# Affine Connection Representation of Gauge Fields

Authors: Zhao-Hui Man
Journal: Advances in High Energy Physics
 (2022)

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## Abstract

There are two ways to unify gravitational field and gauge field. One is to represent gravitational field as principal bundle connection, and the other is to represent gauge field as affine connection. Poincaré gauge theory and metric-affine gauge theory adopt the first approach. This paper adopts the second. In this approach, (i) gauge field and gravitational field can both be represented by affine connection; they can be described by a unified spatial frame. (ii) Time can be regarded as the total metric with respect to all dimensions of internal coordinate space and external coordinate space. On-shell can be regarded as gradient direction. Quantum theory can be regarded as a geometric theory of distribution of gradient directions. Hence, gauge theory, gravitational theory, and quantum theory all reflect intrinsic geometric properties of manifold. (iii) Coupling constants, chiral asymmetry, PMNS mixing, and CKM mixing arise spontaneously as geometric properties in affine connection representation, so they are not necessary to be regarded as direct postulates in the Lagrangian anymore. (iv) The unification theory of gauge fields that are represented by affine connection can avoid the problem that a proton decays into a lepton in theories such asSU5. (v) There exists a geometric interpretation to the color confinement of quarks. In the affine connection representation, we can get better interpretations to the above physical properties; therefore, to represent gauge fields by affine connection is probably a necessary step towards the ultimate theory of physics.


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## Body

## 1. Introduction

### 1.1. Background and Purpose

We know that in gauge theory, the field strength and the gauge-covariant derivative(1)Fμνa=∂μAνa−∂νAμa+gfabcAμbAνc,Dμ=∂μ−igTaAμa,both contain a coupling constant g, which measures the strength of interaction. A problem is that why is there a coupling constant g?If we represent gauge fields by affine connection, we can obtain a nice interpretation. For example, if we useΓMNP to represent gauge potentials, it is not hard to find some specific conditions to turn the curvature tensor RNPQM to
(2)RMNPQ=∂PΓMNQ−∂QΓMNP+ΓMHPΓNQH−ΓNPHΓMHQ=∂PΓMNQ−∂QΓMNP+GRHΓMHPΓRNQ−ΓRNPΓMHQ.Thus,RMNPQ can be used to represent field strength. In addition, for any ρM, we see that
(3)ρM;P=∂PρM−ΓMPHρH=∂PρM−GRHΓRMPρH.Equations (2) and (3) mean that the coupling constant g may have a geometric meaning, which originates from GRH.This implies that only when affine connection is adopted to represent gauge field can some physical properties be better interpreted. On the other hand, in the general relativity theory, gravitational field is also described by affine connection, so it is convenient to describe gravitational field and gauge field uniformly by affine connection. Therefore, it is necessary to study the affine connection representation of gauge fields. This is the basic motivation of this paper.There are the following two ways to unify gravitational field and gauge field.One way is to represent gravitational field as principal bundle connection. We can take the transformation groupGravi3,1 of gravitational field as the structure group of principal bundle to establish a gauge theory of gravitational field, the local transformation group of which is in the form of Gravi3,1⊗Gaugen, e.g., Poincaré gauge theory [1–11] and metric-affine gauge theory [12–23]. This way can be interpreted intuitively as
(4)The other way is to represent gauge field as affine connection. This is the approach adopted by this paper. Gravitational field and gauge field can both be described by affine connection. Besides, we will also establish an affine connection representation of elementary particles. This way can be interpreted intuitively as(5)


### 1.2. Ideas and Methods

We divide the problem of establishing affine connection representation of gauge fields into three parts as follows.(i)
Which affine connection is suitable for describing not only gravitational field, but also gauge field and elementary particle field?(ii)
How to describe the evolution of these fields in affine connection representation?(iii)
What are the concrete forms of electromagnetic, weak, and strong interaction fields in affine connection representation?For the problem (i). On a Riemannian manifold M,G, the metric tensor can be expressed as GMN=δABBMABNB and GMN=δABCAMCBN, where BMA and CAM are semimetrics or to say frame fields. It is evident that semimetric is more fundamental than metric, so we hope BMA or CAM is regarded as a unified frame field of gravitational field and gauge field, and the frame transformation of BMA or CAM is regarded as gauge transformation. Hence, we need a more general manifold M,BMA rather than the Riemannian manifold M,G.Next, we put metric and semimetric together to construct a new connection, which is not only an affine connection, but also a connection on a fibre bundle. In this way, gravitational field and various gauge fields can be unified on a manifoldM,BMA that is defined by semimetric.In addition, we notice that in the theories based on principal bundle connection representation,(1)
Several complex-valued functions, which satisfy the Dirac equation, are sometimes used to refer to a charged lepton fieldl and sometimes a neutrino field ν. It is not clear how to distinguish these field functions l and ν by inherent geometric constructions(2)
Gauge potentials are abstract; they have no inherent geometric constructions. In other words, the Levi-Civita connectionΓνρμ of gravity is constructed by the metric gμν; however, it is not explicit what geometric quantity the connection Aμa of gauge field is constructed byBy contrast, in the affine connection representation of this paper, we are able to use the semimetricsBMA and CAM of internal coordinate space to endow particle fields l and ν and gauge field Aμa with geometric constructions. Thus, they are not only irreducible representations of group but also possessed of concrete geometric entities.For the problem (ii). There is a fundamental difficulty that time is effected by gravitational field, but not effected by gauge field. This leads to an essential difference between the description of evolution of gravitational field and that of gauge field. In this case, it seems difficult to obtain a unified theory of evolution in affine connection representation. Nevertheless, we find that we can define time as the total metric with respect to all dimensions of internal coordinate space and external coordinate space and define evolution as one-parameter group of diffeomorphism, to overcome the above difficulty.Now that gauge field and gravitational field are both represented as affine connection, then the properties that are related to gauge field, such as charge, current, mass, energy, momentum, and action, must have corresponding affine representations. Thus, Yang-Mills equation, energy-momentum equation, and Dirac equation are turned into geometric properties in gradient direction; in other words, on-shell evolution is characterized by gradient direction. Correspondingly, quantum theory can be interpreted as a geometric theory of distribution of gradient directions.For the problem (iii). The basic idea is that on a D-dimensional manifold, the components Bma and Cam of semimetrics BMA and CAM with m,a∈4,5,⋯,D are regarded as the frame field of electromagnetic, weak, and strong interactions. The other components of BMA and CAM are regarded as the frame field of gravitation.We take the affine connection as(6)ΓNPM≜12NPM+NPM=12CAMDPBNA+NPM=12CAMDCBNAbPC+NPM=12CAM∂BNA∂ζC+BCABNBbPC+12GMQ∂GNQ∂xP+∂GPQ∂xN−∂GNP∂xQ=12CAM∂BNA∂xP+CAMBPABNB+12GMQ∂GNQ∂xP+∂GPQ∂xN−∂GNP∂xQ,where bPC≜∂ζC/∂xP is a local coordinate transformation, NPM is Christoffel symbol, GMN=δABBMABNB,
(7)NPM≜CAMDPBNA=CAM∂BNA∂xP+CAMBPABNBis said to be a gauge connection, and ΓNPM is said to be a holonomic connection. BPA≜BCAbPC.
(8)BCA≜12CA′A∂BBA′∂ζC+∂BCA′∂ζB is said to be a torsion-free simple connection. Thus,
(9)ΓMNP=12MNP+MNP=12δADBMD∂BNA∂xP+BPABNB+12∂GNM∂xP+∂GPM∂xN−∂GNP∂xM.For the sake of simplicity, we firstly consider the affine connection representation of gauge fields without gravitation. That is to say, let(10)s,i,j=1,2,3;a,m,n,l,q=4,5,⋯,D;A,B,M,N,P=1,2,⋯,D, and consider a D-dimensional manifold M,BMA that satisfies the following conditions:
(i)
Bis=δis,Bia=0,Bms=0(ii)
Gij=δij,Gmn=const,Gmi=0(iii)
Whenm≠n, Gmn=0Thus,MNP=0, MNP≠0 in general. The components ΓmnP of ΓMNP=1/2MNP with m,n∈4,5,⋯,D describe gauge potentials of electromagnetic, weak, and strong interactions. We also use the affine connection ΓNPM to construct elementary particle fields ρMN. The components ρmn of ρMN with m,n∈4,5,⋯,D describe field functions of leptons and quarks.The componentsGmn of GMN with m,n∈4,5,⋯,D describe coupling constants of particle fields ρmn and gauge potentials ΓmnP. The other components of GMN are the metrics of gravitational field. The other components of ρMN and ΓMNP provide possible candidates for dark matters and their interactions.


### 1.3. Content and Organization

In this paper, we are going to show how to construct the affine connection representation of gauge fields. Sections are organized as follows.Corresponding to the problem (i), in Section2, we make some necessary mathematical preparations and discuss the coordinate transformation and frame transformation of the above connection. Meanwhile, in order to make the languages that are used to describe gauge field and gravitational field unified and harmonized, we generalize the notion of reference-system and give it a strict mathematical definition. The reference-system in conventional sense is just only defined on a local coordinate neighborhood, and it has only 1+3 dimensions. But in this paper, we define the concept of reference-system over the entire manifold. It is possessed of more dimensions but different from Kaluza-Klein theory [24–26] and string theories [27–39]. Thus, both of gravitational field and gauge field are regarded as special cases of such a concept of reference-system.Corresponding to the problem (ii), in Section3, we establish the general theory of evolution in affine connection representation of gauge fields, and in Section 4, we discuss the application of this general theory of evolution to 1+3-dimensional classical spacetime.Corresponding to the problem (iii), in Sections5–7, we show concrete forms of affine connection representations of electromagnetic, weak, and strong interaction fields.Some important topics are organized as follows.(1)
Time is regarded as the total metric with respect to all spatial dimensions including external coordinate space and internal coordinate space (see Definition2 and Remark 35 for detail). The CPT inversion is interpreted as the composition of full inversion of coordinates and full inversion of metrics (see Section 3.7 for detail). The conventional 1+3-dimensional Minkowski coordinate xμ originates from the general D-dimensional coordinate xM. The construction method of extra dimensions is different from those of Kaluza-Klein theory and string theory (see Section 4.2 for detail)(2)
On-shell evolution is characterized by gradient direction field (see Sections3.4–3.6 and 4.3 for detail). Quantum theory is regarded as a geometric theory of distribution of gradient directions. We show two dual descriptions of gradient direction. They just exactly correspond to the Schrödinger picture and the Heisenberg picture. In these points of view, the gravitational theory and quantum theory become coordinated. They have a unified description of evolution, and the definition of Feynman propagator is simplified to a stricter form (see Sections 3.8 and 3.9 for detail)(3)
Yang-Mills equation originates from a geometric property of gradient direction. We show the affine connection representation of Yang-Mills equation (see Sections3.5 and 4.5 for detail)(4)
Energy-momentum equation originates from a geometric property of gradient direction. We show the affine connection representation of mass, energy, momentum, and action (see Section3.6, Definition 37, and Discussion 38 for detail). Furthermore, we also show the affine connection representation of Dirac equation (see Section 4.4 for detail)(5)
Why do not neutrinos participate in the electromagnetic interactions? And why do not right-handed neutrinos participate in the weak interactions withW bosons? In the theory of this paper, they are natural and geometric results of affine connection representation of gauge fields; therefore, they are not necessary to be regarded as postulates anymore (see Propositions 52 and 63 for detail)(6)
In Section7, we give new interpretations to PMNS mixing of leptons, CKM mixing of quarks, and color confinement. That is to say, in affine connection representation of gauge fields, these physical properties can be interpreted as geometric properties on manifold


## 1.1. Background and Purpose

We know that in gauge theory, the field strength and the gauge-covariant derivative(1)Fμνa=∂μAνa−∂νAμa+gfabcAμbAνc,Dμ=∂μ−igTaAμa,both contain a coupling constant g, which measures the strength of interaction. A problem is that why is there a coupling constant g?If we represent gauge fields by affine connection, we can obtain a nice interpretation. For example, if we useΓMNP to represent gauge potentials, it is not hard to find some specific conditions to turn the curvature tensor RNPQM to
(2)RMNPQ=∂PΓMNQ−∂QΓMNP+ΓMHPΓNQH−ΓNPHΓMHQ=∂PΓMNQ−∂QΓMNP+GRHΓMHPΓRNQ−ΓRNPΓMHQ.Thus,RMNPQ can be used to represent field strength. In addition, for any ρM, we see that
(3)ρM;P=∂PρM−ΓMPHρH=∂PρM−GRHΓRMPρH.Equations (2) and (3) mean that the coupling constant g may have a geometric meaning, which originates from GRH.This implies that only when affine connection is adopted to represent gauge field can some physical properties be better interpreted. On the other hand, in the general relativity theory, gravitational field is also described by affine connection, so it is convenient to describe gravitational field and gauge field uniformly by affine connection. Therefore, it is necessary to study the affine connection representation of gauge fields. This is the basic motivation of this paper.There are the following two ways to unify gravitational field and gauge field.One way is to represent gravitational field as principal bundle connection. We can take the transformation groupGravi3,1 of gravitational field as the structure group of principal bundle to establish a gauge theory of gravitational field, the local transformation group of which is in the form of Gravi3,1⊗Gaugen, e.g., Poincaré gauge theory [1–11] and metric-affine gauge theory [12–23]. This way can be interpreted intuitively as
(4)The other way is to represent gauge field as affine connection. This is the approach adopted by this paper. Gravitational field and gauge field can both be described by affine connection. Besides, we will also establish an affine connection representation of elementary particles. This way can be interpreted intuitively as(5)


## 1.2. Ideas and Methods

We divide the problem of establishing affine connection representation of gauge fields into three parts as follows.(i)
Which affine connection is suitable for describing not only gravitational field, but also gauge field and elementary particle field?(ii)
How to describe the evolution of these fields in affine connection representation?(iii)
What are the concrete forms of electromagnetic, weak, and strong interaction fields in affine connection representation?For the problem (i). On a Riemannian manifold M,G, the metric tensor can be expressed as GMN=δABBMABNB and GMN=δABCAMCBN, where BMA and CAM are semimetrics or to say frame fields. It is evident that semimetric is more fundamental than metric, so we hope BMA or CAM is regarded as a unified frame field of gravitational field and gauge field, and the frame transformation of BMA or CAM is regarded as gauge transformation. Hence, we need a more general manifold M,BMA rather than the Riemannian manifold M,G.Next, we put metric and semimetric together to construct a new connection, which is not only an affine connection, but also a connection on a fibre bundle. In this way, gravitational field and various gauge fields can be unified on a manifoldM,BMA that is defined by semimetric.In addition, we notice that in the theories based on principal bundle connection representation,(1)
Several complex-valued functions, which satisfy the Dirac equation, are sometimes used to refer to a charged lepton fieldl and sometimes a neutrino field ν. It is not clear how to distinguish these field functions l and ν by inherent geometric constructions(2)
Gauge potentials are abstract; they have no inherent geometric constructions. In other words, the Levi-Civita connectionΓνρμ of gravity is constructed by the metric gμν; however, it is not explicit what geometric quantity the connection Aμa of gauge field is constructed byBy contrast, in the affine connection representation of this paper, we are able to use the semimetricsBMA and CAM of internal coordinate space to endow particle fields l and ν and gauge field Aμa with geometric constructions. Thus, they are not only irreducible representations of group but also possessed of concrete geometric entities.For the problem (ii). There is a fundamental difficulty that time is effected by gravitational field, but not effected by gauge field. This leads to an essential difference between the description of evolution of gravitational field and that of gauge field. In this case, it seems difficult to obtain a unified theory of evolution in affine connection representation. Nevertheless, we find that we can define time as the total metric with respect to all dimensions of internal coordinate space and external coordinate space and define evolution as one-parameter group of diffeomorphism, to overcome the above difficulty.Now that gauge field and gravitational field are both represented as affine connection, then the properties that are related to gauge field, such as charge, current, mass, energy, momentum, and action, must have corresponding affine representations. Thus, Yang-Mills equation, energy-momentum equation, and Dirac equation are turned into geometric properties in gradient direction; in other words, on-shell evolution is characterized by gradient direction. Correspondingly, quantum theory can be interpreted as a geometric theory of distribution of gradient directions.For the problem (iii). The basic idea is that on a D-dimensional manifold, the components Bma and Cam of semimetrics BMA and CAM with m,a∈4,5,⋯,D are regarded as the frame field of electromagnetic, weak, and strong interactions. The other components of BMA and CAM are regarded as the frame field of gravitation.We take the affine connection as(6)ΓNPM≜12NPM+NPM=12CAMDPBNA+NPM=12CAMDCBNAbPC+NPM=12CAM∂BNA∂ζC+BCABNBbPC+12GMQ∂GNQ∂xP+∂GPQ∂xN−∂GNP∂xQ=12CAM∂BNA∂xP+CAMBPABNB+12GMQ∂GNQ∂xP+∂GPQ∂xN−∂GNP∂xQ,where bPC≜∂ζC/∂xP is a local coordinate transformation, NPM is Christoffel symbol, GMN=δABBMABNB,
(7)NPM≜CAMDPBNA=CAM∂BNA∂xP+CAMBPABNBis said to be a gauge connection, and ΓNPM is said to be a holonomic connection. BPA≜BCAbPC.
(8)BCA≜12CA′A∂BBA′∂ζC+∂BCA′∂ζB is said to be a torsion-free simple connection. Thus,
(9)ΓMNP=12MNP+MNP=12δADBMD∂BNA∂xP+BPABNB+12∂GNM∂xP+∂GPM∂xN−∂GNP∂xM.For the sake of simplicity, we firstly consider the affine connection representation of gauge fields without gravitation. That is to say, let(10)s,i,j=1,2,3;a,m,n,l,q=4,5,⋯,D;A,B,M,N,P=1,2,⋯,D, and consider a D-dimensional manifold M,BMA that satisfies the following conditions:
(i)
Bis=δis,Bia=0,Bms=0(ii)
Gij=δij,Gmn=const,Gmi=0(iii)
Whenm≠n, Gmn=0Thus,MNP=0, MNP≠0 in general. The components ΓmnP of ΓMNP=1/2MNP with m,n∈4,5,⋯,D describe gauge potentials of electromagnetic, weak, and strong interactions. We also use the affine connection ΓNPM to construct elementary particle fields ρMN. The components ρmn of ρMN with m,n∈4,5,⋯,D describe field functions of leptons and quarks.The componentsGmn of GMN with m,n∈4,5,⋯,D describe coupling constants of particle fields ρmn and gauge potentials ΓmnP. The other components of GMN are the metrics of gravitational field. The other components of ρMN and ΓMNP provide possible candidates for dark matters and their interactions.


## 1.3. Content and Organization

In this paper, we are going to show how to construct the affine connection representation of gauge fields. Sections are organized as follows.Corresponding to the problem (i), in Section2, we make some necessary mathematical preparations and discuss the coordinate transformation and frame transformation of the above connection. Meanwhile, in order to make the languages that are used to describe gauge field and gravitational field unified and harmonized, we generalize the notion of reference-system and give it a strict mathematical definition. The reference-system in conventional sense is just only defined on a local coordinate neighborhood, and it has only 1+3 dimensions. But in this paper, we define the concept of reference-system over the entire manifold. It is possessed of more dimensions but different from Kaluza-Klein theory [24–26] and string theories [27–39]. Thus, both of gravitational field and gauge field are regarded as special cases of such a concept of reference-system.Corresponding to the problem (ii), in Section3, we establish the general theory of evolution in affine connection representation of gauge fields, and in Section 4, we discuss the application of this general theory of evolution to 1+3-dimensional classical spacetime.Corresponding to the problem (iii), in Sections5–7, we show concrete forms of affine connection representations of electromagnetic, weak, and strong interaction fields.Some important topics are organized as follows.(1)
Time is regarded as the total metric with respect to all spatial dimensions including external coordinate space and internal coordinate space (see Definition2 and Remark 35 for detail). The CPT inversion is interpreted as the composition of full inversion of coordinates and full inversion of metrics (see Section 3.7 for detail). The conventional 1+3-dimensional Minkowski coordinate xμ originates from the general D-dimensional coordinate xM. The construction method of extra dimensions is different from those of Kaluza-Klein theory and string theory (see Section 4.2 for detail)(2)
On-shell evolution is characterized by gradient direction field (see Sections3.4–3.6 and 4.3 for detail). Quantum theory is regarded as a geometric theory of distribution of gradient directions. We show two dual descriptions of gradient direction. They just exactly correspond to the Schrödinger picture and the Heisenberg picture. In these points of view, the gravitational theory and quantum theory become coordinated. They have a unified description of evolution, and the definition of Feynman propagator is simplified to a stricter form (see Sections 3.8 and 3.9 for detail)(3)
Yang-Mills equation originates from a geometric property of gradient direction. We show the affine connection representation of Yang-Mills equation (see Sections3.5 and 4.5 for detail)(4)
Energy-momentum equation originates from a geometric property of gradient direction. We show the affine connection representation of mass, energy, momentum, and action (see Section3.6, Definition 37, and Discussion 38 for detail). Furthermore, we also show the affine connection representation of Dirac equation (see Section 4.4 for detail)(5)
Why do not neutrinos participate in the electromagnetic interactions? And why do not right-handed neutrinos participate in the weak interactions withW bosons? In the theory of this paper, they are natural and geometric results of affine connection representation of gauge fields; therefore, they are not necessary to be regarded as postulates anymore (see Propositions 52 and 63 for detail)(6)
In Section7, we give new interpretations to PMNS mixing of leptons, CKM mixing of quarks, and color confinement. That is to say, in affine connection representation of gauge fields, these physical properties can be interpreted as geometric properties on manifold


## 2. Mathematical Preparations

### 2.1. Geometric Manifold

In order to make the languages that are used to describe gauge field and gravitational field unified and harmonized, we adopt the following definition.Definition 1.
LetM be a D-dimensional connected smooth real manifold. ∀p∈M, take a coordinate chart Up,φUp on a neighborhood Up of p. They constitute a coordinate covering
(11)φ≜Up,φUpp∈M,which is said to be a point-by-point covering. For the sake of simplicity, Up can be denoted by U and φUp by φU.Letφ and ψ be two point-by-point coverings. For the two coordinate frames φU and ψU on the neighborhood U of point p, if
(12)fp≜φU∘ψU−1:ψUU⟶φUU,ξA↦xMis a smooth homeomorphism, fp is called a local reference-system.If everyp∈M is endowed with a local reference-system fp and we require the semimetrics BMA and CAM in Equation (15) to be smooth real functions on M, then
(13)f:M⟶REF,p↦fpis said to be a reference-system on M, and M,f is said to be a geometric manifold.


