_{QPB}}+{{{\hat{f}}}_{-}}{{\left[s,{{{\hat{g}}}_{+}}\right]}_{QPB}}-{{{\hat{g}}}_{+}}{{\left[s,{{{\hat{f}}}_{-}}\right]}_{QPB}}\right.$
$\displaystyle\begin{matrix}{}&{}&{}&{}\\\
\end{matrix}\left.+{{\left[{{{\hat{f}}}_{+}},{{{\hat{g}}}_{-}}\right]}_{QPB}}+{{{\hat{f}}}_{+}}{{\left[s,{{{\hat{g}}}_{-}}\right]}_{QPB}}-{{{\hat{g}}}_{-}}{{\left[s,{{{\hat{f}}}_{+}}\right]}_{QPB}}\right\\}$
$\displaystyle=\left[{{{\hat{f}}}_{+}},{{{\hat{g}}}_{+}}\right]-\left[{{{\hat{f}}}_{-}},{{{\hat{g}}}_{-}}\right]+\sqrt{-1}\left(\left[{{{\hat{f}}}_{-}},{{{\hat{g}}}_{+}}\right]+\left[{{{\hat{f}}}_{+}},{{{\hat{g}}}_{-}}\right]\right)$

where

$\displaystyle\left[{{{\hat{f}}}_{+}},{{{\hat{g}}}_{+}}\right]$
$\displaystyle={{\left[{{{\hat{f}}}_{+}},{{{\hat{g}}}_{+}}\right]}_{QPB}}+G\left(s,{{{\hat{f}}}_{+}},{{{\hat{g}}}_{+}}\right)$
$\displaystyle={{\left[{{{\hat{f}}}_{+}},{{{\hat{g}}}_{+}}\right]}_{QPB}}+{{{\hat{f}}}_{+}}{{\left[s,{{{\hat{g}}}_{+}}\right]}_{QPB}}-{{{\hat{g}}}_{+}}{{\left[s,{{{\hat{f}}}_{+}}\right]}_{QPB}}$

Similarly,

$\displaystyle\left[\hat{f},\hat{g}\right]$
$\displaystyle=\left[{{{\hat{f}}}_{+}}+\sqrt{-1}{{{\hat{f}}}_{-}},{{{\hat{g}}}_{+}}+\sqrt{-1}{{{\hat{g}}}_{-}}\right]$
$\displaystyle={{\left[{{{\hat{f}}}_{+}},{{{\hat{g}}}_{+}}\right]}_{QPB}}-{{\left[{{{\hat{f}}}_{-}},{{{\hat{g}}}_{-}}\right]}_{QPB}}+\sqrt{-1}\left({{\left[{{{\hat{f}}}_{-}},{{{\hat{g}}}_{+}}\right]}_{QPB}}+{{\left[{{{\hat{f}}}_{+}},{{{\hat{g}}}_{-}}\right]}_{QPB}}\right)$
$\displaystyle+G\left(s,{{{\hat{f}}}_{+}},{{{\hat{g}}}_{+}}\right)-G\left(s,{{{\hat{f}}}_{-}},{{{\hat{g}}}_{-}}\right)+\sqrt{-1}\left(G\left(s,{{{\hat{f}}}_{-}},{{{\hat{g}}}_{+}}\right)+G\left(s,{{{\hat{f}}}_{+}},{{{\hat{g}}}_{-}}\right)\right)$
$\displaystyle={{\left[{{{\hat{f}}}_{+}}+\sqrt{-1}{{{\hat{f}}}_{-}},{{{\hat{g}}}_{+}}+\sqrt{-1}{{{\hat{g}}}_{-}}\right]}_{QPB}}+G\left(s,{{{\hat{f}}}_{+}}+\sqrt{-1}{{{\hat{f}}}_{-}},{{{\hat{g}}}_{+}}+\sqrt{-1}{{{\hat{g}}}_{-}}\right)$

It implies that

$\displaystyle{{\left[{{{\hat{f}}}_{+}}+\sqrt{-1}{{{\hat{f}}}_{-}},{{{\hat{g}}}_{+}}+\sqrt{-1}{{{\hat{g}}}_{-}}\right]}_{QPB}}$
$\displaystyle={{\left[{{{\hat{f}}}_{+}},{{{\hat{g}}}_{+}}\right]}_{QPB}}-{{\left[{{{\hat{f}}}_{-}},{{{\hat{g}}}_{-}}\right]}_{QPB}}+\sqrt{-1}\left({{\left[{{{\hat{f}}}_{-}},{{{\hat{g}}}_{+}}\right]}_{QPB}}+{{\left[{{{\hat{f}}}_{+}},{{{\hat{g}}}_{-}}\right]}_{QPB}}\right)$
$\displaystyle
G\left(s,{{{\hat{f}}}_{+}}+\sqrt{-1}{{{\hat{f}}}_{-}},{{{\hat{g}}}_{+}}+\sqrt{-1}{{{\hat{g}}}_{-}}\right)$
$\displaystyle=G\left(s,{{{\hat{f}}}_{+}},{{{\hat{g}}}_{+}}\right)-G\left(s,{{{\hat{f}}}_{-}},{{{\hat{g}}}_{-}}\right)+\sqrt{-1}\left(G\left(s,{{{\hat{f}}}_{-}},{{{\hat{g}}}_{+}}\right)+G\left(s,{{{\hat{f}}}_{+}},{{{\hat{g}}}_{-}}\right)\right)$

Above derivation reveals that the quantum geobracket as a new quantum
geometric bracket remains the same operation as the QPB does. In other words,
the operation rule of the quantum geobracket is parallel to the operation rule
of the QPB.

##  7 Quantum covariant Hamiltonian system

This section will give the covariant dynamics which contains two different
dynamics: the generalized Heisenberg equation and G-dynamics. It tells that
the generalized Heisenberg equation perfects the Heisenberg equation by
considering the structural function $s$. Firstly, let’s give the definition of
quantum covariant Hamiltonian system (QCHS) defined by the QCPB as definition
6 shown.

######  Definition 8 (QCHS).

The QCHS defined by QCPB in terms of quantum operator $\hat{f},~{}\hat{H}$ is
generally given by

$\left[\hat{f},\hat{H}\right]={{\left[\hat{f},\hat{H}\right]}_{QPB}}+G\left(s,\hat{f},\hat{H}\right)$

where $G\left(s,\hat{f},\hat{H}\right)$ is quantum geobracket .

In the beginning, let us start with the QCPB by taking $\hat{g}=\hat{H}$ into
consideration, then it formally yields below definition.

######  Definition 9 (quantum geobracket).

The quantum geobracket for a Hamiltonian operator $\hat{H}$ is given by

$G\left(s,\hat{f},\hat{H}\right)=\hat{f}{{\left[s,\hat{H}\right]}_{QPB}}-\hat{H}{{\left[s,\hat{f}\right]}_{QPB}}$

and $s$ is a real structural function only associated with the structure
space.

Manifestly, this operator $\hat{f}$ covariantly commutes with $H$. By our
construction, the QCHS derived from the QCPB is a transition for the further
development of a complete and covariant theory that can naturally generalize
the Heisenberg equation. Since $\hat{f}$ is an observable, $s$ is assumed to
be real function as well. Our goal here is to give a description of the method
in general for perfecting the known theory such as Heisenberg equation. With
that in mind, we proceed accordingly.

