A generalization of intrinsic geometry and an affine connection representation of gauge fields


 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. Poincare gauge theory and metric-affine gauge theory adopt the first approach. This paper adopts the second.
 
We show a generalization of Riemannian geometry and a new affine connection, and apply them to establishing a unified coordinate description of gauge field and gravitational field. It has the following advantages.
 
(i) Gauge field and gravitational field have the same affine connection representation, and 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, gravitational theory and quantum theory obtain the same view of time and space and a unified description of evolution in affine connection representation of gauge fields.
 
(iii) Chiral asymmetry, coupling constants, MNS mixing and CKM mixing can appear spontaneously in affine connection representation, while in $U(1) \times SU(2)\times SU(3)$ principal bundle connection representation they can just only be artificially set up. Some principles and postulates of the conventional theories that are based on principal bundle connection representation can be turned into theorems in affine connection representation, so they are not necessary to be regarded as principles or postulates anymore.


Background and purpose
We know that gravitational field can be described by the geometric theory on Riemannian manifold, and that gauge field can be described by the geometric theory on fibre bundle. There have been many theories trying to solve the problem that how to establish a unified description of gravitational field and gauge field.
The main difficulties of this problem in the above two geometric theories are as follows.
(i) The affine connection that describes gravitational field is different from the abstract connection that describes gauge field. Gravitational field and gauge field are not described by the same connection.
(ii) The affine connection of gravitational field has an explicit sense of view of time and space, but the abstract connection of gauge field has not. Affine connection reflects the degrees of freedom of spacetime, but abstract connection just only reflects some abstract degrees of freedom.
(iii) Time is a component of gravitational field, but not a component of gauge field. Minkowski coordinates basically regard 1-dimensional time and 3-dimensional space on an equal footing. The description of evolution of gravitational field is essentially different from that of gauge field.
There are the following two ways to solve the problem (i).
The other way is to represent gauge field as affine connection. This is the approach adopted by this paper. In this way, gravitational field and gauge field can be uniformly described by affine connection, and both the problems (i) and (ii) can be solved. In addition, we find that in affine connection representation, time can be regarded as the total metric with respect to all dimensions of internal coordinate space and external coordinate space, so the problem (iii) can be solved.
The difficulty with the second approach is that, Riemannian manifold is not suitable for establishing an affine connection representation of gauge fields. In other words, an affine connection (e.g. Levi-Civita connection, spin connection, Riemmann-Cartan connection) that is dependent on Christoffel symbol { ρ µν } can describe gravitational field, but cannot describe general gauge field. We propose some new mathematical tools to overcome the above difficulty, and establish a torsion-free affine connection representation of gauge fields in a geometry that is more general than Riemannian geometry. This is the main work of this paper.
In the theory of this paper, gravitational field and gauge field are unified in affine connection representation. Thus, the problems (i), (ii) and (iii) can all be solved. In addition, there is another advantage that some principles and postulates of the conventional theories that are based on principal bundle connection representation can be turned into theorems in affine connection representation, so they are not necessary to be regarded as principles or postulates anymore.

