Affine Connection Representation of Gauge Fields

Man, Zhao-Hui

doi:https://doi.org/10.1155/2022/4573981

Advances in High Energy Physics

On this page

Abstract Introduction Conclusions Data Availability Disclosure Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 4573981 | https://doi.org/10.1155/2022/4573981

Affine Connection Representation of Gauge Fields

Zhao-Hui Man¹

Academic Editor: Sunny Vagnozzi

Received10 Sept 2021

Accepted14 Feb 2022

Published17 May 2022

Abstract

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

1. Introduction

1.1. Background and Purpose

We know that in gauge theory, the field strength and the gauge-covariant derivative both contain a coupling constant , which measures the strength of interaction. A problem is that why is there a coupling constant ?

If we represent gauge fields by affine connection, we can obtain a nice interpretation. For example, if we use to represent gauge potentials, it is not hard to find some specific conditions to turn the curvature tensor to

Thus, can be used to represent field strength. In addition, for any , we see that

Equations (2) and (3) mean that the coupling constant may have a geometric meaning, which originates from .

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 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 , e.g., Poincaré gauge theory [1–11] and metric-affine gauge theory [12–23]. 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

1.2. Ideas and Methods

We divide the problem of establishing affine connection representation of gauge fields into three parts as follows. (i)Which affine connection is suitable for describing not only gravitational field, but also gauge field and elementary particle field?(ii)How to describe the evolution of these fields in affine connection representation?(iii)What are the concrete forms of electromagnetic, weak, and strong interaction fields in affine connection representation?

For the problem (i). On a Riemannian manifold , the metric tensor can be expressed as and , where and are semimetrics or to say frame fields. It is evident that semimetric is more fundamental than metric, so we hope or is regarded as a unified frame field of gravitational field and gauge field, and the frame transformation of or is regarded as gauge transformation. Hence, we need a more general manifold rather than the Riemannian manifold .

Next, we put metric and semimetric together to construct a new connection, which is not only an affine connection, but also a connection on a fibre bundle. In this way, gravitational field and various gauge fields can be unified on a manifold that is defined by semimetric.

In addition, we notice that in the theories based on principal bundle connection representation, (1)Several complex-valued functions, which satisfy the Dirac equation, are sometimes used to refer to a charged lepton field and sometimes a neutrino field . It is not clear how to distinguish these field functions and by inherent geometric constructions(2)Gauge potentials are abstract; they have no inherent geometric constructions. In other words, the Levi-Civita connection of gravity is constructed by the metric ; however, it is not explicit what geometric quantity the connection of gauge field is constructed by

By contrast, in the affine connection representation of this paper, we are able to use the semimetrics and of internal coordinate space to endow particle fields and and gauge field 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 -dimensional manifold, the components and of semimetrics and with are regarded as the frame field of electromagnetic, weak, and strong interactions. The other components of and are regarded as the frame field of gravitation.

We take the affine connection as where is a local coordinate transformation, is Christoffel symbol, , is said to be a gauge connection, and is said to be a holonomic connection. . 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 and consider a -dimensional manifold that satisfies the following conditions: (i)(ii)(iii)When ,

Thus, , in general. The components of with describe gauge potentials of electromagnetic, weak, and strong interactions. We also use the affine connection to construct elementary particle fields . The components of with describe field functions of leptons and quarks.

The components of with describe coupling constants of particle fields and gauge potentials . The other components of are the metrics of gravitational field. The other components of and provide possible candidates for dark matters and their interactions.

1.3. Content and Organization

In this paper, we are going to show how to construct the affine connection representation of gauge fields. Sections are organized as follows.

Corresponding to the problem (i), in 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 dimensions. But in this paper, we define the concept of reference-system over the entire manifold. It is possessed of more dimensions but different from Kaluza-Klein theory [24–26] and string theories [27–39]. Thus, both of gravitational field and gauge field are regarded as special cases of such a concept of reference-system.

Corresponding to the problem (ii), in 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 -dimensional classical spacetime.

Corresponding to the problem (iii), in Sections 5–7, 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 2 and Remark 35 for detail). The CPT inversion is interpreted as the composition of full inversion of coordinates and full inversion of metrics (see Section 3.7 for detail). The conventional -dimensional Minkowski coordinate originates from the general -dimensional coordinate . 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 3.4–3.6 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 Sections 3.5 and 4.5 for detail)(4)Energy-momentum equation originates from a geometric property of gradient direction. We show the affine connection representation of mass, energy, momentum, and action (see Section 3.6, Definition 37, and Discussion 38 for detail). Furthermore, we also show the affine connection representation of Dirac equation (see Section 4.4 for detail)(5)Why do not neutrinos participate in the electromagnetic interactions? And why do not right-handed neutrinos participate in the weak interactions with bosons? In the theory of this paper, they are natural and geometric results of affine connection representation of gauge fields; therefore, they are not necessary to be regarded as postulates anymore (see Propositions 52 and 63 for detail)(6)In 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

2. Mathematical Preparations

2.1. Geometric Manifold

In order to make the languages that are used to describe gauge field and gravitational field unified and harmonized, we adopt the following definition.

Definition 1. Let be a -dimensional connected smooth real manifold. , take a coordinate chart on a neighborhood of . They constitute a coordinate covering which is said to be a point-by-point covering. For the sake of simplicity, can be denoted by and by .

Let and be two point-by-point coverings. For the two coordinate frames and on the neighborhood of point , if is a smooth homeomorphism, is called a local reference-system.

If every is endowed with a local reference-system and we require the semimetrics and in Equation (15) to be smooth real functions on , then is said to be a reference-system on , and is said to be a geometric manifold.

2.2. Metric and Semimetric

In the absence of a special declaration, the indices take values as and . The derivative functions of on define the semimetrics (or to say frame field) and of on the manifold that are

Let and . The metric tensors of are

Similarly, it can also be defined that and corresponding .

2.3. Gauge Transformation in Affine Connection Representation

, induces local reference-system transformations and reference-system transformations on the manifold

We also speak of and as (affine) gauge transformations. (i) and are identical transformations if and only if of is an identity matrix(ii) and are flat transformations if and only if , (iii) and are orthogonal transformations if and only if

The totality of all reference-system transformations on is denoted by , which is a subgroup of , where represents external direct product.

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

Suppose there are reference-systems and on the manifold , denote , and , on the neighborhood of , and satisfy

On the geometric manifold , we define torsion-free simple connection and its coefficients by

Then, we can compute the absolute derivative of the frame field

Thus, it is obtained that

Denote ; thus, we can define on the required gauge connection, which is

It is important that is not only an affine connection on , but also a connection on frame bundle. (i)as an Affine Connection. Under the coordinate transformation , , , . Consequently, the gauge connection is transformed according to Due to Equation (24), under the coordinate transformation, the holonomic connection is transformed according to (ii)as a Connection on Frame Bundle. Under the frame transformation . Consequently, the gauge connection is tranformed according to

Equations (24) and (27) show that is not only an affine connection, but also a connection on frame bundle.

Apply Equations (24)–(27) to the curvature tensors and then, it is obtained that

We see from Equation (29) that the without gravitation is both a curvature tensor of affine connection and a curvature tensor on frame bundle, and that the with gravitation is a curvature tensor of affine connection, but not a curvature tensor on frame bundle. In other words, under the gauge transformation , and represent the same physical state, while and represent different physical states. This shows that the gravitational field in makes the gauge frames and have physical effects.

3. The Evolution in Affine Connection Representation of Gauge Fields

Now that we have the required affine connection, next we have to solve the problem that how to describe the evolution in affine connection representation.

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

3.1. The Relation between Time and Space

Definition 2. Suppose and . Let On a geometric manifold , the and which are defined by are said to be total space metrics or time metrics. We also suppose and are regarded as proper-time metrics. For convenience, is said to be external space and is said to be internal space.

Remark 3. The above definition implies a new viewpoint about time and space. The relation between time and space in this way is different from the Minkowski coordinates . Time and space are not the components on an equal footing anymore, but have a relation of total to component. It can be seen later that time reflects the total evolution in the full-dimensional space, while a specific spatial dimension reflects just a partial evolution in a specific direction.

3.2. Evolution Path as a Submanifold

Definition 4. Let there be reference-systems , , , and on a manifold , such that , on the neighborhood of , Denote and ; then, we say and move relatively and interact mutually, and also we say that evolves in , or evolves on the geometric manifold . Meanwhile, evolves in , or we say evolves on .
From Equation (23), we know that in and , gauge fields originate from and , and gravitational fields and are effected by and , 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 , such that . Thus, determines a smooth tangent vector field on . If is nonzero everywhere, we say is a set of evolution paths, and is an evolution direction field. Let be an interval; then, the regular imbedding is said to be an evolution path through . The tangent vector is called an evolution direction at . For the sake of simplicity, we also denote ; then, is also a regular imbedding. If it is not necessary to emphasize the point , is denoted by concisely.

In order to describe physical evolution, next we are going to strictly describe the mathematical properties of the reference-systems and which are sent onto the evolution path .

Definition 5. Let the time metrics of , , and be , , and , respectively. On , we have parameter equations Take for example, according to Equation (37), on we define Define and , which induce and , such that and . So we can also define They determine the following smooth functions on the entire , similar to Section 2.2, that For convenience, we still use the notations and and have the following smooth functions. It is easy to verify that and are both true on by a simple calculation.

3.3. Evolution Lemma

We have the following two evolution lemmas. The affine connection representations of Yang-Mills equation, energy-momentum equation, and Dirac equation are dependent on them.

Definition 6. , suppose and are tangent spaces, and are cotangent spaces. The regular imbedding induces the tangent map and the cotangent map Evidently, is an injection, and is a surjection. , if and only if are true, we denote

Then, we have the following two propositions that are evidently true.

Proposition 7. If and , then

Proposition 8. The following conclusions are true.

3.4. On-Shell Evolution as a Gradient

Let be a smooth -order tensor field. The restriction on is , where represents the tensor basis generated by several and , and the tensor coefficients of are concisely denoted by .

Let be a holonomic connection. Consider . Denote

, the integral curve of , that is, , is a gradient line of . It can be seen later that the above gradient operator characterizes the on-shell evolution.

For any evolution path , let . Denote and , as well as

Proposition 9. The following conclusions are evidently true. (i) if and only if is an arbitrary evolution path(ii) if and only if is a gradient line of

Remark 10. More generally, suppose there is a tensor . In such a notation, all the indices are concisely ignored except . uniquely determines a characteristic direction .

If the system of 1-order linear partial differential equations has a solution , then it is true that and . Thus, in the evolution direction , the following conclusions are true. where , , and .

Now for any geometric property in the form of tensor , we are able to express its on-shell evolution in the form of .

Next, two important on-shell evolutions are discussed in the following two sections. One is the on-shell evolution of the potential field of a reference-system. The other is the one that a general charge of a reference-system evolves in the potential field of another reference-system.