######  Definition 10.

The covariant equilibrium equation is given by
$\left[\hat{f},\hat{H}\right]=0$, i.e,

${{\left[\hat{f},\hat{H}\right]}_{QPB}}+G\left(s,\hat{f},\hat{H}\right)=0$

for operators $\hat{f},~{}\hat{H}$, then $\hat{f}$ is called quantum covariant
conserved quantity.

Furthermore, there is a certain case given by

$\left[\hat{H},\hat{H}\right]=-\hat{H}{{\left[s,\hat{H}\right]}_{QPB}}+\hat{H}{{\left[s,\hat{H}\right]}_{QPB}}=0$
(8)

when $\hat{f}=\hat{H}$ based on definition 10. It is clear to see that
$\hat{H}$ is a quantum covariant conserved quantity.

###  7.1 Covariant dynamics, generalized Heisenberg equation, G-dynamics

We now illustrate this definition with one theorem below. More precisely, the
time covariant evolution of any observable $\hat{f}$ in the covariant dynamics
is given by both generalized Heisenberg equation of motion and G-dynamics.

Based on the quantization map $Q$ mapping a function $f$ on the classical
phase space to an operator ${\displaystyle Q_{f}}$ on the quantum Hilbert
space, namely, $f\mapsto{{Q}_{f}}\equiv\hat{f}$. Similarly, it goes for the
GSPB and QCPB such that

${\displaystyle Q_{\\{f,g\\}}={\frac{1}{\sqrt{-1}\hbar}}[Q_{f},Q_{g}]}$

or conversely,

$[Q_{f},Q_{g}]=\sqrt{-1}\hbar{\displaystyle Q_{\\{f,g\\}}}$

According to the rule of supplanting Poisson brackets by commutators written
by Dirac in his book as Eq(1) shown. Similarly, we have same generally
corresponding relation

$\displaystyle\left\\{f,g\right\\}={{\left\\{f,g\right\\}}_{GPB}}+G\left(s,f,g\right)$
$\displaystyle\mapsto\frac{1}{\sqrt{-1}\hbar}\left[\hat{f},\hat{g}\right]=\frac{1}{\sqrt{-1}\hbar}{{\left[\widehat{f},\widehat{g}\right]}_{QPB}}+\frac{1}{\sqrt{-1}\hbar}G\left(s,\hat{f},\hat{g}\right)$

As a generalization to the manifolds, it certainly fits the curved spacetime
which means that the $\frac{1}{\sqrt{-1}\hbar}\left[\hat{f},\hat{g}\right]$ is
covariant, and quantum term $G\left(s,\hat{f},\hat{g}\right)$ is necessary to
exist for a complete quantum theory.

As (5) did, analogously, this method of work lead us to propose that the
quantum counterparts ${\hat{f}},{\hat{g}}$ of classical observables $f,g$
satisfy covariant rule

$\left[\hat{f},\hat{g}\right]=\sqrt{-1}\hbar\widehat{\left\\{f,g\right\\}}$

where $\left[\hat{f},\hat{g}\right]$ is the QCPB, while $\left\\{f,g\right\\}$
is the GSPB. More exactly,

${{\left[\hat{f},\hat{g}\right]}_{QPB}}+G\left(s,\hat{f},\hat{g}\right)=\sqrt{-1}\hbar\widehat{\left({{\left\\{f,g\right\\}}_{GPB}}+G\left(s,f,g\right)\right)}$

Actually, in order for the model to work for the quantum mechanics and
elementary particles, we need to address specifically as follows:

######  Theorem 7.

The covariant dynamics, the generalized Heisenberg equation, G-dynamics can be
formally formulated as

covariant dynamics:

$\frac{\mathcal{D}\hat{f}}{dt}=\frac{1}{\sqrt{-1}\hbar}\left[\hat{f},\hat{H}\right]$

generalized Heisenberg equation:

$\frac{d\hat{f}}{dt}=\frac{1}{\sqrt{-1}\hbar}{{\left[\hat{f},\hat{H}\right]}_{QPB}}-\frac{1}{\sqrt{-1}\hbar}\hat{H}{{\left[s,\hat{f}\right]}_{QPB}}$

G-dynamics:

$\hat{w}=\frac{1}{\sqrt{-1}\hbar}{{\left[s,\hat{H}\right]}_{QPB}}$.

respectively, where $\frac{\mathcal{D}}{dt}=\frac{d}{dt}+\hat{w}$ is covariant
time operator, and ${{\hat{H}}}$ is the Hamiltonian and $[\cdot,\cdot]$
denotes the GGC of two operators.

Here, G-dynamics is only for the description of some kind of rotation, it’s
truly a rotation operator which never appears before in quantum mechanics. As
a result of the G-dynamics,

$\hat{w}=\frac{1}{\sqrt{-1}\hbar}{{\left[s,\hat{H}\right]}_{QPB}}.$ (9)

Actually, quantum dynamical variables $\widehat{f}$ covariantly evolves in
time

$\displaystyle\frac{\mathcal{D}}{dt}\hat{f}$
$\displaystyle=\frac{1}{\sqrt{-1}\hbar}\left[\hat{f},\hat{H}\right]$
$\displaystyle=\frac{1}{\sqrt{-1}\hbar}{{\left[\hat{f},\hat{H}\right]}_{QPB}}-\frac{1}{\sqrt{-1}\hbar}\hat{H}{{\left[s,\hat{f}\right]}_{QPB}}$
$\displaystyle\begin{matrix}{}&{}&{}&{}&{}\\\
\end{matrix}+\frac{1}{\sqrt{-1}\hbar}\hat{f}{{\left[s,\hat{H}\right]}_{QPB}}$
$\displaystyle=\frac{d}{dt}\hat{f}+\hat{f}\hat{w}$

Analogously, this is similar to the dynamic equation in ordinary differential
equation when covariant equilibrium $\frac{\mathcal{D}}{dt}\hat{f}=0$ holds,
it also reveals that covariant equation $\frac{d}{dt}\hat{f}+\hat{f}\hat{w}=0$
is a complete expression in quantum mechanics. Thus, any observable that
covariant commutes with the Hamiltonian is not a constant of the motion.

###  7.2 Imaginary geomenergy

In this section, the imaginary geomenergy 555geometric energy is abbreviated
as geomenergy is defined based on the G-dynamics, by using these new concepts,
we can better give a presentation for the covariant dynamics, ect.

######  Definition 11.

The imaginary geomenergy is defined by
${{E}^{\left(\operatorname{Im}\right)}}\left(\hat{w}\right)=\sqrt{-1}\hbar\hat{w}$,
where $\hat{w}$ means G-dynamics.