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?
(II) How to describe the evolution of gauge field and gravitational field 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). We need a new geometry in a new way to integrate Riemannian geometry and gauge field geometry. The new geometry is required to be more general than Riemannian geometry, and more concrete than the existing gauge field geometry.
It has long been known that the coefficients G M N = δ AB B A M B B N of metric tensor remain unchanged when orthogonal transformations act on the semi-metric B A M (or to say frame field) [24,25]. So we can say a Riemannian geometric property is an invariant under identical transformations of metric, and we can also speak of it as an invariant under orthogonal transformations of semimetric.
If we take different orthogonal transformations of semi-metric at different points of a Riemannian manifold, the Riemannian manifold remains unchanged. It is evident that local orthogonal transformations mean more general geometric properties. However, Riemannian manifold cannot reflect these geometric properties.
We see that the semi-metric B A M is more fundamental than the metric G M N . Therefore, we need to generalize the Riemannian manifold (M, G M N ) to a more general manifold (M, B A M ), and then to regard orthogonal transformation of semi-metric as gauge transformation.
Next, we need an affine connection to describe gauge field. Riemann-Cartan geometry is a generalization of Riemannian geometry, its affine connection is independent of metric, but Riemann-Cartan connection and spin connection are both dependent on Christoffel symbol, so they are not suitable for describing gauge field.
We make a further generalization, and adopt a new torsion-free affine connection that is totally independent of Christoffel symbol, and use semi-metric to construct an affine connection representation of gauge fields. In this way, gravitational field and gauge field can be unified on a manifold (M, B A M ) that is defined by semi-metric. Intrinsic geometry should not be confined to the point of view of metric and Christoffel symbol. . It shows that semi-metric is indeed possessed of an intrinsic geometric significance independent of metric, and that it reflects the degree of slackness and tightness of the distribution of a coordinate frame in another coordinate frame. , we are able to use the semi-metric B A M to endow particle fields and gauge fields with intrinsic geometric constructions.
In the theories based on principal bundle connection representation: (1) For particle fields. Several complex-valued functions which satisfy the Dirac equation, are sometimes used to refer to a charged lepton field l, and sometimes a neutrino field ν. It is not clear that how to distinguish these field functions l and ν by inherent mathematical constructions.
By contrast, in the affine connection representation of this paper, various gauge fields including A A A and φ and various particle fields including l and ν are all constructed by semi-metric of internal coordinate space. Thus, they are not only abstract algebraic representations of group, but also possessed of concrete geometric entities.
For the problem (II). There is a fundamental difficulty that time is a component of gravitational field, but not a component of 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 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 connection 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). Suppose Γ M N P is the required affine connection, which is constructed by semi-metric B A M . The basic idea is that the components of Γ M N P with M, N ∈ {4, 5, · · · , D} describe electromagnetic, weak and strong interaction fields. The other components of Γ M N P describe gravitational field.
Corresponding to the problem (II), in section 3 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 sections 5 to 7 we show concrete forms of affine connection representations of electromagnetic, weak and strong interaction fields.
Some 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 Definition 3.1.1 and Discussion 4.3.2 for detail. The CP T inversion is interpreted as the composition of the total inversion of coordinates and the total inversion of metrics, see section 3.8 for detail. The conventional (1 + 3)-dimensional Minkowski coordinate x µ originates from the general D-dimensional coordinate x M . 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 sections 3.4, 3.5, 3.6 and 4.4 for detail.
(3) 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 3.9 and 3.10 for detail.
(4) Yang-Mills equation originates from a geometric property of gradient direction. We show the affine connection representation of Yang-Mills equation. See section 3.5 and section 4.6 for detail.
(5) Energy-momentum equation originates from a geometric property of gradient direction. We show the affine connection representation of mass, energy, momentum and action, see section 3.6, Definition 4.4.1 and Discussion 4.4.1 for detail. Furthermore, we also show the affine connection representation of Dirac equation, see section 4.5 for detail.
(6) Gauge field and particle field are two geometric fields both constructed by semi-metric. This paper is not going to adopt the way of taking several abstract functions ν, l, A K µ and speaking of them directly as a neutrino field, an electric charged lepton field and a gauge field, but to show the inherent and geometric constructions of ν, l, A K µ concretely. See Discussion 3.5.??, sections 5, 6 and Definition 7.2 for detail. (7) The coupling constants of interactions between gauge field and particle field originate from the metric of the internal coordinate space of manifold. See sections 5, 6 and 7 for detail.
(8) Why do not neutrinos participate in the electromagnetic interactions? And why do not right-handed neutrinos participate in the weak interactions with W bosons? In the Standard Model, they are both postulates. However, in the new framework, they are natural and geometric results of affine connection representation of gauge fields, therefore not necessary to be regarded as postulates anymore. See Proposition 5.2 and Proposition 7.1 for detail.
(9) Why do leptons and quarks both have three generations?
We give an answer in section 7. It is a natural geometric result of affine connection representation of gauge fields.

Geometric manifold
In order to establish the affine connection representation of gauge fields, we have to generalize Riemannian manifold to geometric manifold. Definition 2.1.1. Let M be a D-dimensional connected smooth real manifold. ∀p ∈ M , take a coordinate chart (U p , φ U p ) on a neighborhood U p of p. They constitute a coordinate covering which is said to be a point-by-point covering. For the sake of simplicity, U p can be denoted by U , and φ U p by φ U .
Let φ and ψ be two point-by-point coverings. For the two coordinate frames φ U and ψ U on the neighborhood U of point p, if is a smooth homeomorphism, f p is called a local reference-system, where ψ U is said to be the basis coordinate frame of f p , and φ U is said to be the performance coordinate frame of f p . If every p ∈ M is endowed with a local reference-system f (p), and we require the semi-metrics B A M and C M A in section 2.2 to be smooth real functions on M , then is said to be a reference-system on M , and (M, f ) is said to be a geometric manifold.

Metric and semi-metric
In the absence of a special declaration, the indices take values as A, B, C, D, E = 1, 2, · · · , D and M, N, P, Q, R = 1, 2, · · · , D. The derivative functions of f (p) on U p define the semi-metrics (or to say frame field) B A M and C M A of f on the manifold M , that are Similarly, it can also be defined thatb M

Gauge transformation in affine connection representation
and reference-system transformations on manifold M We also speak of L f and R f as (affine) gauge transformations, see Discussion 4.6.3 for reasons. represents external direct product.

Generalized intrinsic geometry
Let the notation F k represent L k or R k . The above transforamtions lead to the following relations of equivalence.
( The totality of all the geometric manifolds on M is denoted by M(M ). The above four relations of equivalence determine the following four classifications.
The largest one is M(M )/ ∼ =, which is said to be (generalized) intrinsic geometry. The smallest one is M(M )/ ∼, which is said to be universal geometry.
Let the notation hX represent a geometric property of X. We have the following conclusions. If detailed physical properties are described by geometric properties on M(M )/ ∼ =, then universal physical properties can necessarily be described by geometric properties on M(M )/ ∼. Therefore, general covariance and gauge invariance are both geometric properties on M(M )/ ∼, they remain unchanged under arbitrary transformations of reference-system. It So (M, f ) and (M, g) are the same Riemannian manifold, but different geometric manifolds.
Under an orthogonal affine gauge transformation L k , the metric G M N and the Christoffel symbol

Simple connection
We do not take into account spin connection and Riemann-Cartan connection, because they are dependent on Christoffel symbol { M N P }. A nice choice is to adopt the following definition, which is independent of Christoffel symbol. Definition 2.5.1. Let there be an affine connection D. It presents as If the affine connection coefficients are defined as D is said to be a simple connection. In addition, denote Γ M N P ≜ G M M ′ Γ M ′ N P . It is easy to verify that and that D is torsion-free. If we do not require it to be torsion-free, it is fine to adopt Γ M N P ≜ C M A ∂B A N ∂x P . Proposition 2.5.1. Suppose there are reference-systems g and k on a manifold M . Let the semimetrics of g be Then let the simple connections of (M, g), (M, k), Proof. We just need to calculate it.
This shows that simple connection reflects more curved-properties of a manifold than Levi-Civita connection.