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

Table I of article [40] proposes a famous correspondence between gauge field terminologies and fibre bundle terminologies. However, it does not find out the corresponding mathematical object to the source . 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 evolves in according to Definition 4, that is, ,

We always take the following notations in the coordinate frame . (i)Let the holonomic connections, which are defined by Equation (25), of geometric manifolds and be and , respectively. The colon “:” and the semicolon “;” are used to express the covariant derivatives on and , respectively, e.g.,(ii)Let the coefficients of curvature tensor of and be and , respectively, i.e.,

Denote . On an arbitrary evolution path , we define

Then, according to Definition 6 and the evolution lemma of Proposition 8, we obtain and

Let . Then, according to Proposition 9, if and only if , , we have

Applying the evolution lemma of Proposition 8 again, we obtain

Denote ; then, if and only if , we have which is said to be (affine) Yang-Mills equation of . It contains effects of gravitation, which makes the gauge frames and have physical effects. According to Equation (29), we know Equation (57) is coordinate covariant, and if gravitation is removed, it is also gauge covariant.

Thus, we have the following two results. (i)The Yang-Mills equation originates from a geometric property in the direction . In other words, the on-shell evolution of gauge field is described by the direction field (ii)We obtain the mathematical origination of charge and current. We know that the evolution path is an imbedding submanifold of . Thus, the charge originates from the pull-back from to , and the current originates from that is associated to

If we let be completely flat, i.e., , , then by calculation, we find can still be nonvanishing. This shows that originates from ultimately.

Definition 11. We speak of the real-valued as the field function of a general charge or speak of it as a charge of for short.

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

In order to be compatible with the affine connection representation of gauge fields, we also have to define mass, energy, momentum, and action in the form associated to affine connection. We are going to show them in this section and Section 4.3.

Let . For the sake of simplicity, denote the charge of by concisely. Let be the holonomic connection of ; then, where and . According to Proposition 9, if and only if , the evolution direction is taken as , we have that is,

Definition 12. For more convenience, the notation is further abbreviated as . In affine connection representation, energy and momentum of are defined as

Proposition 13. At any point on , the equation holds if and only if the evolution direction . Equation (63) is the (affine) energy-momentum equation of .

Proof. According to the above discussion, , is equivalent to Then, due to Proposition 7, we obtain the directional derivative in the gradient direction : i.e., , or .

Proposition 14. At any point on , the equations hold if and only if the evolution direction .

Proof. Due to the evolution lemma of Proposition 8, we immediately obtain Equation (66) from Equation (64).

Remark 15. In the gradient direction , Equation (66) is consistent with the conventional formula 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 is described by the gradient direction field .

Definition 16. Let be the totality of paths from to . And suppose , and the evolution parameter satisfies . The elementary affine action of is defined as

Thus, if and only if is a gradient line of .

In particular, in the case where is orthogonal, we can also define action in the following way.

On , let there be Dirac algebras and such that

In a gradient direction of , from Equation (63), we obtain that

Take without loss of generality, and then, in the gradient direction of , we have

So we can take

Remark 17 and Remark 41 explain the rationality of this definition. We have in the gradient direction of , so and are consistent.

Remark 17. In the Minkowski coordinate frame of Section 4.2, the evolution parameter is replaced by ; then, there still exists a concept of gradient direction . Correspondingly, Equations (68) and (72) present as where is the rest-mass and is the proper-time.

Remark 18. 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 38 for further illustrations).

3.7. Inversion Transformation in Affine Connection Representation

In affine connection representation, inversion is interpreted as a full inversion of coordinates and metrics. Let and .

Let the local coordinate representation of reference-system be , ; then, parity inversion can be represented as

Let the local coordinate representation of reference-system be , ; then, charge conjugate inversion can be represented as

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 be a closed submanifold of , and let its metric be . Denote the totality of closed submanifolds of by ; then, full inversion of metrics can be expressed as

Denote time inversion by and then, the joint transformation of the full inversion of coordinates and the full inversion of metrics is

Summarize the above discussions; then, we have

The invariance in affine connection representation is very clear. Concretely, on , we consider the transformation acting on . Denote and ; then, through simple calculations, we obtain that

Remark 19. 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 .

3.8. Two Dual Descriptions of Gradient Direction Field

Discussion 20. Let and be nonvanishing smooth tangent vector fields on the manifold . And let be the Lie derivative operator induced by the one-parameter group of diffeomorphism . Then, according to a well-known theorem [41], we obtain the Lie derivative equation

Suppose , is a unit-length vector, i.e., . Let the parameter of be . Then, on the evolution path , we have

Thus, Equation (85) can also be represented as

On the other hand, and , and due to (86) and Proposition 7, we have , that is,

Definition 21. Let . It is evident that , . If and only if taking , we speak of (87) and (88) as real-valued (affine) Heisenberg equation and (affine) Schrödinger equation, respectively, that is,

Discussion 22. The above two equations both describe the gradient direction field and thereby reflect on-shell evolution. Such two dual descriptions of gradient direction show the real-valued affine connection representation of Heisenberg picture and Schrödinger picture.

It is not hard to find out several different kinds of complex-valued representations of gradient direction. For examples, one is the affine Dirac equation in Section 4.4, and another is as follows.

Let , where it is fine to take either or from Definition 16. According to Equation (89), it is easy to obtain on that

This is consistent with the conventional Heisenberg equation and Schrödinger equation (taking the natural units that ) and they have a coordinate correspondence

We know that originates from the difference that the evolution parameter is or . The imaginary unit originates from the difference between the regular coordinates and the Minkowski coordinates . That is to say, the regular coordinates satisfy and the Minkowski coordinates satisfy

This causes the appearance of the imaginary unit in the correspondence

So Equations (90) and (91) have exactly the same essence, and their differences only come from different coordinate representations.

The differences between coordinate representations have nothing to do with the geometric essence and the physical essence. We notice that the value of a gradient direction is dependent on geometry, but independent of that the equations are real-valued or complex-valued. Therefore, it is unnecessary for us to confine to such algebraic forms as real-valued or complex-valued forms, but we should focus on such geometric essence as gradient direction.

The essential virtue of complex-valued form is that it is applicable for describing the coherent superposition of propagator. However, this is independent of the above discussions, and we are going to discuss it in Section 3.9.

3.9. Quantum Evolution as a Distribution of Gradient Directions

From Proposition 13, 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 is effected by the bending shape of ; in other words, the distribution of gradient directions can be used to reflect the shape of . 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 connection representation.

Definition 23. Let be a geometric property on , such as a charge of . Then, is a gradient direction field of on .
Let be the totality of all flat transformations defined in Section 2.3. , the flat transformation induces a transformation . Denote , the restriction of at are denoted by .

We say is the total distribution of the gradient direction field .

Remark 24. When is fixed, can reflect the shape of . When is fixed, the extension to can reflect the shape of .

However, when and are both fixed, is a fixed individual gradient direction, which cannot reflect the shape of . In other words, if the momentum and the position of are both definitely observed, the physical reality would be unknowable; therefore, this is unacceptable. This is an embodiment of quantum uncertainty in affine connection representation.

Definition 25. Let be the one-parameter group of diffeomorphisms corresponding to . The parameter of is . , according to Definition 4, let be the evolution path through , such that . , denote , we also denote and , the restriction of at is denoted by .

Remark 26. At the beginning , intuitively, the gradient directions of start from and point to all directions around uniformly. If is not flat, when evolving to a certain time , the distribution of gradient directions on is no longer as uniform as beginning. The following definition precisely characterizes this kind of ununiformity.

Definition 27. Let the transformation act on ; then, we obtain the trivial . Now is sent to a flat , and the gradient direction field of on is sent to a gradient direction field of on . Correspondingly, , is sent to . In a word, induces the following two maps: , deonte . Due to , let be a neighborhood of , with respect to the topology of .

Take ; then,

Let be a Borel measure on the manifold . We know ,

Thus, and are Borel sets, so they are measurable. Denote

When , we have , , , and .

For the sake of simplicity, denote . Thus, we have , and denote .

Because is absolutely continuous with respect to , Radon-Nikodym theorem [42] ensures the existence of the following limit. The Radon-Nikodym derivative is said to be the distribution density of along in position representation.

On a neighborhood of , , denote the normal section of by , that is,

Thus, and . If , we have and . The Radon-Nikodym derivative is said to be the distribution density of along in momentum representation.

In a word, and describe the density of the gradient lines that are adjacent to in two different ways. They have the following property that is evidently true.

Proposition 28. Let be a gradient line. such that , , , and ; then,

Definition 29. If is a gradient line of some , we also say is a gradient line of .

Remark 30. For any and , it anyway makes sense to discuss the gradient line of from to . It is because even if the gradient line of starting from does not pass through , it just only needs to carry out a certain flat transformation defined in Section 2.3 to obtain a ; thus, the gradient line of starting from can just exactly pass through . Due to , we do not distinguish them, and it is just fine to uniformly use . Intuitively speaking, when takes two different initial momentums, presents as and , respectively.

Discussion 31. With the above preparations, we obtain a new way to describe the construction of the propagator strictly.
For any path that starts at and ends at , we denote concisely. Let be the totality of all the paths from to . Denote , we can let and without loss of generality. Thus, is the totality of all the paths from to .

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 that (i) it is difficult to generally define a measure on , 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 to , where is the totality of all the gradient lines of from to . Thus, Equation (108) is turned into

We notice that as long as we take the probability amplitude of the gradient line such that in position representation or take in momentum representation, it can exactly be consistent with the Copenhagen interpretation. This provides the following new constructive definition for the propagator.

Definition 32. Suppose is defined as Definition 23, and denote .
Let be the totality of all the gradient lines of from to . Denote Let be the totality of all the gradient lines of , whose starting direction is and ending direction is . Denote Let be a Borel measure on . In consideration of Remark 41, we let be the affine action in Definition 16. We say the geometric property is the propagator of from to in position representation. If we let be a Borel measure on , then we say is the propagator of from to in momentum representation.

Discussion 33. Now (112) and (113) are strictly defined, but the Feynman path integral (108) has not been possessed of a strict mathematical definition until now. This makes it impossible at present to obtain (e.g., in position representation) a strict mathematical proof of

Fortunately, the following two reasons make us believe that Equation (114) is expected to be regarded as a strict definition of Feynman path integral; that is to say, the integral on the right-hand side of “=” can be regarded as the strict definition of the notation on the left-hand side of “=”.

On the one hand, we notice that the distribution densities and of gradient directions establish an association between probability interpretation and geometric interpretation of quantum evolution. Therefore, we can base on probability interpretation to intuitively consider both sides of “=” in Equation (114) as the same thing.

On the other hand, on the condition of Proposition 28, denote ; then, is expected to be provable according to Equations (106) and (112). However, to obtain a strict proof of Equation (115) from Equations (106) and (112) is not a trivial mathematical problem, which is necessary but not easy, and needs more mathematical research.

Discussion 34. The quantization methods of QFT are successful, and they are also applicable in affine connection representation, but in this paper, we do not discuss them. We try to propose some more ideas to understand the quantization of field in affine connection representation. (1)If we takeaccording to Definition 16, where is the holonomic connection of , then consider the distribution of , and we know that describes the quantization of energy-momentum. Every gradient line in 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 is the holonomic connection of , then consider the distribution of , Denote and take ; then, the wave function 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.