Thus, a dynamical variable $\hat{f}$, which corresponds to a fixed linear
operator, corresponds to a moving linear operator in this new scheme. It is
clear that the covariant dynamics leads us to a complete quantum system in
which the state of the system is represented by two separated dynamic quantum
system, and the dynamical variables are represented by corresponding moving
operators. This covariant dynamics which is outlined in another form below

$\displaystyle\sqrt{-1}\hbar\frac{\mathcal{D}}{dt}\hat{f}$
$\displaystyle=\left[\hat{f},\hat{H}\right]={{\left[\hat{f},\hat{H}\right]}_{QPB}}+G\left(s,\hat{f},\hat{H}\right)$
$\displaystyle=\sqrt{-1}\hbar\frac{d}{dt}\hat{f}+\sqrt{-1}\hbar\hat{f}\hat{w}$

Based on the equation (9), the covariant dynamics can be rewritten as

$\frac{\mathcal{D}}{dt}\hat{f}=\frac{1}{\sqrt{-1}\hbar}\left[\hat{f},\hat{H}\right]=\frac{d}{dt}\hat{f}+\hat{f}\hat{w}$
(10)

where $\hat{w}$ is G-dynamics. We consider the covariant equilibrium equation,
that is, $\frac{\mathcal{D}}{dt}\hat{f}=0$, in other words, it’s an equation
shown by $\frac{d}{dt}\hat{f}+\hat{f}\hat{w}=0$. More precisely, supported by
the (10), we obtain

$\sqrt{-1}\hbar\frac{\mathcal{D}}{dt}\hat{f}=\left[\hat{f},\hat{H}\right]=\sqrt{-1}\hbar\frac{d}{dt}\hat{f}+\hat{f}\sqrt{-1}\hbar\hat{w}$

With the imaginary geomenergy defined above 11, then covariant dynamics is
rewritten in the form

$\sqrt{-1}\hbar\frac{\mathcal{D}}{dt}\hat{f}=\left[\hat{f},\hat{H}\right]=\sqrt{-1}\hbar\frac{d}{dt}\hat{f}+\hat{f}{{E}^{\left(\operatorname{Im}\right)}}\left(\hat{w}\right)$

According to equation (8), the covariant dynamics of motion in terms of
Hamilton $\hat{H}$ can be written as

$\frac{\mathcal{D}}{dt}\hat{H}=\frac{1}{\sqrt{-1}\hbar}\left[\hat{H},\hat{H}\right]=0$

Accordingly, the generalized Heisenberg equation follows

$\frac{d}{dt}\hat{H}={\dot{\hat{H}}}=-\frac{1}{\sqrt{-1}\hbar}\hat{H}\left[s,\hat{H}\right]=-\hat{H}\hat{w}$

Obviously, the generalized Heisenberg equation is a revisionary theory
associated with the structure function $s$ of the known Heisenberg equation,
it means that the classical Heisenberg equation is incomplete for quantum
mechanics, because of the missing term
$-\frac{1}{\sqrt{-1}\hbar}\hat{H}{{\left[s,\hat{f}\right]}_{QPB}}$. On the
other hand, the G-dynamics is completely induced by the covariant dynamics as
an independent dynamics. The equilibrium equation of the generalized
Heisenberg equation is given by $\frac{d\hat{f}}{dt}=0$, in other words,

${{\left[\hat{f},\hat{H}\right]}_{QPB}}=\hat{H}{{\left[s,\hat{f}\right]}_{QPB}}$

In the same way, the equilibrium equation of the covariant dynamics is
$\frac{\mathcal{D}\hat{f}}{dt}=0$, that is, the covariant equilibrium equation
of the covariant dynamics is given by $\left[\hat{f},\hat{H}\right]=0$ which
means $\frac{d\hat{f}}{dt}+\hat{f}\hat{w}=0$, it leads to a quantum covariant
conservative quantity, fundamentally.

###  7.3 Canonical quantum covariant Hamilton equations

For the sake of argument that we manage to consider the position operator and
momentum operator respectively.

######  Theorem 8.

The generalized Heisenberg equations and canonical quantum covariant Hamilton
equations of motion on manifolds can be written respectively

the generalized Heisenberg equations of motion

$\displaystyle\frac{d}{dt}{{{\hat{q}}}_{\alpha}}={\dot{{\widehat{q}}_{\alpha}}}=\frac{1}{\sqrt{-1}\hbar}{{\left[{{{\hat{q}}}_{\alpha}},\hat{H}\right]}_{QPB}}-\frac{1}{\sqrt{-1}\hbar}\hat{H}\left[s,{{{\hat{q}}}_{\alpha}}\right]_{QPB}$
$\displaystyle\frac{d}{dt}{{{\hat{p}}}_{\alpha}}={\dot{{\widehat{p}}_{\alpha}}}=\frac{1}{\sqrt{-1}\hbar}{{\left[{{{\hat{p}}}_{\alpha}},\hat{H}\right]}_{QPB}}-\frac{1}{\sqrt{-1}\hbar}\hat{H}\left[s,{{{\hat{p}}}_{\alpha}}\right]_{QPB}$

canonical quantum covariant Hamilton equations

$\frac{\mathcal{D}}{dt}{{\hat{q}}_{\alpha}}=\frac{1}{\sqrt{-1}\hbar}\left[{{{\hat{q}}}_{\alpha}},\hat{H}\right],~{}~{}~{}\frac{\mathcal{D}}{dt}{{\hat{p}}_{\alpha}}=\frac{1}{\sqrt{-1}\hbar}\left[{{{\hat{p}}}_{\alpha}},\hat{H}\right]$

where
$\left[\cdot,~{}\cdot\right]={{\left[\cdot,~{}\cdot\right]}_{QPB}}+G\left(s;~{}\cdot,~{}\cdot\right)$
is quantum covariant Poisson bracket.

When the system of the spring is very small then we have to use a quantum
mechanical description of the system to correctly take into account quantum
effects such as the quantization of energy levels. We will follow the
procedure of canonical covariant quantization to obtain such a general
description: namely, we impose the equal-time geometric canonical commutation
relation.

######  Corollary 2.

the generalized Heisenberg equations of motion

$\displaystyle\frac{d}{dt}{{{\hat{q}}}_{\alpha}}={\dot{{\widehat{q}}_{\alpha}}}=\frac{1}{\sqrt{-1}\hbar}{{\left[{{{\hat{q}}}_{\alpha}},\hat{H}\right]}_{QPB}}$
$\displaystyle\frac{d}{dt}{{{\hat{p}}}_{\alpha}}={\dot{{\widehat{p}}_{\alpha}}}=\frac{1}{\sqrt{-1}\hbar}{{\left[{{{\hat{p}}}_{\alpha}},\hat{H}\right]}_{QPB}}-\frac{1}{\sqrt{-1}\hbar}\hat{H}\left[s,{{{\hat{p}}}_{\alpha}}\right]_{QPB}$

canonical quantum covariant Hamilton equations

$\displaystyle\frac{\mathcal{D}}{dt}{{{\hat{q}}}_{\alpha}}=\frac{1}{\sqrt{-1}\hbar}\left[{{{\hat{q}}}_{\alpha}},\hat{H}\right]=\frac{d}{dt}{{{\hat{q}}}_{\alpha}}+{{{\hat{q}}}_{\alpha}}\hat{w}$
$\displaystyle\frac{\mathcal{D}}{dt}{{{\hat{p}}}_{\alpha}}=\frac{1}{\sqrt{-1}\hbar}\left[{{{\hat{p}}}_{\alpha}},\hat{H}\right]=\frac{d}{dt}{{{\hat{p}}}_{\alpha}}+{{{\hat{p}}}_{\alpha}}\hat{w}$

where
$\left[\cdot,~{}\cdot\right]={{\left[\cdot,~{}\cdot\right]}_{QPB}}+G\left(s,\cdot,~{}\cdot\right)$
is QCPB.