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 a component of gravitational field, but not a component of 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. s, i = 1, · · · , r ; a, m = r + 1, · · · , D.

The relation between time and space
On the geometric manifold (M, f ), the dξ 0 and dx 0 which are defined by are said to be total space metrics or time metrics. We also suppose dξ (N ) and dx (N ) are regarded as proper-time metrics. For convenience, P is said to be external space and N is said to be internal space. . 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. The relation of the above concept to Minkowski coordinates is discussed detailedly in section 4.

Evolution path as a submanifold
Definition 3.2.1. Let there be two reference-systems f and g on a manifold M . If ∀p ∈ M , f (p) ≜ ϕ U • ψ −1 U and g(p) ≜ ϕ U • ρ −1 U have the same performance coordinate frame ϕ U , namely it can be intuitively expressed as a diagram we say f and g motion relatively and interact mutually, and also say that f evolves in g, or f evolves on the geometric manifold (M, g). Meanwhile, g evolves in f , or to say g evolves on (M, f ).
Let there be a one-parameter group of diffeomorphisms acting on M , such that φ X (p, 0) = p. Thus, φ X determines a smooth tangent vector field X on M . If X is nonzero everywhere, we say φ X is a set of evolution paths, and X is an evolution direction field. Let T ⊆ R be an interval, then the regular imbedding is said to be an evolution path through p. The tangent vector d dt ≜ [L p ] = X(p) is called an evolution direction at p. For the sake of simplicity, we also denote L p ≜ L p (T ) ⊂ M , then is also a regular imbedding. If it is not necessary to emphasize the point p, L p is denoted by L concisely.
In order to describe physical evolution, next we are going to strictly describe the mathematical properties of the reference-systems f and g sending onto the evolution path L.

Definition 3.2.2.
Let the coordinates of (U, ψ U ), (U, ϕ U ), (U, ρ U ) be ξ A , x M , ζ A , respectively, and the corresponding time metrics dξ 0 , dx 0 , dζ 0 . On U L ≜ U ∩ L p we have parameter equations Take f for example, according to Eq.(12), on U L we define They determine the following smooth functions on the entire L, similar to section 2.2: For convenience, we still use the notations ε and δ, and have the following smooth functions.
It is easy to verify that dx 0 = G 00 dx 0 and d dx 0 = G 00 d dx 0 are both true on L by a simple calculation.

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.
are true, we denote Evidently we have f • γ = f L • γ L , and it makes d dt , df = d dt L , df L true. □ Proposition 3.3.2. The following conclusions are true.
The following local discussion can also be applied to the entire manifold. Take the above left-hand side equations for example.
We know π * is an injection, so We know π * is a surjection, so

On-shell evolution as a gradient
Definition 3.4.1. Let T be an smooth n-order tensor field. The restriction on (U, represents the tensor basis generated by several ∂ ∂x M and dx M , and the tensor coefficients of T are concisely denoted by t : ∀p ∈ M , the integral curve of ∇t, that is L p ≜ 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 path L, In such a notation, all the indices are concisely ignored except Q. u Q dx Q uniquely determines a characteristic direction u Q ∂ ∂x Q . If the system of 1-order linear partial differential equations t ;Q = u Q has a solution t, then it is If it does not have a solution, we also denote the notations Dt ≜ u Q dx Q and ∇t ≜ u Q d dx Q in form for convenience.
Thus, no matter whether t ;Q = u Q has a solution, as long as we take the evolution direction [L] as [L] = u Q ∂ ∂x Q , the following conclusions are always true. where Now for any geometric property in the form of tensor U, 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.

On-shell evolution of potential field and affine connection representation of Yang-Mills equation
The table I of the article [76] 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 J K µ . In this section, we give an answer to this problem from the point of view of section 3.4, and show the affine connection representation of Yang-Mills equation.
Suppose f evolves in g according to Definition 3.2.1, that is, ∀p ∈ M , We always take the following notations in the coordinate frame (U, x M ).
(i) Let the simple connections of geometric manifolds (M, f ) and (M, g) be Λ M N P and Γ M N P , respectively. The colon ":" and semicolon ";" are used to express the covariant derivatives on (M, f ) and (M, g), respectively, e.g.: Λ M N P is said to be the potential field of f , and Γ M N P is said to be the potential field of g. (ii) Let the coefficients of Riemannian curvatures of (M, f ) and (M, g) be K M N P Q and R M N P Q , respectively, i.e.
(iii) The values of indices of internal space and external space are taken according to Definition 4.1.
Consider the evolution of f . Denote K M N P Q Then, according to Definition 3.3.1 and the evolution lemma of Proposition 3.3.2, we obtain Applying the evolution lemma of Proposition 3.3.2 again, we obtain which is said to be (affine) Yang-Mills equation of f . Thus, we have the following two results.
(i) We obtain the mathematical origination of charge and current. We know that the evolution path L is an imbedding submanifold of M . Thus, the charge ρ M N 0 originates from the pull-back π * from M to L for the curvature divergence, and the current j M N Q originates from ∇t that is related to ρ M N 0 . Besides, there is a further explanation in section 4.4.
(ii) We obtain the following proposition. This shows that Yang-Mills equation originates from a geometric property in the direction of ∇t.
as the field function of a general charge, or a charge for short.