4. Affine Connection Representation of Gauge Fields in Classical Spacetime

The new framework established in Section 3 is discussed in the -dimensional general coordinate , which is more general than the -dimensional conventional Minkowski coordinate .

is the total metric of internal space and external space, and is the metric of internal space. (i)The evolution parameter of the -dimensional general coordinate is . The parameter equation of an evolution path is represented as (ii)The evolution parameter of the -dimensional Minkowski coordinate is . The parameter equation of is represented as

The coordinate works on the -dimensional classical spacetime submanifold defined as follows.

4.1. Classical Spacetime Submanifold

Let there be a smooth tangent vector field on . If , satisfies that are not all zero and are not all zero, where ; then, we say is internal-directed. For any evolution path , we also say is internal-directed.

Suppose , , and . is a smooth tangent vector field on . Fix a point . If is internal-directed; then, there exist a unique -dimensional imbedding submanifold and a unique smooth tangent vector field on such that (i) is a closed submanifold of (ii)The tangent map satisfies that ,

Such an is said to be a classical spacetime submanifold.

Let and be the one-parameter groups of diffeomorphisms corresponding to and , respectively. Thus, we have

So the evolution in classical spacetime can be described by . It should be noticed that (i) inherits a part of geometric properties of , but not all. The physical properties reflected by are incomplete(ii)The correspondence between and the restriction of to is one-to-one. For convenience, next we are not going to distinguish the notations and on but uniformly denote them by (iii)An arbitrary path on uniquely corresponds to a path on . Evidently, the image sets of and are the same, that is, . For convenience, later we are not going to distinguish the notations and on but uniformly denote them by

4.2. Classical Spacetime Reference-System

Let there be a geometric manifold and its classical spacetime submanifold . And let be an evolution path on . Suppose and is a coordinate neighborhood of . According to Definition 5, suppose the on and the on satisfy that

Thus, it is true that (1)There exists a unique local reference-system on such that(2)If is internal-directed, then the above coordinate frames and of uniquely determine the coordinate frames and such thatand the coordinates satisfy

That is to say, is just exactly the reference system in conventional sense, which has two different coordinate representations (123) and (124).

We speak of 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 . is a transformation induced by . (1)If , then is said to be (Lorentz) orthogonal. In this case, is just exactly a local Lorentz transformation(2)If and are constants on , then is said to be flat(3)If is both orthogonal and flat, then is said to be an inertial-system. In this case, is just exactly a Lorentz transformation

Remark 35. Due to it is easy to know that is orthogonal if and only if , i.e., . It is only in this case that we can denote and uniformly by ; otherwise, we should be aware of the difference between and in nontrivial gravitational field. No matter whether is an inertial system or not, and whether there is a nontrivial gravitation field or not, and are always both true in their respective coordinate frames.

Remark 36. The evolution lemmas in Section 3.3 can be expressed in Minkowski coordinate as follows: (i)If and , then (ii)The following conclusions are true

4.3. Affine Connection Representation of Classical Spacetime Evolution

Let be the holonomic connection on , and denote ; then, the absolute differential and gradient of Section 3.4 can be expressed on in Minkowski coordinate as

Evidently, if and only if is an arbitrary path. if and only if is the gradient line.

Definition 37. Similar to Section 3.6, suppose a charge of evolves on . We have the following definitions. (1)The geometric properties and are said to be the rest mass of (2) and are said to be the energy-momentum of , and and are said to be the energy of (3) and are said to be the canonical rest mass of (4) and are said to be the canonical energy-momentum of , and , are said to be the canonical energy of

Discussion 38. Similar to Proposition 13, , if and only if the evolution direction , the directional derivative is that is, , or which is the affine connection representation of energy-momentum equation.

Similar to Proposition 14, according to the evolution lemma, , if and only if the evolution direction , we have , that is and . This can also be regarded as the origin of .

Similar to Remark 18, denote

Then, for the same reason as Remark 18, based on Definition 37, we can strictly obtain

In the mass-point model, and do not make sense, so Equation (133) turns into

This is the affine connection representation of the force of interaction (e.g., the Lorentz force or of the electrodynamics).

Similar to Definition 16, let be the totality of paths on from point to point . And let and parameter satisfy . The affine connection representation of action in Minkowski coordinates can be defined as

There are more illustrations in Remark 41.

4.4. Affine Connection Representation of Dirac Equation

Discussion 39. Define Dirac algebras and such that

Suppose is orthogonal. According to Remark 35, . Due to Discussion 38, in a gradient direction of , we have

Without loss of generality, take , that is,

Next, denote

From Equation (138), it is obtained that that is,

We speak of the real-valued Equations (138) and (141) as affine Dirac equations.

Discussion 40. Next, we construct a kind of complex-valued representation of affine Dirac equation. The restriction of the charge to is a function with respect to the coordinates . Let

Suppose a function on satisfies that

We define and in the following way.

In the QFT propagator, we usually take in the path integral of a fermion in the form of where and 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 . Meanwhile, in order to remain the covariance, has to be replaced by . Thus, in affine connection representation of gauge fields, we shall consider an action in the form of

Concretely speaking, denote

From Equation (135), we have

And from Equation (140), we know . Then, it is obtained that

Thus, we have obtained a complex-valued representation of gradient direction of .

Remark 41. From the above discussion, we know in the gradient direction of that

This shows that and in Definition 16 and Remark 17 are indeed applicable for constructing propagator by and in affine connection representation of gauge fields. Therefore, the idea in Discussion 34 is reasonable.

4.5. From Classical Spacetime back to Full-Dimensional Space

Discussion 42. Now there is a problem. and cannot totally reflect the geometric properties of internal space of and . Concretely speaking, in the previous section, we discuss the affine Dirac equation on . Similar to Section 3.5, we have the affine Yang-Mills equation on . Suppose there is no gravitational field, then the remaining nonvanishing equations are just only There are multiple internal charges on . We intend to use these to describe leptons and hadrons. However, via encapsulation of classical spacetime, remains only one internal charge , and it falls short. It is impossible for the only one real-valued field function to describe so many leptons and hadrons.

On the premise of not abandoning the -dimensional spacetime, if we want to describe gauge fields, there is a method that to use some noncoordinate abstract degrees of freedom on the phase of of a complex-valued field function . This way is effective, but not natural. It is not satisfactory for a theory to adopt a coordinate representation for external space but a noncoordinate representation for internal space.

A logically more natural way is required to abandon the framework of -dimensional spacetime and . We should put internal space and external space together to describe their unified geometry with the same spatial frame. On and , there are enough real-valued field functions to describe leptons and hadrons and enough internal components of affine connection to describe gauge potentials.

Therefore, only on the full-dimensional and can total advantages of affine connection representation of gauge fields be brought into full play and thereby show complete details of geometric properties of gauge field. So we are going to stop the discussions about the classical spacetime , but to focus on the full-dimensional manifold .

Discussion 43. On , due to , , and , we know that gauge field and gravitational field can both be described by spatial frames and 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 of with to describe the gauge potentials of typical gauge fields such as electromagnetic, weak, and strong interaction fields and adopt the components of with to describe the charges of leptons and hadrons. The physical meanings of the other components of and are not clear at present; maybe they could be used to describe dark matters and their interactions.

On orthogonal and , there are full-dimensional field equations, i.e., affine Dirac equation and affine Yang-Mills equation which reflect the on-shell evolution directions and , respectively. Their quantum evolutions are described by the propagators in Definition 32 or Discussion 34.

Discussion 44. On an orthogonal , Equation (149) presents as a full-dimensional action If and only if is an orthogonal transformation, sends to where is determined by the reference-system but not , so does not vary with the transformation . We see that in affine connection representation of gauge fields, the gauge transformations and essentially boil down to the reference-system transformation .

Remark 45. For a general , is not necessarily orthogonal, so the corresponding action should be described by

In this general case, Definition 16 and the method in Discussion 34 are also available and effective, where we take

Remark 46. We see that the real-valued representation of action is more concise than the complex-valued representation of action. Hence, it is more convenient to adopt real-valued representations for field function, field equation, and action.

In the following sections, we are going to use to show the affine connection representations of electromagnetic, weak, and strong interaction fields and to adopt the real-valued representation to discuss the interactions between gauge fields and elementary particles. They are based on the following definition.

Definition 47. Let , and . Consider and that are defined by Equation (33), that is, , and furthermore, let () and both of and satisfy

In the above extremely simplified case, we use and to show electromagnetic, weak, and strong interactions without gravitation.

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

Definition 48. Suppose and conform to Definition 47. Let and both of and satisfy

Thus, and can describe weak and electromagnetic interactions.

Proposition 49. Let the holonomic connection of be and . And let the coefficients of curvature tensor of be and . Denote And denote . Thus, the following equations hold spontaneously.

Proof. Due to Equation (159), it is obtained that the semimetric of satisfies Then, compute , and we obtain It is obtained from Equation (159) again that the semimetric of satisfies Let . Compute the metric of , and we obtain Compute the holonomic connection of according to , and it is obtained that Compute the coefficients of curvature of , that is, and then, we obtain Hence, Then, and can also be computed similarly.

Remark 50. Comparing the above conclusion and principal bundle theory, we know this proposition shows that the reference-system 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 principal bundle connection representation cannot imply it spontaneously.

Definition 51. According to Definition 11, let the charges of the above reference-system be , where . Then, is said to be an electric charged lepton, and is said to be a neutrino. and are collectively denoted by . Thus, is said to be a left-handed lepton, and is said to be a right-handed lepton, denoted by Denote by concisely. Then, we define on that and say is (affine) electromagnetic potential, while , , and are (affine) weak gauge potentials.

Proposition 52. If satisfies the symmetry condition , then the geometric properties and of satisfy the following conclusions on ,

Proof. Let , . It follows from Equation (168) that

Then, Equations (172) and (173) lead to Equation (174).

