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. Poincar\'{e} 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 as SU(5). (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.


Background and purpose
We know that in gauge theory, the field strength and the gauge-covariant derivative 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 use Γ MN P to represent gauge potentials, it is not hard to find some specific conditions to turn the curvature tensor R M N P Q to Thus, R MN P Q can be used to represent field strength. In addition, for any ρ M , we see that Eq.(1) and Eq.(2) mean that the coupling constant g may have a geometric meaning, which originates from G RH . 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 group Gravi(3, 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 Gravi(3, 1) ⊗ Gauge(n), e.g. Poincaré gauge theory [1][2][3][4][5][6][7][8][9][10][11] and metric-affine gauge theory [12][13][14][15][16][17][18][19][20][21][22][23]. This way can be interpreted intuitively as 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

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 G MN = δ AB B A M B B N and G MN = δ AB C M A C N B , where B A M and C M A are semi-metrics, or to say frame fields. It is evident that semi-metric is more fundamental than metric, so we hope B A M or C M A is regarded as a unified frame field of gravitational field and gauge field, and the frame transformation of B A M or C M A is regarded as gauge transformation. Hence, we need a more general manifold (M, B A M ) rather than the Riemannian manifold (M, G). Next, we put metric and semi-metric 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 manifold (M, B A M ) that is defined by semi-metric. 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 field l, 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 connection Γ µ νρ of gravity is constructed by the metric g µν , however it is not explicit what geometric quantity the connection A a µ of gauge field is constructed by. By contrast, in the affine connection representation of this paper, we are able to use the semi-metrics B A M and C M A of internal coordinate space to endow particle fields l and ν and gauge fields A a µ 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 B a m and C m a of semi-metrics B A M and C M A with m, a ∈ {4, 5, · · · , D} are regarded as the frame field of electromagnetic, weak, and strong interactions. The other components of B A M and C M A are regarded as the frame field of gravitation. We take the affine connection as A BC is said to be a torsion-free simple connection. Thus, For the sake of simplicity, we firstly consider the affine connection representation of gauge fields without gravitation. That is to say, let 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, B A M that satisfies the following conditions: (i) B s i = δ s i , B a i = 0, B s m = 0; (ii) G ij = δ ij , G mn = const, G mi = 0; (iii) When m = n, G mn = 0. Thus, {M N P } = 0, [M N P ] = 0 in general. The components Γ mnP of Γ MN P = 1 2 [M N P ] with m, n ∈ {4, 5, · · · , D} describe gauge potentials of electromagnetic, weak, and strong interactions. We also use the affine connection Γ M N P to construct elementary particle fields ρ MN . The components ρ mn of ρ MN with m, n ∈ {4, 5, · · · , D} describe field functions of leptons and quarks.
The components G mn of G MN with m, n ∈ {4, 5, · · · , D} describe coupling constants of particle fields ρ mn and gauge potentials Γ mnP . The other components of G MN are the metrics of gravitational field. The other components of ρ MN and Γ MN P provide possible candidates for dark matters and their interactions.

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 section 2, 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 diffenrent from Kaluza-Klein theory [24][25][26] and string theories [27][28][29][30][31][32][33][34][35][36][37][38][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 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 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 Definition 3.1.1 and Remark 4.2.1 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 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 4.3 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 section 3.5 and section 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 section 3.6, Definition 4.3.1 and Discussion 4.3.1 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 with W 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 Proposition 5.2 and Proposition 7.1 for detail. (6) In section 7, 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.

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 2.1.1. Let M be a D-dimensional connected smooth real manifold. ∀p ∈ M , take a coordinate chart (U p , ϕ Up ) 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 Up 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 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 Eq.(6) 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 the manifold M We also speak of L f and R f as (affine) gauge transformations. represents external direct product.