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 section 4.4. Let Definition 3.6.1. For more convenience, the notation ρ M N is further abbreviated as ρ. In affine connection representation, energy and momentum of ρ are defined as  Proof. According to the above discussion, ∀p ∈ M , [L p ] = ∇ρ| p is equivalent to Then due to Proposition 3.3.1 we obtain the directional derivative in the gradient direction ∇ρ:   (21). □ Remark. According to Proposition 4.2.2, the light velocity c = 1, therefore in the gradient direction ∇ρ, the conclusion of Proposition 3.6.2 is consistent with the conventional Thus, in affine connection representation, the energy-momentum equation and the conventional definition of momentum both originate from a geometric property in gradient direction. Definition 3.6.2. Let P(b, a) be the totality of paths from a to b. And suppose L ∈ P(b, a), and the evolution parameter The elementary affine action of ρ is defined as Thus, δs(L) = 0 if and only if L is a gradient line of ρ.
Specially, in the case where g is orthogonal, we can also define action in the following way. On (M, g) let there be Dirac algebras γ M and γ N such that In a gradient direction of ρ, from Eq. (20) we obtain that Take γ P ρ ;P = ρ ;0 without loss of generality, then, in the gradient direction of ρ we have So we can define the following (orthogonal affine) action of ρ.
wherem τ is the rest-mass andx τ is the proper-time.

Affine connection representation of interaction
First, we define the following notations.
Then, through some calculations, we can obtain that which are the affine connection representations of general Lorentz force equations. See Discussion 4.4.1 for further illustrations. Furthermore, the affine connection representation of the conservation of energy-momentum can base on the above result to be constructed. We need to take

Inversion transformation in affine connection representation
In affine conection representation, CP T inversion is interpreted as a total inversion of coordinates and metrics. Let i, j = 1, 2, 3 and m, n = 4, 5, · · · , D.
Let the local coordinate representation of reference-system k be then parity inversion can be represented as Let the local coordinate representation of reference-system h be Time coordinate invesion can be represented as

Total inversion of coordinates can be represented as
The positive or negative sign of metric marks two opposite directions of evolution. Let N be a closed submanifold of M , and let its metric be dx (N ) . Denote the totality of closed submanifolds of M by B(M ), then total inversion of metrics can be expressed as Denote time inversion by then the joint transformation of the total inversion of coordinates CP T 0 and the total inversion of metrics Summerize the above discussions, then we have The CP T invariance in affine connection representation is very clear. Concretely, on (M, g) we consider the CP T transformation acting on g. Denote s ≜ L Dρ and D P e is ≜ ∂ ∂x P − i[ρΓ P ] e is , then through simple calculations we obtain that Remark 3.8.1. In quantum mechanics there is a complex conjugation in the time inversion of wave function T : ψ(x, t) → ψ * (x, −t). In affine connection representation, we know the complex conjugation is a straightforward mathematical result of the total inversion of metrics T (M ) .

Two dual descriptions of gradient direction field
Discussion 3.9.1. Let X and Y be non-vanishing smooth tangent vector fields on the manifold M . And let L Y be the Lie derivative operator induced by the one-parameter group of diffeomorphism φ Y . Then, according to a well-known theorem [77], we obtain the Lie derivative equation Thus, Eq. (29) can also be represented as On the other hand, ∀df ∈ T (M ) and df L ≜ π * (df ), due to (30) and Proposition 3.
If and only if taking Y = H, we speak of (31) and (32) as real-valued (affine) Heisenberg equation and (affine) Schrödinger equation, respectively, that is Discussion 3.9.2. 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 section 4.5, another is as follows.
Let ψ ≜ f e is L , where it is fine to take either s L ≜ s(L) or s L ≜ s(L) from Definition 3.6.2. According to Eq.(33), it is easy to obtain on L, that This is consistent with the conventional Heisenberg equation and Schrödinger equation (taking the natural units that ℏ = 1, c = 1) and they have a coordinate correspondence Due to Discussion 4.3.1 we know that ∂ ∂t ↔ d dx 0 originates from the difference that the evolution parameter is x τ or x 0 . Due to Discussion 4.3.2 we know that the imaginary unit i originates from the difference between the regular coordinates x 1 , x 2 , x 3 , x τ and the Minkowski coordinates x 1 , x 2 , x 3 , x 0 . That is to say, the regular coordinates satisfy and the Minkowski coordinates satisfy This causes the appearance of the imaginary unit i in the correspondence So Eq.(34) and Eq.(35) 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 the intrinsic 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 complexvalued forms, but we should focus on such geometric essence as gradient direction.
The advantage of complex-valued form is that it is convenient for describing the coherent superposition of the propagator. However, this is independent of the above discussions, and we are going to discuss it in section 3.10.