### 2.2. Metric and Semimetric

In the absence of a special declaration, the indices take values asA,B,C,D,E=1,2,⋯,D and M,N,P,Q,R=1,2,⋯,D. The derivative functions
(14)bMA≜∂ξA∂xM,cAM≜∂xM∂ξAof fp on Up define the semimetrics (or to say frame field) BMA and CAM of f on the manifold M that are
(15)BMA:M⟶ℝ,p↦BMAp≜bfpMAp,CAM:M⟶ℝ,p↦CAMp≜cfpAMp.LetδAB=δAB=δBA=KroneckerA,B and εMN=εMN=εNM=KroneckerM,N. The metric tensors of f are
(16)GMN=δABBMABNB,HAB=εMNCAMCBN.Similarly, it can also be defined thatb¯AM≜∂ξA/∂xM,c¯MA≜∂xM/∂ξA and corresponding B¯AM,C¯MA.


### 2.3. Gauge Transformation in Affine Connection Representation

∀p∈M, fp≜ρU∘ψU−1 induces local reference-system transformations
(17)Lfp:kp≜ψU∘φU−1↦ρU∘φU−1=fp∘kp,Rfp:hp≜φU∘ρU−1↦φU∘ψU−1=hp∘fp,and reference-system transformations on the manifold M(18)Lf:p↦Lfp,Rf:p↦Rfp.We also speak ofLf and Rf as (affine) gauge transformations.
(i)
Lf and Rf are identical transformations if and only if BMA of f is an identity matrix(ii)
Lf and Rf are flat transformations if and only if ∀p1,p2∈M, BMAp1=BMAp2(iii)
Lf and Rf are orthogonal transformations if and only if δABBMABNB=εMNThe totality of all reference-system transformations onM is denoted by GLM, which is a subgroup of ⊗p∈MGLD,ℝp, where ⊗ represents external direct product.


### 2.4. Coordinate Transformation of Holonomic Connection and Frame Transformation of Gauge Connection

Suppose there are reference-systemsg and g on the manifold M, denote G≜g∘g, and ∀p∈M, on the neighborhood U of p, gp and gp satisfy
(19)U,xM⟵gpU,ζA⟵gpU,βA′.On the geometric manifoldM,g, we define torsion-free simple connection D and its coefficients BCAg by
(20)D∂∂ζB≜ωgBA⊗∂∂ζA=BCAgdζC⊗∂∂ζA=12CgA′A∂BgBA′∂ζC+∂BgCA′∂ζBdζC⊗∂∂ζA.Then, we can compute the absolute derivative of the frame field∂/∂xN(21)D∂∂xN=DBgNB∂∂ζB=dBgNB⊗∂∂ζB+BgNBD∂∂ζB=∂BgNB∂ζCdζC⊗∂∂ζB+BgNBBCAgdζC⊗∂∂ζA=∂BgNA∂ζC+BgNBBCAgdζC⊗∂∂ζA.Thus, it is obtained that(22)DCBgNA=∂BgNA∂ζC+BgNBBCAg.DenoteDP≜bgpPCDC; thus, we can define on M,G the required gauge connection, which is
(23)NPMG≜CgAMDPBgNA=CgAM∂BgNA∂xP+CgAMBPAgBgNB.It is important thatNPMG is not only an affine connection on M,G, but also a connection on frame bundle.
(i)
NPMGas an Affine Connection. Under the coordinate transformation Lkp:U,xM⟶U,xM′, bM′M≜∂xM/∂xM′, cMM′≜∂xM′/∂xM, BgMA↦BgM′A=bM′MBgMA,CgAM↦CgAM′=cMM′CgAM. Consequently, the gauge connection NPMG is transformed according to
(24)Lkp:NPMG↦N′P′M′G=cMM′NPMGbN′NbP′P+cMM′∂bN′M∂xP′.Due to Equation (24), under the coordinate transformation, the holonomic connection
(25)ΓGNPM≜12NPMG+NPMG=12CgAM∂BgNA∂xP+CgAMBPAgBgNB+12GGMQ∂GGNQ∂xP+∂GGPQ∂xN−∂GGNP∂xQis transformed according to
(26)Lkp:ΓGNPM↦ΓGN′P′M′=cMM′ΓGNPMbN′NbP′P+cMM′∂bN′M∂xP′.(ii)
NPMGas a Connection on Frame Bundle. Under the frame transformation Lk:M,G↦M,G′,∂/∂xM↦∂/∂xM′=BkM′M∂/∂xM,BgMA↦Bg′M′A=BkM′MBgMA,CgAM↦Cg′AM′=CkMM′CgAM. Consequently, the gauge connection NPMG is tranformed according to(27)Lk:NPMG↦N′P′M′G′=N′PM′G′bP′P=CkMM′NPMGBkN′N+CkMM′∂BkN′M∂xPbP′P.Equations (24) and (27) show that NPMG is not only an affine connection, but also a connection on frame bundle.Apply Equations (24)–(27) to the curvature tensors
(28)NPQM≜∂NQM∂xP−∂NPM∂xQ+HPMNQH−NPHHQM,NPQM≜∂NQM∂xP−∂NPM∂xQ+HPMNQH−NPHHQM,RNPQM≜∂ΓNQM∂xP−∂ΓNPM∂xQ+ΓHPMΓNQH−ΓNPHΓHQM,and then, it is obtained that
(29)Lk:NPQMG↦N′P′Q′M′G′=N′PQM′G′bP′PbQ′Q=CkMM′NPQMGBkN′NbP′PbQ′Q,Lkp:NPQMG↦N′P′Q′M′G=cMM′NPQMGbN′NbP′PbQ′Q,Lkp:NPQMG↦N′P′Q′M′G=cMM′NPQMGbN′NbP′PbQ′Q,Lkp:RGNPQM↦RGN′P′Q′M′=cMM′RGNPQMbN′NbP′PbQ′Q.We see from Equation (29) that the NPQMG without gravitation is both a curvature tensor of affine connection and a curvature tensor on frame bundle, and that the RGNPQM with gravitation is a curvature tensor of affine connection, but not a curvature tensor on frame bundle. In other words, under the gauge transformation Lk, NPQMG and N′PQM′G′ represent the same physical state, while RGNPQM and RG′N′PQM′ represent different physical states. This shows that the gravitational field in RGNPQM makes the gauge frames BMA and CAM have physical effects.


## 2.1. Geometric Manifold

In order to make the languages that are used to describe gauge field and gravitational field unified and harmonized, we adopt the following definition.Definition 1.
LetM be a D-dimensional connected smooth real manifold. ∀p∈M, take a coordinate chart Up,φUp on a neighborhood Up of p. They constitute a coordinate covering
(11)φ≜Up,φUpp∈M,which is said to be a point-by-point covering. For the sake of simplicity, Up can be denoted by U and φUp by φU.Letφ and ψ be two point-by-point coverings. For the two coordinate frames φU and ψU on the neighborhood U of point p, if
(12)fp≜φU∘ψU−1:ψUU⟶φUU,ξA↦xMis a smooth homeomorphism, fp is called a local reference-system.If everyp∈M is endowed with a local reference-system fp and we require the semimetrics BMA and CAM in Equation (15) to be smooth real functions on M, then
(13)f:M⟶REF,p↦fpis said to be a reference-system on M, and M,f is said to be a geometric manifold.


## 2.2. Metric and Semimetric

In the absence of a special declaration, the indices take values asA,B,C,D,E=1,2,⋯,D and M,N,P,Q,R=1,2,⋯,D. The derivative functions
(14)bMA≜∂ξA∂xM,cAM≜∂xM∂ξAof fp on Up define the semimetrics (or to say frame field) BMA and CAM of f on the manifold M that are
(15)BMA:M⟶ℝ,p↦BMAp≜bfpMAp,CAM:M⟶ℝ,p↦CAMp≜cfpAMp.LetδAB=δAB=δBA=KroneckerA,B and εMN=εMN=εNM=KroneckerM,N. The metric tensors of f are
(16)GMN=δABBMABNB,HAB=εMNCAMCBN.Similarly, it can also be defined thatb¯AM≜∂ξA/∂xM,c¯MA≜∂xM/∂ξA and corresponding B¯AM,C¯MA.


## 2.3. Gauge Transformation in Affine Connection Representation

∀p∈M, fp≜ρU∘ψU−1 induces local reference-system transformations
(17)Lfp:kp≜ψU∘φU−1↦ρU∘φU−1=fp∘kp,Rfp:hp≜φU∘ρU−1↦φU∘ψU−1=hp∘fp,and reference-system transformations on the manifold M(18)Lf:p↦Lfp,Rf:p↦Rfp.We also speak ofLf and Rf as (affine) gauge transformations.
(i)
Lf and Rf are identical transformations if and only if BMA of f is an identity matrix(ii)
Lf and Rf are flat transformations if and only if ∀p1,p2∈M, BMAp1=BMAp2(iii)
Lf and Rf are orthogonal transformations if and only if δABBMABNB=εMNThe totality of all reference-system transformations onM is denoted by GLM, which is a subgroup of ⊗p∈MGLD,ℝp, where ⊗ represents external direct product.


## 2.4. Coordinate Transformation of Holonomic Connection and Frame Transformation of Gauge Connection

Suppose there are reference-systemsg and g on the manifold M, denote G≜g∘g, and ∀p∈M, on the neighborhood U of p, gp and gp satisfy
(19)U,xM⟵gpU,ζA⟵gpU,βA′.On the geometric manifoldM,g, we define torsion-free simple connection D and its coefficients BCAg by
(20)D∂∂ζB≜ωgBA⊗∂∂ζA=BCAgdζC⊗∂∂ζA=12CgA′A∂BgBA′∂ζC+∂BgCA′∂ζBdζC⊗∂∂ζA.Then, we can compute the absolute derivative of the frame field∂/∂xN(21)D∂∂xN=DBgNB∂∂ζB=dBgNB⊗∂∂ζB+BgNBD∂∂ζB=∂BgNB∂ζCdζC⊗∂∂ζB+BgNBBCAgdζC⊗∂∂ζA=∂BgNA∂ζC+BgNBBCAgdζC⊗∂∂ζA.Thus, it is obtained that(22)DCBgNA=∂BgNA∂ζC+BgNBBCAg.DenoteDP≜bgpPCDC; thus, we can define on M,G the required gauge connection, which is
(23)NPMG≜CgAMDPBgNA=CgAM∂BgNA∂xP+CgAMBPAgBgNB.It is important thatNPMG is not only an affine connection on M,G, but also a connection on frame bundle.
(i)
NPMGas an Affine Connection. Under the coordinate transformation Lkp:U,xM⟶U,xM′, bM′M≜∂xM/∂xM′, cMM′≜∂xM′/∂xM, BgMA↦BgM′A=bM′MBgMA,CgAM↦CgAM′=cMM′CgAM. Consequently, the gauge connection NPMG is transformed according to
(24)Lkp:NPMG↦N′P′M′G=cMM′NPMGbN′NbP′P+cMM′∂bN′M∂xP′.Due to Equation (24), under the coordinate transformation, the holonomic connection
(25)ΓGNPM≜12NPMG+NPMG=12CgAM∂BgNA∂xP+CgAMBPAgBgNB+12GGMQ∂GGNQ∂xP+∂GGPQ∂xN−∂GGNP∂xQis transformed according to
(26)Lkp:ΓGNPM↦ΓGN′P′M′=cMM′ΓGNPMbN′NbP′P+cMM′∂bN′M∂xP′.(ii)
NPMGas a Connection on Frame Bundle. Under the frame transformation Lk:M,G↦M,G′,∂/∂xM↦∂/∂xM′=BkM′M∂/∂xM,BgMA↦Bg′M′A=BkM′MBgMA,CgAM↦Cg′AM′=CkMM′CgAM. Consequently, the gauge connection NPMG is tranformed according to(27)Lk:NPMG↦N′P′M′G′=N′PM′G′bP′P=CkMM′NPMGBkN′N+CkMM′∂BkN′M∂xPbP′P.Equations (24) and (27) show that NPMG is not only an affine connection, but also a connection on frame bundle.Apply Equations (24)–(27) to the curvature tensors
(28)NPQM≜∂NQM∂xP−∂NPM∂xQ+HPMNQH−NPHHQM,NPQM≜∂NQM∂xP−∂NPM∂xQ+HPMNQH−NPHHQM,RNPQM≜∂ΓNQM∂xP−∂ΓNPM∂xQ+ΓHPMΓNQH−ΓNPHΓHQM,and then, it is obtained that
(29)Lk:NPQMG↦N′P′Q′M′G′=N′PQM′G′bP′PbQ′Q=CkMM′NPQMGBkN′NbP′PbQ′Q,Lkp:NPQMG↦N′P′Q′M′G=cMM′NPQMGbN′NbP′PbQ′Q,Lkp:NPQMG↦N′P′Q′M′G=cMM′NPQMGbN′NbP′PbQ′Q,Lkp:RGNPQM↦RGN′P′Q′M′=cMM′RGNPQMbN′NbP′PbQ′Q.We see from Equation (29) that the NPQMG without gravitation is both a curvature tensor of affine connection and a curvature tensor on frame bundle, and that the RGNPQM with gravitation is a curvature tensor of affine connection, but not a curvature tensor on frame bundle. In other words, under the gauge transformation Lk, NPQMG and N′PQM′G′ represent the same physical state, while RGNPQM and RG′N′PQM′ represent different physical states. This shows that the gravitational field in RGNPQM makes the gauge frames BMA and CAM have physical effects.


## 3. The Evolution in Affine Connection Representation of Gauge Fields

Now that we have the required affine connection, next we have to solve the problem that how to describe the evolution in affine connection representation.In the existing theories, time is effected by gravitational field, but not effected by gauge field. This leads to an essential difference between the description of evolution of gravitational field and that of gauge field. In this case, it is difficult to obtain a unified theory of evolution in affine connection representation. We adopt the following way to overcome this difficulty.


### 3.1. The Relation between Time and Space

Definition 2.
SupposeM=P×N and r≜dimP=3. Let
(30)A,B,M,N=1,⋯,D;s,i=1,⋯,r;a,m=r+1,⋯,D.
On a geometric manifoldM,f, the dξ0 and dx0 which are defined by
(31)dξ02≜∑A=1DdξA2=δABdξAdξB=GMNdxMdxN,dx02≜∑M=1DdxM2=εMNdxMdxN=HABdξAdξB,are said to be total space metrics or time metrics. We also suppose
(32)dξP2≜∑s=1rdξs2,dξN2≜∑a=r+1Ddξa2,dxP2≜∑i=1rdxi2,dxN2≜∑m=r+1Ddxm2.
dξN and dxN are regarded as proper-time metrics. For convenience, P is said to be external space and N is said to be internal space.Remark 3.
The above definition implies a new viewpoint about time and space. The relation between time and space in this way is different from the Minkowski coordinatesxμμ=0,1,2,3. Time and space are not the components on an equal footing anymore, but have a relation of total to component. It can be seen later that time reflects the total evolution in the full-dimensional space, while a specific spatial dimension reflects just a partial evolution in a specific direction.


### 3.2. Evolution Path as a Submanifold

Definition 4.
Let there be reference-systemsf, g, f, and g on a manifold M, such that ∀p∈M, on the neighborhood U of p,
(33)U,αA′⟶fpU,ξA⟶fpU,xM⟵gpU,ζA⟵gpU,βA′.
DenoteF≜f∘f and G≜g∘g; then, we say F and Gmove relatively and interact mutually, and also we say that F evolves in G, or F evolves on the geometric manifold M,G. Meanwhile, G evolves in F, or we say G evolves on M,F.
From Equation (23), we know that in F and G, gauge fields originate from f and g, and gravitational fields GFMN and GGMN are effected by f and g, respectively. We are going to describe their evolutions step by step in the following sections.
Let there be a one-parameter group of diffeomorphisms(34)φX:M×ℝ⟶M,acting on M, such that φXp,0=p. Thus, φX determines a smooth tangent vector field X on M. If X is nonzero everywhere, we say φX is a set of evolution paths, and X is an evolution direction field. Let T⊆ℝ be an interval; then, the regular imbedding
(35)Lp≜φX,p:T⟶M,t↦φXp,tis said to be an evolution path through p. The tangent vector d/dt≜Lp=Xp is called an evolution direction at p. For the sake of simplicity, we also denote Lp≜LpT⊂M; then,
(36)π:Lp⟶M,q↦qis also a regular imbedding. If it is not necessary to emphasize the point p, Lp is denoted by L concisely.In order to describe physical evolution, next we are going to strictly describe the mathematical properties of the reference-systemsf and g which are sent onto the evolution path L.Definition 5.
Let the time metrics ofU,ξA, U,xM, and U,ζA be dξ0, dx0, and dζ0, respectively. On UL≜U∩Lp, we have parameter equations
(37)ξA=ξAx0,xM=xMξ0,ζA=ζAx0,ξ0=ξ0x0,x0=x0ξ0,ζ0=ζ0x0.
Takef for example, according to Equation (37), on UL we define
(38)b0A≜dξAdx0,b00≜dξ0dx0,ε0M≜dxMdx0=b00c0M=b0AcAM,c0M≜dxMdξ0,c00≜dx0dξ0,δ0A≜dξAdξ0=c00b0A=c0MbMA.
Definedξ0≜dx0/dξ0dx0 and dx0≜dξ0/dx0dξ0, which induce d/dξ0 and d/dx0, such that d/dξ0,dξ0=1 and d/dx0,dx0=1. So we can also define
(39)b¯A0≜dξAdx0,b¯00≜dξ0dx0,ε¯M0≜dxMdx0=b¯00c¯M0=b¯A0c¯MA,c¯M0≜dxMdξ0,c¯00≜dx0dξ0,δ¯A0≜dξAdξ¯0=c¯00b¯A0=c¯M0b¯AM.
They determine the following smooth functions on the entireL, similar to Section 2.2, that
(40)B0A:L⟶ℝ,p↦B0Ap≜bfp0Ap,C0M:L⟶ℝ,p↦C0Mp≜cfp0Mp,B¯A0:L⟶ℝ,p↦B¯A0p≜b¯fpA0p,C¯M0:L⟶ℝ,p↦C¯M0p≜c¯fpM0p,B00:L⟶ℝ,p↦B00p≜bfp00p,C00:L⟶ℝ,p↦C00p≜cfp00p,B¯00:L⟶ℝ,p↦B¯00p≜b¯fp00p,C¯00:L⟶ℝ,p↦C¯00p≜c¯fp00p.
For convenience, we still use the notationsε and δ and have the following smooth functions.
(41)ε0M≜B00C0M=B0ACAM,δ0A≜C00B0A=C0MBMA,G00≜B00B00=GMNε0Mε0N,ε¯M0≜B¯00C¯M0=B¯A0C¯MA,δ¯A0≜C¯00B¯A0=C¯M0B¯AM,G00≜C00C00=GMNε¯M0ε¯N0.
It is easy to verify thatdx0=G00dx0 and d/dx0=G00d/dx0 are both true on L by a simple calculation.


### 3.3. Evolution Lemma

We have the following two evolution lemmas. The affine connection representations of Yang-Mills equation, energy-momentum equation, and Dirac equation are dependent on them.Definition 6.
∀p∈L, suppose TpM and TpL are tangent spaces, Tp∗M and Tp∗L are cotangent spaces. The regular imbedding π:L⟶M,q↦q induces the tangent map and the cotangent map
(42)π∗:TpL⟶TpM,γL↦π∘γL,π∗:Tp∗M⟶Tp∗L,df↦df∘π.
Evidently,π∗ is an injection, and π∗ is a surjection. ∀d/dtL∈TpL,d/dt∈TpM,df∈Tp∗M,dfL∈Tp∗L, if and only if
(43)ddt=π∗ddtL,dfL=π∗dfare true, we denote
(44)ddt≅ddtL,df≃dfL.Then, we have the following two propositions that are evidently true.Proposition 7.
Ifd/dt≅d/dtL and df≃dfL, then
(45)ddt,df=ddtL,dfL.Proposition 8.
The following conclusions are true.(46)wM∂∂xM≅w0ddx0⟺wM=w0ε0M,wMdxM≃w0dx0⟺wMε0M=w0,w¯M∂∂xM≅w¯0ddx0⟺w¯M=w¯0ε¯M0,w¯MdxM≃w¯0dx0⟺w¯Mε¯M0=w¯0.


### 3.4. On-Shell Evolution as a Gradient

LetT be a smooth n-order tensor field. The restriction on U,xM is T≜t∂/∂x⊗dx, where ∂/∂x⊗dx represents the tensor basis generated by several ∂/∂xM and dxM, and the tensor coefficients of T are concisely denoted by t:U⟶ℝ.LetD be a holonomic connection. Consider DT≜t;QdxQ⊗∂/∂x⊗dx. Denote
(47)Dt≜t;QdxQ,∇t≜t;Q∂∂xQ.∀p∈M, the integral curve of ∇t, that is, Lp≜φ∇t,p, is a gradient line of T. It can be seen later that the above gradient operator ∇ characterizes the on-shell evolution.For any evolution pathL, let UL≜U∩L. Denote tL≜tUL and tL;0≜t;Qε0Q, as well as
(48)DLtL≜tL;0dx0,∇LtL≜tL;0ddx0.Proposition 9.
The following conclusions are evidently true.(i)
Dt≃DLtL if and only if L is an arbitrary evolution path(ii)
∇t≅∇LtL if and only if L is a gradient line of TRemark 10.
More generally, suppose there is a tensorU≜uQdxQ⊗∂/∂x⊗dx. In such a notation, all the indices are concisely ignored except Q. uQdxQ uniquely determines a characteristic direction uQ∂/∂xQ.If the system of 1-order linear partial differential equationst;Q=uQ has a solution t, then it is true that Dt=uQdxQ and ∇t=uQ∂/∂xQ. Thus, in the evolution direction L=uQ∂/∂xQ, the following conclusions are true.
(49)Dt≃DLtL,∇t≅∇LtL,where DLtL≜u0dx0, ∇LtL≜u0d/dx0, and u0≜uQε0Q.Now for any geometric property in the form of tensorU, we are able to express its on-shell evolution in the form of ∇t.Next, two important on-shell evolutions are discussed in the following two sections. One is the on-shell evolution of the potential field of a reference-system. The other is the one that a general charge of a reference-system evolves in the potential field of another reference-system.