Coordinate transformation of holonomic connection and frame transformation of gauge connection
Suppose there are reference-systems g and g on the manifold M , denote G g•g, and ∀p ∈ M , on the neighborhood U of p, g(p) and g(p) satisfy . On the geometric manifold (M, g) we define torsion-free simple connection D and its coefficients ( A BC ) g by Then, we can compute the absolute derivative of the frame field ∂ Thus, it is obtained that Denote D P (b g(p) ) C P D C ; thus we can define on (M, G) the required gauge connection, which is It is important that M N P G is not only an affine connection on (M, G), but also a connection on frame bundle. (I). M N P G as an affine connection. Under the coordinate transformation L k(p) : Consequently, the gauge connection M N P G is transformed according to Due to Eq. (11), under the coordinate transformation, the holonomic connection (II). M N P G as a connection on frame bundle. Under the frame transformation L k : (M, G) → (M, G ′ ), Consequently, the gauge connection M N P G is tranformed according to Eq.(11) and Eq. (14) show that M N P G is not only an affine connection, but also a connection on frame bundle.
Apply Eq.(11)∼ (14) to the curvature tensors then it is obtained that We see from Eq.(15) that the M N P Q G without gravitation is both a curvature tensor of affine connection and a curvature tensor on frame bundle, and that the (R G ) M N P Q with gravitation is a curvature tensor of affine connection, but not a curvature tensor on frame bundle. In other words, under the gauge transformation L k , M N P Q G and states. This shows that the gravitational field in (R G ) M N P Q makes the gauge frames B A M and C M A have physical effects.

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. s, i = 1, · · · , r ; a, m = r + 1, · · · , D.

The relation between time and space
On a 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.
Remark 3.1.1. 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 coordinates x µ (µ = 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.

Evolution path as a submanifold
Definition 3.2.1. Let there be reference-systems f , g, f, g on a manifold M , such that ∀p ∈ M , on the neighborhood U of p, Denote F f • f and G g • g, then we say F and G motion 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 Eq.(10) we know that in F and G, gauge fields originate from f and g, and gravitational fields (G F ) MN and (G G ) MN 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 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 which are sent onto the evolution path L.

Definition 3.2.2.
Let the time metrics of (U, ξ A ), (U, x M ), and (U, ζ A ) be dξ 0 , dx 0 , and dζ 0 , respectively. On U L U ∩ L p we have parameter equations Take f for example, according to Eq. (20), on U L we define Define dξ 0 dx 0 dξ 0 dx 0 and dx 0 dξ 0 dx 0 dξ 0 , which induce d dξ0 and d dx0 , such that d dξ0 , dξ 0 = 1, d dx0 , dx 0 = 1. So we can also defineb They determine the following smooth functions on the entire L, similar to section 2.2, that 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 dx0 = 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 Then, we have the following two propositions that are evidently true.
Proposition 3.3.2. The following conclusions are true.

On-shell evolution as a gradient
Let T be a smooth n-order tensor field. The restriction on where ∂ ∂x ⊗ dx represents the tensor basis generated by several ∂ ∂x M and dx M , and the tensor coefficients of T are concisely denoted by t : U → R.
Let D be a holonomic connection.
∀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, let U L U ∩ L. Denote t L t| UL and t L;0 t ;Q ε Q 0 , as well as More generally, suppose there is a tensor U u Q dx Q ⊗ ∂ ∂x ⊗ dx . In such a notation, all the indices are concisely ignored except Q. u Q dx Q uniquely determines a characteristic direction u Q ∂ ∂xQ .
If the system of 1-order linear partial differential equations t ;Q = u Q has a solution t, then it is true that where , and u 0 u Q ε Q 0 . 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 [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 J K µ . 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. 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 ).
(ii) Let the coefficients of curvature tensor of (M, F ) and (M, G) be K M N P Q and R M N P Q , respectively, i.e.
On an arbitrary evolution path L, we define Then, according to Applying the evolution lemma of Proposition 3.3.2 again, we obtain which is said to be (affine) Yang-Mills equation of F . It contains effects of gravitation, which makes the gauge frames (B f ) A M and (C f ) M A have physical effects. According to Eq.(15), we know Eq. (28) 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 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, and the current then by calculation, we find ρ M N 0 can still be non-vanishing. This shows that ρ M N 0 originates from (M, f) ultimately. Definition 3.5.1. We speak of the real-valued as the field function of a general charge or speak of it as a charge of F 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 4.3. Let For the sake of simplicity, denote the charge ρ MN 0 of F by ρ MN concisely. Let D be the holonomic connection of (M, G); then, ∂xQ . According to Proposition 3.4.1, if and only if ∀p ∈ L, the evolution direction is taken as that is, Definition 3.6.1. For more convenience, the notation ρ MN is further abbreviated as ρ. In affine connection representation, energy and momentum of ρ are defined as Proposition 3.6.1. At any point p on M , the equation holds if and only if the evolution direction 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: At any point p on M , the equations 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 field ρ is described by the gradient direction field ∇ρ. 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.
In particular, 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.(32), we obtain that Take γ P ρ ;P = ρ ;0 without loss of generality, and then, in the gradient direction of ρ, we have So we can take wherem τ is the rest-mass andx τ is the proper-time.
Remark 3.6.2. Define the following notations.
Then, through some calculations, we can obtain that which is the affine connection representation of general Lorentz force equation. See Discussion 4.3.1 for further illustrations.