Quantum evolution as a distribution of gradient directions
From Proposition 3.6.1 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 manifold (M, 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 manifold.
Quantum 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 connetion representation.
We say |H| is the total distribution of the gradient direction field H. ∀T ∈ T, we say H T is a position distribution of gradient directions in momentum representation.
∀a ∈ M , we say |H(a)| is a momentum distribution of gradient directions in position representation. Remark 3.10.1. When T is fixed, H T can reflect the shape of (M, g). When a is fixed, |H(a)| can reflect the shape of (M, g).
However, when T and a are both fixed, H T (a) is a fixed individual gradient direction, which cannot reflect the shape of (M, g). In other words, if the momentum p T and the position x a of ρ are both definitely observed, the physical reality g would be unknowable, therefore this is unacceptable. This is the embodiment of quantum uncertainty in affine connection representation.   ). Correspondingly, ∀t ∈ R, φ |H|,a (t) is sent to φ |O|,a (t).
In a word, L g −1 induces the following two maps: ∀T ∈ T, deonte N ≜ {N ∈ T | det N = det T }. Due to T ∼ = GL(D, R), let U be a neighborhood of T , with respect to the topology of GL(D, R).
Take Ω = N ∩ U, then Let µ be a Borel measure on the manifold M . ∀t ∈ R, we know Thus, φ |H Ω |,a (t) ⊆ φ |H N |,a (t) and φ |O Ω |,a (t) ⊆ φ |O N |,a (t) are Borel sets, so they are measurable. Denote For the sake of simplicity, denote L ≜ φ H T ,a . Thus, we have a = L(0), b = L(t), and denote Because µ a is absolutely continuous with respect to µ, Radon-Nikodym theorem [78] ensures the existence of the following limit. The Radon-Nikodym derivative is called the distribution density of |H| along L in position representation.
On a neighborhood U of a, ∀T ∈ T, denote the normal section of H T (a) by N H T ,a , that is is called the distribution density of |H| along L in momentum representation.
In a word, W L (b, a) and Z L (p b , p a ) describe the density of the gradient lines that are adjacent to b in two different ways. They have the following evident property.
Proof. Directly compute 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, 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 3.10.1. With the above preparations, we obtain a new way to describe the construction of the propagator strictly.
For any path L that starts at a and ends at b, we denote ||L|| ≜ L dx 0 concisely. Let P(b, a) be the totality of all the paths from a to b. Denote However, there are so many redundant paths in , (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 from P(b,   Let L(p b , p a ) be the totality of all the gradient lines of |ρ|, whose starting-direction is p a and ending-direction is p b . Denote Let dL be a Borel measure on H(b, x 0 b ; a, x 0 a ). In consideration of Remark 4.5.1, we let s be the affine action s(L) in Definition 3.6.2. We say the intrinsic geometric property is the propagator of |ρ| from (a, x 0 a ) to (b, x 0 b ) in position representation. If we let dL be a Borel measure on H(p b , x 0 b ; p a , x 0 a ), then we say is the propagator of |ρ| from (p a , x 0 a ) to (p b , x 0 b ) in momentum representation. Discussion 3.10.2. Now (39) and (40) are strictly defined, but the Feynman path integral (38) 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 We notice that the distribution densities W L (b, a) and Z L (b, 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 "=" as the same thing.
Discussion 3.10.3. The quantization methods of QFT are successful, and they are also applicable in affine connection representation for orthogonal f and g, but in this paper we do not discuss them. We try to propose some more ideas for general f and g to understand the quantization of field in affine connection representation.
(1) If we take according to Definition 3.6.2, where D is the simple connection of (M, g), then consider the distribution of H ≜ ∇ρ, we know that describe the quantization of energy-momentum. Every gradient line in ∇ρ(b, x 0 b ; a, x 0 a ) corresponds to a set of eigenvalues of energy and momentum.
(2) If we take according to section 3.5, where D is the simple connection of (M, f ), then consider the distribution of H ≜ ∇t, we know that describe the quantizations 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.

Affine connection representation of gauge fields in classical spacetime
The new framework established in section 3 is discussed in the D-dimensional general coordinate x M , which is more general than the (1 + 3)-dimensional conventional Minkowski coordinate x µ . This is different from the extra dimensions in Kaluza-Klein theory and string theory.

Existence and uniqueness of classical spacetime submanifold
Step 1. To constructM . We define a closed submanifold P × {o} on M through o via the parameter equation x m = x m o . Then let φ X : M × R → M be the one-parameter group of diffeomorphisms corresponding to X. Restrict φ X to P × {o} and we obtain where points o and o ′ are on the same orbit L o ≜ φ X,o . P ×{o} and P ′ ×{o ′ } are both homeomorphic to P . In this sense we do not distinguish P and P ′ , we have Then consider all of such {o} on the entire orbit L o , and we obtain a map DenoteM ≜ P × L o . So we know thatM is determined by X and the fixed point o, hence it is unique. Thus, φ X |M ×R :M × R →M constitutes a one-parameter group of diffeomorphisms onM .
Step 2. To construct γ :M → M andX. Because X is internal-directed, the restriction of X to L o ≜ φ X,o : R → M is non-vanishing everywhere and L o is an injection. The image set of L o can also be denoted by L o . ∀t ∈ R, q ≜ φ X,o (t) ∈ L o , we can define a closed submanifold N q on M through q via the parameter equation x i = x i q , and N q is homeomorphic to N . Due to the one-to-one correspondence between q and N q , L o → N is a regular imbedding. Furthermore: (1) γ : P × L o → P × N is a regular imbedding, that is, γ :M → M . Hence the tangent map γ * : T (M ) → T (M ) is an injection. Therefore, the smooth tangent vectorX which satisfies ∀q ∈M , γ * :X(q) → X(q) is uniquely defined by X via γ −1 * . (2) We notice that o ∈M , hence L o ≜ φ X,o is an orbit of φ X |M ×R . In consideration of that L o uniquely determines γ, and γ uniquely determines γ * , and γ −1 * uniquely determinesX, so finallỹ X is uniquely determined by φ X |M ×R . Thus, we have φX = φ X |M ×R . □ (3) The correspondence betweenX and the restriction of X toM is one-to-one. For convenience, next we are not going to distinguish the notations X andX onM , but uniformly denote them by X.
(4) An arbitrary pathL : T →M , t → p onM uniquely corresponds to a path L ≜ γ •L : T → M, t → p on M . Evidently the image sets of L andL are the same, that is, L(T ) =L(T ). For convenience, later we are not going to distinguish the notations L andL onM , but uniformly denote them by L.