###  7.4 The QCPB for quantum harmonic oscillator

As a certainly example based on the theorem 7, we will now start by briefly
reviewing the quantum mechanics of a one-dimensional harmonic oscillator (2),
and see how the QCPB can be incorporated using GGC in the covariant
quantization procedure. With the Hamiltonian given by (2), let’s use the QCPB
to recalculate the (3), we can see how difference emerges.

$\displaystyle\frac{\mathcal{D}}{dt}x\left(t\right)$
$\displaystyle=\frac{\sqrt{-1}}{\hbar}\left[\hat{H},x\left(t\right)\right]$
$\displaystyle=\frac{\sqrt{-1}}{\hbar}{{\left[\hat{H},x\left(t\right)\right]}_{QPB}}+\frac{\sqrt{-1}}{\hbar}G\left(s,\hat{H},x\left(t\right)\right)$
$\displaystyle=\frac{\sqrt{-1}}{\hbar}{{\left[\hat{H},x\left(t\right)\right]}_{QPB}}+\frac{\sqrt{-1}}{\hbar}x\left(t\right){{\left[\hat{H},s\right]}_{QPB}}$

where the quantum geobracket related to the position gives

$G\left(s,\hat{H},x\left(t\right)\right)=\hat{H}{{\left[s,x\left(t\right)\right]}_{QPB}}-x\left(t\right){{\left[s,\hat{H}\right]}_{QPB}}=-x\left(t\right){{\left[s,\hat{H}\right]}_{QPB}}$

By direct computation, we obtain

${{\left[\hat{H},s\right]}_{QPB}}=\frac{{{\hat{p}}^{\left(cl\right)2}}s+2{{\hat{p}}^{\left(cl\right)}}s{{\hat{p}}^{\left(cl\right)}}}{2m}$

The quantum geobracket is certainly shown

$G\left(s,\hat{H},x\left(t\right)\right)=x\left(t\right){{\left[\hat{H},s\right]}_{QPB}}=x\left(t\right)\frac{{{\hat{p}}^{\left(cl\right)2}}s+2{{\hat{p}}^{\left(cl\right)}}s{{\hat{p}}^{\left(cl\right)}}}{2m}$
$\frac{\mathcal{D}}{dt}x\left(t\right)=\frac{\sqrt{-1}}{\hbar}\left[\hat{H},x\left(t\right)\right]=\frac{{{\hat{p}}^{\left(cl\right)}}\left(t\right)}{m}+\frac{\sqrt{-1}}{\hbar}x\left(t\right)\frac{{{\hat{p}}^{\left(cl\right)2}}s+2{{\hat{p}}^{\left(cl\right)}}s{{\hat{p}}^{\left(cl\right)}}}{2m}$

G-dynamics is given by

$\hat{w}=\frac{\sqrt{-1}}{\hbar}\frac{{{\hat{p}}^{\left(cl\right)2}}s+2{{\hat{p}}^{\left(cl\right)}}s{{\hat{p}}^{\left(cl\right)}}}{2m}$
(11)

More specifically, the covariant dynamics reads

$\frac{\mathcal{D}}{dt}x\left(t\right)=\frac{\sqrt{-1}}{\hbar}\left[\hat{H},x\left(t\right)\right]=\frac{{{\hat{p}}^{\left(cl\right)}}\left(t\right)}{m}+x\left(t\right)\hat{w}$

And the generalized Heisenberg equation follows

$\displaystyle\frac{d}{dt}x\left(t\right)$
$\displaystyle=\frac{\sqrt{-1}}{\hbar}{{\left[\hat{H},x\left(t\right)\right]}_{QPB}}+\frac{\sqrt{-1}}{\hbar}\hat{H}{{\left[s,x\left(t\right)\right]}_{QPB}}$
$\displaystyle=\frac{\sqrt{-1}}{\hbar}{{\left[\hat{H},x\left(t\right)\right]}_{QPB}}=\frac{{{\widehat{p}}^{\left(cl\right)}}\left(t\right)}{m}$

In the same way, the covariant dynamics for the momentum operator is

$\displaystyle\frac{\mathcal{D}}{dt}{{\hat{p}}^{\left(cl\right)}}\left(t\right)$
$\displaystyle=\frac{\sqrt{-1}}{\hbar}\left[\hat{H},{{\hat{p}}^{\left(cl\right)}}\left(t\right)\right]$
$\displaystyle=\frac{\sqrt{-1}}{\hbar}{{\left[\hat{H},{{\hat{p}}^{\left(cl\right)}}\left(t\right)\right]}_{QPB}}+\frac{\sqrt{-1}}{\hbar}G\left(s,\hat{H},{{\hat{p}}^{\left(cl\right)}}\left(t\right)\right)$

where the quantum geobracket in terms of momentum operators is given by

$G\left(s,\hat{H},{{\hat{p}}^{\left(cl\right)}}\left(t\right)\right)=\hat{H}{{\left[s,{{\hat{p}}^{\left(cl\right)}}\left(t\right)\right]}_{QPB}}-{{\hat{p}}^{\left(cl\right)}}\left(t\right){{\left[s,\hat{H}\right]}_{QPB}}$

Further evaluation leads to

$G\left(s,\hat{H},{{{\hat{p}}}^{\left(cl\right)}}\left(t\right)\right)=\sqrt{-1}\hbar\hat{H}\frac{ds}{dx}+\frac{{{\widehat{p}}^{\left(cl\right)}}\left(t\right)}{2m}\left({{\widehat{p}}^{\left(cl\right)2}}s+2{{\widehat{p}}^{\left(cl\right)}}s{{\widehat{p}}^{\left(cl\right)}}\right)$

Subsequently, it yields

$\frac{\sqrt{-1}}{\hbar}G\left(s,\hat{H},{{{\hat{p}}}^{\left(cl\right)}}\left(t\right)\right)=-\hat{H}\frac{ds}{dx}+\frac{\sqrt{-1}}{\hbar}\frac{{{\widehat{p}}^{\left(cl\right)}}\left(t\right)}{2m}\left({{\widehat{p}}^{\left(cl\right)2}}s+2{{\widehat{p}}^{\left(cl\right)}}s{{\widehat{p}}^{\left(cl\right)}}\right)$

As a result, we obtain

$\displaystyle\frac{\mathcal{D}}{dt}{{{\hat{p}}}^{\left(cl\right)}}\left(t\right)$
$\displaystyle=\frac{\sqrt{-1}}{\hbar}\left[\hat{H},{{{\hat{p}}}^{\left(cl\right)}}\left(t\right)\right]$
$\displaystyle=-m{{\omega}^{2}}x-\hat{H}\frac{ds}{dx}+\frac{\sqrt{-1}}{\hbar}{{\widehat{p}}^{\left(cl\right)}}\left(t\right)\frac{\left({{\widehat{p}}^{\left(cl\right)2}}s+2{{\widehat{p}}^{\left(cl\right)}}s{{\widehat{p}}^{\left(cl\right)}}\right)}{2m}$

Accordingly, the generalized Heisenberg equation appears

$\displaystyle\frac{d}{dt}{{{\hat{p}}}^{\left(cl\right)}}\left(t\right)$
$\displaystyle=\frac{\sqrt{-1}}{\hbar}{{\left[\hat{H},{{{\hat{p}}}^{\left(cl\right)}}\left(t\right)\right]}_{QPB}}+\frac{\sqrt{-1}}{\hbar}\hat{H}{{\left[s,{{{\hat{p}}}^{\left(cl\right)}}\left(t\right)\right]}_{QPB}}$
$\displaystyle=-m{{\omega}^{2}}x-\hat{H}\frac{ds}{dx}$

While G-dynamics remains as (11) given.