Inversion transformation in affine connection representation
In affine conection representation, CP T 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-system k be Let the local coordinate representation of reference-system h be Time coordinate inversion can be represented as

Full 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, full inversion of metrics can be expressed as Denote time inversion by T T (M) T 0 , and then, the joint transformation of the full inversion of coordinates CP T 0 and the full 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 3.7.1. In quantum mechanics there is a complex conjugation in the time inversion of wave function In affine connection representation, we know the complex conjugation can be interpreted as a straightforward mathematical result of the full inversion of metrics T (M) .

Two dual descriptions of gradient direction field
Discussion 3.8.1. 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 [41], we obtain the Lie derivative equation Suppose ∀p ∈ M , Y (p) is a unit-length vector, i.e., ||Y (p)|| = 1. Let the parameter of ϕ Y be x 0 . Then, on the evolution path L ϕ Y,p , we have Thus, Eq. (41) can also be represented as On the other hand, ∀df ∈ T (M ) and df L π * (df ), due to (42) and Proposition 3.
If and only if taking Y = H, we speak of (43) and (44) as real-valued (affine) Heisenberg equation and (affine) Schrödinger equation, respectively, that is, Let ψ f e isL , where it is fine to take either s L s(L) or s L s(L) from Definition 3.6.2. According to Eq.(45), it is easy to obtain on L that This is consistent with the conventional Heisenberg equation and Schrödinger equation (taking the natural units that = 1, c = 1) and they have a coordinate correspondence We know that ∂ ∂t ↔ d dx 0 originates from the difference that the evolution parameter is x τ or x 0 . 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.(46) and Eq.(47) 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 section 3.9.