Gravitational field on classical spacetime submanifold Proposition 4.2.1.
Let there be a geometric manifold (M, f ) and its classical spacetime submanifold M . And let L ≜ φX ,a be an evolution path onM . Suppose p ∈ L and U is a coordinate neighborhood of p. According to Defnition 3.2.2, suppose the f (p) on U and the f L (p) on Thus, according to the notations in Definition 4.1, it is true that: (1) There exists a unique local reference-systemf (p) onŨ ≜ U ∩M such that and (2) If L is internal-directed, then the above coordinate frames (Ũ , ξ U ) and (Ũ , x K ) off (p) uniquely determine the coordinate frames (Ũ ,ξ α ) and (Ũ ,x µ ) such that and the coordinates satisfỹ That is to say,f (p),f L (p) have two coordinate representations: (42) and (43). Proof.
(1) According to the proof of the previous proposition, L is a regular submanifold of N . Let the metrics on N be Review Definition 3.2.2 and we know the regular imbedding π : L → N, q → q induces the parameter equations x m = x m τ (x τ ) and ξ a = ξ a τ (ξ τ ) of L. Then substitute them into f (p), f L (p) and we obtain (2) As same as the above, we also obtain x K = x K (ξ U ) = x K (ξ 0 ). The relation between two parameters x 0 and x τ of L can be expressed as x τ = x τ L (ξ 0 (x 0 )), and the relation between two parameters ξ 0 and ξ τ can be expressed as ξ τ = ξ τ L (x 0 (ξ 0 )). Substitute them intof (p),f L (p), then which can be abbreviated to In fact,f (p) is a reference system in conventional sense. We sayf is a classical spacetime reference-system onM , and (M ,f ) is a gravitational manifold.
(2) If the semi-metricsB α µ andC µ α ofg are constants onM , then we sayg is flat.   (Ũ , x K ),Ũ L can be described by equations x i = x i (x 0 ) and x τ = const with respect to the parameter x 0 . We notice that x τ = const, so there does not exist an equation ofŨ L with respect to the parameterx τ in Minkowski coordinate frame (Ũ ,x µ ). Therefore, we always have dx τ = dx τ = 0 onŨ L . Thus, (ii) According to Definition 4.2.2, an orthogonal transformation satisfies dζ τ = dx τ = 0, hence (iii) It is naturally true for the Lorentz transformation as an orthogonal one. This conclusion reflects the principle of constancy of light velocity.
The semi-metrics of (M ,f ) with respect to the Minkowski coordinatex µ areB α µ andC µ α , such that . (46) Applying the chain rule of differentiation, it is not hard to obtain the following relations between the regular semi-metrics and the Minkowski semi-metrics via the formula (44).
The evolution lemma of Proposition 3.3.2 can be expressed in Minkowski coordinate as Thus, we obtain proper-time metrics dξ τ and dx τ in Minkowski coordinates as below: It is easy to know there are relations between the regular metric G IJ and the Minkowski metricG µν as below: DenoteG τ τ ≜B τ τB τ τ ,G τ τ ≜C τ τC τ τ , then it is easy to obtainG τ τ =

Affine connection representation of classical spacetime evolution
LetD be the simple connection onM , and denotet L;τ ≜t ;σε σ τ , then the absolute differential and gradient of section 3.4 can be expressed onM in Minkowski coordinate as Evidently,Dt D LtL if and only if L is an arbitrary path.∇t ∼ =∇ LtL if and only if L is the gradient line. (1)m τ ≜ρ ;τ andm τ ≜ρ ;τ are said to be the rest mass ofρ.
This can also be regarded as the origin of p = mv. Similar to discussions of section 3.7, denote [ρR ρσ ] ≜ρ µχR χ νρσ +ρ χνR χ µρσ , Then for the same reason as section 3.7, based on Definition 4.4.1, we can strictly prove the affine connection representation of general Lorentz force equation ofρ, which is And similarly, we also haveT µν ;µ = 0. In the mass-point model,m τ ;ρ andε σ τ ;ρ do not make sense, so Eq.(47) turns intof ρ = [ρR ρσ ]ε σ τ . This is the affine connection representation of the force of interaction (e.g. the Lorentz force  L ∈P(b, a), and parameterx τ satisfy τ a ≜x τ (a) <x τ (b) ≜ τ b . The affine connection representation of action in Minkowski coordinates can be defined as There are more illustrations in Remark 4.5.1.