Remark 53. From the above proposition, we see that some constraint conditions make the general linear group broken to a smaller group , i.e., 14.44" height="0.9025em" version="1.1" viewBox="0 0 236.84 14.44" width="14.8025em" x="0px" xml:space="preserve" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" y="0px"> c-0.322,0-1.123,0.029-1.445,0.029c-0.293,0-1.005-0.029-1.299-0.029c-0.068,0-0.185,0-0.185,0.195 c0,0.107,0.087,0.107,0.273,0.107c0.02,0,0.206,0,0.371,0.02c0.176,0.02,0.264,0.029,0.264,0.156 c0,0.039-0.01,0.068-0.039,0.186l-1.309,5.244c-0.098,0.381-0.117,0.459-0.889,0.459c-0.166,0-0.264,0-0.264,0.196 c0,0.107,0.088,0.107,0.264,0.107h4.521c0.234,0,0.244,0,0.303-0.166l0.771-2.109c0.04-0.107,0.04-0.126,0.04-0.136L13.35,9.48 c-0.088,0-0.098,0.049-0.167,0.205c-0.332,0.898-0.762,2.011-2.451,2.011H9.814c-0.137,0-0.156,0-0.215-0.01 c-0.098-0.009-0.126-0.02-0.126-0.098c0-0.029,0-0.049,0.049-0.224L10.84,6.091z M6.933,5.213L6.826,5.115 c-0.029,0-0.039,0.01-0.146,0.117L5.996,5.984C5.908,5.847,5.458,5.115,4.375,5.115C2.197,5.115,0,7.273,0,9.539 c0,1.552,1.084,2.675,2.666,2.675c0.43,0,0.869-0.088,1.221-0.234c0.488-0.195,0.674-0.4,0.85-0.596 c0.088,0.245,0.341,0.606,0.439,0.606l0.068-0.04c0.02-0.02,0.117-0.39,0.166-0.596l0.186-0.751 c0.039-0.166,0.087-0.332,0.126-0.499C5.83,9.666,5.839,9.646,6.396,9.636c0.049,0,0.157-0.01,0.157-0.195L6.425,9.333 c-0.225,0-0.801,0.029-1.025,0.029c-0.303,0-1.065-0.029-1.367-0.029c-0.088,0-0.206,0-0.206,0.195 c0,0.107,0.078,0.107,0.293,0.107c0.01,0,0.293,0,0.518,0.02c0.253,0.029,0.302,0.059,0.302,0.185 c0,0.088-0.107,0.527-0.205,0.889c-0.273,1.074-1.543,1.182-1.885,1.182c-0.938,0-1.963-0.557-1.963-2.05 c0-0.303,0.098-1.914,1.123-3.183c0.527-0.665,1.475-1.26,2.441-1.26c0.997,0,1.573,0.752,1.573,1.885 c0,0.39-0.03,0.4-0.03,0.498s0.107,0.098,0.146,0.098c0.127,0,0.127-0.02,0.176-0.195L6.933,5.213z"> c-1.104,0-1.826,0.899-1.826,1.768c0,0.547,0.489,0.547,0.518,0.547c0.166,0,0.507-0.117,0.507-0.518 c0-0.253-0.175-0.508-0.518-0.508l-0.127,0.01c0.225-0.635,0.752-0.997,1.318-0.997c0.889,0,1.309,0.791,1.309,1.592 c0,0.781-0.488,1.553-1.025,2.158l-1.875,2.089c-0.108,0.107-0.108,0.127-0.108,0.361h3.623l0.273-1.699H21.77 c-0.049,0.293-0.118,0.723-0.215,0.87c-0.069,0.078-0.713,0.078-0.928,0.078H18.87z M17.064,14.343c0-0.029,0-0.049-0.166-0.215 c-1.221-1.23-1.533-3.076-1.533-4.57c0-1.699,0.371-3.398,1.572-4.619c0.126-0.117,0.126-0.136,0.126-0.166l-0.098-0.098 c-0.098,0-0.976,0.665-1.552,1.905c-0.498,1.074-0.615,2.158-0.615,2.978c0,0.762,0.107,1.943,0.645,3.047 c0.586,1.201,1.425,1.836,1.523,1.836L17.064,14.343z"> c0.117,0,0.244-0.039,0.342-0.126l0.049-0.029c0.009,0,0.02,0.009,0.02,0.146c0,0.722-0.342,1.308-0.665,1.63 c-0.107,0.107-0.107,0.127-0.107,0.156l0.098,0.107C23.712,13.884,24.494,13.132,24.494,11.99z"> M28.14,11.658c0.079-0.176,0.079-0.45,0.079-0.538V6.189c0-0.098,0-0.361-0.079-0.537h1.114 c-0.127,0.156-0.127,0.352-0.127,0.498v4.97c0,0.098,0,0.362,0.079,0.538H28.14z M29.468,6.18c0-0.206,0-0.527,0.605-0.527 c0.83,0,1.24,0.341,1.24,1.494c0,1.26-0.303,1.445-1.845,1.445V6.18z M31.353,8.455c0.273-0.361,0.303-0.879,0.303-1.309 c0-0.468-0.059-0.986-0.371-1.387c0.4,0.088,1.367,0.391,1.367,1.377C32.651,7.771,32.358,8.269,31.353,8.455z M29.468,8.933 h0.351l1.465,2.266c0.098,0.156,0.332,0.537,0.439,0.683c0.068,0.117,0.097,0.117,0.322,0.117h1.348 c0.175,0,0.322,0,0.322-0.175l-0.146-0.166c-0.361-0.078-0.839-0.723-1.074-1.035c-0.068-0.098-0.556-0.742-1.21-1.797 c0.869-0.156,1.709-0.527,1.709-1.689c0-1.358-1.436-1.826-2.587-1.826h-3.066c-0.176,0-0.332,0-0.332,0.175 c0,0.167,0.186,0.167,0.264,0.167c0.556,0,0.605,0.068,0.605,0.556v4.893c0,0.488-0.049,0.557-0.605,0.557 c-0.078,0-0.264,0-0.264,0.166c0,0.175,0.156,0.175,0.332,0.175h2.675c0.176,0,0.323,0,0.323-0.175 c0-0.166-0.167-0.166-0.274-0.166c-0.557,0-0.595-0.078-0.595-0.557V8.933z"> c0,0.03,0,0.049,0.186,0.225c0.957,0.966,1.514,2.52,1.514,4.56c0,1.67-0.361,3.389-1.572,4.619 c-0.127,0.117-0.127,0.137-0.127,0.166l0.098,0.098c0.098,0,0.977-0.664,1.553-1.904C36.607,11.462,36.724,10.378,36.724,9.558z "> c0,1.06,0.807,1.927,2.119,1.927c0.601,0,1.237-0.157,1.592-0.54c0.068,0.15,0.253,0.396,0.335,0.396 c0.062,0,0.075-0.027,0.109-0.171l0.13-0.526c0.034-0.123,0.13-0.533,0.15-0.615c0.082-0.307,0.089-0.314,0.492-0.321 c0.048,0,0.15,0,0.15-0.144l-0.103-0.103c-0.225,0-0.512,0.027-0.751,0.027c-0.171,0-0.396,0-0.567-0.007 c-0.17-0.007-0.369-0.021-0.54-0.021c-0.048,0-0.15,0-0.15,0.15c0,0.096,0.075,0.096,0.246,0.096c0.13,0,0.267,0,0.396,0.014 c0.219,0.021,0.226,0.048,0.226,0.13c0,0.027,0,0.041-0.048,0.218c-0.048,0.191-0.123,0.5-0.15,0.567 c-0.198,0.417-0.833,0.602-1.367,0.602c-0.683,0-1.579-0.328-1.579-1.483c0-0.731,0.342-1.647,0.848-2.16 c0.704-0.697,1.435-0.806,1.818-0.806c0.792,0,1.257,0.574,1.257,1.339c0,0.191-0.021,0.239-0.021,0.28 c0,0.088,0.089,0.088,0.124,0.088c0.109,0,0.109-0.021,0.143-0.15L58.38,0.089z"> l-0.04-0.063c-0.863-0.687-1.019-1.677-1.019-2.306C59.805,3.698,60.355,2.914,60.848,2.572z"> c0,0.322-0.551,0.468-0.936,0.468c-0.258,0.181-0.473,0.4-0.629,0.648l0.098,0.078c0.136-0.185,0.361-0.39,0.624-0.39 c0.166,0,0.336,0.039,0.497,0.078c0.249,0.063,0.444,0.117,0.688,0.195c0.195,0.063,0.531,0.171,0.653,0.171 c0.346,0,0.931-0.512,1.297-0.849c0.2-0.434,0.322-0.814,0.322-1.282c0-0.581-0.483-1.112-1.136-1.268 c-0.385-0.087-0.687-0.098-1.043-0.098c-1.053,0-2.028,0.556-2.028,1.521l0.127,0.054c0.015-0.288,0.132-0.609,0.39-0.824 c0.346-0.292,0.863-0.371,1.35-0.371c0.551,0,1.107,0.112,1.497,0.439c0.307,0.253,0.404,0.614,0.404,0.96 c0,0.316-0.039,0.677-0.244,0.936c-0.205,0.259-0.327,0.415-0.688,0.415c-0.117,0-0.541-0.131-0.805-0.209 c-0.239-0.073-0.526-0.151-0.78-0.195c0.366-0.131,0.868-0.366,0.868-0.711c0-0.234-0.527-0.61-0.527-0.805 c0-0.185,0.351-0.438,0.498-0.517L63.97,3.386z"> c0,0.137,0.132,0.137,0.209,0.137H71.442z"> l-0.039-0.063c-0.863-0.687-1.019-1.677-1.019-2.306C79.514,3.698,80.065,2.914,80.558,2.572z M77.669,4.893 c0-1.653-1.252-2.438-1.35-2.438l-0.078,0.068l0.029,0.063c0.731,0.58,1.024,1.428,1.024,2.306c0,0.439-0.083,0.897-0.249,1.297 c-0.234,0.556-0.541,0.824-0.78,1.019l-0.024,0.053l0.078,0.068C76.417,7.33,77.669,6.55,77.669,4.893z M74.381,3.011 c0-0.146-0.015-0.146-0.176-0.146c-0.346,0.312-0.887,0.312-0.984,0.312h-0.083v0.214h0.083c0.112,0,0.429-0.015,0.711-0.136 v2.438c0,0.151,0,0.204-0.517,0.204h-0.244v0.215c0.268-0.02,0.702-0.02,0.985-0.02s0.716,0,0.984,0.02V5.896h-0.244 c-0.517,0-0.517-0.053-0.517-0.204V3.011z"> c0,0.322-0.551,0.468-0.936,0.468c-0.258,0.181-0.473,0.4-0.629,0.648l0.097,0.078c0.137-0.185,0.361-0.39,0.624-0.39 c0.166,0,0.336,0.039,0.497,0.078c0.249,0.063,0.444,0.117,0.688,0.195c0.195,0.063,0.531,0.171,0.653,0.171 c0.346,0,0.931-0.512,1.297-0.849c0.2-0.434,0.322-0.814,0.322-1.282c0-0.581-0.482-1.112-1.136-1.268 c-0.385-0.087-0.688-0.098-1.043-0.098c-1.053,0-2.028,0.556-2.028,1.521l0.126,0.054c0.015-0.288,0.132-0.609,0.39-0.824 c0.347-0.292,0.863-0.371,1.351-0.371c0.551,0,1.106,0.112,1.497,0.439c0.307,0.253,0.405,0.614,0.405,0.96 c0,0.316-0.039,0.677-0.244,0.936C85,5.722,84.878,5.877,84.517,5.877c-0.117,0-0.541-0.131-0.804-0.209 c-0.239-0.073-0.527-0.151-0.78-0.195C83.298,5.341,83.8,5.107,83.8,4.761c0-0.234-0.526-0.61-0.526-0.805 c0-0.185,0.351-0.438,0.497-0.517L83.678,3.386z"> c0,0.137,0.132,0.137,0.21,0.137H91.151z"> c0,0.439-0.083,0.897-0.248,1.297c-0.234,0.556-0.542,0.824-0.78,1.019L95.95,7.262l0.078,0.068 C96.125,7.33,97.378,6.55,97.378,4.893z M94.09,3.011c0-0.146-0.015-0.146-0.176-0.146c-0.346,0.312-0.887,0.312-0.985,0.312 h-0.083v0.214h0.083c0.112,0,0.429-0.015,0.712-0.136v2.438c0,0.151,0,0.204-0.517,0.204H92.88v0.215 c0.268-0.02,0.702-0.02,0.985-0.02s0.717,0,0.985,0.02V5.896h-0.244c-0.516,0-0.516-0.053-0.516-0.204V3.011z"> c0,0.164,0.164,0.164,0.253,0.164H103.859z M103.853,2.508c0.103,0,0.26,0,0.26-0.171c0-0.164-0.164-0.164-0.253-0.164h-4.523 c-0.089,0-0.253,0-0.253,0.164c0,0.171,0.157,0.171,0.259,0.171H103.853z"> 6,1.469-3.286,3.014c0,1.06,0.806,1.927,2.118,1.927c0.602,0,1.237-0.157,1.592-0.54 c0.068,0.15,0.253,0.396,0.335,0.396c0.062,0,0.076-0.027,0.109-0.171l0.13-0.526c0.035-0.123,0.131-0.533,0.15-0.615 