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. Remark 3.9.1. When T is fixed, H T can reflect the shape of (M, G). When a is fixed, the extension to |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 an embodiment of quantum uncertainty in affine connection representation. Definition 3.9.2. Let ϕ H be the one-parameter group of diffeomorphisms corresponding to H. The parameter of ϕ H is x 0 . ∀a ∈ M , according to Definition 3.2.1, let ϕ H,a be the evolution path through a, such that ϕ H,a (0) = a. ∀t ∈ R, denote ϕ |H|,a {ϕ X,a |X ∈ |H|}, ϕ |H|,a (t) {ϕ X,a (t)|X ∈ |H|}.
∀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).
For the sake of simplicity, denote L ϕ HT ,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 [42] ensures the existence of the following limit. The Radon-Nikodym derivative is said to be 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 HT ,a , that is, is said to be 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 property that is evidently true.
Definition 3.9.4. If L is a gradient line of some ρ ′ ∈ |ρ|, we also say L is a gradient line of |ρ|.
Remark 3.9.3. For any a 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, 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 3.9.1. 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 ∀L ∈ P(b, x 0 b ; a, x 0 a ), we can let L(x 0 a ) = a and L(x 0 b ) = b without loss of generality. Thus, P(b, x 0 b ; a, x 0 a ) is the totality of all the paths from L(x 0 a ) = a to L(x 0 b ) = 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 However, there are so many redundant paths in P(b, x 0 b ; a, x 0 a ) that (i) it is difficult to generally define a measure dL on P(b, x 0 b ; a, x 0 a ), 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 from P(b, We notice that as long as we take the probability amplitude Ψ be the totality of all the gradient lines of |ρ|, whose starting-direction is p a and ending-direction is In consideration of Remark 4.4.1, we let s be the affine action s(L) in Definition 3.6.2. We say the 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 3.9.2. Now (52) and (53) are strictly defined, but the Feynman path integral (51) 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 Fortunately, the following two reasons make us believe that Eq.(54) 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 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 "=" in Eq.(54) as the same thing.
On the other hand, on the condition of Proposition 3.9.1, denote is expected to be provable according to Eq.(50) and Eq.(52). However, to obtain a strict proof of Eq.(55) from Eq.(50) and Eq.(52) is not a trivial mathematical problem, which is necessary but not easy, and needs more mathematical research. Discussion 3.9.3. 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 according to Definition 3.6.2, where D is the holonomic connection of (M, G), then consider the distribution of H ∇ρ, and 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. 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 according to section 3.5, where D is the holonomic connection of (M, F ), then consider the distribution of H ∇t, and take H = ∇t; then, the wave function ψ(b, x 0 b ) that is defined by the equation describes 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.

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 .
The coordinate x µ works on the (1 + 3)-dimensional classical spacetime submanifold defined as follows.

Classical spacetime submanifold
Let there be a smooth tangent vector field X on (M, f ). If ∀p ∈ M , X(p) = b A ∂ ∂ξ A p = c M ∂ ∂x M p satisfies that b a are not all zero and c m are not all zero, where a, m = r + 1, · · · , D, then we say X is internal-directed. For any evolution path L ϕ X,p , we also say L is internal-directed.
Suppose M = 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 fieldX onM such that: (i) P × {o} is a closed submanifold ofM .
Let ϕ X : M ×R → M and ϕX :M ×R →M be the one-parameter groups of diffeomorphisms corresponding to X andX, respectively. Thus, we have ϕX = ϕ X |M ×R .
So the evolution in classical spacetime can be described by ϕX . It should be noticed that (i)M inherits a part of geometric properties of M , but not all. The physical properties reflected byM are incomplete.
(ii) 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.
(iii) 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.

Classical spacetime reference-system
Let there be a geometric manifold (M, f ) and its classical spacetime submanifoldM . 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 U L U ∩ L satisfy that Thus, it is true that: (1) There exists a unique local reference-systemf (p) onŨ U ∩M such that (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 satisfyξ That is to say,f (p) is just exactly the reference system in conventional sense, which has two different coordinate representations (58) and (59). We speak off :M → REFM , p →f (p) ∈ REF p as 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 byg.
(2) IfB α µ andC µ α are constants onM , theng is said to be flat. (3) Ifg is both orthogonal and flat, theng is said to be an inertial-system. In this case, Fg is just exactly a Lorentz transformation.

Remark 4.2.1. Due to
it is easy to know thatg is orthogonal if and only if dξ τ = dx τ , i.e.,G τ τ B τ τB τ τ = 1,G τ τ C τ τC τ τ = 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 non-trivial gravitational field. No matter whetherg is an inertial-system or not, and whether there is a non-trivial gravitation field or not, always both true in their respective coordinate frames.

Affine connection representation of classical spacetime evolution
LetD be the holonomic connection on (M ,G), 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) The geometric propertiesm τ ρ ;τ andm τ ρ ;τ are said to be the rest mass ofρ. (2)p µ −ρ ;µ andp µ −ρ ;µ are said to be the energy-momentum ofρ, andẼ 0 ρ ;0 ,Ẽ 0 ρ ;0 are said to be the energy ofρ.   Then for the same reason as Remark 3.6.2, based on Definition 4.3.1, we can strictly obtaiñ In the mass-point model,m τ ;ρ andε σ τ ;ρ do not make sense, so Eq.(60) turns intõ This is the affine connection representation of the force of interaction (e.g. the Lorentz force There are more illustrations in Remark 4.4.1.