Affine connection representation of Dirac equation
Discussion 4.5.1. Define Dirac algebras γ µ and γ α such that Next, denote From Eq. (49), it is obtained that that is denoted concisely by . We speak of Eq. (49) and (51) We define ψ ων andM ωντ in the following way.
In the QFT propagator, we usually take S in the path integral e iS Dψ of a fermion in the form of where S and d 4x are both covariant. We believe that the external spatial integral is not an essential part for evolution, so for the sake of simplicity, we do not take into account the external spatial part Concretely speaking, denotẽ Thus, we have obtained a complex-valued representation of gradient direction ofρ ων . It can be denoted concisely by iγ µD µ ψ =M τ ψ. on (M, f ). We intend to use these ρ mn to describe leptons and hadrons. However, via encapsulation of classical spacetime, (M ,f ) remains only one internal chargeρ 00 , it falls short. It is impossible for the only one real-valued field functionρ 00 to describe so many leptons and hadrons.
On the premise of not abandoning the (1 + 3)-dimensional spacetime, if we want to describe gauge fields, there is a method that to use some non-coordinate abstract degrees of freedom that is irrelevant to view of time and space on the phase e iTaθ a of a complex-valued field function ψ, just like that in Standard Model. This way is effective, but not natural. It is not satisfying for a theory to adopt a coordinate representation for external space but a non-coordinate representation for internal space.
A logically more natural way is required to abandon the framework of (1 + 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 realvalued field functions ρ mn to describe leptons and hadrons, and enough internal components Γ m nP of affine connection to describe gauge potentials. Therefore, only on the full-dimensional (M, 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 spacetimeM , but to focus on the full-dimensional manifold M . and G M N = δ AB B A M B B N we know that gauge field and gravitational field can both be described by the unified spatial frames B A M and C M A of a reference-system f or g. Reference-system is the common origination of gauge field and gravitational field.
We adopt the components of Γ M N P with M, N ∈ {4, 5, · · · , D} to describe typical gauge fields such as electromagnetic, weak and strong interaction fields. The other components of Γ M N P describe gravitational field. Simple connection Γ M N P is the common affine connection representation of gauge field and gravitational field.
We adopt the components of ρ M N with M, N ∈ {4, 5, · · · , D} to describe the charges of leptons and hadrons. The meanings of the other components of ρ M N are not clear at present, maybe they could be used to describe dark matter.
On orthogonal (M, g) and (M, f ), there are full-dimensional field equations, i.e. affine Dirac equation and affine Yang-Mills equation which reflect the on-shell evolution directions ∇ρ and ∇t, respectively. Their quantum evolutions are described by the propagators in Definition 3.10.5 or Discussion 3.10.3.
If and only if L k : g → g ′ is an orthogonal transformation, L k sends s ρ (L) to It is evident that (i) Because ρ M N is determined by the reference-system f but not g, ρ M N remains unchanged under the transformation L k : g → g ′ .
(ii) L k makes ρ M N ;P , ψ M N and D M N P changed, i.e.
Therefore, in affine connection representation of gauge fields, the gauge transformations ψ → ψ ′ and D → D ′ essentially boil down to the reference-system transformation L k .
(iii) From the real-valued representation of action s M N (L) = L γ P ρ M N ;P + ε P 0 ρ M N ;P dx 0 , it can be seen evidently that L k : s ρ (L) → s ′ ρ (L) = s ρ (L). Remark 1. The above (ii) and (iii) show the unified gauge invariance and general covariance under the reference-system transformation L k . Remark 2. For a general (M, g), g is not necessarily orthogonal, so the corresponding action should be described by In this general case, Definition 3.6.2 and the method in Discussion 3.10.3 are also available and effective, where we take

Remark 3.
We see that the real-valued representation of action is more concise than the complexvalued representation of action. Hence, it is more convenient to adopt real-valued representations for field function, field equation and action. The research objects in the following sections are based on the following definition. We are going to use the simple connection Γ M N P to show the affine connection representations of electromagnetic, weak and strong interaction fields, and to adopt the real-valued representation ρ M N ;P to discuss the interactions between gauge fields and elementary particles.
We say f is a typical gauge field, and L [f ] is a typical gauge transformation.
We say f is a typical gauge field with gravitation, and L f is a gravitational gauge transformation.

Affine connection representation of the gauge field of weak-electromagnetic interaction
We say f is a weak and electromagnetic unified field. The reason for such naming lies in the following proposition.
Proof. According to the definition, the semi-metric of f satisfies that The metric of f satisfies that Concretely: Calculate the simple connection of f , that is Calculate the coefficients of curvature of f , that is then we obtain Hence, Similarly we also obtain F 1 Remark 5.1. Comparing the above conclusion and U (1) × SU (2) principal bundle theory, we know this proposition indicates that the reference-system f indeed can describe weak and electromagnetic field.
The following proposition shows an advantage of affine connection representation, that is to say, affine connection representation implies the chiral asymmetry of weak interaction, but U (1)×SU (2) principal bundle connection representation cannot imply it automatically.
and say the intrinsic geometric property A P is the electromagnetic potential, Z P is Z potential, W 1 P and W 2 P are W potentials. Then denote     □ Remark 5.2. The above proposition shows that the chiral asymmetry of leptons is a direct mathematical result of affine connection representation, and that it is not necessary to be postulated artificially like that in U Y (1) × SU L (2) theory. Discussion 5.1. According to section 2.4, the intrinsic geometry is the largest geometry on geometric manifold, and its geometric properties are the richest, so that any irregular smooth shape can be precisely characterized by B A M and C M A . Its equivalent transformation group is the trivial subgroup {e} that has only one element. We notice that: (1) Principal bundle connection representation starts from a very large group, and reduces symmetries in way of "symmetry breaking" to approach the target geometry.
(2) Affine connection representation starts from the smallest group {e}, and adds symmetries in way of "symmetry condition" to approach the target geometry.
Such two ways must lead to the same destination. They both go towards the same specific geometry.