### 3.5. On-Shell Evolution of Potential Field and Affine Connection Representation of Yang-Mills Equation

Table I of article [40] proposes a famous correspondence between gauge field terminologies and fibre bundle terminologies. However, it does not find out the corresponding mathematical object to the source JμK. In this section, we give an answer to this problem and show the affine connection representation of Yang-Mills equation.In order to obtain the general Yang-Mills equation with gravitation, we have to adopt holonomic connection to construct it. SupposeF evolves in G according to Definition 4, that is, ∀p∈M,
(50)U,αA′⟶fpU,ξA⟶fpU,xM⟵gpU,ζA⟵gpU,βA′.We always take the following notations in the coordinate frameU,xM.
(i)
Let the holonomic connections, which are defined by Equation (25), of geometric manifolds M,F and M,G be ΓFNPM and ΓGNPM, respectively. The colon “:” and the semicolon “;” are used to express the covariant derivatives on M,F and M,G, respectively, e.g.,(51)uQ:P=∂uQ∂xP+ΓFHPQuH,uQ;P=∂uQ∂xP+ΓGHPQuH.(ii)
Let the coefficients of curvature tensor ofM,F and M,G be KNPQM and RNPQM, respectively, i.e.,(52)KNPQM≜∂ΓFNQM∂xP−∂ΓFNPM∂xQ+ΓFNQHΓFHPM−ΓFNPHΓFHQM,RNPQM≜∂ΓGNQM∂xP−∂ΓGNPM∂xQ+ΓGNQHΓGHPM−ΓGNPHΓGHQM.DenoteKNPQM:P≜GFPP′KNPQ:P′M. On an arbitrary evolution path L, we define
(53)ρN0Mdx0≜π∗KNPQM:PdxQ∈T∗L.Then, according to Definition6 and the evolution lemma of Proposition 8, we obtain ρN0M=KNPQM:Pε0Q and
(54)KNPQM:PdxQ≃ρN0Mdx0.Let∇t=KNPQM:P∂/∂xQ. Then, according to Proposition 9, if and only if ∀p∈M, Lp=∇tp, we have
(55)KNPQM:P∂∂xQ≅ρN0Mddx0.Applying the evolution lemma of Proposition8 again, we obtain
(56)KNPQM:P=ρN0Mε¯Q0.DenotejNQM≜ρN0Mε¯Q0; then, if and only if Lp=∇tp, we have
(57)KNPQM:P=jNQM,which is said to be (affine) Yang-Mills equation of F. It contains effects of gravitation, which makes the gauge frames BfMA and CfAM have physical effects. According to Equation (29), we know Equation (57) is coordinate covariant, and if gravitation is removed, it is also gauge covariant.Thus, we have the following two results.(i)
The Yang-Mills equation originates from a geometric property in the direction∇t. In other words, the on-shell evolution of gauge field is described by the direction field ∇t(ii)
We obtain the mathematical origination of charge and current. We know that the evolution pathL is an imbedding submanifold of M. Thus, the charge ρN0M originates from the pull-back π∗ from M to L, and the current jNQM originates from ∇t that is associated to ρN0MIf we letM,f be completely flat, i.e., BfMA=δMA, CfAM=δAM, then by calculation, we find ρN0M can still be nonvanishing. This shows that ρN0M originates from M,f ultimately.Definition 11.
We speak of the real-valued(58)ρMN0≜GMHρN0Has the field function of a general charge or speak of it as a charge of F for short.


### 3.6. On-Shell Evolution of General Charge and Affine Connection Representation of Mass, Energy, Momentum, and Action

In order to be compatible with the affine connection representation of gauge fields, we also have to define mass, energy, momentum, and action in the form associated to affine connection. We are going to show them in this section and Section4.3.LetF0≜ρMN0dxM⊗dxN. For the sake of simplicity, denote the charge ρMN0 of F by ρMN concisely. Let D be the holonomic connection of M,G; then,
(59)DF0≜DρMN⊗dxM⊗dxN,∇F0≜∇ρMN⊗dxM⊗dxN,where DρMN≜ρMN;QdxQ and ∇ρMN≜ρMN;Q∂/∂xQ. According to Proposition 9, if and only if ∀p∈L, the evolution direction is taken as Lp=∇ρMNp, we have
(60)DρMN≃DLρMN,∇ρMN≅∇LρMN,that is,
(61)ρMN;QdxQ≃ρMN;0dx0,ρMN;Q∂∂xQ≅ρMN;0ddx0.Definition 12.
For more convenience, the notationρMN is further abbreviated as ρ. In affine connection representation, energy and momentum of ρ are defined as
(62)E0≜ρ;0≜ρ;Qε0Q,pQ≜ρ;Q,H0≜dρdx0,PQ≜∂ρ∂xQ,E0≜ρ;0≜ρ;Qε¯Q0,pQ≜ρ;Q,H0≜dρdx0,PQ≜∂ρ∂xQ.Proposition 13.
At any pointp on M, the equation
(63)E0E0=pQpQholds if and only if the evolution direction Lp=∇ρp. Equation (63) is the (affine) energy-momentum equation of ρ.Proof.
According to the above discussion,∀p∈M, Lp=∇ρp is equivalent to
(64)pQdxQ≃E0dx0,pQ∂∂xQ≅E0ddx0.
Then, due to Proposition7, we obtain the directional derivative in the gradient direction ∇ρ:
(65)pQ∂∂xQ,pMdxM=E0ddx0,E0dx0,
i.e.,GQMpQpM=G00E0E0, or pQpQ=E0E0.Proposition 14.
At any pointp on M, the equations
(66)pQ=E0dxQdx0,pQ=E0dxQdx0hold if and only if the evolution direction Lp=∇ρp.Proof.
Due to the evolution lemma of Proposition8, we immediately obtain Equation (66) from Equation (64).Remark 15.
In the gradient direction∇ρ, Equation (66) is consistent with the conventional formula
(67)p=mv.
Thus, in affine connection representation, the energy-momentum equation and the conventional definition of momentum both originate from a geometric property in gradient direction. In other words, the on-shell evolution of the particle fieldρ is described by the gradient direction field ∇ρ.Definition 16.
LetPb,a be the totality of paths from a to b. And suppose L∈Pb,a, and the evolution parameter x0 satisfies ta≜x0a<x0b≜tb. The elementary affine action of ρ is defined as
(68)sL≜∫LDρ=∫LpQdxQ=∫tatbE0dx0.Thus,δsL=0 if and only if L is a gradient line of ρ.In particular, in the case whereG is orthogonal, we can also define action in the following way.OnM,G, let there be Dirac algebras γM and γN such that
(69)γMγN+γNγM=2GMN,γMγN+γNγM=2GMN,γMγM=1.In a gradient direction ofρ, from Equation (63), we obtain that
(70)pQpQ=E0E0⟺ρ;Qρ;Q=ρ;0ρ;0⟺GPQρ;Pρ;Q=G00ρ;0ρ;0⟺γPγQ+γQγPρ;Pρ;Q=2ρ;0ρ;0⟺γPρ;PγQρ;Q+γQρ;QγPρ;P=2ρ;0ρ;0⟺γPρ;P2=ρ;02.TakeγPρ;P=ρ;0 without loss of generality, and then, in the gradient direction of ρ, we have
(71)γPρ;Pdx0=ρ;0dx0=ε0Pρ;Pdx0=Dρ.So we can take(72)sL≜∫LγPρ;Pdx0+Dρ=∫tatbγPρ;P+ε0Pρ;Pdx0=∫tatbγPρ;P+E0dx0.Remark17 and Remark 41 explain the rationality of this definition. We have sL=2sL in the gradient direction of ρ, so sL and sL are consistent.Remark 17.
In the Minkowski coordinate frame of Section4.2, the evolution parameter x0 is replaced by x~τ; then, there still exists a concept of gradient direction ∇~ρ~. Correspondingly, Equations (68) and (72) present as
(73)s~L≜∫LD~ρ~=∫Lp~μdx~μ=∫τaτbm~τdx~τ,s~L=∫τaτbγμρ~;μ+m~τdx~τ,where m~τ is the rest-mass and x~τ is the proper-time.Remark 18.
Define the following notations.(74)ρΓG≜∂ρMN∂xG−ρMN;G=ρMHΓNGH+ρHNΓMGH,ρRPQ≜ρMHRNPQH+ρHNRMPQH.
Then, through some calculations, we can obtain that(75)fP≜pP;0=E0;P−pQε0;PQ+ρRPQε0Q,which is the affine connection representation of general Lorentz force equation (see Discussion 38 for further illustrations).


### 3.7. Inversion Transformation in Affine Connection Representation

In affine connection representation,CPT inversion is interpreted as a full inversion of coordinates and metrics. Let i,j=1,2,3 and m,n=4,5,⋯,D.Let the local coordinate representation of reference-systemk be x′j=−δijxi, x′n=δmnxm; then, parity inversion can be represented as
(76)P≜Lk:xi⟶−xi,xm⟶xm.Let the local coordinate representation of reference-systemh be x′j=δijxi, x′n=−δmnxm; then, charge conjugate inversion can be represented as
(77)C≜Lh:xi⟶xi,xm⟶−xm.Time coordinate inversion can be represented as(78)T0:x0⟶−x0.Full inversion of coordinates can be represented as(79)CPT0:xQ⟶−xQ,x0⟶−x0.The positive or negative sign of metric marks two opposite directions of evolution. LetN be a closed submanifold of M, and let its metric be dxN. Denote the totality of closed submanifolds of M by BM; then, full inversion of metrics can be expressed as
(80)TM≜∏N∈BMdxN⟶−dxN.Denote time inversion by(81)T≜TMT0,and then, the joint transformation of the full inversion of coordinates CPT0 and the full inversion of metrics TM is
(82)CPT0TM=CPT.Summarize the above discussions; then, we have(83)CPT0:xQ⟶−xQ,x0⟶−x0,dxQ⟶dxQ,dx0⟶dx0,TM:xQ⟶xQ,x0⟶x0,dxQ⟶−dxQ,dx0⟶−dx0,CPT:xQ⟶−xQ,x0⟶−x0,dxQ⟶−dxQ,dx0⟶−dx0.TheCPT invariance in affine connection representation is very clear. Concretely, on M,G, we consider the CPT transformation acting on G. Denote s≜∫LDρ and DPeis≜∂/∂xP−iρΓPeis; then, through simple calculations, we obtain that
(84)CPT:Dρ⟶Dρ,DPeis⟶−DPe−is.Remark 19.
In quantum mechanics, there is a complex conjugation in the time inversion of wave functionT:ψx,t⟶ψ∗x,−t. In affine connection representation, we know the complex conjugation can be interpreted as a straightforward mathematical result of the full inversion of metrics TM.


### 3.8. Two Dual Descriptions of Gradient Direction Field

Discussion 20.
LetX and Y be nonvanishing smooth tangent vector fields on the manifold M. And let LY be the Lie derivative operator induced by the one-parameter group of diffeomorphism φY. Then, according to a well-known theorem [41], we obtain the Lie derivative equation
(85)X,Y=LYX.Suppose∀p∈M, Yp is a unit-length vector, i.e., Yp=1. Let the parameter of φY be x0. Then, on the evolution path L≜φY,p, we have
(86)Y≅ddx0.Thus, Equation (85) can also be represented as
(87)X,Y=ddx0X.On the other hand,∀df∈TM and dfL≜π∗df, and due to (86) and Proposition 7, we have Y,df=d/dx0,dfL, that is,
(88)Yf=ddx0fL.Definition 21.
LetH≜∇ρ−1∇ρ=ε0M∂/∂xM≅d/dx0. It is evident that ∀p∈M, Hp=1. If and only if taking Y=H, we speak of (87) and (88) as real-valued (affine) Heisenberg equation and (affine) Schrödinger equation, respectively, that is,
(89)X,H=ddx0X,Hf=ddx0fL.Discussion 22.
The above two equations both describe the gradient direction field and thereby reflect on-shell evolution. Such two dual descriptions of gradient direction show the real-valued affine connection representation of Heisenberg picture and Schrödinger picture.It is not hard to find out several different kinds of complex-valued representations of gradient direction. For examples, one is the affine Dirac equation in Section4.4, and another is as follows.Letψ≜feisL, where it is fine to take either sL≜sL or sL≜sL from Definition 16. According to Equation (89), it is easy to obtain on L that
(90)X,H=ddx0X,Hψ=dψdx0.This is consistent with the conventional Heisenberg equation and Schrödinger equation (taking the natural units thatℏ=1,c=1)
(91)X,−iH=∂∂tX,−iHψ=∂ψ∂t,and they have a coordinate correspondence
(92)∂∂ixk⟷∂∂xk,∂∂t↔ddx0.We know that∂/∂t⟷d/dx0 originates from the difference that the evolution parameter is xτ or x0. The imaginary unit i originates from the difference between the regular coordinates x1,x2,x3,xτ and the Minkowski coordinates x1,x2,x3,x0. That is to say, the regular coordinates satisfy
(93)dx02=dx12+dx22+dx32+dxτ2,and the Minkowski coordinates satisfy
(94)dxτ2=dx02−dx12−dx22−dx32=dx02+dix12+dix22+dix32.This causes the appearance of the imaginary uniti in the correspondence
(95)ixk⟷xk.So Equations (90) and (91) have exactly the same essence, and their differences only come from different coordinate representations.The differences between coordinate representations have nothing to do with the geometric essence and the physical essence. We notice that the value of a gradient direction is dependent on geometry, but independent of that the equations are real-valued or complex-valued. Therefore, it is unnecessary for us to confine to such algebraic forms as real-valued or complex-valued forms, but we should focus on such geometric essence as gradient direction.The essential virtue of complex-valued form is that it is applicable for describing the coherent superposition of propagator. However, this is independent of the above discussions, and we are going to discuss it in Section3.9.


### 3.9. Quantum Evolution as a Distribution of Gradient Directions

From Proposition13, we see that, in affine connection representation, the classical on-shell evolution is described by gradient direction. Then, naturally, quantum evolution should be described by the distribution of gradient directions.The distribution of gradient directions on a geometric manifoldM,G is effected by the bending shape of M,G; in other words, the distribution of gradient directions can be used to reflect the shape of M,G. This is the way that the quantum theory in affine connection representation describes physical reality.In order to know the full picture of physical reality, it is necessary to fully describe the shape of the geometric manifold. For a single observation,(1)
It is the reference-system, not a point, that is used to describe the physical reality, so the coordinate of an individual point is not enough to fully describe the location information about the physical reality(2)
Through a single observation of momentum, we can only obtain information about an individual gradient direction; this cannot reflect the full picture of the shape of the geometric manifoldQuantum evolution provides us with a guarantee that we can obtain the distribution of gradient directions through multiple observations, so that we can describe the full picture of the shape of the geometric manifold.Next, we are going to carry out strict mathematical descriptions for the quantum evolution in affine connection representation.Definition 23.
Letρ be a geometric property on M, such as a charge of F. Then, H≜∇ρ is a gradient direction field of ρ on M,G.
LetT be the totality of all flat transformations Lk defined in Section 2.3. ∀T∈T, the flat transformation T:f↦Tf induces a transformation T∗:ρ↦T∗ρ. Denote
(96)ρ≜ρT≜T∗ρ∣T∈T,H≜HT≜∇ρT∣T∈T.
∀a∈M, the restriction of H at a are denoted by Ha≜HTa∣T∈T.We sayH is the total distribution of the gradient direction field H.Remark 24.
WhenT is fixed, HT can reflect the shape of M,G. When a is fixed, the extension to Ha can reflect the shape of M,G.However, whenT and a are both fixed, HTa is a fixed individual gradient direction, which cannot reflect the shape of M,G. In other words, if the momentum pT and the position xa of ρ are both definitely observed, the physical reality G would be unknowable; therefore, this is unacceptable. This is an embodiment of quantum uncertainty in affine connection representation.Definition 25.
LetφH be the one-parameter group of diffeomorphisms corresponding to H. The parameter of φH is x0. ∀a∈M, according to Definition 4, let φH,a be the evolution path through a, such that φH,a0=a. ∀t∈ℝ, denote
(97)φ∣H∣,a≜φX,a∣X∈H,φH,at≜φX,at∣X∈H.
∀Ω⊆T, we also denote HΩ≜HT∣T∈Ω⊆H and
(98)φ∣HΩ∣,a≜φX,a∣X∈HΩ⊆φH,a,φHΩ,at≜φX,at∣X∈HΩ⊆φH,at.
∀a∈M, the restriction of HΩ at a is denoted by HΩa≜HTa∣T∈Ω.Remark 26.
At the beginningt=0, intuitively, the gradient directions Ha of ρ start from a and point to all directions around a uniformly. If M,G is not flat, when evolving to a certain time t>0, the distribution of gradient directions on φH,at is no longer as uniform as beginning. The following definition precisely characterizes this kind of ununiformity.Definition 27.
Let the transformationLG−1 act on G; then, we obtain the trivial e≜LG−1G. Now M,G is sent to a flat M,e, and the gradient direction field H of ρ on M,G is sent to a gradient direction field O of ρ on M,e. Correspondingly, ∀t∈ℝ, φH,at is sent to φO,at. In a word, LG−1 induces the following two maps:
(99)G∗−1:H⟶O,G∗∗−1:φH,a⟶φO,a.
∀T∈T, deonte N≜N∈T∣detN=detT. Due to T≅GLD,ℝ, let U be a neighborhood of T, with respect to the topology of GLD,ℝ.TakeΩ=N∩U; then,
(100)OΩ=G∗−1HΩ,φOΩ,a=G∗∗−1φHΩ,a.Letμ be a Borel measure on the manifold M. We know ∀t∈ℝ,
(101)φHN,at≃φON,at≃SD−1.Thus,φHΩ,at⊆φ∣HN∣,at and φOΩ,at⊆φON,at are Borel sets, so they are measurable. Denote
(102)μaφHΩ,at≜μG∗∗−1φHΩ,at=μφOΩ,at.WhenU⟶T, we have Ω⟶T, HΩ⟶HT, HΩa⟶HTa, and φHΩ,at⟶b≜φHT,at.For the sake of simplicity, denoteL≜φHT,a. Thus, we have a=L0,b=Lt, and denote pa≜La=HTa,pb≜Lb=HTb.Becauseμa is absolutely continuous with respect to μ, Radon-Nikodym theorem [42] ensures the existence of the following limit. The Radon-Nikodym derivative
(103)WLb,a≜dμadμb≜limU⟶TμaφHΩ,atμφHΩ,at=limU⟶TμG∗∗−1φHΩ,atμφHΩ,at=limU⟶TμφOΩ,atμφHΩ,at is said to be the distribution density of H along Lin position representation.On a neighborhoodU of a, ∀T∈T, denote the normal section of HTa by NHT,a, that is,
(104)NHT,a≜n∈U∣HTa·n−a=0,NHT,at≜φHT,xt∣x∈NHT,a.Thus,NHT,a=NHT,a0 and NHT,b≜NHT,at. If U⟶a, we have NHT,a⟶a and NHT,at⟶b≜φHT,at. The Radon-Nikodym derivative
(105)ZLb,a≜dμadμb≜limU⟶aμNHT,aμNHT,b=limU⟶aμNHT,aμNHT,atis said to be the distribution density of H along Lin momentum representation.In a word,WLb,a and ZLpb,pa describe the density of the gradient lines that are adjacent to b in two different ways. They have the following property that is evidently true.Proposition 28.
LetL be a gradient line. ∀a,b,c∈L such that Lxa0=a, Lxb0=b, Lxc0=c, and xb0>xc0>xa0; then,
(106)WLb,a=WLb,cWLc,a,ZLb,a=ZLb,cZLc,a.Definition 29.
IfL is a gradient line of some ρ′∈∣ρ∣, we also say L is a gradient line of ∣ρ∣.Remark 30.
For anya and b, it anyway makes sense to discuss the gradient line of ∣ρ∣ from a to b. It is because even if the gradient line of ρ starting from a does not pass through b, it just only needs to carry out a certain flat transformation T defined in Section 2.3 to obtain a ρ′≜T∗ρ; thus, the gradient line of ρ′ starting from a can just exactly pass through b. Due to ρ,ρ′∈ρ, we do not distinguish them, and it is just fine to uniformly use ρ. Intuitively speaking, when ρ takes two different initial momentums, ρ presents as ρ and ρ′, respectively.Discussion 31.
With the above preparations, we obtain a new way to describe the construction of the propagator strictly.
For any pathL that starts at a and ends at b, we denote L≜∫Ldx0 concisely. Let Pb,a be the totality of all the paths from a to b. Denote
(107)Pb,xb0;a,xa0≜L∣L∈Pb,a,L=xb0−xa0.
∀L∈Pb,xb0;a,xa0, we can let Lxa0=a and Lxb0=b without loss of generality. Thus, Pb,xb0;a,xa0 is the totality of all the paths from Lxa0=a to Lxb0=b.Abstractly, the propagator is defined as the Green function of the evolution equation. Concretely, the propagator still needs a constructive definition. One method is the Feynman path integral(108)Kb,xb0;a,xa0≜∫Pb,xb0;a,xa0eisdL.However, there are so many redundant paths inPb,xb0;a,xa0 that (i) it is difficult to generally define a measure dL on Pb,xb0;a,xa0, and (ii) it may cause unnecessary infinities when carrying out some calculations.In order to solve this problem, we try to reduce the scope of summation fromPb,xb0;a,xa0 to Hb,xb0;a,xa0, where Hb,xb0;a,xa0 is the totality of all the gradient lines of ρ from Lxa0=a to Lxb0=b. Thus, Equation (108) is turned into
(109)Kb,xb0;a,xa0=∫Hb,xb0;a,xa0ΨLeisdL.We notice that as long as we take the probability amplitudeΨL of the gradient line L such that ΨL2=WLb,a in position representation or take ΨL2=ZLb,a in momentum representation, it can exactly be consistent with the Copenhagen interpretation. This provides the following new constructive definition for the propagator.Definition 32.
Supposeρ is defined as Definition 23, and denote H≜∇ρ.
LetLb,a be the totality of all the gradient lines of ρ from a to b. Denote
(110)Hb,xb0;a,xa0≜L∣L∈Lb,a,L=xb0−xa0.
LetLpb,pa be the totality of all the gradient lines of ∣ρ∣, whose starting direction is pa and ending direction is pb. Denote
(111)Hpb,xb0;pa,xa0≜L∣L∈Lpb,pa,L=xb0−xa0.
LetdL be a Borel measure on Hb,xb0;a,xa0. In consideration of Remark 41, we let s be the affine action sL in Definition 16. We say the geometric property
(112)Kb,xb0;a,xa0≜∫Hb,xb0;a,xa0WLb,aeisdLis the propagator of ρ from a,xa0 to b,xb0in position representation. If we let dL be a Borel measure on Hpb,xb0;pa,xa0, then we say
(113)Kpb,xb0;pa,xa0≜∫Hpb,xb0;pa,xa0ZLb,aeisdLis the propagator of ρ from pa,xa0 to pb,xb0in momentum representation.Discussion 33.
Now (112) and (113) are strictly defined, but the Feynman path integral (108) has not been possessed of a strict mathematical definition until now. This makes it impossible at present to obtain (e.g., in position representation) a strict mathematical proof of
(114)∫Pb,xb0;a,xa0eisdL=∫Hb,xb0;a,xa0WLb,aeisdL.Fortunately, the following two reasons make us believe that Equation (114) is expected to be regarded as a strict definition of Feynman path integral; that is to say, the integral on the right-hand side of “=” can be regarded as the strict definition of the notation on the left-hand side of “=”.On the one hand, we notice that the distribution densitiesWLb,a and ZLb,a of gradient directions establish an association between probability interpretation and geometric interpretation of quantum evolution. Therefore, we can base on probability interpretation to intuitively consider both sides of “=” in Equation (114) as the same thing.On the other hand, on the condition of Proposition28, denote Hxc0≜Lxc0L∈Hb,xb0;a,xa0; then,
(115)Kb,xb0;a,xa0=∫Hxc0Kb,xb0;c,xc0Kc,xc0;a,xa0dcis expected to be provable according to Equations (106) and (112). However, to obtain a strict proof of Equation (115) from Equations (106) and (112) is not a trivial mathematical problem, which is necessary but not easy, and needs more mathematical research.Discussion 34.
The quantization methods of QFT are successful, and they are also applicable in affine connection representation, but in this paper, we do not discuss them. We try to propose some more ideas to understand the quantization of field in affine connection representation.(1)
If we take(116)s=∫LDρ=∫LpQdxQ=∫LE0dx0,according to Definition 16, where D is the holonomic connection of M,G, then consider the distribution of H≜∇ρ, and we know that
(117)Kb,xb0;a,xa0≜∫∇ρb,xb0;a,xa0WLb,aeisdL,Kpb,xb0;pa,xa0≜∫∇ρpb,xb0;pa,xa0ZLb,aeisdLdescribes the quantization of energy-momentum. Every gradient line in ∇ρb,xb0;a,xa0 corresponds to a set of eigenvalues of energy and momentum. This is consistent with conventional theories, and this inspires us to consider the following new ideas to carry out the quantization of charge and current of gauge field.
(2)
In an analogous manner, if we take(118)s=∫LDt=∫LKNPQM:PdxQ=∫LρN0Mdx0, according to Section 3.5, where D is the holonomic connection of M,F, then consider the distribution of H≜∇t,
(119)Kb,xb0;a,xa0≜∫∇tb,xb0;a,xa0WLb,aeisdL,Kpb,xb0;pa,xa0≜∫∇tpb,xb0;pa,xa0ZLb,aeisdL.
DenoteHb,xb0;xc0≜c∈M∀L∈Hb,xb0;c,xc0,L=xb0−xc0 and take H=∇t; then, the wave function ψb,xb0 that is defined by the equation
(120)ψb,xb0=∫Hb,xb0;xc0Kb,xb0;c,xc0ψc,xc0dcdescribes the quantization of charge and current. It should be emphasized that this is not the quantization of the energy-momentum of the field, but the quantization of the field itself, which presents as quantized (e.g., discrete) charges and currents.