c0.082-0.307,0.089-0.314,0.492-0.321c0.048,0,0.15,0,0.15-0.144l-0.103-0.103c-0.226,0-0.513,0.027-0.751,0.027 c-0.172,0-0.397,0-0.568-0.007s-0.369-0.021-0.539-0.021c-0.048,0-0.15,0-0.15,0.15c0,0.096,0.074,0.096,0.246,0.096 c0.129,0,0.266,0,0.396,0.014c0.219,0.021,0.227,0.048,0.227,0.13c0,0.027,0,0.041-0.048,0.218 c-0.048,0.191-0.124,0.5-0.151,0.567c-0.198,0.417-0.834,0.602-1.367,0.602c-0.683,0-1.578-0.328-1.578-1.483 c0-0.731,0.342-1.647,0.848-2.16c0.704-0.697,1.436-0.806,1.818-0.806c0.792,0,1.257,0.574,1.257,1.339 c0,0.191-0.021,0.239-0.021,0.28c0,0.088,0.089,0.088,0.123,0.088c0.109,0,0.109-0.021,0.144-0.15L110.358,0.089z"> c0,0.322-0.552,0.468-0.936,0.468c-0.259,0.18-0.474,0.399-0.629,0.648l0.098,0.078c0.136-0.186,0.36-0.39,0.624-0.39 c0.165,0,0.336,0.039,0.496,0.078c0.249,0.063,0.444,0.117,0.688,0.195c0.195,0.063,0.532,0.171,0.653,0.171 c0.347,0,0.932-0.512,1.297-0.849c0.2-0.434,0.321-0.814,0.321-1.282c0-0.581-0.482-1.112-1.136-1.268 c-0.385-0.087-0.688-0.098-1.043-0.098c-1.053,0-2.028,0.556-2.028,1.521l0.128,0.053c0.014-0.288,0.131-0.609,0.389-0.824 c0.347-0.292,0.863-0.371,1.351-0.371c0.552,0,1.106,0.112,1.497,0.439c0.308,0.253,0.405,0.614,0.405,0.96 c0,0.316-0.04,0.677-0.244,0.936c-0.205,0.258-0.327,0.414-0.688,0.414c-0.117,0-0.542-0.131-0.805-0.209 c-0.238-0.073-0.526-0.151-0.78-0.195c0.365-0.131,0.868-0.366,0.868-0.711c0-0.234-0.526-0.61-0.526-0.804 c0-0.186,0.352-0.439,0.498-0.517L119.114,3.12z M113.304,3.12c-0.41,0.176-0.906,0.453-0.906,0.834l0.312,0.414 c0.107,0.103,0.189,0.224,0.189,0.361c0,0.322-0.55,0.468-0.936,0.468c-0.259,0.18-0.473,0.399-0.629,0.648l0.098,0.078 c0.137-0.186,0.361-0.39,0.624-0.39c0.166,0,0.337,0.039,0.498,0.078c0.248,0.063,0.442,0.117,0.687,0.195 c0.195,0.063,0.531,0.171,0.654,0.171c0.345,0,0.93-0.512,1.296-0.849c0.199-0.434,0.322-0.814,0.322-1.282 c0-0.581-0.483-1.112-1.136-1.268c-0.386-0.087-0.688-0.098-1.044-0.098c-1.053,0-2.027,0.556-2.027,1.521l0.126,0.053 c0.016-0.288,0.132-0.609,0.391-0.824c0.346-0.292,0.862-0.371,1.351-0.371c0.55,0,1.106,0.112,1.495,0.439 c0.308,0.253,0.405,0.614,0.405,0.96c0,0.316-0.039,0.677-0.243,0.936c-0.205,0.258-0.327,0.414-0.688,0.414 c-0.116,0-0.54-0.131-0.804-0.209c-0.239-0.073-0.526-0.151-0.779-0.195c0.365-0.131,0.867-0.366,0.867-0.711 c0-0.234-0.527-0.61-0.527-0.804c0-0.186,0.352-0.439,0.498-0.517L113.304,3.12z"> c0,0.144-0.034,0.321-0.034,0.458c0,0.055,0.02,0.116,0.115,0.116c0.103,0,0.11-0.061,0.124-0.177l0.122-1.189 c0.021-0.205-0.026-0.205-0.198-0.205h-3.437c-0.13,0-0.219,0-0.219,0.15c0,0.096,0.082,0.096,0.219,0.096 c0.007,0,0.136,0,0.26,0.013c0.144,0.014,0.156,0.027,0.156,0.096l-0.026,0.15l-0.896,3.574 c-0.068,0.267-0.075,0.314-0.608,0.314c-0.116,0-0.205,0-0.205,0.144l0.109,0.103c0.198,0,0.704-0.027,0.901-0.027l0.526,0.007 c0.179,0,0.37,0.021,0.54,0.021c0.048,0,0.15,0,0.15-0.151c0-0.096-0.081-0.096-0.239-0.096c-0.014,0-0.184,0-0.355-0.013 c-0.144-0.014-0.211-0.021-0.211-0.116l0.02-0.123L137.691,0.642z M124.227,4.695c0,0.376-0.069,0.813-0.533,1.25l-0.048,0.082 l0.096,0.096c0.096,0,0.677-0.547,0.677-1.36c0-0.424-0.165-0.745-0.479-0.745c-0.227,0-0.39,0.178-0.39,0.39 c0,0.218,0.157,0.396,0.396,0.396C124.11,4.804,124.22,4.695,124.227,4.695z"> l-0.039-0.063c-0.863-0.687-1.019-1.677-1.019-2.306C140.715,3.698,141.266,2.914,141.758,2.572z"> c0,0.322-0.552,0.468-0.937,0.468c-0.259,0.181-0.473,0.4-0.629,0.648l0.098,0.078c0.136-0.185,0.36-0.39,0.624-0.39 c0.166,0,0.336,0.039,0.497,0.078c0.249,0.063,0.443,0.117,0.687,0.195c0.195,0.063,0.531,0.171,0.654,0.171 c0.346,0,0.931-0.512,1.297-0.849c0.199-0.434,0.321-0.814,0.321-1.282c0-0.581-0.482-1.112-1.136-1.268 c-0.385-0.087-0.688-0.098-1.044-0.098c-1.053,0-2.027,0.556-2.027,1.521l0.127,0.054c0.015-0.288,0.131-0.609,0.39-0.824 c0.347-0.292,0.863-0.371,1.351-0.371c0.551,0,1.106,0.112,1.497,0.439c0.307,0.253,0.404,0.614,0.404,0.96 c0,0.316-0.039,0.677-0.244,0.936c-0.205,0.259-0.326,0.415-0.687,0.415c-0.118,0-0.542-0.131-0.806-0.209 c-0.238-0.073-0.526-0.151-0.779-0.195c0.365-0.131,0.868-0.366,0.868-0.711c0-0.234-0.526-0.61-0.526-0.805 c0-0.185,0.35-0.438,0.496-0.517L144.88,3.386z"> c0,0.137,0.132,0.137,0.21,0.137H152.353z"> c0,0.439-0.084,0.897-0.249,1.297c-0.233,0.556-0.541,0.824-0.78,1.019l-0.023,0.053l0.077,0.068 C157.326,7.33,158.579,6.55,158.579,4.893z M155.29,3.011c0-0.146-0.014-0.146-0.175-0.146 c-0.347,0.312-0.887,0.312-0.984,0.312h-0.084v0.214h0.084c0.111,0,0.429-0.015,0.711-0.136v2.438c0,0.151,0,0.204-0.516,0.204 h-0.244v0.215c0.268-0.02,0.701-0.02,0.984-0.02c0.282,0,0.717,0,0.985,0.02V5.896h-0.244c-0.518,0-0.518-0.053-0.518-0.204 V3.011z"> c0,0.322-0.552,0.468-0.937,0.468c-0.258,0.181-0.473,0.4-0.628,0.648l0.097,0.078c0.137-0.185,0.36-0.39,0.624-0.39 c0.166,0,0.336,0.039,0.497,0.078c0.249,0.063,0.444,0.117,0.688,0.195c0.195,0.063,0.531,0.171,0.653,0.171 c0.346,0,0.932-0.512,1.297-0.849c0.2-0.434,0.321-0.814,0.321-1.282c0-0.581-0.482-1.112-1.136-1.268 c-0.385-0.087-0.688-0.098-1.043-0.098c-1.053,0-2.028,0.556-2.028,1.521l0.127,0.054c0.015-0.288,0.132-0.609,0.39-0.824 c0.347-0.292,0.863-0.371,1.351-0.371c0.551,0,1.106,0.112,1.497,0.439c0.307,0.253,0.403,0.614,0.403,0.96 c0,0.316-0.038,0.677-0.242,0.936c-0.205,0.259-0.327,0.415-0.688,0.415c-0.118,0-0.542-0.131-0.805-0.209 c-0.239-0.073-0.526-0.151-0.78-0.195c0.365-0.131,0.868-0.366,0.868-0.711c0-0.234-0.526-0.61-0.526-0.805 c0-0.185,0.351-0.438,0.497-0.517L161.943,3.386z"> 069,0.522-0.347,0.722c-0.277,0.2-0.658,0.22-0.854,0.22h-0.921L167.452,3.162z M167.102,4.634h1.063 c0.751,0,1.628-0.458,1.628-1.063c0-0.454-0.507-0.79-1.213-0.79h-1.921c-0.098,0-0.171,0-0.171,0.136 c0,0.083,0.058,0.083,0.181,0.083c0.049,0,0.209,0,0.312,0.029l-0.014,0.122l-0.64,2.554c-0.039,0.146-0.044,0.186-0.419,0.186 c-0.122,0-0.2,0-0.2,0.136l0.088,0.083c0.103,0,0.219-0.015,0.326-0.015c0.123,0,0.239-0.005,0.361-0.005 s0.239,0.005,0.36,0.005c0.112,0,0.234,0.015,0.342,0.015c0.028,0,0.121,0,0.121-0.137c0-0.083-0.067-0.083-0.16-0.083 l-0.21-0.01c-0.121-0.01-0.126-0.015-0.126-0.054l0.014-0.068L167.102,4.634z"> c0,0.164,0.164,0.164,0.253,0.164H175.991z M175.984,2.508c0.103,0,0.26,0,0.26-0.171c0-0.164-0.165-0.164-0.253-0.164h-4.524 c-0.089,0-0.253,0-0.253,0.164c0,0.171,0.157,0.171,0.26,0.171H175.984z"> c0,0.144-0.034,0.321-0.034,0.458c0,0.055,0.021,0.116,0.116,0.116c0.103,0,0.109-0.061,0.123-0.177l0.123-1.189 c0.021-0.205-0.027-0.205-0.198-0.205h-3.438c-0.13,0-0.218,0-0.218,0.15c0,0.096,0.081,0.096,0.218,0.096 c0.007,0,0.137,0,0.261,0.013c0.143,0.014,0.156,0.027,0.156,0.096l-0.027,0.15l-0.895,3.574 c-0.069,0.267-0.076,0.314-0.609,0.314c-0.115,0-0.204,0-0.204,0.144l0.108,0.103c0.198,0,0.705-0.027,0.902-0.027l0.526,0.007 c0.178,0,0.369,0.021,0.54,0.021c0.048,0,0.15,0,0.15-0.151c0-0.096-0.082-0.096-0.239-0.096c-0.014,0-0.185,0-0.355-0.013 c-0.144-0.014-0.212-0.021-0.212-0.116l0.021-0.123L179.559,0.642z"> c0,0.322-0.551,0.468-0.937,0.468c-0.258,0.181-0.473,0.4-0.628,0.648l0.098,0.078c0.137-0.185,0.36-0.39,0.624-0.39 c0.165,0,0.336,0.039,0.497,0.078c0.248,0.063,0.443,0.117,0.688,0.195c0.194,0.063,0.531,0.171,0.652,0.171 c0.347,0,0.932-0.512,1.297-0.849c0.2-0.434,0.321-0.814,0.321-1.282c0-0.581-0.482-1.112-1.135-1.268 c-0.386-0.087-0.688-0.098-1.043-0.098c-1.054,0-2.028,0.556-2.028,1.521l0.127,0.054c0.014-0.288,0.131-0.609,0.389-0.824 c0.347-0.292,0.863-0.371,1.351-0.371c0.552,0,1.107,0.112,1.497,0.439c0.308,0.253,0.404,0.614,0.404,0.96 c0,0.316-0.039,0.677-0.243,0.936c-0.205,0.259-0.327,0.415-0.688,0.415c-0.117,0-0.541-0.131-0.804-0.209 c-0.239-0.073-0.527-0.151-0.78-0.195c0.365-0.131,0.867-0.366,0.867-0.711c0-0.234-0.526-0.61-0.526-0.805 c0-0.185,0.352-0.438,0.498-0.517L184.098,3.386z"> l-0.038-0.063c-0.863-0.687-1.02-1.677-1.02-2.306C188.388,3.698,188.938,2.914,189.431,2.572z"> c0,0.322-0.55,0.468-0.936,0.468c-0.259,0.181-0.473,0.4-0.629,0.648l0.098,0.078c0.137-0.185,0.36-0.39,0.624-0.39 c0.165,0,0.336,0.039,0.497,0.078c0.248,0.063,0.443,0.117,0.688,0.195c0.195,0.063,0.531,0.171,0.652,0.171 c0.347,0,0.932-0.512,1.297-0.849c0.2-0.434,0.322-0.814,0.322-1.282c0-0.581-0.482-1.112-1.136-1.268 c-0.386-0.087-0.688-0.098-1.043-0.098c-1.054,0-2.027,0.556-2.027,1.521l0.126,0.054c0.015-0.288,0.132-0.609,0.39-0.824 