###  7.5 The GSPB and the QCPB

In total, we emphasize that the GSPB in an abstract form is given as
definition shown

$\left\\{f,g\right\\}={{\left\\{f,g\right\\}}_{GPB}}+G\left(s,f,g\right)$

While the QCPB is taken the same pattern to reflect the GSPB

$\left[\hat{f},\hat{g}\right]={{\left[\hat{f},\hat{g}\right]}_{QPB}}+G\left(s,\hat{f},\hat{g}\right)$

as a similar pattern for classical and quantum brackets. In the view of the
relation between quantum mechanics and classic mechanics. Notice that in this
representation the fundamental equations are similar to the classical
equations with the substitution.

$\displaystyle\left\\{f,g\right\\}={{\left\\{f,g\right\\}}_{GPB}}+G\left(s,f,g\right)$
$\displaystyle\begin{matrix}{}&{}&{}&{}&{}\\\
\end{matrix}\mapsto\frac{1}{\sqrt{-1}\hbar}\left[\hat{f},\hat{g}\right]=\frac{1}{\sqrt{-1}\hbar}\left({{\left[\hat{f},\hat{g}\right]}_{QPB}}+G\left(s,\hat{f},\hat{g}\right)\right)$

With the solid foundation of the GSPB and the QCPB in abstract, then the
covariant dynamical equation certainly follows based on the GSPB and the QCPB
respectively. From the GCHS

$\frac{\mathcal{D}f}{dt}=\left\\{f,H\right\\}={{\left\\{f,H\right\\}}_{GPB}}+G\left(s,f,H\right)$

to the quantum covariant dynamics. Furthermore, the covariant equation of the
motion of an arbitrary operator $\hat{f}$ or the observables $\hat{f}$ that
satisfies the quantum covariant evolution

$\frac{\mathcal{D}\hat{f}}{dt}=\frac{1}{\sqrt{-1}\hbar}\left[\hat{f},\hat{H}\right]=\frac{1}{\sqrt{-1}\hbar}\left({{\left[\hat{f},\hat{H}\right]}_{QPB}}+G\left(s,\hat{f},\hat{H}\right)\right)$

Namely, the time evolution of the operator $\hat{f}$ is then given by the
corresponding equation

$\frac{\mathcal{D}f}{dt}=\left\\{f,H\right\\}\to\frac{\mathcal{D}\hat{f}}{dt}=\frac{1}{\sqrt{-1}\hbar}\left[\hat{f},\hat{H}\right]$

Note that above descriptions are formally abstract,

_The GSPB and the QCPB :
$\displaystyle\left\\{f,g\right\\}={{\left\\{f,g\right\\}}_{GPB}}+G\left(s,f,g\right)$
$\displaystyle\left[\hat{f},\hat{g}\right]={{\left[\hat{f},\hat{g}\right]}_{QPB}}+G\left(s,\hat{f},\hat{g}\right)$
_
---

As a consequence, the equations of motion are respectively obtained by
applying generalized Heisenberg equations of motion and canonical quantum
covariant Hamilton equations.

According to the rule of quantization, we have the GSPB to QCPB given by

$\left\\{\cdot,\cdot\right\\}\to\left[\cdot,\cdot\right]$

More precisely,

$\left\\{\cdot,\cdot\right\\}={{\left\\{\cdot,\cdot\right\\}}_{GPB}}+G\left(s,\cdot,\cdot\right)$

is transformed to

$\frac{1}{\sqrt{-1}\hbar}\left[\cdot,\cdot\right]=\frac{1}{\sqrt{-1}\hbar}{{\left[\cdot,\cdot\right]}_{QPB}}+\frac{1}{\sqrt{-1}\hbar}G\left(s,\cdot,\cdot\right)$

More generally, this technique leads to a complete covariant quantization that
can be accurately computed and analyzed, mathematically.

######  Theorem 9.

There is a quantization map $Q$ such that

${\displaystyle\quad Q_{\\{f,g\\}}={\frac{1}{\sqrt{-1}\hbar}}[Q_{f},Q_{g}]}$
(12)

where $\\{\cdot,\cdot\\}$ and $[\cdot,\cdot]$ are GSPB and QCPB respectively.

More accurately, the equation (12) is clearly expressed as

$\displaystyle{{Q}_{{{\left\\{f,g\right\\}}_{GPB}}+G\left(s,f,g\right)}}$
$\displaystyle=\frac{1}{\sqrt{-1}\hbar}\left({{\left[{{Q}_{f}},{{Q}_{g}}\right]}_{QPB}}+G\left(s,{{Q}_{f}},{{Q}_{g}}\right)\right)$
$\displaystyle=\frac{1}{\sqrt{-1}\hbar}{{\left[{{Q}_{f}},{{Q}_{g}}\right]}_{QPB}}+\frac{1}{\sqrt{-1}\hbar}G\left(s,{{Q}_{f}},{{Q}_{g}}\right)$

or conversely, we have

${{\left[{{Q}_{f}},{{Q}_{g}}\right]}_{QPB}}+G\left(s,{{Q}_{f}},{{Q}_{g}}\right)=\sqrt{-1}\hbar{{Q}_{{{\left\\{f,g\right\\}}_{GPB}}+G\left(s,f,g\right)}}$

This follows through use of the quantum correspondence, since we refer to the
(6).

###  7.6 Geobracket and quantum geobracket

In this section, we put the geobracket and quantum geobracket together to
contrastively analyze their connections. Especially, the covariant correction
term–the quantum geobracket $G\left(s,\hat{f},\hat{g}\right)$ is unknown, but
by logically derivation, the specific formula of the geobracket
$G\left(s,f,g\right)$ is surely concretized in GSPB, what we do is to copy
this mode for the quantum geobracket $G\left(s,\hat{f},\hat{g}\right)$. As
described previously, the geobracket $G\left(s,f,g\right)$ in GSPB is defined
as

$G\left(s,f,g\right)=f{{\left\\{s,g\right\\}}_{GPB}}-g{{\left\\{s,f\right\\}}_{GPB}}$

In a similar mode for quantum mechanics, quantum geobracket
$G\left(s,\hat{f},\hat{g}\right)$ can be basically concretized in QCPB, that
is given by as defined

$G\left(s,\hat{f},\hat{g}\right)=\hat{f}{{\left[s,\hat{g}\right]}_{QPB}}-\hat{g}{{\left[s,\hat{f}\right]}_{QPB}}$

With the help of the structural function, the QCPB is well defined for
covariant quantum mechanics666Notes: GCC: Geometric canonical commutation;
CCHE:canonical covariant Hamilton equations; CD:covariant dynamics; GHE:
generalized Heisenberg equations; CTHE: Canonical thorough Hamilton
equations..