## 3.1. The Relation between Time and Space

Definition 2.
SupposeM=P×N and r≜dimP=3. Let
(30)A,B,M,N=1,⋯,D;s,i=1,⋯,r;a,m=r+1,⋯,D.
On a geometric manifoldM,f, the dξ0 and dx0 which are defined by
(31)dξ02≜∑A=1DdξA2=δABdξAdξB=GMNdxMdxN,dx02≜∑M=1DdxM2=εMNdxMdxN=HABdξAdξB,are said to be total space metrics or time metrics. We also suppose
(32)dξP2≜∑s=1rdξs2,dξN2≜∑a=r+1Ddξa2,dxP2≜∑i=1rdxi2,dxN2≜∑m=r+1Ddxm2.
dξN and dxN are regarded as proper-time metrics. For convenience, P is said to be external space and N is said to be internal space.Remark 3.
The above definition implies a new viewpoint about time and space. The relation between time and space in this way is different from the Minkowski coordinatesxμμ=0,1,2,3. Time and space are not the components on an equal footing anymore, but have a relation of total to component. It can be seen later that time reflects the total evolution in the full-dimensional space, while a specific spatial dimension reflects just a partial evolution in a specific direction.


## 3.2. Evolution Path as a Submanifold

Definition 4.
Let there be reference-systemsf, g, f, and g on a manifold M, such that ∀p∈M, on the neighborhood U of p,
(33)U,αA′⟶fpU,ξA⟶fpU,xM⟵gpU,ζA⟵gpU,βA′.
DenoteF≜f∘f and G≜g∘g; then, we say F and Gmove relatively and interact mutually, and also we say that F evolves in G, or F evolves on the geometric manifold M,G. Meanwhile, G evolves in F, or we say G evolves on M,F.
From Equation (23), we know that in F and G, gauge fields originate from f and g, and gravitational fields GFMN and GGMN are effected by f and g, respectively. We are going to describe their evolutions step by step in the following sections.
Let there be a one-parameter group of diffeomorphisms(34)φX:M×ℝ⟶M,acting on M, such that φXp,0=p. Thus, φX determines a smooth tangent vector field X on M. If X is nonzero everywhere, we say φX is a set of evolution paths, and X is an evolution direction field. Let T⊆ℝ be an interval; then, the regular imbedding
(35)Lp≜φX,p:T⟶M,t↦φXp,tis said to be an evolution path through p. The tangent vector d/dt≜Lp=Xp is called an evolution direction at p. For the sake of simplicity, we also denote Lp≜LpT⊂M; then,
(36)π:Lp⟶M,q↦qis also a regular imbedding. If it is not necessary to emphasize the point p, Lp is denoted by L concisely.In order to describe physical evolution, next we are going to strictly describe the mathematical properties of the reference-systemsf and g which are sent onto the evolution path L.Definition 5.
Let the time metrics ofU,ξA, U,xM, and U,ζA be dξ0, dx0, and dζ0, respectively. On UL≜U∩Lp, we have parameter equations
(37)ξA=ξAx0,xM=xMξ0,ζA=ζAx0,ξ0=ξ0x0,x0=x0ξ0,ζ0=ζ0x0.
Takef for example, according to Equation (37), on UL we define
(38)b0A≜dξAdx0,b00≜dξ0dx0,ε0M≜dxMdx0=b00c0M=b0AcAM,c0M≜dxMdξ0,c00≜dx0dξ0,δ0A≜dξAdξ0=c00b0A=c0MbMA.
Definedξ0≜dx0/dξ0dx0 and dx0≜dξ0/dx0dξ0, which induce d/dξ0 and d/dx0, such that d/dξ0,dξ0=1 and d/dx0,dx0=1. So we can also define
(39)b¯A0≜dξAdx0,b¯00≜dξ0dx0,ε¯M0≜dxMdx0=b¯00c¯M0=b¯A0c¯MA,c¯M0≜dxMdξ0,c¯00≜dx0dξ0,δ¯A0≜dξAdξ¯0=c¯00b¯A0=c¯M0b¯AM.
They determine the following smooth functions on the entireL, similar to Section 2.2, that
(40)B0A:L⟶ℝ,p↦B0Ap≜bfp0Ap,C0M:L⟶ℝ,p↦C0Mp≜cfp0Mp,B¯A0:L⟶ℝ,p↦B¯A0p≜b¯fpA0p,C¯M0:L⟶ℝ,p↦C¯M0p≜c¯fpM0p,B00:L⟶ℝ,p↦B00p≜bfp00p,C00:L⟶ℝ,p↦C00p≜cfp00p,B¯00:L⟶ℝ,p↦B¯00p≜b¯fp00p,C¯00:L⟶ℝ,p↦C¯00p≜c¯fp00p.
For convenience, we still use the notationsε and δ and have the following smooth functions.
(41)ε0M≜B00C0M=B0ACAM,δ0A≜C00B0A=C0MBMA,G00≜B00B00=GMNε0Mε0N,ε¯M0≜B¯00C¯M0=B¯A0C¯MA,δ¯A0≜C¯00B¯A0=C¯M0B¯AM,G00≜C00C00=GMNε¯M0ε¯N0.
It is easy to verify thatdx0=G00dx0 and d/dx0=G00d/dx0 are both true on L by a simple calculation.


## 3.3. Evolution Lemma

We have the following two evolution lemmas. The affine connection representations of Yang-Mills equation, energy-momentum equation, and Dirac equation are dependent on them.Definition 6.
∀p∈L, suppose TpM and TpL are tangent spaces, Tp∗M and Tp∗L are cotangent spaces. The regular imbedding π:L⟶M,q↦q induces the tangent map and the cotangent map
(42)π∗:TpL⟶TpM,γL↦π∘γL,π∗:Tp∗M⟶Tp∗L,df↦df∘π.
Evidently,π∗ is an injection, and π∗ is a surjection. ∀d/dtL∈TpL,d/dt∈TpM,df∈Tp∗M,dfL∈Tp∗L, if and only if
(43)ddt=π∗ddtL,dfL=π∗dfare true, we denote
(44)ddt≅ddtL,df≃dfL.Then, we have the following two propositions that are evidently true.Proposition 7.
Ifd/dt≅d/dtL and df≃dfL, then
(45)ddt,df=ddtL,dfL.Proposition 8.
The following conclusions are true.(46)wM∂∂xM≅w0ddx0⟺wM=w0ε0M,wMdxM≃w0dx0⟺wMε0M=w0,w¯M∂∂xM≅w¯0ddx0⟺w¯M=w¯0ε¯M0,w¯MdxM≃w¯0dx0⟺w¯Mε¯M0=w¯0.


## 3.4. On-Shell Evolution as a Gradient

LetT be a smooth n-order tensor field. The restriction on U,xM is T≜t∂/∂x⊗dx, where ∂/∂x⊗dx represents the tensor basis generated by several ∂/∂xM and dxM, and the tensor coefficients of T are concisely denoted by t:U⟶ℝ.LetD be a holonomic connection. Consider DT≜t;QdxQ⊗∂/∂x⊗dx. Denote
(47)Dt≜t;QdxQ,∇t≜t;Q∂∂xQ.∀p∈M, the integral curve of ∇t, that is, Lp≜φ∇t,p, is a gradient line of T. It can be seen later that the above gradient operator ∇ characterizes the on-shell evolution.For any evolution pathL, let UL≜U∩L. Denote tL≜tUL and tL;0≜t;Qε0Q, as well as
(48)DLtL≜tL;0dx0,∇LtL≜tL;0ddx0.Proposition 9.
The following conclusions are evidently true.(i)
Dt≃DLtL if and only if L is an arbitrary evolution path(ii)
∇t≅∇LtL if and only if L is a gradient line of TRemark 10.
More generally, suppose there is a tensorU≜uQdxQ⊗∂/∂x⊗dx. In such a notation, all the indices are concisely ignored except Q. uQdxQ uniquely determines a characteristic direction uQ∂/∂xQ.If the system of 1-order linear partial differential equationst;Q=uQ has a solution t, then it is true that Dt=uQdxQ and ∇t=uQ∂/∂xQ. Thus, in the evolution direction L=uQ∂/∂xQ, the following conclusions are true.
(49)Dt≃DLtL,∇t≅∇LtL,where DLtL≜u0dx0, ∇LtL≜u0d/dx0, and u0≜uQε0Q.Now for any geometric property in the form of tensorU, we are able to express its on-shell evolution in the form of ∇t.Next, two important on-shell evolutions are discussed in the following two sections. One is the on-shell evolution of the potential field of a reference-system. The other is the one that a general charge of a reference-system evolves in the potential field of another reference-system.


## 3.5. On-Shell Evolution of Potential Field and Affine Connection Representation of Yang-Mills Equation

Table I of article [40] proposes a famous correspondence between gauge field terminologies and fibre bundle terminologies. However, it does not find out the corresponding mathematical object to the source JμK. In this section, we give an answer to this problem and show the affine connection representation of Yang-Mills equation.In order to obtain the general Yang-Mills equation with gravitation, we have to adopt holonomic connection to construct it. SupposeF evolves in G according to Definition 4, that is, ∀p∈M,
(50)U,αA′⟶fpU,ξA⟶fpU,xM⟵gpU,ζA⟵gpU,βA′.We always take the following notations in the coordinate frameU,xM.
(i)
Let the holonomic connections, which are defined by Equation (25), of geometric manifolds M,F and M,G be ΓFNPM and ΓGNPM, respectively. The colon “:” and the semicolon “;” are used to express the covariant derivatives on M,F and M,G, respectively, e.g.,(51)uQ:P=∂uQ∂xP+ΓFHPQuH,uQ;P=∂uQ∂xP+ΓGHPQuH.(ii)
Let the coefficients of curvature tensor ofM,F and M,G be KNPQM and RNPQM, respectively, i.e.,(52)KNPQM≜∂ΓFNQM∂xP−∂ΓFNPM∂xQ+ΓFNQHΓFHPM−ΓFNPHΓFHQM,RNPQM≜∂ΓGNQM∂xP−∂ΓGNPM∂xQ+ΓGNQHΓGHPM−ΓGNPHΓGHQM.DenoteKNPQM:P≜GFPP′KNPQ:P′M. On an arbitrary evolution path L, we define
(53)ρN0Mdx0≜π∗KNPQM:PdxQ∈T∗L.Then, according to Definition6 and the evolution lemma of Proposition 8, we obtain ρN0M=KNPQM:Pε0Q and
(54)KNPQM:PdxQ≃ρN0Mdx0.Let∇t=KNPQM:P∂/∂xQ. Then, according to Proposition 9, if and only if ∀p∈M, Lp=∇tp, we have
(55)KNPQM:P∂∂xQ≅ρN0Mddx0.Applying the evolution lemma of Proposition8 again, we obtain
(56)KNPQM:P=ρN0Mε¯Q0.DenotejNQM≜ρN0Mε¯Q0; then, if and only if Lp=∇tp, we have
(57)KNPQM:P=jNQM,which is said to be (affine) Yang-Mills equation of F. It contains effects of gravitation, which makes the gauge frames BfMA and CfAM have physical effects. According to Equation (29), we know Equation (57) is coordinate covariant, and if gravitation is removed, it is also gauge covariant.Thus, we have the following two results.(i)
The Yang-Mills equation originates from a geometric property in the direction∇t. In other words, the on-shell evolution of gauge field is described by the direction field ∇t(ii)
We obtain the mathematical origination of charge and current. We know that the evolution pathL is an imbedding submanifold of M. Thus, the charge ρN0M originates from the pull-back π∗ from M to L, and the current jNQM originates from ∇t that is associated to ρN0MIf we letM,f be completely flat, i.e., BfMA=δMA, CfAM=δAM, then by calculation, we find ρN0M can still be nonvanishing. This shows that ρN0M originates from M,f ultimately.Definition 11.
We speak of the real-valued(58)ρMN0≜GMHρN0Has the field function of a general charge or speak of it as a charge of F for short.


## 3.6. On-Shell Evolution of General Charge and Affine Connection Representation of Mass, Energy, Momentum, and Action

In order to be compatible with the affine connection representation of gauge fields, we also have to define mass, energy, momentum, and action in the form associated to affine connection. We are going to show them in this section and Section4.3.LetF0≜ρMN0dxM⊗dxN. For the sake of simplicity, denote the charge ρMN0 of F by ρMN concisely. Let D be the holonomic connection of M,G; then,
(59)DF0≜DρMN⊗dxM⊗dxN,∇F0≜∇ρMN⊗dxM⊗dxN,where DρMN≜ρMN;QdxQ and ∇ρMN≜ρMN;Q∂/∂xQ. According to Proposition 9, if and only if ∀p∈L, the evolution direction is taken as Lp=∇ρMNp, we have
(60)DρMN≃DLρMN,∇ρMN≅∇LρMN,that is,
(61)ρMN;QdxQ≃ρMN;0dx0,ρMN;Q∂∂xQ≅ρMN;0ddx0.Definition 12.
For more convenience, the notationρMN is further abbreviated as ρ. In affine connection representation, energy and momentum of ρ are defined as
(62)E0≜ρ;0≜ρ;Qε0Q,pQ≜ρ;Q,H0≜dρdx0,PQ≜∂ρ∂xQ,E0≜ρ;0≜ρ;Qε¯Q0,pQ≜ρ;Q,H0≜dρdx0,PQ≜∂ρ∂xQ.Proposition 13.
At any pointp on M, the equation
(63)E0E0=pQpQholds if and only if the evolution direction Lp=∇ρp. Equation (63) is the (affine) energy-momentum equation of ρ.Proof.
According to the above discussion,∀p∈M, Lp=∇ρp is equivalent to
(64)pQdxQ≃E0dx0,pQ∂∂xQ≅E0ddx0.
Then, due to Proposition7, we obtain the directional derivative in the gradient direction ∇ρ:
(65)pQ∂∂xQ,pMdxM=E0ddx0,E0dx0,
i.e.,GQMpQpM=G00E0E0, or pQpQ=E0E0.Proposition 14.
At any pointp on M, the equations
(66)pQ=E0dxQdx0,pQ=E0dxQdx0hold if and only if the evolution direction Lp=∇ρp.Proof.
Due to the evolution lemma of Proposition8, we immediately obtain Equation (66) from Equation (64).Remark 15.
In the gradient direction∇ρ, Equation (66) is consistent with the conventional formula
(67)p=mv.
Thus, in affine connection representation, the energy-momentum equation and the conventional definition of momentum both originate from a geometric property in gradient direction. In other words, the on-shell evolution of the particle fieldρ is described by the gradient direction field ∇ρ.Definition 16.
LetPb,a be the totality of paths from a to b. And suppose L∈Pb,a, and the evolution parameter x0 satisfies ta≜x0a<x0b≜tb. The elementary affine action of ρ is defined as
(68)sL≜∫LDρ=∫LpQdxQ=∫tatbE0dx0.Thus,δsL=0 if and only if L is a gradient line of ρ.In particular, in the case whereG is orthogonal, we can also define action in the following way.OnM,G, let there be Dirac algebras γM and γN such that
(69)γMγN+γNγM=2GMN,γMγN+γNγM=2GMN,γMγM=1.In a gradient direction ofρ, from Equation (63), we obtain that
(70)pQpQ=E0E0⟺ρ;Qρ;Q=ρ;0ρ;0⟺GPQρ;Pρ;Q=G00ρ;0ρ;0⟺γPγQ+γQγPρ;Pρ;Q=2ρ;0ρ;0⟺γPρ;PγQρ;Q+γQρ;QγPρ;P=2ρ;0ρ;0⟺γPρ;P2=ρ;02.TakeγPρ;P=ρ;0 without loss of generality, and then, in the gradient direction of ρ, we have
(71)γPρ;Pdx0=ρ;0dx0=ε0Pρ;Pdx0=Dρ.So we can take(72)sL≜∫LγPρ;Pdx0+Dρ=∫tatbγPρ;P+ε0Pρ;Pdx0=∫tatbγPρ;P+E0dx0.Remark17 and Remark 41 explain the rationality of this definition. We have sL=2sL in the gradient direction of ρ, so sL and sL are consistent.Remark 17.
In the Minkowski coordinate frame of Section4.2, the evolution parameter x0 is replaced by x~τ; then, there still exists a concept of gradient direction ∇~ρ~. Correspondingly, Equations (68) and (72) present as
(73)s~L≜∫LD~ρ~=∫Lp~μdx~μ=∫τaτbm~τdx~τ,s~L=∫τaτbγμρ~;μ+m~τdx~τ,where m~τ is the rest-mass and x~τ is the proper-time.Remark 18.
Define the following notations.(74)ρΓG≜∂ρMN∂xG−ρMN;G=ρMHΓNGH+ρHNΓMGH,ρRPQ≜ρMHRNPQH+ρHNRMPQH.
Then, through some calculations, we can obtain that(75)fP≜pP;0=E0;P−pQε0;PQ+ρRPQε0Q,which is the affine connection representation of general Lorentz force equation (see Discussion 38 for further illustrations).


## 3.7. Inversion Transformation in Affine Connection Representation

In affine connection representation,CPT inversion is interpreted as a full inversion of coordinates and metrics. Let i,j=1,2,3 and m,n=4,5,⋯,D.Let the local coordinate representation of reference-systemk be x′j=−δijxi, x′n=δmnxm; then, parity inversion can be represented as
(76)P≜Lk:xi⟶−xi,xm⟶xm.Let the local coordinate representation of reference-systemh be x′j=δijxi, x′n=−δmnxm; then, charge conjugate inversion can be represented as
(77)C≜Lh:xi⟶xi,xm⟶−xm.Time coordinate inversion can be represented as(78)T0:x0⟶−x0.Full inversion of coordinates can be represented as(79)CPT0:xQ⟶−xQ,x0⟶−x0.The positive or negative sign of metric marks two opposite directions of evolution. LetN be a closed submanifold of M, and let its metric be dxN. Denote the totality of closed submanifolds of M by BM; then, full inversion of metrics can be expressed as
(80)TM≜∏N∈BMdxN⟶−dxN.Denote time inversion by(81)T≜TMT0,and then, the joint transformation of the full inversion of coordinates CPT0 and the full inversion of metrics TM is
(82)CPT0TM=CPT.Summarize the above discussions; then, we have(83)CPT0:xQ⟶−xQ,x0⟶−x0,dxQ⟶dxQ,dx0⟶dx0,TM:xQ⟶xQ,x0⟶x0,dxQ⟶−dxQ,dx0⟶−dx0,CPT:xQ⟶−xQ,x0⟶−x0,dxQ⟶−dxQ,dx0⟶−dx0.TheCPT invariance in affine connection representation is very clear. Concretely, on M,G, we consider the CPT transformation acting on G. Denote s≜∫LDρ and DPeis≜∂/∂xP−iρΓPeis; then, through simple calculations, we obtain that
(84)CPT:Dρ⟶Dρ,DPeis⟶−DPe−is.Remark 19.
In quantum mechanics, there is a complex conjugation in the time inversion of wave functionT:ψx,t⟶ψ∗x,−t. In affine connection representation, we know the complex conjugation can be interpreted as a straightforward mathematical result of the full inversion of metrics TM.