c0.347-0.292,0.863-0.371,1.351-0.371c0.551,0,1.106,0.112,1.496,0.439c0.308,0.253,0.405,0.614,0.405,0.96 c0,0.316-0.039,0.677-0.244,0.936c-0.205,0.259-0.326,0.415-0.688,0.415c-0.116,0-0.54-0.131-0.804-0.209 c-0.239-0.073-0.526-0.151-0.78-0.195c0.366-0.131,0.867-0.366,0.867-0.711c0-0.234-0.526-0.61-0.526-0.805 c0-0.185,0.352-0.438,0.498-0.517L192.553,3.386z"> c0,0.137,0.131,0.137,0.21,0.137H200.024z"> c0,0.439-0.082,0.897-0.247,1.297c-0.235,0.556-0.542,0.824-0.78,1.019l-0.025,0.053l0.079,0.068 C204.999,7.33,206.252,6.55,206.252,4.893z M202.964,3.011c0-0.146-0.015-0.146-0.176-0.146 c-0.346,0.312-0.888,0.312-0.985,0.312h-0.082v0.214h0.082c0.112,0,0.43-0.015,0.713-0.136v2.438c0,0.151,0,0.204-0.518,0.204 h-0.244v0.215c0.269-0.02,0.703-0.02,0.985-0.02c0.283,0,0.717,0,0.984,0.02V5.896h-0.243c-0.517,0-0.517-0.053-0.517-0.204 V3.011z"> 069,0.522-0.347,0.722c-0.277,0.2-0.658,0.22-0.853,0.22h-0.922L209.314,3.162z M208.964,4.634h1.063 c0.75,0,1.628-0.458,1.628-1.063c0-0.454-0.508-0.79-1.214-0.79h-1.921c-0.098,0-0.17,0-0.17,0.136 c0,0.083,0.058,0.083,0.18,0.083c0.049,0,0.21,0,0.312,0.029l-0.014,0.122l-0.64,2.554c-0.038,0.146-0.043,0.186-0.419,0.186 c-0.122,0-0.199,0-0.199,0.136l0.087,0.083c0.103,0,0.22-0.015,0.327-0.015c0.122,0,0.238-0.005,0.36-0.005 s0.239,0.005,0.36,0.005c0.113,0,0.234,0.015,0.342,0.015c0.029,0,0.122,0,0.122-0.137c0-0.083-0.068-0.083-0.161-0.083 l-0.21-0.01c-0.121-0.01-0.126-0.015-0.126-0.054l0.015-0.068L208.964,4.634z"> 7,0.117,0.195,0.117c0.166,0,0.176-0.02,0.215-0.196c0.225-0.957,0.801-1.777,1.904-2.226 c0.117-0.039,0.146-0.059,0.146-0.127l-0.078-0.107c-0.43-0.166-1.611-0.654-1.982-2.294c-0.029-0.117-0.039-0.147-0.205-0.147 c-0.078,0-0.195,0-0.195,0.117c0,0.02,0.088,0.635,0.498,1.289c0.195,0.293,0.479,0.635,0.897,0.947h-7.254 c-0.176,0-0.353,0-0.353,0.195c0,0.196,0.177,0.196,0.353,0.196H224.533z M218.912,9.753c0.165,0,0.342,0,0.342-0.196 c0-0.195-0.177-0.195-0.342-0.195h-5.283c-0.165,0-0.342,0-0.342,0.195c0,0.196,0.177,0.196,0.342,0.196H218.912z M213.485,9.753c0.166,0,0.342,0,0.342-0.196c0-0.195-0.176-0.195-0.342-0.195h-5.282c-0.167,0-0.342,0-0.342,0.195 c0,0.196,0.175,0.196,0.342,0.196H213.485z M208.069,9.753c0.165,0,0.341,0,0.341-0.196c0-0.195-0.176-0.195-0.341-0.195h-5.283 c-0.166,0-0.342,0-0.342,0.195c0,0.196,0.176,0.196,0.342,0.196H208.069z M202.643,9.753c0.165,0,0.342,0,0.342-0.196 c0-0.195-0.177-0.195-0.342-0.195h-5.283c-0.166,0-0.342,0-0.342,0.195c0,0.196,0.176,0.196,0.342,0.196H202.643z M197.216,9.753c0.166,0,0.342,0,0.342-0.196c0-0.195-0.176-0.195-0.342-0.195h-5.282c-0.167,0-0.342,0-0.342,0.195 c0,0.196,0.175,0.196,0.342,0.196H197.216z M191.79,9.753c0.165,0,0.342,0,0.342-0.196c0-0.195-0.177-0.195-0.342-0.195h-5.283 c-0.166,0-0.342,0-0.342,0.195c0,0.196,0.176,0.196,0.342,0.196H191.79z M186.373,9.753c0.165,0,0.342,0,0.342-0.196 c0-0.195-0.177-0.195-0.342-0.195h-5.283c-0.166,0-0.342,0-0.342,0.195c0,0.196,0.176,0.196,0.342,0.196H186.373z M180.946,9.753c0.166,0,0.342,0,0.342-0.196c0-0.195-0.176-0.195-0.342-0.195h-5.282c-0.167,0-0.342,0-0.342,0.195 c0,0.196,0.175,0.196,0.342,0.196H180.946z M175.52,9.753c0.166,0,0.343,0,0.343-0.196c0-0.195-0.177-0.195-0.343-0.195h-5.282 c-0.166,0-0.343,0-0.343,0.195c0,0.196,0.177,0.196,0.343,0.196H175.52z M170.093,9.753c0.167,0,0.342,0,0.342-0.196 c0-0.195-0.175-0.195-0.342-0.195h-5.282c-0.166,0-0.342,0-0.342,0.195c0,0.196,0.176,0.196,0.342,0.196H170.093z M164.667,9.753c0.166,0,0.341,0,0.341-0.196c0-0.195-0.175-0.195-0.341-0.195h-5.283c-0.166,0-0.342,0-0.342,0.195 c0,0.196,0.176,0.196,0.342,0.196H164.667z M159.25,9.753c0.166,0,0.342,0,0.342-0.196c0-0.195-0.176-0.195-0.342-0.195h-5.283 c-0.165,0-0.342,0-0.342,0.195c0,0.196,0.177,0.196,0.342,0.196H159.25z M153.823,9.753c0.167,0,0.342,0,0.342-0.196 c0-0.195-0.175-0.195-0.342-0.195h-5.282c-0.166,0-0.342,0-0.342,0.195c0,0.196,0.176,0.196,0.342,0.196H153.823z M148.397,9.753c0.165,0,0.341,0,0.341-0.196c0-0.195-0.176-0.195-0.341-0.195h-5.283c-0.166,0-0.342,0-0.342,0.195 c0,0.196,0.176,0.196,0.342,0.196H148.397z M142.971,9.753c0.165,0,0.342,0,0.342-0.196c0-0.195-0.177-0.195-0.342-0.195h-5.283 c-0.165,0-0.342,0-0.342,0.195c0,0.196,0.177,0.196,0.342,0.196H142.971z M137.544,9.753c0.166,0,0.342,0,0.342-0.196 c0-0.195-0.176-0.195-0.342-0.195h-5.282c-0.167,0-0.342,0-0.342,0.195c0,0.196,0.175,0.196,0.342,0.196H137.544z M132.128,9.753c0.165,0,0.341,0,0.341-0.196c0-0.195-0.176-0.195-0.341-0.195h-5.283c-0.166,0-0.342,0-0.342,0.195 c0,0.196,0.176,0.196,0.342,0.196H132.128z M126.701,9.753c0.165,0,0.342,0,0.342-0.196c0-0.195-0.177-0.195-0.342-0.195h-5.283 c-0.166,0-0.342,0-0.342,0.195c0,0.196,0.176,0.196,0.342,0.196H126.701z M121.274,9.753c0.166,0,0.342,0,0.342-0.196 c0-0.195-0.176-0.195-0.342-0.195h-5.282c-0.167,0-0.342,0-0.342,0.195c0,0.196,0.175,0.196,0.342,0.196H121.274z M115.848,9.753c0.166,0,0.343,0,0.343-0.196c0-0.195-0.177-0.195-0.343-0.195h-5.282c-0.166,0-0.342,0-0.342,0.195 c0,0.196,0.176,0.196,0.342,0.196H115.848z M110.432,9.753c0.165,0,0.342,0,0.342-0.196c0-0.195-0.177-0.195-0.342-0.195h-5.283 c-0.166,0-0.342,0-0.342,0.195c0,0.196,0.176,0.196,0.342,0.196H110.432z M105.004,9.753c0.167,0,0.343,0,0.343-0.196 c0-0.195-0.176-0.195-0.343-0.195h-5.282c-0.167,0-0.342,0-0.342,0.195c0,0.196,0.175,0.196,0.342,0.196H105.004z M99.579,9.753 c0.166,0,0.341,0,0.341-0.196c0-0.195-0.175-0.195-0.341-0.195h-5.283c-0.166,0-0.341,0-0.341,0.195 c0,0.196,0.175,0.196,0.341,0.196H99.579z M94.151,9.753c0.166,0,0.342,0,0.342-0.196c0-0.195-0.176-0.195-0.342-0.195h-5.282 c-0.166,0-0.342,0-0.342,0.195c0,0.196,0.176,0.196,0.342,0.196H94.151z M88.725,9.753c0.166,0,0.342,0,0.342-0.196 c0-0.195-0.176-0.195-0.342-0.195h-5.283c-0.166,0-0.342,0-0.342,0.195c0,0.196,0.176,0.196,0.342,0.196H88.725z M83.309,9.753 c0.166,0,0.342,0,0.342-0.196c0-0.195-0.176-0.195-0.342-0.195h-5.283c-0.166,0-0.342,0-0.342,0.195 c0,0.196,0.176,0.196,0.342,0.196H83.309z M77.882,9.753c0.166,0,0.342,0,0.342-0.196c0-0.195-0.176-0.195-0.342-0.195h-5.283 c-0.166,0-0.342,0-0.342,0.195c0,0.196,0.176,0.196,0.342,0.196H77.882z M72.456,9.753c0.166,0,0.342,0,0.342-0.196 c0-0.195-0.176-0.195-0.342-0.195h-5.283c-0.166,0-0.342,0-0.342,0.195c0,0.196,0.176,0.196,0.342,0.196H72.456z M67.029,9.753 c0.166,0,0.342,0,0.342-0.196c0-0.195-0.176-0.195-0.342-0.195h-5.283c-0.166,0-0.342,0-0.342,0.195 c0,0.196,0.176,0.196,0.342,0.196H67.029z M61.612,9.753c0.166,0,0.342,0,0.342-0.196c0-0.195-0.176-0.195-0.342-0.195H56.33 c-0.166,0-0.342,0-0.342,0.195c0,0.196,0.176,0.196,0.342,0.196H61.612z M56.186,9.753c0.166,0,0.342,0,0.342-0.196 c0-0.195-0.176-0.195-0.342-0.195h-5.283c-0.166,0-0.342,0-0.342,0.195c0,0.196,0.176,0.196,0.342,0.196H56.186z M50.759,9.753 c0.166,0,0.342,0,0.342-0.196c0-0.195-0.176-0.195-0.342-0.195h-5.282c-0.166,0-0.342,0-0.342,0.195 c0,0.196,0.176,0.196,0.342,0.196H50.759z M46.846,9.753c0.166,0,0.342,0,0.342-0.196c0-0.195-0.176-0.195-0.342-0.195h-5.283 c-0.166,0-0.341,0-0.341,0.195c0,0.196,0.175,0.196,0.341,0.196H46.846z"> c0.118,0,0.244-0.039,0.342-0.126l0.049-0.029c0.01,0,0.021,0.009,0.021,0.146c0,0.722-0.342,1.308-0.664,1.63 c-0.107,0.107-0.107,0.127-0.107,0.156l0.098,0.107C236.06,13.884,236.84,13.132,236.84,11.99z M235.15,5.213l-0.107-0.098 c-0.049,0-0.06,0.01-0.176,0.146l-0.47,0.557c-0.253-0.459-0.762-0.703-1.396-0.703c-1.24,0-2.411,1.123-2.411,2.305 c0,0.791,0.518,1.24,1.016,1.386l1.045,0.273c0.36,0.088,0.898,0.234,0.898,1.035c0,0.878-0.801,1.796-1.758,1.796 c-0.625,0-1.709-0.215-1.709-1.425c0-0.235,0.048-0.469,0.059-0.527l0.02-0.069c0-0.097-0.068-0.107-0.117-0.107l-0.098,0.039 l-0.586,2.295l0.107,0.098c0.049,0,0.059-0.009,0.176-0.146l0.479-0.556c0.42,0.566,1.084,0.703,1.65,0.703 c1.327,0,2.479-1.298,2.479-2.509c0-0.674-0.332-1.006-0.479-1.143c-0.224-0.225-0.371-0.264-1.24-0.488 c-0.214-0.059-0.565-0.156-0.653-0.176c-0.265-0.088-0.597-0.371-0.597-0.889c0-0.791,0.782-1.621,1.71-1.621 c0.811,0,1.405,0.419,1.405,1.513c0,0.313-0.039,0.488-0.039,0.547c0,0.01,0,0.098,0.118,0.098c0.098,0,0.106-0.029,0.146-0.195 L235.15,5.213z"> viewBox="0 0 25.428 11.44" width="25.428pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">so that the chiral asymmetry of leptons arises in Equation (174) spontaneously.