Remark 5.3. Proposition 5.2 shows that:
(1) In affine connection representation of gauge fields, the coupling constant g is possessed of a geometric meaning, that is in fact the metric of internal space. But it does not have such a clear geometric meaning in U (1) × SU (2) principal bundle connection representation.
(2) At the most fundamental level, the coupling constant of Z P and that of A P are equal, i.e.
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. Then we have coupling constantsg Z =g A in the medium, and the Weinberg angle appears.
It is quite reasonable to consider a Higgs boson as a zero-spin pair of neutrinos, because in Standard Model, Higgs boson only couples with Z 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 leptons of three generations do not appear in Proposition 5.2, but it can spontaneously appear in Proposition 7.1 due to the affine connection representation of the gauge field that is given by Definition 7.1. We say f is a strong interaction field.

Affine connection representation of the gauge field of strong interaction
We say d 1 On (M, g) we denote We notice that there are just only three independent ones in U 1 where the coefficients matrix is non-singular. Proposition 6.1. Let λ a (a = 1, 2, · · · , 8) be the Gell-Mann matrices, and T a ≜ 1 2 λ a the generators of SU (3) group. When (M, g) satisfies the symmetry condition Γ (D−2)(D−2)P + Γ (D−1)(D−1)P + Γ DDP = 0, denote Thus, We just need to substitute the Gell-Mann matrices into A P = T a A a P and directly verify them. We say f is a typical unified gauge field.
Definition 7.3. Define the symmetry condition of unification: (1) Basic conditions, No.1:    Thus, the intrinsic geometric properties l and ν of f satisfy the following conclusions on (M, g).          l L;P = ∂ P l L − gl L Z P − gl R A P − gν ′ L W 1 P , l R;P = ∂ P l R − gl R Z P − gl L A P , ν L;P = ∂ P ν L − gν L Z P − gl ′ L W 1 P , ν R;P = ∂ P ν R − gν R Z P . (56) Proof. First, we compute the covariant differential of ρ mn of f . Substitute them into the previous equations, and we obtain that l L;P = ∂ P l L − gl L Z P − gl R A P − gν ′ L W 1 P , l R;P = ∂ P l R − gl R Z P − gl L A P , , ν L;P = ∂ P ν L − gν L Z P − gl ′ L W 1 P , ν R;P = ∂ P ν R − gν R Z P . □ Remark 7.1. Reviewing Discussion 5.1, we know the above proposition shows the geometric origin of MNS mixing of weak interaction from the perspective of intrinsic geometry. In affine connection representation of gauge fields, MNS mixing spontaneously appears as an intrinsic geometric property, therefore it is not necessary to be postulated artificially like that in the Standard Model.
In conventional physics, e, µ and τ have just only ontological differences, but they have no difference in mathematical connotation. By contrast, Proposition 7.1 tells us that leptons of three   In addition, this proposition also shows that the new framework has an advantage that it never causes the problem that a proton decays into a lepton like some GUTs do. For an individual up-type quark, this paper has not made progress on the proof yet. Anyway, it is significant that Proposition 7.3 shows the geometric connotation of the color confinement. It involves a natural geometric constraint of the curvatures among different dimensions. On the other hand, For the cases containing the gravitational effects, we just need to take the (2) of Definition 4.6.1 or a general reference-system defined in Definition 2.1.1.

Conclusions
1. An affine connection representation of gauge fields is established in this paper. It has the following main points of view.
(i) Geometric manifold in Definition 2.1.1 is more suitable than Riemannian manifold for describing physical space.
(ii) Simple connection in Definition 2.5.1 contains more geometric information than Levi-Civita connection, and is independent of Christoffel symbol. It can uniformly describe gauge field and gravitational field.
(iii) Time is the total metric with respect to all dimensions of internal coordinate space and external coordinate space.
(iv) On-shell evolution is the evolution along gradient direction.
(v) Quantum evolution is the evolution along distribution of gradient directions. It has a geometric meaning discussed in section 3.10.
2. In the affine connection representation of gauge fields, the following unified mathematical descriptions are achieved.
(i) Gauge field and gravitational field have a unified coordinate description. They can be represented by the same affine connection. The components of Γ M N P with M, N ∈ {4, 5, · · · , D} describe electromagnetic, weak and strong interaction fields. The other components of Γ M N P describe gravitational field.
(ii) Gauge field and elementary particle field have a unified geometric construction. They are both geometric entities constructed with semi-metric.
(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) 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.
(v) Several postulates in Standard Model are turned into geometric results that appears spontaneously in affine connection representation, so they are not necessary to be regarded as postulates anymore.