## 3.8. Two Dual Descriptions of Gradient Direction Field

Discussion 20.
LetX and Y be nonvanishing smooth tangent vector fields on the manifold M. And let LY be the Lie derivative operator induced by the one-parameter group of diffeomorphism φY. Then, according to a well-known theorem [41], we obtain the Lie derivative equation
(85)X,Y=LYX.Suppose∀p∈M, Yp is a unit-length vector, i.e., Yp=1. Let the parameter of φY be x0. Then, on the evolution path L≜φY,p, we have
(86)Y≅ddx0.Thus, Equation (85) can also be represented as
(87)X,Y=ddx0X.On the other hand,∀df∈TM and dfL≜π∗df, and due to (86) and Proposition 7, we have Y,df=d/dx0,dfL, that is,
(88)Yf=ddx0fL.Definition 21.
LetH≜∇ρ−1∇ρ=ε0M∂/∂xM≅d/dx0. It is evident that ∀p∈M, Hp=1. If and only if taking Y=H, we speak of (87) and (88) as real-valued (affine) Heisenberg equation and (affine) Schrödinger equation, respectively, that is,
(89)X,H=ddx0X,Hf=ddx0fL.Discussion 22.
The above two equations both describe the gradient direction field and thereby reflect on-shell evolution. Such two dual descriptions of gradient direction show the real-valued affine connection representation of Heisenberg picture and Schrödinger picture.It is not hard to find out several different kinds of complex-valued representations of gradient direction. For examples, one is the affine Dirac equation in Section4.4, and another is as follows.Letψ≜feisL, where it is fine to take either sL≜sL or sL≜sL from Definition 16. According to Equation (89), it is easy to obtain on L that
(90)X,H=ddx0X,Hψ=dψdx0.This is consistent with the conventional Heisenberg equation and Schrödinger equation (taking the natural units thatℏ=1,c=1)
(91)X,−iH=∂∂tX,−iHψ=∂ψ∂t,and they have a coordinate correspondence
(92)∂∂ixk⟷∂∂xk,∂∂t↔ddx0.We know that∂/∂t⟷d/dx0 originates from the difference that the evolution parameter is xτ or x0. The imaginary unit i originates from the difference between the regular coordinates x1,x2,x3,xτ and the Minkowski coordinates x1,x2,x3,x0. That is to say, the regular coordinates satisfy
(93)dx02=dx12+dx22+dx32+dxτ2,and the Minkowski coordinates satisfy
(94)dxτ2=dx02−dx12−dx22−dx32=dx02+dix12+dix22+dix32.This causes the appearance of the imaginary uniti in the correspondence
(95)ixk⟷xk.So Equations (90) and (91) have exactly the same essence, and their differences only come from different coordinate representations.The differences between coordinate representations have nothing to do with the geometric essence and the physical essence. We notice that the value of a gradient direction is dependent on geometry, but independent of that the equations are real-valued or complex-valued. Therefore, it is unnecessary for us to confine to such algebraic forms as real-valued or complex-valued forms, but we should focus on such geometric essence as gradient direction.The essential virtue of complex-valued form is that it is applicable for describing the coherent superposition of propagator. However, this is independent of the above discussions, and we are going to discuss it in Section3.9.


## 3.9. Quantum Evolution as a Distribution of Gradient Directions

From Proposition13, we see that, in affine connection representation, the classical on-shell evolution is described by gradient direction. Then, naturally, quantum evolution should be described by the distribution of gradient directions.The distribution of gradient directions on a geometric manifoldM,G is effected by the bending shape of M,G; in other words, the distribution of gradient directions can be used to reflect the shape of M,G. This is the way that the quantum theory in affine connection representation describes physical reality.In order to know the full picture of physical reality, it is necessary to fully describe the shape of the geometric manifold. For a single observation,(1)
It is the reference-system, not a point, that is used to describe the physical reality, so the coordinate of an individual point is not enough to fully describe the location information about the physical reality(2)
Through a single observation of momentum, we can only obtain information about an individual gradient direction; this cannot reflect the full picture of the shape of the geometric manifoldQuantum evolution provides us with a guarantee that we can obtain the distribution of gradient directions through multiple observations, so that we can describe the full picture of the shape of the geometric manifold.Next, we are going to carry out strict mathematical descriptions for the quantum evolution in affine connection representation.Definition 23.
Letρ be a geometric property on M, such as a charge of F. Then, H≜∇ρ is a gradient direction field of ρ on M,G.
LetT be the totality of all flat transformations Lk defined in Section 2.3. ∀T∈T, the flat transformation T:f↦Tf induces a transformation T∗:ρ↦T∗ρ. Denote
(96)ρ≜ρT≜T∗ρ∣T∈T,H≜HT≜∇ρT∣T∈T.
∀a∈M, the restriction of H at a are denoted by Ha≜HTa∣T∈T.We sayH is the total distribution of the gradient direction field H.Remark 24.
WhenT is fixed, HT can reflect the shape of M,G. When a is fixed, the extension to Ha can reflect the shape of M,G.However, whenT and a are both fixed, HTa is a fixed individual gradient direction, which cannot reflect the shape of M,G. In other words, if the momentum pT and the position xa of ρ are both definitely observed, the physical reality G would be unknowable; therefore, this is unacceptable. This is an embodiment of quantum uncertainty in affine connection representation.Definition 25.
LetφH be the one-parameter group of diffeomorphisms corresponding to H. The parameter of φH is x0. ∀a∈M, according to Definition 4, let φH,a be the evolution path through a, such that φH,a0=a. ∀t∈ℝ, denote
(97)φ∣H∣,a≜φX,a∣X∈H,φH,at≜φX,at∣X∈H.
∀Ω⊆T, we also denote HΩ≜HT∣T∈Ω⊆H and
(98)φ∣HΩ∣,a≜φX,a∣X∈HΩ⊆φH,a,φHΩ,at≜φX,at∣X∈HΩ⊆φH,at.
∀a∈M, the restriction of HΩ at a is denoted by HΩa≜HTa∣T∈Ω.Remark 26.
At the beginningt=0, intuitively, the gradient directions Ha of ρ start from a and point to all directions around a uniformly. If M,G is not flat, when evolving to a certain time t>0, the distribution of gradient directions on φH,at is no longer as uniform as beginning. The following definition precisely characterizes this kind of ununiformity.Definition 27.
Let the transformationLG−1 act on G; then, we obtain the trivial e≜LG−1G. Now M,G is sent to a flat M,e, and the gradient direction field H of ρ on M,G is sent to a gradient direction field O of ρ on M,e. Correspondingly, ∀t∈ℝ, φH,at is sent to φO,at. In a word, LG−1 induces the following two maps:
(99)G∗−1:H⟶O,G∗∗−1:φH,a⟶φO,a.
∀T∈T, deonte N≜N∈T∣detN=detT. Due to T≅GLD,ℝ, let U be a neighborhood of T, with respect to the topology of GLD,ℝ.TakeΩ=N∩U; then,
(100)OΩ=G∗−1HΩ,φOΩ,a=G∗∗−1φHΩ,a.Letμ be a Borel measure on the manifold M. We know ∀t∈ℝ,
(101)φHN,at≃φON,at≃SD−1.Thus,φHΩ,at⊆φ∣HN∣,at and φOΩ,at⊆φON,at are Borel sets, so they are measurable. Denote
(102)μaφHΩ,at≜μG∗∗−1φHΩ,at=μφOΩ,at.WhenU⟶T, we have Ω⟶T, HΩ⟶HT, HΩa⟶HTa, and φHΩ,at⟶b≜φHT,at.For the sake of simplicity, denoteL≜φHT,a. Thus, we have a=L0,b=Lt, and denote pa≜La=HTa,pb≜Lb=HTb.Becauseμa is absolutely continuous with respect to μ, Radon-Nikodym theorem [42] ensures the existence of the following limit. The Radon-Nikodym derivative
(103)WLb,a≜dμadμb≜limU⟶TμaφHΩ,atμφHΩ,at=limU⟶TμG∗∗−1φHΩ,atμφHΩ,at=limU⟶TμφOΩ,atμφHΩ,at is said to be the distribution density of H along Lin position representation.On a neighborhoodU of a, ∀T∈T, denote the normal section of HTa by NHT,a, that is,
(104)NHT,a≜n∈U∣HTa·n−a=0,NHT,at≜φHT,xt∣x∈NHT,a.Thus,NHT,a=NHT,a0 and NHT,b≜NHT,at. If U⟶a, we have NHT,a⟶a and NHT,at⟶b≜φHT,at. The Radon-Nikodym derivative
(105)ZLb,a≜dμadμb≜limU⟶aμNHT,aμNHT,b=limU⟶aμNHT,aμNHT,atis said to be the distribution density of H along Lin momentum representation.In a word,WLb,a and ZLpb,pa describe the density of the gradient lines that are adjacent to b in two different ways. They have the following property that is evidently true.Proposition 28.
LetL be a gradient line. ∀a,b,c∈L such that Lxa0=a, Lxb0=b, Lxc0=c, and xb0>xc0>xa0; then,
(106)WLb,a=WLb,cWLc,a,ZLb,a=ZLb,cZLc,a.Definition 29.
IfL is a gradient line of some ρ′∈∣ρ∣, we also say L is a gradient line of ∣ρ∣.Remark 30.
For anya and b, it anyway makes sense to discuss the gradient line of ∣ρ∣ from a to b. It is because even if the gradient line of ρ starting from a does not pass through b, it just only needs to carry out a certain flat transformation T defined in Section 2.3 to obtain a ρ′≜T∗ρ; thus, the gradient line of ρ′ starting from a can just exactly pass through b. Due to ρ,ρ′∈ρ, we do not distinguish them, and it is just fine to uniformly use ρ. Intuitively speaking, when ρ takes two different initial momentums, ρ presents as ρ and ρ′, respectively.Discussion 31.
With the above preparations, we obtain a new way to describe the construction of the propagator strictly.
For any pathL that starts at a and ends at b, we denote L≜∫Ldx0 concisely. Let Pb,a be the totality of all the paths from a to b. Denote
(107)Pb,xb0;a,xa0≜L∣L∈Pb,a,L=xb0−xa0.
∀L∈Pb,xb0;a,xa0, we can let Lxa0=a and Lxb0=b without loss of generality. Thus, Pb,xb0;a,xa0 is the totality of all the paths from Lxa0=a to Lxb0=b.Abstractly, the propagator is defined as the Green function of the evolution equation. Concretely, the propagator still needs a constructive definition. One method is the Feynman path integral(108)Kb,xb0;a,xa0≜∫Pb,xb0;a,xa0eisdL.However, there are so many redundant paths inPb,xb0;a,xa0 that (i) it is difficult to generally define a measure dL on Pb,xb0;a,xa0, and (ii) it may cause unnecessary infinities when carrying out some calculations.In order to solve this problem, we try to reduce the scope of summation fromPb,xb0;a,xa0 to Hb,xb0;a,xa0, where Hb,xb0;a,xa0 is the totality of all the gradient lines of ρ from Lxa0=a to Lxb0=b. Thus, Equation (108) is turned into
(109)Kb,xb0;a,xa0=∫Hb,xb0;a,xa0ΨLeisdL.We notice that as long as we take the probability amplitudeΨL of the gradient line L such that ΨL2=WLb,a in position representation or take ΨL2=ZLb,a in momentum representation, it can exactly be consistent with the Copenhagen interpretation. This provides the following new constructive definition for the propagator.Definition 32.
Supposeρ is defined as Definition 23, and denote H≜∇ρ.
LetLb,a be the totality of all the gradient lines of ρ from a to b. Denote
(110)Hb,xb0;a,xa0≜L∣L∈Lb,a,L=xb0−xa0.
LetLpb,pa be the totality of all the gradient lines of ∣ρ∣, whose starting direction is pa and ending direction is pb. Denote
(111)Hpb,xb0;pa,xa0≜L∣L∈Lpb,pa,L=xb0−xa0.
LetdL be a Borel measure on Hb,xb0;a,xa0. In consideration of Remark 41, we let s be the affine action sL in Definition 16. We say the geometric property
(112)Kb,xb0;a,xa0≜∫Hb,xb0;a,xa0WLb,aeisdLis the propagator of ρ from a,xa0 to b,xb0in position representation. If we let dL be a Borel measure on Hpb,xb0;pa,xa0, then we say
(113)Kpb,xb0;pa,xa0≜∫Hpb,xb0;pa,xa0ZLb,aeisdLis the propagator of ρ from pa,xa0 to pb,xb0in momentum representation.Discussion 33.
Now (112) and (113) are strictly defined, but the Feynman path integral (108) has not been possessed of a strict mathematical definition until now. This makes it impossible at present to obtain (e.g., in position representation) a strict mathematical proof of
(114)∫Pb,xb0;a,xa0eisdL=∫Hb,xb0;a,xa0WLb,aeisdL.Fortunately, the following two reasons make us believe that Equation (114) is expected to be regarded as a strict definition of Feynman path integral; that is to say, the integral on the right-hand side of “=” can be regarded as the strict definition of the notation on the left-hand side of “=”.On the one hand, we notice that the distribution densitiesWLb,a and ZLb,a of gradient directions establish an association between probability interpretation and geometric interpretation of quantum evolution. Therefore, we can base on probability interpretation to intuitively consider both sides of “=” in Equation (114) as the same thing.On the other hand, on the condition of Proposition28, denote Hxc0≜Lxc0L∈Hb,xb0;a,xa0; then,
(115)Kb,xb0;a,xa0=∫Hxc0Kb,xb0;c,xc0Kc,xc0;a,xa0dcis expected to be provable according to Equations (106) and (112). However, to obtain a strict proof of Equation (115) from Equations (106) and (112) is not a trivial mathematical problem, which is necessary but not easy, and needs more mathematical research.Discussion 34.
The quantization methods of QFT are successful, and they are also applicable in affine connection representation, but in this paper, we do not discuss them. We try to propose some more ideas to understand the quantization of field in affine connection representation.(1)
If we take(116)s=∫LDρ=∫LpQdxQ=∫LE0dx0,according to Definition 16, where D is the holonomic connection of M,G, then consider the distribution of H≜∇ρ, and we know that
(117)Kb,xb0;a,xa0≜∫∇ρb,xb0;a,xa0WLb,aeisdL,Kpb,xb0;pa,xa0≜∫∇ρpb,xb0;pa,xa0ZLb,aeisdLdescribes the quantization of energy-momentum. Every gradient line in ∇ρb,xb0;a,xa0 corresponds to a set of eigenvalues of energy and momentum. This is consistent with conventional theories, and this inspires us to consider the following new ideas to carry out the quantization of charge and current of gauge field.
(2)
In an analogous manner, if we take(118)s=∫LDt=∫LKNPQM:PdxQ=∫LρN0Mdx0, according to Section 3.5, where D is the holonomic connection of M,F, then consider the distribution of H≜∇t,
(119)Kb,xb0;a,xa0≜∫∇tb,xb0;a,xa0WLb,aeisdL,Kpb,xb0;pa,xa0≜∫∇tpb,xb0;pa,xa0ZLb,aeisdL.
DenoteHb,xb0;xc0≜c∈M∀L∈Hb,xb0;c,xc0,L=xb0−xc0 and take H=∇t; then, the wave function ψb,xb0 that is defined by the equation
(120)ψb,xb0=∫Hb,xb0;xc0Kb,xb0;c,xc0ψc,xc0dcdescribes the quantization of charge and current. It should be emphasized that this is not the quantization of the energy-momentum of the field, but the quantization of the field itself, which presents as quantized (e.g., discrete) charges and currents.


## 4. Affine Connection Representation of Gauge Fields in Classical Spacetime

The new framework established in Section3 is discussed in the D-dimensional general coordinate xM, which is more general than the 1+3-dimensional conventional Minkowski coordinate xμ.dx02=∑M=1DdxM2 is the total metric of internal space and external space, and dxτ2=∑m=4Ddxm2 is the metric of internal space.
(i)
The evolution parameter of theD-dimensional general coordinate xMM=1,2,⋯,D is x0. The parameter equation of an evolution path L is represented as xM=xMx0(ii)
The evolution parameter of the1+3-dimensional Minkowski coordinate xμμ=0,1,2,3 is xτ. The parameter equation of L is represented as xμ=xμxτThe coordinatexμ works on the 1+3-dimensional classical spacetime submanifold defined as follows.


### 4.1. Classical Spacetime Submanifold

Let there be a smooth tangent vector fieldX on M,f. If ∀p∈M, Xp=bA∂/∂ξAp=cM∂/∂xMp satisfies that ba are not all zero and cm are not all zero, where a,m=r+1,⋯,D; then, we say X is internal-directed. For any evolution path L≜φX,p, we also say L is internal-directed.SupposeM=P×N, D≜dimM, and r≜dimP=3. X is a smooth tangent vector field on M. Fix a point o∈M. If X is internal-directed; then, there exist a unique 1+3-dimensional imbedding submanifold γ:M~⟶M,p↦p and a unique smooth tangent vector field X~ on M~ such that
(i)
P×o is a closed submanifold of M~(ii)
The tangent mapγ∗:TM~⟶TM satisfies that ∀q∈M~, γ∗:X~q↦XqSuch anM~ is said to be a classical spacetime submanifold.LetφX:M×ℝ⟶M and φX~:M~×ℝ⟶M~ be the one-parameter groups of diffeomorphisms corresponding to X and X~, respectively. Thus, we have
(121)φX~=φXM~×ℝ.So the evolution in classical spacetime can be described byφX~. It should be noticed that
(i)
M~ inherits a part of geometric properties of M, but not all. The physical properties reflected by M~ are incomplete(ii)
The correspondence betweenX~ and the restriction of X to M~ is one-to-one. For convenience, next we are not going to distinguish the notations X and X~ on M~ but uniformly denote them by X(iii)
An arbitrary pathL~:T⟶M~,t↦p on M~ uniquely corresponds to a path L≜γ∘L~:T⟶M,t↦p on M. Evidently, the image sets of L and L~ are the same, that is, LT=L~T. For convenience, later we are not going to distinguish the notations L and L~ on M~ but uniformly denote them by L


### 4.2. Classical Spacetime Reference-System

Let there be a geometric manifoldM,f and its classical spacetime submanifold M~. And let L≜φX~,a be an evolution path on M~. Suppose p∈L and U is a coordinate neighborhood of p. According to Definition 5, suppose the fp on U and the fLp on UL≜U∩L satisfy that
(122)fp:ξA=ξAxM=ξAx0,ξ0=ξ0x0,A,M=1,2,⋯,D.Thus, it is true that(1)
There exists a unique local reference-systemf~p on U~≜U∩M~ such that(123)f~p:ξU=ξUxK=ξUx0,ξ0=ξ0x0,U,K=1,2,3,τ.(2)
IfL is internal-directed, then the above coordinate frames U~,ξU and U~,xK of f~p uniquely determine the coordinate frames U~,ξ~α and U~,x~μ such that(124)f~p:ξ~α=ξ~αx~μ=ξ~αx~τ,ξ~τ=ξ~τx~τ,α,μ=0,1,2,3,and the coordinates satisfy
(125)ξ~s=ξs,ξ~τ=ξτ,ξ~0=ξ0,x~i=xi,x~τ=xτ,x~0=x0.That is to say,f~p is just exactly the reference system in conventional sense, which has two different coordinate representations (123) and (124).We speak of(126)f~:M~⟶REFM~,p↦f~p∈REFpas a classical spacetime reference-system. Thus, inertial system can be strictly defined as follows, no need for Newton's first law. Suppose we have a geometric manifold M~,g~. Fg~ is a transformation induced by g~.
(1)
Ifδ~αβB~μαB~νβ=ε~μν, then g~ is said to be (Lorentz) orthogonal. In this case, Fg~ is just exactly a local Lorentz transformation(2)
IfB~μα and C~αμ are constants on M~, then g~ is said to be flat(3)
Ifg~ is both orthogonal and flat, then g~ is said to be an inertial-system. In this case, Fg~ is just exactly a Lorentz transformationRemark 35.
Due to(127)dξ~τ2=dξ02−∑s=13dξs2=δ~αβdξ~αdξ~β=G~μνdx~μdx~ν,G~μν≜δ~αβB~μαB~νβ,dx~τ2=dx02−∑i=13dxi2=ε~μνdx~μdx~ν=H~αβdξ~αdξ~β,H~αβ≜ε~μνC~αμC~βν,it is easy to know that g~ is orthogonal if and only if dξ~τ=dx~τ, i.e., G~ττ≜B~ττB~ττ=1,G~ττ≜C~ττC~ττ=1. It is only in this case that we can denote dξ~τ and dx~τ uniformly by dτ; otherwise, we should be aware of the difference between dξ~τ and dx~τ in nontrivial gravitational field. No matter whether g~ is an inertial system or not, and whether there is a nontrivial gravitation field or not, dξ~τ2=dξ02−∑s=13dξs2 and dx~τ2=dx02−∑i=13dxi2 are always both true in their respective coordinate frames.Remark 36.
The evolution lemmas in Section3.3 can be expressed in Minkowski coordinate as follows:
(i)
Ifd/dt~≅d/dt~L and df~≃df~L, then d/dt~,df~=d/dt~L,df~L(ii)
The following conclusions are true(128)wμ∂∂x~μ≅wτddx~τ⟺wμ=wτε~τμ,w¯μ∂∂x~μ≅w¯τddx~τ⟺w¯μ=w¯τε¯~μτ,wμdx~μ≃wτdx~τ⟺ε~τμwμ=wτ,w¯μdx~μ≃w¯τdx~τ⟺ε¯~μτw¯μ=w¯τ.


### 4.3. Affine Connection Representation of Classical Spacetime Evolution

LetD~ be the holonomic connection on M~,G~, and denote t~L;τ≜t~;σε~τσ; then, the absolute differential and gradient of Section 3.4 can be expressed on M~ in Minkowski coordinate as
(129)D~t~≜t~;σdx~σ,D~Lt~L≜t~L;τdx~τ,∇~t~≜t~;σ∂∂x~σ,∇~Lt~L≜t~L;τddx~τ.Evidently,D~t~≃D~Lt~L if and only if L is an arbitrary path. ∇~t~≅∇~Lt~L if and only if L is the gradient line.Definition 37.
Similar to Section3.6, suppose a charge ρ~ of F~ evolves on M~,G~. We have the following definitions.
(1)
The geometric propertiesm~τ≜ρ~;τ and m~τ≜ρ~;τ are said to be the rest mass of ρ~(2)
p~μ≜−ρ~;μ and p~μ≜−ρ~;μ are said to be the energy-momentum of ρ~, and E~0≜ρ~;0 and E~0≜ρ~;0 are said to be the energy of ρ~(3)
M~τ≜dρ~/dx~τ and M~τ≜dρ~/dx~τ are said to be the canonical rest mass of ρ~(4)
P~μ≜−∂ρ~/∂x~μ and P~μ≜−∂ρ~/∂x~μ are said to be the canonical energy-momentum of ρ~, and H~0≜∂ρ~/∂x~0, H~0≜∂ρ~/∂x~0 are said to be the canonical energy of ρ~Discussion 38.
Similar to Proposition13, ∀p∈M~, if and only if the evolution direction Lp=∇~ρ~p, the directional derivative is
(130)m~τddx~τ,m~τdx~τ=p~μ∂∂x~μ,p~μdx~μ,that is, G~ττm~τm~τ=G~μνp~μp~ν, or
(131)m~τm~τ=p~μp~μ,which is the affine connection representation of energy-momentum equation.Similar to Proposition14, according to the evolution lemma, ∀p∈M~, if and only if the evolution direction Lp=∇~ρ~p, we have p~μ=−m~τdx~μ/dx~τ, that is E~0=m~τdx~0/dx~τ=m~τdx0/dxτ and p~i=−m~τdx~i/dx~τ=m~τ−dx~i/dx~τ=m~τdxi/dxτ=E~0dxi/dx0. This can also be regarded as the origin of p=mv.Similar to Remark18, denote
(132)ρ~Γ~ω≜∂ρ~μν∂x~ω−ρ~μν;ω=ρ~μχΓ~νωχ+ρ~χνΓ~μωχ,ρ~R~ρσ≜ρ~μχR~νρσχ+ρ~χνR~μρσχ.Then, for the same reason as Remark18, based on Definition 37, we can strictly obtain
(133)f~ρ≜p~ρ;τ=m~τ;ρ−p~σε~τ;ρσ+ρ~R~ρσε~τσ.In the mass-point model,m~τ;ρ and ε~τ;ρσ do not make sense, so Equation (133) turns into
(134)f~ρ=ρ~R~ρσε~τσ.This is the affine connection representation of the force of interaction (e.g., the Lorentz forcef=qE+v×B or fρ=jσFρσ of the electrodynamics).Similar to Definition16, let P~b,a be the totality of paths on M~ from point a to point b. And let L∈P~b,a and parameter x~τ satisfy τa≜x~τa<x~τb≜τb. The affine connection representation of action in Minkowski coordinates can be defined as
(135)s~L≜∫LD~ρ~=∫Lp~μdx~μ=∫τaτbm~τdx~τ,s~L≜∫τaτbγμρ~;μ+m~τdx~τ.There are more illustrations in Remark41.