Remark 54. Proposition 52 shows that (1)In affine connection representation of gauge fields, the coupling constant is possessed of a geometric meaning that it is in fact the metric of internal space. But it does not have such a clear geometric meaning in principal bundle connection representation(2)At the most fundamental level, the coupling constant of and that of are equal, i.e.,Suppose there is a kind of medium. boson and photon move in it. Suppose field has interaction with the medium, but electromagnetic field 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 field and 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 52, but it can spontaneously arise in Proposition 63 due to the affine connection representation of the gauge field that is given by Definition 59

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

Definition 55. Suppose and conform to Definition 47. Let and both of and satisfy

Thus, and can describe strong interaction.

Definition 56. According to Definition 11, let the charges of be , where . Define We say and are red color charges, and are blue color charges, and and are green color charges. Then, , , and are said to be down-type color charges, and , , and are said to be up-type color charges. Their left-handed and right-handed charges are On , we denote We notice that there are just only three independent ones in , , , , , and . Without loss of generality, let where the coefficients matrix is nonsingular. Thus, it is not hard to find the following proposition true.

Proposition 57. Let be the Gell-Mann matrices and the generators of group. When satisfies the symmetry condition , denote where Thus, if and only if

Remark 58. On the one hand, the above proposition shows that Definition 55 is an affine connection representation of strong interaction field. It does not define the gauge potentials as abstractly as that in principal -bundle theory but endows gauge potentials with concrete geometric constructions.

On the other hand, the above proposition implies that if we take appropriate symmetry conditions, the algebraic properties of group can be described by the transformation group of internal space of . In other words, the exponential map defines a homomorphism, and is a subgroup of . Therefore, Definition 55 is compatible with theory.

7. Affine Connection Representation of the Unified Gauge Field

Definition 59. Suppose and conform to Definition 47. Let and both of and satisfy Thus, and can describe the unified field of electromagnetic, weak, and strong interactions.

Definition 60. According to Definition 11, let the charges of be , where . Define And denote On , we denote

Discussion 61. We know from Section 2.3 that the gauge frame matrix ; therefore, when are without any constraints, we can obtain a gauge theory. In consideration of the fact that the exponential map
defines a homomorphism and is a subgroup of . So there must exist some constraint conditions of to make reduce to , i.e.,

More generally, suppose we do not know what the symmetry that can exactly describe “the real world” is, we just denote it by ; then, the map makes us be able to turn the problem of seeking for into the problem of seeking for a set of constraint conditions of . “To describe ” and “to describe the constraint conditions of ” are equivalent to each other.

Because gauge potentials and particle fields are both constructed from the gauge frame field , clearly here, it is more flexible and convenient “to describe the constraint conditions of ” than “to describe .”

Next, we have no idea what the best constraint conditions look like, but we can try to define a set of constraint conditions to see what can be obtained.

Definition 62. Similar to Remark 53, we define the constraint conditions as follows. (1)1st basic conditions (2)2nd basic conditions (3)1st conditions of PMNS mixing of leptons (4)2nd conditions of PMNS mixing of leptons (5)1st conditions of CKM mixing of quarks (6)2nd conditions of CKM mixing of quarks where are constants.

Proposition 63. When and satisfy the symmetry conditions (1), (2), (3), and (4) of Definition 62, denote

Then, the geometric properties and of satisfy the following conclusions on .

Proof. First, we compute the covariant differential of of . According to Definitions 60 and 62, by calculation, we obtain that Then, according to definitions of and , we obtain that Substitute them into the previous equations, and we obtain that

Remark 64. The above proposition shows the geometric origin of PMNS mixing of weak interaction. In affine connection representation of gauge fields, PMNS mixing arises as a geometric property on manifold.

In conventional physics, , , and have just only ontological differences, but they have no difference in mathematical connotation. By contrast, Proposition 63 tells us that leptons of three generations should be constructed by different linear combinations of . Thus, , , and may have concrete and distinguishable mathematical connotations. For example, let , , , , , , , and be constants; then, we might suppose that

Proposition 65. When and satisfy the symmetry conditions (1), (2), (5), and (6) of Definition 62, denote Then, the geometric properties , , , , , of satisfy the following conclusions on .

Proof. Substitute Definition 60 into and consider Definition 62, then compute them, and then substitute into them, and we finally obtain the results.

Remark 66. The above proposition shows a geometric origin of CKM mixing. We see that, in affine connection representation of gauge fields, arise as geometric properties on manifold. Detailed equations of CKM mixing can be obtained on an additional condition such as

Definition 67. A particle is not an existence at the place of an individual point, and its concept is defined on the entire manifold. Concretely speaking, if the reference-system satisfies we say is a lepton; otherwise, is a hadron.

Suppose is a hadron. For , , , , , , if satisfies that five of them are zero and the other one is nonzero, we say is an individual quark.

Proposition 68. There does not exist an individual quark. In other words, if any five ones of are zero, then .