covariant quantum mechanics
---
QCPB:
$\left[\hat{f},\hat{g}\right]={{\left[\hat{f},\hat{g}\right]}_{QPB}}+G\left(s,\hat{f},\hat{g}\right)$,
$G\left(s,\hat{f},\hat{g}\right)=\hat{f}{{\left[s,\hat{g}\right]}_{QPB}}-\hat{g}{{\left[s,\hat{f}\right]}_{QPB}}$
GCC:
$\left[\overset{\wedge}{\mathop{{{q}_{i}}}}\,,\overset{\wedge}{\mathop{{{p}_{j}}}}\,\right]=\sqrt{-1}\hbar{{D}_{j}}{{q}_{i}}$,
${{D}_{i}}={{\partial}_{i}}+{{\partial}_{i}}s,{{\partial}_{i}}=\frac{\partial}{\partial{{q}_{i}}}$.
Hamiltonian : $\widehat{H}$
CD:
$\frac{\mathcal{D}}{dt}{{\hat{q}}_{\alpha}}=\frac{1}{\sqrt{-1}\hbar}\left[{{{\hat{q}}}_{\alpha}},\hat{H}\right],~{}~{}~{}\frac{\mathcal{D}}{dt}{{\hat{p}}_{\alpha}}=\frac{1}{\sqrt{-1}\hbar}\left[{{{\hat{p}}}_{\alpha}},\hat{H}\right]$
GHE:
${\dot{{\widehat{q}}_{\alpha}}}=\frac{1}{\sqrt{-1}\hbar}{{\left[{{{\hat{q}}}_{\alpha}},\hat{H}\right]}_{QPB}},~{}~{}{\dot{{\widehat{p}}_{\alpha}}}=\frac{1}{\sqrt{-1}\hbar}\left({{\left[{{{\hat{p}}}_{\alpha}},\hat{H}\right]}_{QPB}}-\hat{H}\left[s,{{{\widehat{p}}}_{\alpha}}\right]_{QPB}\right)$

where $\left[\hat{f},\hat{g}\right]_{QPB}=\hat{f}\hat{g}-\hat{g}\hat{f}$ is
quantum Poisson brackets for operators.

##  8 Covariant quantization of Field

In this section, we only depend on the QCPB to obtain the covariant
quantization of field, generally, it helps us better understand how
specifically the fields move and evolve.

For the quantum of a Bose field, the canonical coordinate ${{\varphi}_{i}}$
and the canonical conjugate momentum ${{p}_{i}}$ of the field satisfies the
QCPB, it’s depicted as

$\left[{{\varphi}_{i}},{{p}_{j}}\right]={{\left[{{\varphi}_{i}},{{p}_{j}}\right]}_{QPB}}+G\left(s,{{\varphi}_{i}},{{p}_{j}}\right)$

Subsequently, the QCHS for Bose field follows

$\left[\varphi,\hat{H}\right]={{\left[\varphi,\hat{H}\right]}_{QPB}}+G\left(s,\varphi,\hat{H}\right)$

Then canonical quantum covariant Hamilton equations emerge accordingly,

$\frac{\mathcal{D}\varphi}{dt}=\frac{1}{\sqrt{-1}\hbar}\left[\varphi,\hat{H}\right]=\frac{1}{\sqrt{-1}\hbar}\left({{\left[\varphi,\hat{H}\right]}_{QPB}}+G\left(s,\varphi,\hat{H}\right)\right)$

Similarly, meanwhile, for the quantum of Fermi field $\Psi\left(t,x\right)$,
and the canonical coordinates and canonical conjugate momenta of the field are
$\Psi\left(t,x\right)$, $\pi\left(t,x\right)$ respectively. The same procedure
as Bose field does goes for Fermi field, we list it as follows, the QCHS for
Fermi field $\Psi\left(t,x\right)$ is given by

$\left[\Psi,\hat{H}\right]={{\left[\Psi,\hat{H}\right]}_{QPB}}+G\left(s,\Psi,\hat{H}\right)$

and quantum covariant Hamilton equation appears

$\frac{\mathcal{D}\Psi}{dt}=\frac{1}{\sqrt{-1}\hbar}\left[\Psi,\hat{H}\right]=\frac{1}{\sqrt{-1}\hbar}\left({{\left[\Psi,\hat{H}\right]}_{QPB}}+G\left(s,\Psi,\hat{H}\right)\right)$
$\frac{\mathcal{D}\pi}{dt}=\frac{1}{\sqrt{-1}\hbar}\left[\pi,\hat{H}\right]=\frac{1}{\sqrt{-1}\hbar}\left({{\left[\pi,\hat{H}\right]}_{QPB}}+G\left(s,\pi,\hat{H}\right)\right)$

In general, the quantum geobracket is taken in the form

$G\left(s,\cdot,\hat{H}\right)\equiv\cdot{{\left[s,\hat{H}\right]}_{QPB}}-\hat{H}{{\left[s,\cdot\right]}_{QPB}}$

where based on the G dynamics

${{\left[s,\hat{H}\right]}_{QPB}}=\sqrt{-1}\hbar\hat{w}$

More specifically,

$G\left(s,\varphi,\hat{H}\right)\equiv\varphi{{\left[s,\hat{H}\right]}_{QPB}}-\hat{H}{{\left[s,\varphi\right]}_{QPB}}$

and

$G\left(s,\Psi,\hat{H}\right)\equiv\Psi{{\left[s,\hat{H}\right]}_{QPB}}-\hat{H}{{\left[s,\Psi\right]}_{QPB}}$

This procedure can be applied to the covariant quantization of any field
theory: whether of fermions or bosons. All other fields can be quantized by
QCPB of this procedure.

###  8.1 Canonical covariant quantization of Bose field

The quantum of a Bose field, the spin is integer or zero, obeying the Bose-
Einstein statistical method, known as bosons, the canonical coordinate
${{\varphi}_{i}}$ and the canonical conjugate momentum ${{p}_{i}}$ of the
field are regarded as the operator of Hilbert space, and the corresponding
geometric canonical commutation relation in discrete field form is

$\left[{{\varphi}_{i}},{{p}_{j}}\right]=\sqrt{-1}\hbar{{D}_{j}}{{\varphi}_{i}},~{}~{}\left[{{\varphi}_{i}},~{}~{}{{\varphi}_{j}}\right]=\left[{{p}_{i}},{{p}_{j}}\right]=0$

Accordingly, the generalized canonical equations of motion for corresponding
quantum fields are shown as

$\overset{\centerdot}{\mathop{\varphi}}\,=\frac{\sqrt{-1}}{\hbar}\left[\hat{H},\varphi\right]_{QPB}+\frac{\sqrt{-1}}{\hbar}\hat{H}\left[s,\varphi\right]_{QPB}$
$\overset{\centerdot}{\mathop{\pi}}\,=\frac{\sqrt{-1}}{\hbar}\left[\hat{H},\pi\right]_{QPB}+\frac{\sqrt{-1}}{\hbar}\hat{H}\left[s,\pi\right]_{QPB}$

The quantization of the field depends entirely on the commutators. And
canonical quantum covariant Hamilton equations are accordingly shown as