### 4.4. Affine Connection Representation of Dirac Equation

Discussion 39.
Define Dirac algebrasγμ and γα such that
(136)γμ=C~αμγα,γαγβ+γβγα=2δ~αβ,γμγν+γνγμ=2G~μν.SupposeM~,G~ is orthogonal. According to Remark 35, G~ττ=1. Due to Discussion 38, in a gradient direction of ρ~≜ρ~ων, we have
(137)ρ~;μρ~;μ=ρ~;τρ~;τ⟺G~μνρ~;μρ~;ν=m~τ2⟺γμρ~;μγνρ~;ν+γνρ~;νγμρ~;μ=2m~τ2⟺γμρ~;μγνρ~;ν=m~τ2⟺γμρ~;μ2=m~τ2.Without loss of generality, takeγμρ~;μ=m~τ, that is,
(138)γμρ~ων;μ=m~ωντ.Next, denote(139)gΓ~μων≜∑σG~νν′Γ~ν′σμ+∑κG~ωω′Γ~ω′κμ,D~μων≜∂μ−gΓ~μων.From Equation (138), it is obtained that
(140)∑ω,νγμρ~ων;μ=∑ω,νm~ωντ⟺∑ω,νγμ∂μρων−ρ~ωχΓ~νμχ−ρ~χνΓ~ωμχ=∑ω,νm~ωντ⟺∑ω,νγμ∂μρων−ρ~ων∑σΓ~σμν−ρ~ων∑κΓ~κμω=∑ω,νm~ωντ⟺∑ω,νγμ∂μρων−ρ~ωνgΓ~μων=∑ω,νm~ωντ⟺∑ω,νγμ∂μ−gΓ~μωνρ~ων=∑ω,νm~ωντ,that is,
(141)∑ω,νγμD~μωνρ~ων=∑ω,νm~ωντ,D~μων≜∂μ−gΓ~μων.We speak of the real-valued Equations (138) and (141) as affine Dirac equations.Discussion 40.
Next, we construct a kind of complex-valued representation of affine Dirac equation. The restriction of the chargeρ~ων to U~,x~μ is a function ρ~ωνx~μ with respect to the coordinates x~μ≜x~0,x~1,x~2,x~3. Let
(142)P~ωνx~0≜∫x~1,x~2,x~3ρ~ωνx~μd3x~.Suppose a functionfων=fωνx~μ on U~,x~μ satisfies that
(143)ρ~ων=fων2P~ων,∫x~1,x~2,x~3fων2d3x~=1,ξ~τμ∂fων∂x~μ=0,γμ∂fων∂x~μ=0.We defineψων and M~ωντ in the following way.
(144)y~ων≜∫Ldρ~ων=∫Ldρ~ωνdx~τdx~τ=∫Ldfων2dx~τP~ων+fων2dP~ωνdx~τdx~τ=fων2∫LdP~ωνdx~τdx~τ≜fων2Y~ων,ψων≜fωνeiY~ων,m~ωντ≜ρ~ων;τ=fων2,τP~ων+fων2P~ων;τ=fων2P~ων;τ≜fων2M~ωντ.In the QFT propagator, we usually takeS in the path integral ∫eiSDψ of a fermion in the form of
(145)−∫iψ¯γμDμψ−ψ¯M~τψd4x~,where S and d4x~ are both covariant. We believe that the external spatial integral ∫x~1,x~2,x~3d3x~ is not an essential part for evolution, so for the sake of simplicity, we do not take into account the external spatial part ∫x~1,x~2,x~3d3x~ but only consider the evolution part ∫Ldx~0. Meanwhile, in order to remain the covariance, ∫Ldx~0 has to be replaced by ∫Ldx~τ. Thus, in affine connection representation of gauge fields, we shall consider an action in the form of
(146)−∫Liψ¯γμDμψ−ψ¯M~τψdx~τ.Concretely speaking, denote(147)D~ωνμ≜∂∂x~μ−iP~Γ~μων,P~Γ~μων≜∑σP~ωνΓ~σμν+∑κP~ωνΓ~κμω.From Equation (135), we have
(148)s~ωνL≜∫Lγμρ~ων;μ+m~ωντdx~τ.And from Equation (140), we know ∑ω,νγμρ~ων;μ=∑ω,νγμD~μωνρ~ων. Then, it is obtained that
(149)s~ρ~L≜∑ω,νs~ωνL=∫L∑ω,νγμρ~ων;μ+m~ωντdx~τ=∫L∑ω,νγμD~μωνρ~ων+m~ωντdx~τ=∫L∑ω,νγμ∂μρ~ων−gΓ~μωνρ~ων+m~ωντdx~τ=∫L∑ω,νγμ∂μy~ων−gΓ~μωνρ~ων+m~ωντdx~τ=∫L∑ω,νγμ∂μfων2Y~ων−gΓ~μωνfων2P~ων+fων2M~ωντdx~τ=∫L∑ω,νγμ∂μY~ων−P~Γ~μωνfων2+fων2M~ωντdx~τ=∫L∑ω,νfωνe−iY~ωνγμfωνeiY~ων∂μY~ων−P~Γ~μωνfωνeiY~ων+fωνe−iY~ωνM~ωντfωνeiY~ωνdx~τ=∫L∑ω,ν−ψ¯ωνiγμeiY~ων∂μfων+fωνeiY~ωνi∂μY~ων−iP~Γ~μωνψων+ψ¯ωνM~ωντψωνdx~τ=∫L∑ω,ν−ψ¯ωνiγμ∂μfωνeiY~ων−iP~Γ~μωνψων+ψ¯ωνM~ωντψωνdx~τ=∫L∑ω,ν−ψ¯ωνiγμ∂μ−iP~Γ~μωνψων+ψ¯ωνM~ωντψωνdx~τ=∫L∑ω,ν−ψ¯ωνiγμD~ωνμψων+ψ¯ωνM~ωντψωνdx~τ=−∫L∑ω,νψ¯ωνiγμD~ωνμ−M~ωντψωνdx~τ.Thus, we have obtained a complex-valued representation of gradient direction ofρ~ων.Remark 41.
From the above discussion, we know in the gradient direction ofρων that
(150)−∑ω,νψ¯ωνiγμD~ωνμψωνdx~τ=∑ω,νD~ρ~ων.This shows thatsL and s~L in Definition 16 and Remark 17 are indeed applicable for constructing propagator by eisL and eis~L in affine connection representation of gauge fields. Therefore, the idea in Discussion 34 is reasonable.


### 4.5. From Classical Spacetime back to Full-Dimensional Space

Discussion 42.
Now there is a problem.M~,F~ and M~,G~ cannot totally reflect the geometric properties of internal space of M,F and M,G. Concretely speaking, in the previous section, we discuss the affine Dirac equation γμρ~ων;μ=m~ωντ on M~,G~. Similar to Section 3.5, we have the affine Yang-Mills equation K~νρσμ:ρ=ρ~νμγσ on M~,F~. Suppose there is no gravitational field, then the remaining nonvanishing equations are just only
(151)γμρ~00;μ=m~00τ,K~0ρσ0:ρ=ρ~00γσ.
There are multiple internal charges(152)ρmnm,n=4,5,⋯,D,on M,F. We intend to use these ρmn to describe leptons and hadrons. However, via encapsulation of classical spacetime, M~,F~ remains only one internal charge ρ~00, and it falls short. It is impossible for the only one real-valued field function ρ~00 to describe so many leptons and hadrons.On the premise of not abandoning the1+3-dimensional spacetime, if we want to describe gauge fields, there is a method that to use some noncoordinate abstract degrees of freedom on the phase of eiTaθa of a complex-valued field function ψ. This way is effective, but not natural. It is not satisfactory for a theory to adopt a coordinate representation for external space but a noncoordinate representation for internal space.A logically more natural way is required to abandon the framework of1+3-dimensional spacetime M~,F~ and M~,G~. We should put internal space and external space together to describe their unified geometry with the same spatial frame. On M,F and M,G, there are enough real-valued field functions ρmn to describe leptons and hadrons and enough internal components mnP of affine connection to describe gauge potentials.Therefore, only on the full-dimensionalM,F and M,G can total advantages of affine connection representation of gauge fields be brought into full play and thereby show complete details of geometric properties of gauge field. So we are going to stop the discussions about the classical spacetime M~, but to focus on the full-dimensional manifold M.Discussion 43.
OnM, due to ΓMNP=1/2MNP+MNP, MNP=δADBMD∂BNA/∂xP+BPABNB, and GMN=δABBMABNB, we know that gauge field and gravitational field can both be described by spatial frames BMA and CAM in a reference-system. Reference-system is the common origination of gauge field and gravitational field. The invariance under reference-system transformation is the common origination of gauge covariance and general covariance.We adopt the componentsmnP of MNP with m,n∈4,5,⋯,D to describe the gauge potentials of typical gauge fields such as electromagnetic, weak, and strong interaction fields and adopt the components ρmn of ρMN with m,n∈4,5,⋯,D to describe the charges of leptons and hadrons. The physical meanings of the other components of ρMN and MNP are not clear at present; maybe they could be used to describe dark matters and their interactions.On orthogonalM,G and M,F, there are full-dimensional field equations, i.e., affine Dirac equation and affine Yang-Mills equation
(153)γPρMN;P=ρMN;0,KNPQM:P=ρNMγQ,which reflect the on-shell evolution directions ∇ρ and ∇t, respectively. Their quantum evolutions are described by the propagators in Definition 32 or Discussion 34.Discussion 44.
On an orthogonalM,G, Equation (149) presents as a full-dimensional action
(154)sρL=∫L∑M,NγPρMN;P+ε0PρMN;Pdx0=−i∫L∑M,Nψ¯MNγPDMNP+ε0PDMNPψMNdx0.
If and only ifLk:g⟶g′ is an orthogonal transformation, Lk sends sρL to
(155)sρ′L=∫L∑M,NγP′ρMN;P′+ε0′P′ρMN;P′dx0′=−i∫L∑M,Nψ′¯MNγP′DMNP′′+ε0′P′DMNP′′ψMN′dx0′,where ρMN is determined by the reference-system f∘f but not g∘g, so ρMN does not vary with the transformation Lk:g⟶g′. We see that in affine connection representation of gauge fields, the gauge transformations ψ↦ψ′ and D↦D′ essentially boil down to the reference-system transformation Lk.Remark 45.
For a generalM,G, G is not necessarily orthogonal, so the corresponding action should be described by
(156)sMNL=∫LB00γPρMN;P+ε0PρMN;Pdx0.In this general case, Definition16 and the method in Discussion 34 are also available and effective, where we take
(157)sMNL=∫LDρMN.Remark 46.
We see that the real-valued representation of action is more concise than the complex-valued representation of action. Hence, it is more convenient to adopt real-valued representations for field function, field equation, and action.In the following sections, we are going to useMNP to show the affine connection representations of electromagnetic, weak, and strong interaction fields and to adopt the real-valued representation ρMN;P to discuss the interactions between gauge fields and elementary particles. They are based on the following definition.Definition 47.
LetM=P×N, r≜dimP=3 and D≜dimM=5or6or8. Consider F=f∘f and G=g∘g that are defined by Equation (33), that is, ∀p∈M,
(158)U,αA′⟶fpU,ξA⟶fpU,xM⟵gpU,ζA⟵gpU,βA′,and furthermore, let
(159)fp:ξa=ξaxm,ξs=δisxi,fp:αa′=αa′ξa,αs′=δss′ξs,gp:ζa=ζaxm,ζs=δisxi,gp:βa′=βa′ζa,βs′=δss′ζs,
(s′,s,i=1,2,3;a′,a,m,n=4,5,⋯,D) and both of F and G satisfy
(160)iGmn=const,iiwhenm≠n,Gmn=0.In the above extremely simplified case, we useF and G to show electromagnetic, weak, and strong interactions without gravitation.


## 4.1. Classical Spacetime Submanifold

Let there be a smooth tangent vector fieldX on M,f. If ∀p∈M, Xp=bA∂/∂ξAp=cM∂/∂xMp satisfies that ba are not all zero and cm are not all zero, where a,m=r+1,⋯,D; then, we say X is internal-directed. For any evolution path L≜φX,p, we also say L is internal-directed.SupposeM=P×N, D≜dimM, and r≜dimP=3. X is a smooth tangent vector field on M. Fix a point o∈M. If X is internal-directed; then, there exist a unique 1+3-dimensional imbedding submanifold γ:M~⟶M,p↦p and a unique smooth tangent vector field X~ on M~ such that
(i)
P×o is a closed submanifold of M~(ii)
The tangent mapγ∗:TM~⟶TM satisfies that ∀q∈M~, γ∗:X~q↦XqSuch anM~ is said to be a classical spacetime submanifold.LetφX:M×ℝ⟶M and φX~:M~×ℝ⟶M~ be the one-parameter groups of diffeomorphisms corresponding to X and X~, respectively. Thus, we have
(121)φX~=φXM~×ℝ.So the evolution in classical spacetime can be described byφX~. It should be noticed that
(i)
M~ inherits a part of geometric properties of M, but not all. The physical properties reflected by M~ are incomplete(ii)
The correspondence betweenX~ and the restriction of X to M~ is one-to-one. For convenience, next we are not going to distinguish the notations X and X~ on M~ but uniformly denote them by X(iii)
An arbitrary pathL~:T⟶M~,t↦p on M~ uniquely corresponds to a path L≜γ∘L~:T⟶M,t↦p on M. Evidently, the image sets of L and L~ are the same, that is, LT=L~T. For convenience, later we are not going to distinguish the notations L and L~ on M~ but uniformly denote them by L


## 4.2. Classical Spacetime Reference-System

Let there be a geometric manifoldM,f and its classical spacetime submanifold M~. And let L≜φX~,a be an evolution path on M~. Suppose p∈L and U is a coordinate neighborhood of p. According to Definition 5, suppose the fp on U and the fLp on UL≜U∩L satisfy that
(122)fp:ξA=ξAxM=ξAx0,ξ0=ξ0x0,A,M=1,2,⋯,D.Thus, it is true that(1)
There exists a unique local reference-systemf~p on U~≜U∩M~ such that(123)f~p:ξU=ξUxK=ξUx0,ξ0=ξ0x0,U,K=1,2,3,τ.(2)
IfL is internal-directed, then the above coordinate frames U~,ξU and U~,xK of f~p uniquely determine the coordinate frames U~,ξ~α and U~,x~μ such that(124)f~p:ξ~α=ξ~αx~μ=ξ~αx~τ,ξ~τ=ξ~τx~τ,α,μ=0,1,2,3,and the coordinates satisfy
(125)ξ~s=ξs,ξ~τ=ξτ,ξ~0=ξ0,x~i=xi,x~τ=xτ,x~0=x0.That is to say,f~p is just exactly the reference system in conventional sense, which has two different coordinate representations (123) and (124).We speak of(126)f~:M~⟶REFM~,p↦f~p∈REFpas a classical spacetime reference-system. Thus, inertial system can be strictly defined as follows, no need for Newton's first law. Suppose we have a geometric manifold M~,g~. Fg~ is a transformation induced by g~.
(1)
Ifδ~αβB~μαB~νβ=ε~μν, then g~ is said to be (Lorentz) orthogonal. In this case, Fg~ is just exactly a local Lorentz transformation(2)
IfB~μα and C~αμ are constants on M~, then g~ is said to be flat(3)
Ifg~ is both orthogonal and flat, then g~ is said to be an inertial-system. In this case, Fg~ is just exactly a Lorentz transformationRemark 35.
Due to(127)dξ~τ2=dξ02−∑s=13dξs2=δ~αβdξ~αdξ~β=G~μνdx~μdx~ν,G~μν≜δ~αβB~μαB~νβ,dx~τ2=dx02−∑i=13dxi2=ε~μνdx~μdx~ν=H~αβdξ~αdξ~β,H~αβ≜ε~μνC~αμC~βν,it is easy to know that g~ is orthogonal if and only if dξ~τ=dx~τ, i.e., G~ττ≜B~ττB~ττ=1,G~ττ≜C~ττC~ττ=1. It is only in this case that we can denote dξ~τ and dx~τ uniformly by dτ; otherwise, we should be aware of the difference between dξ~τ and dx~τ in nontrivial gravitational field. No matter whether g~ is an inertial system or not, and whether there is a nontrivial gravitation field or not, dξ~τ2=dξ02−∑s=13dξs2 and dx~τ2=dx02−∑i=13dxi2 are always both true in their respective coordinate frames.Remark 36.
The evolution lemmas in Section3.3 can be expressed in Minkowski coordinate as follows:
(i)
Ifd/dt~≅d/dt~L and df~≃df~L, then d/dt~,df~=d/dt~L,df~L(ii)
The following conclusions are true(128)wμ∂∂x~μ≅wτddx~τ⟺wμ=wτε~τμ,w¯μ∂∂x~μ≅w¯τddx~τ⟺w¯μ=w¯τε¯~μτ,wμdx~μ≃wτdx~τ⟺ε~τμwμ=wτ,w¯μdx~μ≃w¯τdx~τ⟺ε¯~μτw¯μ=w¯τ.


## 4.3. Affine Connection Representation of Classical Spacetime Evolution

LetD~ be the holonomic connection on M~,G~, and denote t~L;τ≜t~;σε~τσ; then, the absolute differential and gradient of Section 3.4 can be expressed on M~ in Minkowski coordinate as
(129)D~t~≜t~;σdx~σ,D~Lt~L≜t~L;τdx~τ,∇~t~≜t~;σ∂∂x~σ,∇~Lt~L≜t~L;τddx~τ.Evidently,D~t~≃D~Lt~L if and only if L is an arbitrary path. ∇~t~≅∇~Lt~L if and only if L is the gradient line.Definition 37.
Similar to Section3.6, suppose a charge ρ~ of F~ evolves on M~,G~. We have the following definitions.
(1)
The geometric propertiesm~τ≜ρ~;τ and m~τ≜ρ~;τ are said to be the rest mass of ρ~(2)
p~μ≜−ρ~;μ and p~μ≜−ρ~;μ are said to be the energy-momentum of ρ~, and E~0≜ρ~;0 and E~0≜ρ~;0 are said to be the energy of ρ~(3)
M~τ≜dρ~/dx~τ and M~τ≜dρ~/dx~τ are said to be the canonical rest mass of ρ~(4)
P~μ≜−∂ρ~/∂x~μ and P~μ≜−∂ρ~/∂x~μ are said to be the canonical energy-momentum of ρ~, and H~0≜∂ρ~/∂x~0, H~0≜∂ρ~/∂x~0 are said to be the canonical energy of ρ~Discussion 38.
Similar to Proposition13, ∀p∈M~, if and only if the evolution direction Lp=∇~ρ~p, the directional derivative is
(130)m~τddx~τ,m~τdx~τ=p~μ∂∂x~μ,p~μdx~μ,that is, G~ττm~τm~τ=G~μνp~μp~ν, or
(131)m~τm~τ=p~μp~μ,which is the affine connection representation of energy-momentum equation.Similar to Proposition14, according to the evolution lemma, ∀p∈M~, if and only if the evolution direction Lp=∇~ρ~p, we have p~μ=−m~τdx~μ/dx~τ, that is E~0=m~τdx~0/dx~τ=m~τdx0/dxτ and p~i=−m~τdx~i/dx~τ=m~τ−dx~i/dx~τ=m~τdxi/dxτ=E~0dxi/dx0. This can also be regarded as the origin of p=mv.Similar to Remark18, denote
(132)ρ~Γ~ω≜∂ρ~μν∂x~ω−ρ~μν;ω=ρ~μχΓ~νωχ+ρ~χνΓ~μωχ,ρ~R~ρσ≜ρ~μχR~νρσχ+ρ~χνR~μρσχ.Then, for the same reason as Remark18, based on Definition 37, we can strictly obtain
(133)f~ρ≜p~ρ;τ=m~τ;ρ−p~σε~τ;ρσ+ρ~R~ρσε~τσ.In the mass-point model,m~τ;ρ and ε~τ;ρσ do not make sense, so Equation (133) turns into
(134)f~ρ=ρ~R~ρσε~τσ.This is the affine connection representation of the force of interaction (e.g., the Lorentz forcef=qE+v×B or fρ=jσFρσ of the electrodynamics).Similar to Definition16, let P~b,a be the totality of paths on M~ from point a to point b. And let L∈P~b,a and parameter x~τ satisfy τa≜x~τa<x~τb≜τb. The affine connection representation of action in Minkowski coordinates can be defined as
(135)s~L≜∫LD~ρ~=∫Lp~μdx~μ=∫τaτbm~τdx~τ,s~L≜∫τaτbγμρ~;μ+m~τdx~τ.There are more illustrations in Remark41.