Affine connection representation of Dirac equation
Discussion 4.4.1. Define Dirac algebras γ µ and γ α such that Without loss of generality, take γ µρ ;µ =m τ , that is Next, denote From Eq.(62), it is obtained that that is, We speak of the real-valued Eq.(62) and (64) as affine Dirac equations.

Discussion 4.4.2.
Next, we construct a kind of complex-valued representation of affine Dirac equation. The restriction of the chargeρ ων to (Ũ ,x µ ) is a functionρ ων (x µ ) with respect to the coordinates (x µ ) (x 0 ,x 1 ,x 2 ,x 3 ). LetP Suppose a function f ων = f ων (x µ ) on (Ũ ,x µ ) satisfies that 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 but only consider the evolution part L dx 0 . Meanwhile, in order to remain the covariance, L dx 0 has to be replaced by L dx τ . Thus, in affine connection representation of gauge fields, we shall consider an action in the form of Concretely speaking, denotẽ (65) Thus, we have obtained a complex-valued representation of gradient direction ofρ ων . This shows that s(L) ands(L) in Definition 3.6.2 and Remark 3.6.1 are indeed applicable for constructing propagator by e is(L) and e is(L) in affine connection representation of gauge fields. Therefore, the idea in Discussion 3.9.3 is reasonable. There are multiple internal charges ρ mn (m, n = 4, 5, · · · , D)

From classical spacetime back to full-dimensional space
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 , and 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 on the phase of e iTaθ a 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 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  and C M A 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 components [mnP ] of [M N P ] 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 ρ mn of ρ MN with m, n ∈ {4, 5, · · · , D} to describe the charges of leptons and hadrons. The physical meanings of the other components of ρ MN and [M N P ] are not clear at present; maybe they could be used to describe dark matters and their interactions.
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 3.9.5 or Discussion 3.9.3. If and only if L k : g → g ′ is an orthogonal transformation, L k sends s ρ (L) to where ρ MN is determined by the reference-system f • f but not g • g, so ρ MN does not vary with the transformation L k : 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 L k . For a general (M, G), G is not necessarily orthogonal, so the corresponding action should be described by

Remark 1.
In this general case, Definition 3.6.2 and the method in Discussion 3.9.3 are also available and effective, where we take s MN (L) = L Dρ MN .