$\frac{\mathcal{D}}{dt}\varphi=\frac{\sqrt{-1}}{\hbar}\left[\hat{H},\varphi\right]$
$\frac{\mathcal{D}}{dt}\pi=\frac{\sqrt{-1}}{\hbar}\left[\hat{H},\pi\right]$

###  8.2 Canonical covariant quantization of Fermi field

The quantum of Fermi field has half spin and obeys the Fermi-Dirac statistical
method. Suppose that $\Psi\left(t,x\right)$ is the field function of the Fermi
field, and the canonical coordinates and canonical conjugate momenta of the
field are $\Psi\left(t,x\right)$, $\pi\left(t,x\right)$ respectively.
Accordingly, the generalized canonical equations of motion for corresponding
quantum fields are shown as

$\overset{\centerdot}{\mathop{\Psi}}\,=\frac{\sqrt{-1}}{\hbar}\left[\hat{H},\Psi\right]_{QPB}+\frac{\sqrt{-1}}{\hbar}\hat{H}\left[s,\Psi\right]_{QPB}$
$\overset{\centerdot}{\mathop{\pi}}\,=\frac{\sqrt{-1}}{\hbar}\left[\hat{H},\pi\right]_{QPB}+\frac{\sqrt{-1}}{\hbar}\hat{H}\left[s,\pi\right]_{QPB}$

The canonical covariant motion equation of quantum field is

$\displaystyle\frac{\mathcal{D}}{dt}\Psi\left(t,x\right)=\frac{\sqrt{-1}}{\hbar}\left[\hat{H}\left(t\right),\Psi\left(t,x\right)\right]=\overset{\centerdot}{\mathop{\Psi}}\,+\Psi\hat{w}$
$\displaystyle\frac{\mathcal{D}}{dt}\pi\left(t,x\right)=\frac{\sqrt{-1}}{\hbar}\left[\hat{H}\left(t\right),\pi\left(t,x\right)\right]=\overset{\centerdot}{\mathop{\pi}}\,+\pi\hat{w}$

##  9 Geomentum operator and geometric canonical commutation

######  Definition 12.

Let $M$ be a smooth function represented by structural function $s$, then
geomentum operator is defined as

$\hat{p}=-\sqrt{-1}\hbar D$

where $D=\nabla+\nabla s$. The component is
${{\hat{p}}_{j}}=-\sqrt{-1}\hbar{{D}_{j}}$ in which
${{D}_{j}}={{\partial}_{j}}+{{\partial}_{j}}s$ holds,
${{\partial}_{j}}=\frac{\partial}{\partial{{x}_{j}}}$.

For instance, let’s consider the canonical commutation relation Eq(4), namely,
let $\hat{f}=\hat{{{x}_{i}}},~{}\hat{g}=\hat{{{p}_{j}}}$ be given for
discussions. As a result, geometric canonical commutation relation can be
compactly written as

$\left[\hat{{{x}_{i}}},\hat{{{p}_{j}}}\right]=\sqrt{-1}\hbar{{D}_{j}}{{x}_{i}}$

Let $\hat{f}={{x}_{i}},~{}\hat{g}={{\hat{p}}_{j}}=-\sqrt{-1}\hbar{{D}_{j}}$,
by using QCPB, we have

$\left[{{x}_{i}},-\sqrt{-1}\hbar{{D}_{j}}\right]={{\left[{{x}_{i}},-\sqrt{-1}\hbar{{D}_{j}}\right]}_{QPB}}+G(s,{{x}_{i}},-\sqrt{-1}\hbar{{D}_{j}})$

More precisely, it shows

$\left[{{x}_{i}},-\sqrt{-1}\hbar{{D}_{j}}\right]={{\left[{{x}_{i}},-\sqrt{-1}\hbar{{D}_{j}}\right]}_{QPB}}+\sqrt{-1}\hbar{{D}_{j}}{{\left[s,{{x}_{i}}\right]}_{QPB}}+{{x}_{i}}{{\left[s,-\sqrt{-1}\hbar{{D}_{j}}\right]}_{QPB}}$

Further, put the wave function onto it, it leads to the result

$\displaystyle\left[{{x}_{i}},-\sqrt{-1}\hbar{{D}_{j}}\right]\psi$
$\displaystyle=-\sqrt{-1}\hbar{{\left[{{x}_{i}},{{D}_{j}}\right]}_{QPB}}\psi-\sqrt{-1}\hbar{{x}_{i}}{{\left[s,{{D}_{j}}\right]}_{QPB}}\psi$
$\displaystyle=-\sqrt{-1}\hbar\left\\{{{\left[{{x}_{i}},{{D}_{j}}\right]}_{QPB}}\psi+{{x}_{i}}{{\left[s,{{D}_{j}}\right]}_{QPB}}\psi\right\\}$
$\displaystyle=\sqrt{-1}\hbar\left({{\delta}_{ij}}+{{x}_{i}}{{\partial}_{j}}s\right)\psi$
$\displaystyle=\sqrt{-1}\hbar\psi{{D}_{j}}{{x}_{i}}$

As a result, we obtain

$\left[{{x}_{i}},{{\hat{p}}_{j}}\right]=\left[{{x}_{i}},-\sqrt{-1}\hbar{{D}_{j}}\right]=\sqrt{-1}\hbar{{D}_{j}}{{x}_{i}}$

Equivalently, as defined above, using QCPB with
${{\hat{{{p}_{j}}}}^{\left(cl\right)}}=-\sqrt{-1}\hbar\frac{\partial}{\partial{{x}_{j}}},~{}~{}\hat{{{x}_{i}}}\equiv{{x}_{i}}$,
then the QCPB is

$\left[\hat{{{x}_{i}}},{{\hat{{{p}_{j}}}}^{\left(cl\right)}}\right]={{\left[\hat{{{x}_{i}}},{{\hat{{{p}_{j}}}}^{\left(cl\right)}}\right]}_{QPB}}+\hat{{{x}_{i}}}{{\left[s,{{\hat{{{p}_{j}}}}^{\left(cl\right)}}\right]}_{QPB}}-{{\hat{{{p}_{j}}}}^{\left(cl\right)}}{{\left[s,\hat{{{x}_{i}}}\right]}_{QPB}}$

One only needs to calculate the quantum geobracket

$G\left(s,\hat{{{x}_{i}}},{{\hat{{{p}_{j}}}}^{\left(cl\right)}}\right)\psi=\hat{{{x}_{i}}}{{\left[s,{{\hat{{{p}_{j}}}}^{\left(cl\right)}}\right]}_{QPB}}\psi-{{\hat{{{p}_{j}}}}^{\left(cl\right)}}{{\left[s,\hat{{{x}_{i}}}\right]}_{QPB}}\psi$

Due to ${{\left[s,\hat{{{x}_{i}}}\right]}_{QPB}}=0$ is obvious, then

$\displaystyle\hat{{{x}_{i}}}{{\left[s,{{\hat{{{p}_{j}}}}^{\left(cl\right)}}\right]}_{QPB}}\psi$
$\displaystyle={{x}_{i}}s{{\hat{{{p}_{j}}}}^{\left(cl\right)}}\psi-{{x}_{i}}{{\hat{{{p}_{j}}}}^{\left(cl\right)}}\left(s\psi\right)$
$\displaystyle=-{{x}_{i}}s\sqrt{-1}\hbar\frac{\partial\psi}{\partial{{x}_{j}}}+{{x}_{i}}\sqrt{-1}\hbar\frac{\partial}{\partial{{x}_{j}}}\left(s\psi\right)$
$\displaystyle={{x}_{i}}\psi\sqrt{-1}\hbar\frac{\partial}{\partial{{x}_{j}}}s$