## 4.4. Affine Connection Representation of Dirac Equation

Discussion 39.
Define Dirac algebrasγμ and γα such that
(136)γμ=C~αμγα,γαγβ+γβγα=2δ~αβ,γμγν+γνγμ=2G~μν.SupposeM~,G~ is orthogonal. According to Remark 35, G~ττ=1. Due to Discussion 38, in a gradient direction of ρ~≜ρ~ων, we have
(137)ρ~;μρ~;μ=ρ~;τρ~;τ⟺G~μνρ~;μρ~;ν=m~τ2⟺γμρ~;μγνρ~;ν+γνρ~;νγμρ~;μ=2m~τ2⟺γμρ~;μγνρ~;ν=m~τ2⟺γμρ~;μ2=m~τ2.Without loss of generality, takeγμρ~;μ=m~τ, that is,
(138)γμρ~ων;μ=m~ωντ.Next, denote(139)gΓ~μων≜∑σG~νν′Γ~ν′σμ+∑κG~ωω′Γ~ω′κμ,D~μων≜∂μ−gΓ~μων.From Equation (138), it is obtained that
(140)∑ω,νγμρ~ων;μ=∑ω,νm~ωντ⟺∑ω,νγμ∂μρων−ρ~ωχΓ~νμχ−ρ~χνΓ~ωμχ=∑ω,νm~ωντ⟺∑ω,νγμ∂μρων−ρ~ων∑σΓ~σμν−ρ~ων∑κΓ~κμω=∑ω,νm~ωντ⟺∑ω,νγμ∂μρων−ρ~ωνgΓ~μων=∑ω,νm~ωντ⟺∑ω,νγμ∂μ−gΓ~μωνρ~ων=∑ω,νm~ωντ,that is,
(141)∑ω,νγμD~μωνρ~ων=∑ω,νm~ωντ,D~μων≜∂μ−gΓ~μων.We speak of the real-valued Equations (138) and (141) as affine Dirac equations.Discussion 40.
Next, we construct a kind of complex-valued representation of affine Dirac equation. The restriction of the chargeρ~ων to U~,x~μ is a function ρ~ωνx~μ with respect to the coordinates x~μ≜x~0,x~1,x~2,x~3. Let
(142)P~ωνx~0≜∫x~1,x~2,x~3ρ~ωνx~μd3x~.Suppose a functionfων=fωνx~μ on U~,x~μ satisfies that
(143)ρ~ων=fων2P~ων,∫x~1,x~2,x~3fων2d3x~=1,ξ~τμ∂fων∂x~μ=0,γμ∂fων∂x~μ=0.We defineψων and M~ωντ in the following way.
(144)y~ων≜∫Ldρ~ων=∫Ldρ~ωνdx~τdx~τ=∫Ldfων2dx~τP~ων+fων2dP~ωνdx~τdx~τ=fων2∫LdP~ωνdx~τdx~τ≜fων2Y~ων,ψων≜fωνeiY~ων,m~ωντ≜ρ~ων;τ=fων2,τP~ων+fων2P~ων;τ=fων2P~ων;τ≜fων2M~ωντ.In the QFT propagator, we usually takeS in the path integral ∫eiSDψ of a fermion in the form of
(145)−∫iψ¯γμDμψ−ψ¯M~τψd4x~,where S and d4x~ are both covariant. We believe that the external spatial integral ∫x~1,x~2,x~3d3x~ is not an essential part for evolution, so for the sake of simplicity, we do not take into account the external spatial part ∫x~1,x~2,x~3d3x~ but only consider the evolution part ∫Ldx~0. Meanwhile, in order to remain the covariance, ∫Ldx~0 has to be replaced by ∫Ldx~τ. Thus, in affine connection representation of gauge fields, we shall consider an action in the form of
(146)−∫Liψ¯γμDμψ−ψ¯M~τψdx~τ.Concretely speaking, denote(147)D~ωνμ≜∂∂x~μ−iP~Γ~μων,P~Γ~μων≜∑σP~ωνΓ~σμν+∑κP~ωνΓ~κμω.From Equation (135), we have
(148)s~ωνL≜∫Lγμρ~ων;μ+m~ωντdx~τ.And from Equation (140), we know ∑ω,νγμρ~ων;μ=∑ω,νγμD~μωνρ~ων. Then, it is obtained that
(149)s~ρ~L≜∑ω,νs~ωνL=∫L∑ω,νγμρ~ων;μ+m~ωντdx~τ=∫L∑ω,νγμD~μωνρ~ων+m~ωντdx~τ=∫L∑ω,νγμ∂μρ~ων−gΓ~μωνρ~ων+m~ωντdx~τ=∫L∑ω,νγμ∂μy~ων−gΓ~μωνρ~ων+m~ωντdx~τ=∫L∑ω,νγμ∂μfων2Y~ων−gΓ~μωνfων2P~ων+fων2M~ωντdx~τ=∫L∑ω,νγμ∂μY~ων−P~Γ~μωνfων2+fων2M~ωντdx~τ=∫L∑ω,νfωνe−iY~ωνγμfωνeiY~ων∂μY~ων−P~Γ~μωνfωνeiY~ων+fωνe−iY~ωνM~ωντfωνeiY~ωνdx~τ=∫L∑ω,ν−ψ¯ωνiγμeiY~ων∂μfων+fωνeiY~ωνi∂μY~ων−iP~Γ~μωνψων+ψ¯ωνM~ωντψωνdx~τ=∫L∑ω,ν−ψ¯ωνiγμ∂μfωνeiY~ων−iP~Γ~μωνψων+ψ¯ωνM~ωντψωνdx~τ=∫L∑ω,ν−ψ¯ωνiγμ∂μ−iP~Γ~μωνψων+ψ¯ωνM~ωντψωνdx~τ=∫L∑ω,ν−ψ¯ωνiγμD~ωνμψων+ψ¯ωνM~ωντψωνdx~τ=−∫L∑ω,νψ¯ωνiγμD~ωνμ−M~ωντψωνdx~τ.Thus, we have obtained a complex-valued representation of gradient direction ofρ~ων.Remark 41.
From the above discussion, we know in the gradient direction ofρων that
(150)−∑ω,νψ¯ωνiγμD~ωνμψωνdx~τ=∑ω,νD~ρ~ων.This shows thatsL and s~L in Definition 16 and Remark 17 are indeed applicable for constructing propagator by eisL and eis~L in affine connection representation of gauge fields. Therefore, the idea in Discussion 34 is reasonable.


## 4.5. From Classical Spacetime back to Full-Dimensional Space

Discussion 42.
Now there is a problem.M~,F~ and M~,G~ cannot totally reflect the geometric properties of internal space of M,F and M,G. Concretely speaking, in the previous section, we discuss the affine Dirac equation γμρ~ων;μ=m~ωντ on M~,G~. Similar to Section 3.5, we have the affine Yang-Mills equation K~νρσμ:ρ=ρ~νμγσ on M~,F~. Suppose there is no gravitational field, then the remaining nonvanishing equations are just only
(151)γμρ~00;μ=m~00τ,K~0ρσ0:ρ=ρ~00γσ.
There are multiple internal charges(152)ρmnm,n=4,5,⋯,D,on M,F. We intend to use these ρmn to describe leptons and hadrons. However, via encapsulation of classical spacetime, M~,F~ remains only one internal charge ρ~00, and it falls short. It is impossible for the only one real-valued field function ρ~00 to describe so many leptons and hadrons.On the premise of not abandoning the1+3-dimensional spacetime, if we want to describe gauge fields, there is a method that to use some noncoordinate abstract degrees of freedom on the phase of eiTaθa of a complex-valued field function ψ. This way is effective, but not natural. It is not satisfactory for a theory to adopt a coordinate representation for external space but a noncoordinate representation for internal space.A logically more natural way is required to abandon the framework of1+3-dimensional spacetime M~,F~ and M~,G~. We should put internal space and external space together to describe their unified geometry with the same spatial frame. On M,F and M,G, there are enough real-valued field functions ρmn to describe leptons and hadrons and enough internal components mnP of affine connection to describe gauge potentials.Therefore, only on the full-dimensionalM,F and M,G can total advantages of affine connection representation of gauge fields be brought into full play and thereby show complete details of geometric properties of gauge field. So we are going to stop the discussions about the classical spacetime M~, but to focus on the full-dimensional manifold M.Discussion 43.
OnM, due to ΓMNP=1/2MNP+MNP, MNP=δADBMD∂BNA/∂xP+BPABNB, and GMN=δABBMABNB, we know that gauge field and gravitational field can both be described by spatial frames BMA and CAM in a reference-system. Reference-system is the common origination of gauge field and gravitational field. The invariance under reference-system transformation is the common origination of gauge covariance and general covariance.We adopt the componentsmnP of MNP with m,n∈4,5,⋯,D to describe the gauge potentials of typical gauge fields such as electromagnetic, weak, and strong interaction fields and adopt the components ρmn of ρMN with m,n∈4,5,⋯,D to describe the charges of leptons and hadrons. The physical meanings of the other components of ρMN and MNP are not clear at present; maybe they could be used to describe dark matters and their interactions.On orthogonalM,G and M,F, there are full-dimensional field equations, i.e., affine Dirac equation and affine Yang-Mills equation
(153)γPρMN;P=ρMN;0,KNPQM:P=ρNMγQ,which reflect the on-shell evolution directions ∇ρ and ∇t, respectively. Their quantum evolutions are described by the propagators in Definition 32 or Discussion 34.Discussion 44.
On an orthogonalM,G, Equation (149) presents as a full-dimensional action
(154)sρL=∫L∑M,NγPρMN;P+ε0PρMN;Pdx0=−i∫L∑M,Nψ¯MNγPDMNP+ε0PDMNPψMNdx0.
If and only ifLk:g⟶g′ is an orthogonal transformation, Lk sends sρL to
(155)sρ′L=∫L∑M,NγP′ρMN;P′+ε0′P′ρMN;P′dx0′=−i∫L∑M,Nψ′¯MNγP′DMNP′′+ε0′P′DMNP′′ψMN′dx0′,where ρMN is determined by the reference-system f∘f but not g∘g, so ρMN does not vary with the transformation Lk:g⟶g′. We see that in affine connection representation of gauge fields, the gauge transformations ψ↦ψ′ and D↦D′ essentially boil down to the reference-system transformation Lk.Remark 45.
For a generalM,G, G is not necessarily orthogonal, so the corresponding action should be described by
(156)sMNL=∫LB00γPρMN;P+ε0PρMN;Pdx0.In this general case, Definition16 and the method in Discussion 34 are also available and effective, where we take
(157)sMNL=∫LDρMN.Remark 46.
We see that the real-valued representation of action is more concise than the complex-valued representation of action. Hence, it is more convenient to adopt real-valued representations for field function, field equation, and action.In the following sections, we are going to useMNP to show the affine connection representations of electromagnetic, weak, and strong interaction fields and to adopt the real-valued representation ρMN;P to discuss the interactions between gauge fields and elementary particles. They are based on the following definition.Definition 47.
LetM=P×N, r≜dimP=3 and D≜dimM=5or6or8. Consider F=f∘f and G=g∘g that are defined by Equation (33), that is, ∀p∈M,
(158)U,αA′⟶fpU,ξA⟶fpU,xM⟵gpU,ζA⟵gpU,βA′,and furthermore, let
(159)fp:ξa=ξaxm,ξs=δisxi,fp:αa′=αa′ξa,αs′=δss′ξs,gp:ζa=ζaxm,ζs=δisxi,gp:βa′=βa′ζa,βs′=δss′ζs,
(s′,s,i=1,2,3;a′,a,m,n=4,5,⋯,D) and both of F and G satisfy
(160)iGmn=const,iiwhenm≠n,Gmn=0.In the above extremely simplified case, we useF and G to show electromagnetic, weak, and strong interactions without gravitation.


## 5. Affine Connection Representation of the Gauge Field of Weak-Electromagnetic Interaction

Definition 48.
SupposeM,F and M,G conform to Definition 47. Let D=r+2=5 and both of F and G satisfy
(161)GD−1D−1=GDD.Thus,F and G can describe weak and electromagnetic interactions.Proposition 49.
Let the holonomic connection ofM,F be ΓNPM and ΓMNP. And let the coefficients of curvature tensor of M,F be KNPQM and KMNPQ. Denote
(162)BP≜12ΓDDP+ΓD−1D−1P,AP3≜12ΓDDP−ΓD−1D−1P,BPQ≜12KDDPQ+KD−1D−1PQ,FPQ3≜12KDDPQ−KD−1D−1PQ,AP1≜12ΓD−1DP+ΓDD−1P,AP2≜12ΓD−1DP−ΓDD−1P,FPQ1≜12KD−1DPQ+KDD−1PQ,FPQ2≜12KD−1DPQ−KDD−1PQ.
And denoteg≜GD−1D−12+GDD2. Thus, the following equations hold spontaneously.
(163)BPQ=∂BQ∂xP−∂BP∂xQ,FPQ3=∂AQ3∂xP−∂AP3∂xQ+gAP1AQ2−AP2AQ1,FPQ1=∂AQ1∂xP−∂AP1∂xQ+gAP2AQ3−AP3AQ2,FPQ2=∂AQ2∂xP−∂AP2∂xQ+gAP1AQ3−AP3AQ1.Proof.
Due to Equation (159), it is obtained that the semimetric of M,f satisfies
(164)Bfas′=0,Cfs′a=0,Bfsa′=0,Cfa′s=0,Bfss′=δss′,Cfs′s=δs′s.
Then, computeBCAf≜1/2CfA′A∂BfBA′/∂ξC+∂BfCA′/∂ξB, and we obtain
(165)BCsf=0,tuaf=0,bCaf≠0;s,t,u=1,2,3;a,b=4,5,⋯,D;A,B,C=1,2,⋯,D.
It is obtained from Equation (159) again that the semimetric of M,f satisfies
(166)Bms=0,Csm=0,Bia=0,Cai=0,Bis=δis,Csi=δsi.
Lets′,t′,i,j,k=1,2,3;a′,b′,m,n,p=4,5,⋯,D. Compute the metric of M,F, and we obtain
(167)Gij=δs′t′Bis′Bjt′+δa′b′Bia′Bjb′=δs′t′δis′δjt′=δij,Gin=δs′t′Bis′Bnt′+δa′b′Bia′Bnb′=0,Gmj=δs′t′Bms′Bjt′+δa′b′Bma′Bjb′=0,Gmn=BmD−1BnD−1+BmDBnD=const,Gij=δs′t′Cs′iCt′j=δs′t′δs′iδt′j=δij,Gin=δs′t′Cs′iCt′n=0,Gmj=δs′t′Cs′mCt′j=0,Gmn=CD−1mCD−1n+CDmCDn=const.
Compute the holonomic connection ofF according to ΓNPM≜1/2NPM+NPM=1/2CAM∂BNA/∂xP+CAMBPAfBNB, and it is obtained that
(168)ΓNPi=0,Γjkm=0,ΓnPm=12Cam∂Bna∂xP+CambPafBnb,ΓNpm=12Cam∂BNa∂xp+CamBpafBNB,ΓiNP=GiM′ΓNPM′=Gii′ΓNPi′=0,Γmjk=GmM′ΓjkM′=Gmm′Γjkm′=0,ΓmnP=12δabBmb∂Bna∂xP+bPafBnb,ΓmNp=12δabBmb∂BNa∂xp+BpafBNB.
Compute the coefficients of curvature ofF, that is,
(169)KnPQm≜∂ΓnQm∂xP−∂ΓnPm∂xQ+ΓHPmΓnQH−ΓnPHΓHQm,KmnPQ≜GmM′KnPQM′=Gmm′KnPQm′,and then, we obtain
(170)KD−1D−1PQ=∂ΓD−1D−1Q∂xP−∂ΓD−1D−1P∂xQ+GDDΓD−1DPΓDD−1Q−ΓDD−1PΓD−1DQ,KDD−1PQ=∂ΓDD−1Q∂xP−∂ΓDD−1P∂xQ+GDDΓDDPΓDD−1Q−ΓDD−1PΓDDQ+GD−1D−1ΓDD−1PΓD−1D−1Q−ΓD−1D−1PΓDD−1Q,KD−1DPQ=∂ΓD−1DQ∂xP−∂ΓD−1DP∂xQ+GDDΓD−1DPΓDDQ−ΓDDPΓD−1DQ+GD−1D−1ΓD−1D−1PΓD−1DQ−ΓD−1DPΓD−1D−1Q,KDDPQ=∂ΓDDQ∂xP−∂ΓDDP∂xQ+GD−1D−1ΓDD−1PΓD−1DQ−ΓD−1DPΓDD−1Q.
Hence,(171)BPQ≜12KDDPQ+KD−1D−1PQ=12∂ΓDDQ+ΓD−1D−1Q∂xP−12∂ΓDDP+ΓD−1D−1P∂xQ=∂BQ∂xP−∂BP∂xQ.FPQ3≜12KDDPQ−KD−1D−1PQ=12∂ΓDDQ∂xP−∂ΓDDP∂xQ+GD−1D−1ΓDD−1PΓD−1DQ−ΓD−1DPΓDD−1Q−12∂ΓD−1D−1Q∂xP−∂ΓD−1D−1P∂xQ+GDDΓD−1DPΓDD−1Q−ΓDD−1PΓD−1DQ=∂AQ3∂xP−∂AP3∂xQ+gΓDD−1PΓD−1DQ−ΓD−1DPΓDD−1Q=∂AQ3∂xP−∂AP3∂xQ+gAP1AQ2−AP2AQ1.
Then,FPQ1 and FPQ2 can also be computed similarly.Remark 50.
Comparing the above conclusion andU1×SU2 principal bundle theory, we know this proposition shows that the reference-system F indeed can describe weak and electromagnetic field.The following proposition shows an advantage of affine connection representation, that is, affine connection representation spontaneously implies the chiral asymmetry of neutrinos, butU1×SU2 principal bundle connection representation cannot imply it spontaneously.Definition 51.
According to Definition11, let the charges of the above reference-system F be ρmn, where m,n∈D−1,D=4,5. Then, l≜ρD−1D−1,ρDDT is said to be an electric charged lepton, and ν≜ρDD−1,ρD−1DT is said to be a neutrino. l and ν are collectively denoted by L. Thus, 1/21,1L is said to be a left-handed lepton, and 1/21,−1L is said to be a right-handed lepton, denoted by
(172)lL≜12ρD−1D−1+ρDD,lR≜12ρD−1D−1−ρDD,νL≜12ρDD−1+ρD−1D,νR≜12ρDD−1−ρD−1D.
DenoteΓGMNP by ΓMNP concisely. Then, we define on M,G that
(173)ZP≜12ΓD−1D−1P+ΓDDP,AP≜12ΓD−1D−1P−ΓDDP,WP1≜12ΓD−1DP+ΓDD−1P,WP2≜12ΓD−1DP−ΓDD−1P,and say AP is (affine) electromagnetic potential, while ZP, WP1, and WP2 are (affine) weak gauge potentials.Proposition 52.
IfM,G satisfies the symmetry condition ΓD−1DP=ΓDD−1P, then the geometric properties l and ν of F satisfy the following conclusions on M,G,
(174)lL;P=∂PlL−glLZP−glRAP−gνLWP1,lR;P=∂PlR−glRZP−glLAP,νL;P=∂PνL−gνLZP−glLWP1,νR;P=∂PνR−gνRZP.Proof.
LetH∈1,2,3,4,5, h∈4,5. It follows from Equation (168) that
(175)ρmn;P=∂Pρmn−ρHnΓmPH−ρmHΓnPH=∂Pρmn−ρhnΓmPh−ρmhΓnPh.Then, Equations (172) and (173) lead to Equation (174).Remark 53.
From the above proposition, we see that some constraint conditions make the general linear groupGL2,ℝ broken to a smaller group S, i.e.,
(176)so that the chiral asymmetry of leptons arises in Equation (174) spontaneously.Remark 54.
Proposition52 shows that
(1)
In affine connection representation of gauge fields, the coupling constantg is possessed of a geometric meaning that it is in fact the metric of internal space. But it does not have such a clear geometric meaning in U1×SU2 principal bundle connection representation(2)
At the most fundamental level, the coupling constant ofZP and that of AP are equal, i.e.,(177)gZ=gA=g.
Suppose there is a kind of medium.Z boson and photon move in it. Suppose Z field has interaction with the medium, but electromagnetic field A has no interaction with the medium. Thus, we have coupling constants
(178)g~Z≠gA=g,in the medium, and the Weinberg angle arises.
It is quite reasonable to consider a Higgs boson as a zero-spin pair of neutrinos, because in the Lagrangian, Higgs boson only couples withZ field and W field but does not couple with electromagnetic field and gluon field. If so, Higgs boson would lose its fundamentality and it would not have enough importance in a theory at the most fundamental level.
(3)
The mixing of three generations of leptons does not appear in Proposition52, but it can spontaneously arise in Proposition 63 due to the affine connection representation of the gauge field that is given by Definition 59


## 6. Affine Connection Representation of the Gauge Field of Strong Interaction

Definition 55.
SupposeM,F and M,G conform to Definition 47. Let D=r+3=6 and both of F and G satisfy
(179)GD−2D−2=GD−1D−1=GDD.Thus,F and G can describe strong interaction.Definition 56.
According to Definition11, let the charges of F be ρmn, where m,n=4,5,⋯,D. Define
(180)d1≜ρD−2D−2,ρD−1D−1T,d2≜ρD−1D−1,ρDDT,d3≜ρDD,ρD−2D−2T,u1≜ρD−2D−1,ρD−1D−2T,u2≜ρD−1D,ρDD−1T,u3≜ρDD−2,ρD−2DT.
We sayd1 and u1 are red color charges, d2 and u2 are blue color charges, and d3 and u3 are green color charges. Then, d1, d2, and d3 are said to be down-type color charges, and u1, u2, and u3 are said to be up-type color charges. Their left-handed and right-handed charges are
(181)d1L≜12ρD−2D−2+ρD−1D−1,d2L≜12ρD−1D−1+ρDD,d3L≜12ρDD+ρD−2D−2,d1R≜12ρD−2D−2−ρD−1D−1,d2R≜12ρD−1D−1−ρDD,d3R≜12ρDD−ρD−2D−2,u1L≜12ρD−2D−1+ρD−1D−2,u2L≜12ρD−1D+ρDD−1,u3L≜12ρDD−2+ρD−2D,u1R≜12ρD−2D−1−ρD−1D−2,u2R≜12ρD−1D−ρDD−1,u3R≜12ρDD−2−ρD−2D.
OnM,G, we denote
(182)gs≜GD−1D−12+GDD2=GD−1D−12+GD−2D−22=GD−2D−22+GDD2.UP1≜12ΓD−2D−2P+ΓD−1D−1P,VP1≜12ΓD−2D−2P−ΓD−1D−1P,XP23≜12ΓD−2D−1P+ΓD−1D−2P,YP23≜12ΓD−2D−1P−ΓD−1D−2P,UP2≜12ΓD−1D−1P+ΓDDP,VP2≜12ΓD−1D−1P−ΓDDP,XP31≜12ΓD−1DP+ΓDD−1P,YP31≜12ΓD−1DP−ΓDD−1P,UP3≜12ΓDDP+ΓD−2D−2P,VP3≜12ΓDDP−ΓD−2D−2P,XP12≜12ΓDD−2P+ΓD−2DP,YP12≜12ΓDD−2P−ΓD−2DP.
We notice that there are just only three independent ones inUP1, UP2, UP3, VP1, VP2, and VP3. Without loss of generality, let
(183)RP≜aRUP1+bRUP2+cRUP3,SP≜aSUP1+bSUP2+cSUP3,TP≜aTUP1+bTUP2+cTUP3,UP1≜αRRP+αSSP+αTTP,UP2≜βRRP+βSSP+βTTP,UP3≜γRRP+γSSP+γTTP,where the coefficients matrix is nonsingular. Thus, it is not hard to find the following proposition true.Proposition 57.
Letλaa=1,2,⋯,8 be the Gell-Mann matrices and Ta≜1/2λa the generators of SU3 group. When M,G satisfies the symmetry condition ΓD−2D−2P+ΓD−1D−1P+ΓDDP=0, denote
(184)AP≜12AP11AP12AP13AP21AP22AP23AP31AP32AP33,where
(185)AP11≜SP+16TP,AP12≜XP12−iYP12,AP13≜XP31−iYP31,AP21≜XP12+iYP12,AP22≜−SP+16TP,AP23≜XP23−iYP23,AP31≜XP31+iYP31,AP32≜XP23+iYP23,AP33≜−26TP.
Thus,AP=TaAPa if and only if
(186)AP1=XP12,AP2=YP12,AP3=SP,AP4=XP31,AP5=YP31,AP6=XP23,AP7=YP23,AP8=TP.Remark 58.
On the one hand, the above proposition shows that Definition55 is an affine connection representation of strong interaction field. It does not define the gauge potentials as abstractly as that in principal SU3-bundle theory but endows gauge potentials with concrete geometric constructions.On the other hand, the above proposition implies that if we take appropriate symmetry conditions, the algebraic properties ofSU3 group can be described by the transformation group GL3,ℝ of internal space of G. In other words, the exponential map
(187)exp:GL3,ℝ⟶U3,Bma↦eiTamBmadefines a homomorphism, and SU3 is a subgroup of U3. Therefore, Definition 55 is compatible with SU3 theory.