Remark 2.
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 use [M N P ] to show the affine connection representations of electromagnetic, weak, and strong interaction fields and to adopt the real-valued representation ρ MN ;P to discuss the interactions between gauge fields and elementary particles. They are based on the following definition.
In the above extremely simplified case, we use F and G to show electromagnetic, weak, and strong interactions without gravitation.
It is obtained from Eq.(68) again that the semi-metric of (M, f ) satisfies Let s ′ , t ′ , i, j, k = 1, 2, 3; a ′ , b ′ , m, n, p = 4, 5, · · · , D. Compute the metric of (M, F ) and we obtain Compute the holonomic connection of F according to Γ M N P 1 Compute the coefficients of curvature of F , that is, then we obtain Hence, Then, F 1 P Q and F 2 P Q can also be computed similarly. ⊓ ⊔ Remark 5.1. Comparing the above conclusion and U (1)×SU (2) 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, but U (1) × SU (2) principal bundle connection representation cannot imply it spontaneously.
Denote (Γ G ) MN P by Γ MN P concisely. Then, we define on (M, G) that and say A P is (affine) electromagnetic potential, while Z P , W 1 P , and W 2 P are (affine) weak gauge potentials. Proposition 5.2. If (M, G) satisfies the symmetry condition Γ (D−1)DP = Γ D(D−1)P , then the geometric properties l and ν of F satisfy the following conclusions on (M, G), (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. Thus, we have coupling constants 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 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 three generations of leptons does not appear in Proposition 5.2, but it can spontaneously arise in Proposition 7.1 due to the affine connection representation of the gauge field that is given by Definition 7.1. Thus, F and G can describe strong interaction. Definition 6.2. According to Definition 3.5.1, let the charges of F be ρ mn , where m, n = 4, 5, · · · , D. Define We say d 1 and u 1 are red color charges, d 2 and u 2 are blue color charges, and d 3 and u 3 are green color charges. Then, d 1 , d 2 , d 3 are said to be down-type color charges, and u 1 , u 2 , u 3 are said to be up-type color charges. Their left-handed and right-handed charges are 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. Thus, it is not hard to find the following proposition true.
Thus, A P = T a A a P if and only if Thus, F and G can describe the unified field of electromagnetic, weak, and strong interactions.
Definition 7.2. According to Definition 3.5.1, let the charges of F be ρ mn , where m, n = 4, 5, · · · , D. Define On (M, G) we denote More generally, suppose we do not know what the symmetry that can exactly describe "the real world" is, we just denote it by S; then, the map makes us be able to turn the problem of seeking for S into the problem of seeking for a set of constraint conditions of B a m . "To describe S" and "to describe the constraint conditions of B a m " are equivalent to each other. Because gauge potentials Γ mnP and particle fields ρ mn are both constructed from the gauge frame field B a m , clearly here, it is more flexible and convenient "to describe the constraint conditions of B a m " 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 7.3. Similar to Remark 5.2, we define the constraint conditions as follows.
(1) 1st basic conditions: 2nd conditions of PMNS mixing of leptons: 1st conditions of CKM mixing of quarks: Then, the geometric properties l and ν of F satisfy the following conclusions on (M, G).
Proof. First, we compute the covariant differential of ρ mn of F .
According to Definition 7.2 and Definition 7.3, by calculation, we obtain that Then, according to definitions of l ′ and ν ′ , we obtain that   Then the geometric properties d 1 , d 2 , d 3 , u 1 , u 2 , u 3 of F satisfy the following conclusions on (M, G).
− g s d 3L X 12 P + g s d 1L Y 12 P − g s d 2L Y 12 P − gd ′ 3L W 1 P , u 1R;P = ∂ P u 1R − g s u 1R U 1 P + g s 2 u 2R X 12 P + g s 2 u 2R Y 12 P + g s 2 u 3R X 31 P − g s 2 u 3R Y 31 P , u 2R;P = ∂ P u 2R − g s u 2R U 2 P + g s 2 u 3R X 23 P + Proof. Substitute Definition 7.2 into ρ mn and consider Definition 7.3, then compute them, and then substitute d ′ 1L , d ′ 2L , d ′ 3L , u ′ 1L , u ′ 2L , u ′ 3L into them, we finally obtain the results. ⊓ ⊔ Remark 7.2. The above proposition shows a geometric origin of CKM mixing. We see that, in affine connection representation of gauge fields, d ′ 1L , d ′ 2L , d ′ 3L , u ′ 1L , u ′ 2L , u ′ 3L arise as geometric properties on manifold. Detailed equations of CKM mixing can be obtained on an additional condition such as we say F is a lepton, otherwise F is a hadron. Suppose F is a hadron. For d 1 , d 2 , d 3 , u 1 , u 2 , u 3 , if F satisfies that five of them are zero and the other one is non-zero, we say F is an individual quark. For an individual down-type quark, the above proposition is evidently true. Without loss of generality, let u 1 = u 2 = u 3 = 0 and d 1 = d 2 = 0; thus, ρ (D−2)(D−2) = ρ (D−1)(D−1) = ρ DD = 0; hence, we must have d 3 = 0.
For an individual up-type quark, this paper has not made progress on the proof yet. Nevertheless, Proposition 7.3 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.

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 Eq.(3) 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 section 3.9. 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 of Γ M N P describe gauge fields such as electromagnetic, weak, and strong interaction fields. The other parts of Γ M N P describe gravitational field.
(ii) Gauge field and elementary particle field are both geometric entities constructed from semi-metric. The components ρ mn of ρ MN with m, n ∈ {4, 5, · · · , D} describe leptons and quarks, the other components of ρ MN 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 quarks. In 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.