Thus, it leads to the result

$G\left(s,\hat{{{x}_{i}}},{{\hat{{{p}_{j}}}}^{\left(cl\right)}}\right)\psi={{x}_{i}}\psi\sqrt{-1}\hbar\frac{\partial}{\partial{{x}_{j}}}s$

which also deduces
$G\left(s,\hat{{{x}_{i}}},{{\hat{{{p}_{j}}}}^{\left(cl\right)}}\right)=\sqrt{-1}\hbar{{x}_{i}}\frac{\partial}{\partial{{x}_{j}}}s$.
As a result, one can conclude that

$\displaystyle\left[\hat{{{x}_{i}}},{{\hat{{{p}_{j}}}}^{\left(cl\right)}}\right]$
$\displaystyle={{\left[\hat{{{x}_{i}}},{{\hat{{{p}_{j}}}}^{\left(cl\right)}}\right]}_{QPB}}+\hat{{{x}_{i}}}{{\left[s,{{\hat{{{p}_{j}}}}^{\left(cl\right)}}\right]}_{QPB}}$
$\displaystyle=\sqrt{-1}\hbar{{\delta}_{ij}}+G\left(s,\hat{{{x}_{i}}},{{\hat{{{p}_{j}}}}^{\left(cl\right)}}\right)$
$\displaystyle=\sqrt{-1}\hbar\left({{\delta}_{ij}}+{{x}_{i}}\frac{\partial}{\partial{{x}_{j}}}s\right)$

The situation for this same result is because
${{\left[s,-\sqrt{-1}\hbar{{D}_{j}}\right]}_{QPB}}={{\left[s,-\sqrt{-1}\hbar{{\partial}_{j}}\right]}_{QPB}}$
holds.

Geometric canonical commutation relation can be called geometric canonical
quantization rules.

######  Theorem 10 (Geometric canonical quantization rules).

Geometric equal-time canonical commutation relation is

$\left[\hat{{{x}_{i}}},\hat{{{p}_{j}}}\right]=\sqrt{-1}\hbar{{D}_{j}}{{x}_{i}}$
$\left[\hat{{{x}_{i}}},\hat{{{x}_{j}}}\right]=0$
$\left[\hat{{{p}_{i}}},\hat{{{p}_{j}}}\right]=0$

###### Proof.

$\displaystyle\left[-\sqrt{-1}\hbar{{D}_{i}},-\sqrt{-1}\hbar{{D}_{j}}\right]$
$\displaystyle={{\left[-\sqrt{-1}\hbar{{D}_{i}},-\sqrt{-1}\hbar{{D}_{j}}\right]}_{QPB}}+\sqrt{-1}\hbar{{D}_{j}}{{\left[s,-\sqrt{-1}\hbar{{D}_{i}}\right]}_{QPB}}$
$\displaystyle\begin{matrix}{}&{}&{}\\\
\end{matrix}-\sqrt{-1}\hbar{{D}_{i}}{{\left[s,-\sqrt{-1}\hbar{{D}_{j}}\right]}_{QPB}}$
$\displaystyle={{\hbar}^{2}}\left\\{{{D}_{j}}{{\left[s,{{D}_{i}}\right]}_{QPB}}-{{D}_{i}}{{\left[s,{{D}_{j}}\right]}_{QPB}}-{{\left[{{D}_{i}},{{D}_{j}}\right]}_{QPB}}\right\\}$
$\displaystyle={{\hbar}^{2}}\left({{\partial}_{j}}s{{\partial}_{i}}-{{\partial}_{i}}s{{\partial}_{j}}\right)=0$

where
${{\left[{{\partial}_{j}},{{\partial}_{i}}\right]}_{QPB}}=\left({{\partial}_{j}}{{\partial}_{i}}-{{\partial}_{i}}{{\partial}_{j}}\right)=0$.

∎

Geometric canonical commutation relation can be expressed in a specific form

$\left[\hat{{{x}_{i}}},\hat{{{p}_{j}}}\right]=\sqrt{-1}\hbar{{D}_{j}}{{x}_{i}}=\sqrt{-1}\hbar\left({{\delta}_{ij}}+{{x}_{i}}\frac{\partial}{\partial{{x}_{j}}}s\right)$

In other words, it also can be rewritten as
$\left[{{x}_{i}},\hat{{{p}_{j}}}\right]=\sqrt{-1}\hbar{{\theta}_{ij}}$, where
${{\theta}_{ij}}={{\delta}_{ij}}+{{x}_{i}}{{\partial}_{j}}s$, and
${{\partial}_{j}}=\frac{\partial}{\partial{{x}_{j}}}$. Accordingly, the
quantum geobracket relates to the below

$\frac{1}{\sqrt{-1}\hbar}G\left(s,{{x}_{i}},{{\hat{{{p}_{j}}}}}\right)={{\theta}_{ij}}-{{\delta}_{ij}}$

or
$G\left(s,{{x}_{i}},{{\hat{{{p}_{j}}}}}\right)=\sqrt{-1}\hbar\left({{\theta}_{ij}}-{{\delta}_{ij}}\right)$.

##  10 Conclusions

The conclusions can be assembled here. In this paper, by bringing the
structure function determined by the spacetime itself into the commutator,
then a generalized geometric commutator totally relied on structure function
forms, in other words, generalized geometric commutator is constructed by the
commutator equipped with geomutator. Accordingly, quantum covariant Poisson
bracket and quantum geometric bracket are respectively defined by generalized
geometric commutator and geomutator.

In theory, with the definition of quantum covariant Poisson bracket (QCPB) and
generalized structure Possion bracket(GSPB), it well defines the covariant
theoretical framework for the quantum mechanics and generalized Hamiltonian
mechanics.

Firstly, by using the structural function $s$ determined by the manifolds, one
generalized the QPB to more general form QCPB or GGC, it’s formally shown as

$\left[\cdot,~{}\cdot\right]={{\left[\cdot,~{}\cdot\right]}_{QPB}}+G\left(s,\cdot,~{}\cdot\right)$

where
$G\left(s,\cdot,~{}\cdot\right)=\cdot{{\left[s,~{}\cdot\right]}_{QPB}}+\cdot{{\left[\cdot~{},s\right]}_{QPB}}$
is quantum geobracket as a correctional term for the QPB, which is the vital
part of the QCPB to the quantum mechanics. Geometric canonical commutation
relation follows as well. Secondly, using QCPB to define QCHS, and the
covariant dynamics, the generalized Heisenberg equation, G-dynamics are
emerged one by one. In the end, the quantum covariant Poisson bracket (QCPB)
is used for quantization of field including Bose field and Fermi field, we
obtain covariant quantization of field, we revise the classical theory for
both Bose field and Fermi field. As for the application of quantum covariant
Poisson bracket, the classical canonical commutation relation and quantization
of field are considered under the QCPB, we obtain some different useful
results.

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