## 7. Affine Connection Representation of the Unified Gauge Field

Definition 59.
SupposeM,F and M,G conform to Definition 47. Let D=r+5=8 and both of F and G satisfy
(188)GD−4D−4=GD−3D−3,GD−2D−2=GD−1D−1=GDD.
Thus,F and G can describe the unified field of electromagnetic, weak, and strong interactions.Definition 60.
According to Definition11, let the charges of F be ρmn, where m,n=4,5,⋯,D. Define
(189)l≜ρD−4D−4,ρD−3D−3T,d1≜ρD−2D−2,ρD−1D−1T,d2≜ρD−1D−1,ρDDT,d3≜ρDD,ρD−2D−2T,ν≜ρD−3D−4,ρD−4D−3T,u1≜ρD−2D−1,ρD−1D−2T,u2≜ρD−1D,ρDD−1T,u3≜ρDD−2,ρD−2DT.
And denote(190)lL≜12ρD−4D−4+ρD−3D−3,lR≜12ρD−4D−4−ρD−3D−3,νL≜12ρD−3D−4+ρD−4D−3,νR≜12ρD−3D−4−ρD−4D−3,d1L≜12ρD−2D−2+ρD−1D−1,d2L≜12ρD−1D−1+ρDD,d3L≜12ρDD+ρD−2D−2,d1R≜12ρD−2D−2−ρD−1D−1,d2R≜12ρD−1D−1−ρDD,d3R≜12ρDD−ρD−2D−2,u1L≜12ρD−2D−1+ρD−1D−2,u2L≜12ρD−1D+ρDD−1,u3L≜12ρDD−2+ρD−2D,u1R≜12ρD−2D−1−ρD−1D−2,u2R≜12ρD−1D−ρDD−1,u3R≜12ρDD−2−ρD−2D.
OnM,G, we denote
(191)g≜GD−4D−42+GD−3D−32,gs≜GD−1D−12+GDD2=GD−1D−12+GD−2D−22=GD−2D−22+GDD2,ZP≜12ΓD−4D−4P+ΓD−3D−3P,AP≜12ΓD−4D−4P−ΓD−3D−3P,WP1≜12ΓD−4D−3P+ΓD−3D−4P,WP2≜12ΓD−4D−3P−ΓD−3D−4P,UP1≜12ΓD−2D−2P+ΓD−1D−1P,VP1≜12ΓD−2D−2P−ΓD−1D−1P,XP23≜12ΓD−2D−1P+ΓD−1D−2P,YP23≜12ΓD−2D−1P−ΓD−1D−2P,UP2≜12ΓD−1D−1P+ΓDDP,VP2≜12ΓD−1D−1P−ΓDDP,XP31≜12ΓD−1DP+ΓDD−1P,YP31≜12ΓD−1DP−ΓDD−1P,UP3≜12ΓDDP+ΓD−2D−2P,VP3≜12ΓDDP−ΓD−2D−2P,XP12≜12ΓDD−2P+ΓD−2DP,YP12≜12ΓDD−2P−ΓD−2DP.Discussion 61.
We know from Section2.3 that the gauge frame matrix Bma∈GL5,ℝ,a,m=4,5,⋯,8; therefore, when Bma are without any constraints, we can obtain a GL5,ℝ gauge theory. In consideration of the fact that the exponential map
(192)exp:GL5,ℝ⟶U5,Bma↦eiTamBmadefines a homomorphism and U1×SU2×SU3 is a subgroup of U5. So there must exist some constraint conditions of Bma to make GL5,ℝ reduce to U1×SU2×SU3, i.e.,
(193)More generally, suppose we do not know what the symmetry that can exactly describe “the real world” is, we just denote it byS; then, the map
(194)makes us be able to turn the problem of seeking for S into the problem of seeking for a set of constraint conditions of Bma. “To describe S” and “to describe the constraint conditions of Bma” are equivalent to each other.Because gauge potentialsΓmnP and particle fields ρmn are both constructed from the gauge frame field Bma, clearly here, it is more flexible and convenient “to describe the constraint conditions of Bma” than “to describe S.”Next, we have no idea what the best constraint conditions look like, but we can try to define a set of constraint conditions to see what can be obtained.Definition 62.
Similar to Remark53, we define the constraint conditions as follows.
(1)
1st basic conditions(195)GD−4D−4=GD−3D−3,GD−2D−2=GD−1D−1=GDD.(2)
2nd basic conditions(196)ΓD−3D−4P=ΓD−4D−3P,ΓD−2D−2P+ΓD−1D−1P+ΓDDP=0.(3)
1st conditions of PMNS mixing of leptons(197)ΓD−4PD−2=cD−3D−2ΓD−4PD−3,ΓD−4PD−1=cD−3D−1ΓD−4PD−3,ΓD−4PD=cD−3DΓD−4PD−3,ΓD−3PD−2=cD−4D−2ΓD−3PD−4,ΓD−3PD−1=cD−4D−1ΓD−3PD−4,ΓD−3PD=cD−4DΓD−3PD−4,cD−3D−2=cD−4D−2,cD−3D−1=cD−4D−1,cD−3D=cD−4D.(4)
2nd conditions of PMNS mixing of leptons(198)ρD−2D−3=ρD−2D−4,ρD−1D−3=ρD−1D−4,ρDD−3=ρDD−4,ρD−3D−2=ρD−4D−2,ρD−3D−1=ρD−4D−1,ρD−3D=ρD−4D.(5)
1st conditions of CKM mixing of quarks(199)ΓD−2PD−3=cD−2D−4ΓD−4PD−3,ΓD−1PD−3=cD−1D−4ΓD−4PD−3,ΓDPD−3=cDD−4ΓD−4PD−3,ΓD−2PD−4=cD−2D−3ΓD−3PD−4,ΓD−1PD−4=cD−1D−3ΓD−3PD−4,ΓDPD−4=cDD−3ΓD−3PD−4,cD−2D−4=cD−1D−4=cDD−4,cD−2D−3=cD−1D−3=cDD−3.(6)
2nd conditions of CKM mixing of quarks(200)ρD−2D−3=ρD−1D−3=ρDD−3,ρD−2D−4=ρD−1D−4=ρDD−4,ρD−3D−2=ρD−3D−1=ρD−3D,ρD−4D−2=ρD−4D−1=ρD−4D,where cnm are constants.Proposition 63.
WhenM,F and M,G satisfy the symmetry conditions (1), (2), (3), and (4) of Definition 62, denote
(201)l′≜ρD−4D−4+cD−4D−22ρD−2D−4+ρD−4D−2+cD−4D−12ρD−1D−4+ρD−4D−1+cD−4D2ρDD−4+ρD−4D,ρD−3D−3+cD−3D−22ρD−2D−3+ρD−3D−2+cD−3D−12ρD−1D−3+ρD−3D−1+cD−3D2ρDD−3+ρD−3DTν′≜ρD−3D−4+cD−3D−22ρD−2D−4+ρD−4D−2+cD−3D−12ρD−1D−4+ρD−4D−1+cD−3D2ρDD−4+ρD−4D,ρD−4D−3+cD−4D−22ρD−2D−3+ρD−3D−2+cD−4D−12ρD−1D−3+ρD−3D−1+cD−4D2ρDD−3+ρD−3DTThen, the geometric propertiesl and ν of F satisfy the following conclusions on M,G.
(202)lL;P=∂PlL−glLZP−glRAP−gvL′WP1,lR;P=∂PlR−glRZP−glLAP,νL;P=∂PνL−gνLZP−glL′WP1,νR;P=∂PνR−gνRZP.Proof.
First, we compute the covariant differential ofρmn of F.
(203)ρmn;P=∂Pρmn−ρHnΓmPH−ρmHΓnPH=∂Pρmn−ρD−4nΓmPD−4−ρD−3nΓmPD−3−ρD−2nΓmPD−2−ρD−1nΓmPD−1−ρDnΓmPD−ρmD−4ΓnPD−4−ρmD−3ΓnPD−3−ρmD−2ΓnPD−2−ρmD−1ΓnPD−1−ρmDΓnPD.
According to Definitions60 and 62, by calculation, we obtain that
(204)lL;P=∂PlL−glLZP−glRAP−gνLWP1−12cD−4D−2ρD−2D−3+ρD−3D−2+cD−3D−2ρD−2D−4+ρD−4D−2g2WP1−12cD−4D−1ρD−1D−3+ρD−3D−1+cD−3D−1ρD−1D−4+ρD−4D−1g2WP1−12cD−4DρDD−3+ρD−3D+cD−3DρDD−4+ρD−4Dg2WP1,lR;P=∂PlR−glRZP−glLAP,νL;P=∂PνL−gνLZP−glLWP1−12cD−4D−2ρD−2D−4+ρD−4D−2+cD−3D−2ρD−3D−2+ρD−2D−3g2WP1−12cD−4D−1ρD−1D−4+ρD−4D−1+cD−3D−1ρD−3D−1+ρD−1D−3g2WP1−12cD−4DρDD−4+ρD−4D+cD−3DρD−3D+ρDD−3g2WP1,νR;P=∂PνR−gνRZP.
Then, according to definitions ofl′ and ν′, we obtain that
(205)l′L=lL+cD−4D−222ρD−2D−4+ρD−4D−2+cD−4D−122ρD−1D−4+ρD−4D−1+cD−4D22ρDD−4+ρD−4D+cD−3D−222ρD−2D−3+ρD−3D−2+cD−3D−122ρD−1D−3+ρD−3D−1+cD−3D22ρDD−3+ρD−3D,νL′=νL+cD−3D−222ρD−2D−4+ρD−4D−2+cD−3D−122ρD−1D−4+ρD−4D−1+cD−3D22ρDD−4+ρD−4D+cD−4D−222ρD−2D−3+ρD−3D−2+cD−4D−122ρD−1D−3+ρD−3D−1+cD−4D22ρDD−3+ρD−3D.
Substitute them into the previous equations, and we obtain that(206)lL;P=∂PlL−glLZP−glRAP−gvL′WP1,lR;P=∂PlR−glRZP−glLAP,νL;P=∂PνL−gνLZP−glL′WP1,νR;P=∂PνR−gνRZP.Remark 64.
The above proposition shows the geometric origin of PMNS mixing of weak interaction. In affine connection representation of gauge fields, PMNS mixing arises as a geometric property on manifold.In conventional physics,e, μ, and τ have just only ontological differences, but they have no difference in mathematical connotation. By contrast, Proposition 63 tells us that leptons of three generations should be constructed by different linear combinations of ρpq,ρqpp=4,5;q=6,7,8. Thus, e, μ, and τ may have concrete and distinguishable mathematical connotations. For example, let aμ, bμ, aμnm, bμnm, aτ, bτ, aτnm, and bτnm be constants; then, we might suppose that
(207)e≜l=ρD−4D−4,ρD−3D−3T,νe≜ν=ρD−3D−4,ρD−4D−3T,μ≜aμe+12aμD−4D−2ρD−2D−4+aμD−4D−1ρD−1D−4+aμD−4DρDD−4,aμD−3D−2ρD−2D−3+aμD−3D−1ρD−1D−3+aμD−3DρDD−3T,νμ≜bμνe+12bμD−3D−2ρD−2D−4+bμD−3D−1ρD−1D−4+bμD−3DρDD−4,bμD−4D−2ρD−2D−3+bμD−4D−1ρD−1D−3+bμD−4DρDD−3T,τ≜aτμ+12aτD−4D−2ρD−4D−2+aτD−4D−1ρD−4D−1+aτD−4DρD−4D,aτD−3D−2ρD−3D−2+aτD−3D−1ρD−3D−1+aτD−3DρD−3DT,ντ≜bτνμ+12bτD−3D−2ρD−4D−2+bτD−3D−1ρD−4D−1+bτD−3DρD−4D,bτD−4D−2ρD−3D−2+bτD−4D−1ρD−3D−1+bτD−4DρD−3DT.Proposition 65.
WhenM,F and M,G satisfy the symmetry conditions (1), (2), (5), and (6) of Definition 62, denote
(208)d′1L≜122cD−1D−3ρD−4D−2+ρD−2D−4+122cD−2D−3ρD−4D−1+ρD−1D−4+122cD−1D−4ρD−3D−2+ρD−2D−3+122cD−2D−4ρD−3D−1+ρD−1D−3,d′2L≜122cDD−3ρD−4D−1+ρD−1D−4+122cD−1D−3ρD−4D+ρDD−4+122cDD−4ρD−3D−1+ρD−1D−3+122cD−1D−4ρD−3D+ρDD−3,d′3L≜122cD−2D−3ρD−4D+ρDD−4+122cDD−3ρD−4D−2+ρD−2D−4+122cD−2D−4ρD−3D+ρDD−3+122cDD−4ρD−3D−2+ρD−2D−3,u′1L≜122cD−2D−3ρD−4D−2+ρD−2D−4+122cD−2D−4ρD−3D−2+ρD−2D−3+122cD−1D−3ρD−4D−1+ρD−1D−4+122cD−1D−4ρD−3D−1+ρD−1D−3,u2L′≜122cD−1D−3ρD−4D−1+ρD−1D−4+122cD−1D−4ρD−3D−1+ρD−1D−3+122cDD−3ρD−4D+ρDD−4+122cDD−4ρD−3D+ρDD−3,u′3L≜122cDD−3ρD−4D+ρDD−4+122cDD−4ρD−3D+ρDD−3+122cD−2D−3ρD−4D−2+ρD−2D−4+122cD−2D−4ρD−3D−2+ρD−2D−3.
Then, the geometric propertiesd1, d2, d3, u1, u2, u3 of F satisfy the following conclusions on M,G.
(209)d1L;P=∂Pd1L−gsd1LUP1+gsd2LVP1−gsd3LVP1−gsu1LXP23−gs2u2LXP31+gs2u2LYP31−gs2u3LXP12−gs2u3LYP12−gu1L′WP1,d2L;P=∂Pd2L−gsd2LUP2+gsd3LVP2−gsd1LVP2−gsu2LXP31−gs2u3LXP12+gs2u3LYP12−gs2u1LXP23−gs2u1LYP23−gu2L′WP1,d3L;P=∂Pd3L−gsd3LUP3+gsd1LVP3−gsd2LVP3−gsu3LXP12−gs2u1LXP23+gs2u1LYP23−gs2u2LXP31−gs2u2LYP31−gu3L′WP1,d1R;P=∂Pd1R−gsd1LVP1+gsd2LUP1−gsd3LUP1+gsu1LYP23+gs2u2LXP31−gs2u2LYP31−gs2u3LXP12−gs2u3LYP12,d2R;P=∂Pd2R−gsd2LVP2+gsd3LUP2−gsd1LUP2+gsu2LYP31+gs2u3LXP12−gs2u3LYP12−gs2u1LXP23−gs2u1LYP23,d3R;P=∂Pd3R−gsd3LVP3+gsd1LUP3−gsd2LUP3+gsu3LYP12+gs2u1LXP23−gs2u1LYP23−gs2u2LXP31−gs2u2LYP31,u1L;P=∂Pu1L−gsu1LUP1−gs2u2LXP12−gs2u2LYP12−gs2u3LXP31+gs2u3LYP31−gsd1LXP23+gsd2LYP23−gsd3LYP23−gd1L′WP1,u2L;P=∂Pu2L−gsu2LUP2−gs2u3LXP23−gs2u3LYP23−gs2u1LXP12+gs2u1LYP12−gsd2LXP31+gsd3LYP31−gsd1LYP31−gd2L′WP1,u3L;P=∂Pu3L−gsu3LUP3−gs2u1LXP31−gs2u1LYP31−gs2u2LXP23+gs2u2LYP23−gsd3LXP12+gsd1LYP12−gsd2LYP12−gd3L′WP1,u1R;P=∂Pu1R−gsu1RUP1+gs2u2RXP12+gs2u2RYP12+gs2u3RXP31−gs2u3RYP31,u2R;P=∂Pu2R−gsu2RUP2+gs2u3RXP23+gs2u3RYP23+gs2u1RXP12−gs2u1RYP12,u3R;P=∂Pu3R−gsu3RUP3+gs2u1RXP31+gs2u1RYP31+gs2u2RXP23−gs2u2RYP23.Proof.
Substitute Definition60 into ρmn and consider Definition 62, then compute them, and then substitute d1L′,d2L′,d3L′,u1L′,u2L′,u3L′ into them, and we finally obtain the results.Remark 66.
The above proposition shows a geometric origin of CKM mixing. We see that, in affine connection representation of gauge fields,d1L′,d2L′,d3L′,u1L′,u2L′,u3L′ arise as geometric properties on manifold. Detailed equations of CKM mixing can be obtained on an additional condition such as
(210)ρD−2D−2=a23ρD−2D−3+a24ρD−2D−4+a32ρD−3D−2+a42ρD−4D−2,ρD−1D−1=a13ρD−1D−3+a14ρD−1D−4+a31ρD−3D−1+a41ρD−4D−1,ρDD=a03ρDD−3+a04ρDD−4+a30ρD−3D+a40ρD−4D,ρD−2D−1=b23ρD−2D−3+b13ρD−1D−3+b24ρD−2D−4+b14ρD−1D−4,ρD−1D−2=b32ρD−3D−2+b31ρD−3D−1+b42ρD−4D−2+b41ρD−4D−1,ρD−2D=b23ρD−2D−3+b03ρDD−3+b24ρD−2D−4+b04ρDD−4,ρD−1D=b13ρD−1D−3+b03ρDD−3+b14ρD−1D−4+b04ρDD−4,ρDD−1=b31ρD−3D−1+b30ρD−3D+b41ρD−4D−1+b40ρD−4D.Definition 67.
A particle is not an existence at the place of an individual point, and its concept is defined on the entire manifold. Concretely speaking, if the reference-systemF satisfies
(211)ρD−2D−2=ρD−1D−1=ρDD=ρD−2D−1=ρD−1D−2=ρD−1D=ρDD−1=ρDD−2=ρD−2D=0,ΓD−2D−2P=ΓD−1D−1P=ΓDDP=ΓD−2D−1P=ΓD−1D−2P=ΓD−1DP=ΓDD−1P=ΓDD−2P=ΓD−2DP=0,we say F is a lepton; otherwise, F is a hadron.SupposeF is a hadron. For d1, d2, d3, u1, u2, u3, if F satisfies that five of them are zero and the other one is nonzero, we say F is an individual quark.Proposition 68.
There does not exist an individual quark. In other words, if any five ones ofd1,d2,d3,u1,u2,u3 are zero, then d1=d2=d3=u1=u2=u3=0.For an individual down-type quark, the above proposition is evidently true. Without loss of generality, letu1=u2=u3=0 and d1=d2=0; thus, ρD−2D−2=ρD−1D−1=ρDD=0; hence, we must have d3=0.For an individual up-type quark, this paper has not made progress on the proof yet. Nevertheless, Proposition68 provides the color confinement with a new geometric interpretation, which is significant in itself. It involves a natural geometric constraint of the curvatures among different dimensions.


## 8. Conclusions

(1)
An affine connection representation of gauge fields is established in this paper. It has the following main points of view(i)
The holonomic connection Equation (6) contains more geometric information than Levi-Civita connection. It can uniformly describe gauge field and gravitational field(ii)
Time is the total spatial metric with respect to all dimensions of internal coordinate space and external coordinate space(iii)
Energy is the total momentum with respect to all dimensions of internal coordinate space and external coordinate space(iv)
On-shell evolution is described by gradient direction(v)
Quantum theory is a geometric theory of distribution of gradient directions. It has a geometric meaning discussed in Section3.9(2)
In the affine connection representation of gauge fields, some physical objects are incorporated into the same geometric framework(i)
Gauge field and gravitational field can both be represented by affine connection. They have a unified coordinate description. Some parts ofΓNPM describe gauge fields such as electromagnetic, weak, and strong interaction fields. The other parts of ΓNPM describe gravitational field(ii)
Gauge field and elementary particle field are both geometric entities constructed from semimetric. The componentsρmn of ρMN with m,n∈4,5,⋯,D describe leptons and quarks, and the other components of ρMN may describe particle fields of dark matters(iii)
Physical evolutions of gauge field and elementary particle field have a unified geometric description. Their on-shell evolution and quantum evolution both present as geometric properties about gradient direction(iv)
CPT inversion can be geometrically interpreted as a joint transformation of full inversion of coordinates and full inversion of metrics(v)
Rest-mass is the total momentum with respect to internal space. It originates from geometric property of internal space. Energy, momentum, and mass have no essential difference in geometric sense(vi)
Quantum theory and gravitational theory have a unified geometric interpretation and the same view of time and space. They both reflect intrinsic geometric properties of manifold(vii)
The origination of coupling constants of interactions can be interpreted geometrically(viii)
Chiral asymmetry, PMNS mixing, and CKM mixing arise as geometric properties on manifold(ix)
There exists a geometric interpretation to the color confinement of quarksIn the affine connection representation, we can get better interpretations to these physical properties. Therefore, to represent gauge fields by affine connection is probably a necessary step towards the ultimate theory of physics.



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*Source: 4573981-2022-05-17.xml*