For an individual down-type quark, the above proposition is evidently true. Without loss of generality, let and ; thus, ; hence, we must have .

For an individual up-type quark, this paper has not made progress on the proof yet. Nevertheless, Proposition 68 provides the color confinement with a new geometric interpretation, which is significant in itself. It involves a natural geometric constraint of the curvatures among different dimensions.

8. Conclusions

(1)An affine connection representation of gauge fields is established in this paper. It has the following main points of view(i)The holonomic connection Equation (6) contains more geometric information than Levi-Civita connection. It can uniformly describe gauge field and gravitational field(ii)Time is the total spatial metric with respect to all dimensions of internal coordinate space and external coordinate space(iii)Energy is the total momentum with respect to all dimensions of internal coordinate space and external coordinate space(iv)On-shell evolution is described by gradient direction(v)Quantum theory is a geometric theory of distribution of gradient directions. It has a geometric meaning discussed in 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 describe gauge fields such as electromagnetic, weak, and strong interaction fields. The other parts of describe gravitational field(ii)Gauge field and elementary particle field are both geometric entities constructed from semimetric. The components of with describe leptons and quarks, and the other components of 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.

Data Availability

No data were used to support this study.

Disclosure

Preprints have previously been published [43–47].

Conflicts of Interest

The author declares no conflicts of interest.

Acknowledgments

Many thanks are due to the preprint platforms for the opportunity of exhibiting the manuscript in the modification stage of this work. I would like to be extremely grateful to my dear family for their patience, tolerance, consideration, and love all through these years, which give me great confidence in completing this work. It is all worth it.

References

T. W. B. Kibble, “Lorentz invariance and the gravitational field,” Journal of Mathematical Physics, vol. 2, no. 2, pp. 212–221, 1961.
View at: Publisher Site | Google Scholar
D. Sciama, Recent Developments in General Relativity, Festschrift for Infeld, Pergamon, Oxford, UK, 1962.
J. P. Harnad and R. B. Pettitt, “Gauge theories for space-time symmetries. I,” Journal of Mathematical Physics, vol. 17, no. 10, pp. 1827–1837, 1976.
View at: Publisher Site | Google Scholar
P. Von der Heyde, “The field equations of the Poincaré gauge theory of gravitation,” Physics Letters A, vol. 58, no. 3, pp. 141–143, 1976.
View at: Publisher Site | Google Scholar
Y. M. Cho, “Gauge theory, gravitation, and symmetry,” Physical Review D, vol. 14, no. 12, pp. 3341–3344, 1976.
View at: Publisher Site | Google Scholar
F. W. Hehl, G. D. Kerlick, and J. M. Nester, “General relativity with spin and torsion: foundations and prospects,” Reviews of Modern Physics, vol. 48, no. 3, pp. 393–416, 1976.
View at: Publisher Site | Google Scholar
S. W. MacDowell and F. Mansouri, “Unified geometric theory of gravity and supergravity,” Physical Review Letters, vol. 38, no. 14, pp. 739–742, 1977.
View at: Publisher Site | Google Scholar
J. L. Chkareuli, “On the origin of Poincaré gauge gravity,” Physics Letters B, vol. 769, pp. 377–384, 2017.
View at: Publisher Site | Google Scholar
J. L. Chkareuli, “Poincaré gauge gravity an emergent scenario,” Physical Review D, vol. 95, no. 8, article 084051, 2017.
View at: Publisher Site | Google Scholar
Y. N. Obukhov, “Poincaré gauge gravity: an overview,” International Journal of Geometric Methods in Modern Physics, vol. 15, pp. 1840005–1840342, 2018.
View at: Publisher Site | Google Scholar
Y. N. Obukhov and F. W. Hehl, “General relativity as a special case of Poincaré gauge gravity,” Physical Review D, vol. 102, no. 4, article 044058, 2020.
View at: Publisher Site | Google Scholar
F. W. Hehl, G. D. Kerlick, and P. von der Heyde, “On hypermomentum in general relativity I. The notion of hypermomentum,” Ztschrift Fur Naturforschung A, vol. 31, no. 2, pp. 111–114, 1976.
View at: Publisher Site | Google Scholar
F. W. Hehl, G. D. Kerlick, and P. von der Heyde, “On hypermomentum in general relativity. II. The geometry of space-time,” Zeitschrift Fur Naturforschung A, vol. 31, no. 6, pp. 524–527, 1976.
View at: Publisher Site | Google Scholar
F. W. Hehl, G. David, H. Kerlick, and P. von der Heyde, “On hypermomentum in general relativity III. Coupling hypermomentum to geometry,” Zeitschrift Fur Naturforschung A, vol. 31, no. 7, pp. 823–827, 1976.
View at: Publisher Site | Google Scholar
F. W. Hehl, G. D. Kerlick, and P. von der Heyde, “On a new metric affine theory of gravitation,” Physics Letters B, vol. 63, no. 4, pp. 446–448, 1976.
View at: Publisher Site | Google Scholar
F. W. Hehl, E. A. Lord, and Y. Ne'eman, “Hadron dilation, shear and spin as components of the intrinsic hypermomentum current and metric-affine theory of gravitation,” Physics Letters B, vol. 71, no. 2, pp. 432–434, 1977.
View at: Publisher Site | Google Scholar
E. A. Lord, “The metric-affine gravitational theory as the gauge theory of the affine group,” Physics Letters A, vol. 65, no. 1, pp. 1–4, 1978.
View at: Publisher Site | Google Scholar
F. W. Hehl, J. D. Mccrea, and E. W. Mielke, “Metric-affine gauge theory of gravity: field equations, Noether identities, world spinors, and breaking of dilation invariance,” Physics Reports-review Section of Physics Letters, vol. 258, no. 1-2, pp. 1–171, 1995.
View at: Publisher Site | Google Scholar
R. F. Sobreiro and V. J. V. Otoya, “Affine gauge theory of gravity and its reduction to the Riemann-Cartan geometry,” Journal of Physics Conference, vol. 283, no. 1, 2011.
View at: Publisher Site | Google Scholar
R. F. Sobreiro and V. J. Vasquez Otoya, “On the topological reduction from the affine to the orthogonal gauge theory of gravity,” Journal of Geometry and Physics, vol. 61, no. 1, pp. 137–150, 2011.
View at: Publisher Site | Google Scholar
D. J. Cirilo-Lombardo, “Unified field theoretical models from generalized affine geometries,” International Journal of Theoretical Physics, vol. 49, no. 6, pp. 1288–1301, 2010.
View at: Publisher Site | Google Scholar
D. J. Cirilo-Lombardo, “Unified field theoretical models from generalized affine geometries II,” International Journal of Theoretical Physics, vol. 50, no. 6, pp. 1699–1708, 2011.
View at: Publisher Site | Google Scholar
D. J. Cirilo-Lombardo, “Unified field theoretical models from generalized affine geometries iii,” International Journal of Theoretical Physics, vol. 50, no. 11, pp. 3621–3634, 2011.
View at: Publisher Site | Google Scholar
T. Kaluza, “On the unification problem in physics,” International Journal of Modern Physics D, vol. 27, no. 4, pp. 966–972, 2018.
View at: Publisher Site | Google Scholar
O. Klein, “Quantentheorie und fünfdimensionale relativitätstheorie,” Zeitschrift für Physik A, vol. 37, no. 12, pp. 895–906, 1986.
View at: Publisher Site | Google Scholar
O. Klein, “The atomicity of electricity as a quantum theory law,” Nature, vol. 118, no. 2971, p. 516, 1926.
View at: Publisher Site | Google Scholar
P. Goddard, J. Goldstone, C. Rebbi, and C. B. Thorn, “Quantum dynamics of a massless relativistic string,” Nuclear Physics, vol. 56, no. 1, pp. 109–135, 1973.
View at: Publisher Site | Google Scholar
J. Polchinski, S. Chaudhuri, and C. V. Johnson, “Notes on D-branes,” 1996, https://arxiv.org/abs/hep-th/9602052.
View at: Google Scholar
W. Taylor, “Lectures on D-branes, Gauge Theory and M(atrices),” 1998, https://arxiv.org/abs/hep-th/9801182v2.
View at: Google Scholar
C. P. Bachas, “Lectures on D-branes,” 1999, https://arxiv.org/abs/hep-th/9806199v2.
View at: Google Scholar
A. Giveon and D. Kutasov, “Brane dynamics and gauge theory,” Review of Modern Physics, vol. 71, no. 4, pp. 310–318, 1999.
View at: Publisher Site | Google Scholar
J. Wess and B. Zumino, “Supergauge invariant extension of quantum electrodynamics,” Nuclear Physics B, vol. 78, no. 1, pp. 1–13, 1974.
View at: Publisher Site | Google Scholar
A. Hanany and E. Witten, “Type IIB superstrings, BPS monopoles, and three-dimensional gauge dynamics,” Nuclear Physics B, vol. 492, no. 1-2, pp. 152–190, 1997.
View at: Publisher Site | Google Scholar
M. R. Douglas, S. Govindarajan, T. Jayaraman, and A. Tomasiello, “D-branes on Calabi–Yau manifolds and superpotentials,” Communications in Mathematical Physics, vol. 248, no. 1, pp. 85–118, 2004.
View at: Publisher Site | Google Scholar
H. E. Haber and G. L. Kane, “The search for supersymmetry: probing physics beyond the standard model,” Physics Reports, vol. 117, no. 2-4, pp. 75–263, 1985.
View at: Publisher Site | Google Scholar
J. F. Gunion and H. E. Haber, “Higgs bosons in supersymmetric models (I),” Nuclear Physics B, Particle Physics, vol. 272, no. 1, pp. 1–76, 1986.
View at: Publisher Site | Google Scholar
J. F. Gunion and H. E. Haber, “Higgs bosons in supersymmetric models (II). Implications for phenomenology,” Physics, vol. 278, no. 3, pp. 449–492, 1986.
View at: Publisher Site | Google Scholar
J. F. Gunion and H. E. Haber, “Higgs bosons in supersymmetric models (III). Decays into neutralinos and charginos,” Physics, vol. 307, no. 3, pp. 445–475, 1988.
View at: Publisher Site | Google Scholar
J. F. Gunion and H. E. Haber, “Errata for Higgs bosons in supersymmetric models: I, II and III,” 1993, https://www.arxiv.org/abs/hep-ph/9301205v1.
View at: Google Scholar
T. T. Wu and C. N. Yang, “Concept of nonintegrable phase factors and global formulation of gauge fields,” Physical Review D, vol. 12, no. 12, pp. 3845–3857, 1975.
View at: Publisher Site | Google Scholar
S. S. Chern, W. H. Chen, and K. S. Lam, Lecture on Differential Geometry, World Scientific Publishing Co. Pte. Ltd., 1999.
O. Nikodym, “Sur une généralisation des intégrales de M. J. Radon,” Fundamenta Mathematicae, vol. 15, no. 1, pp. 131–179, 1930.
View at: Publisher Site | Google Scholar
Z. H. Man, “Affine connection representation of gauge fields,” 2020, https://arxiv.org/abs/2004.02223.
View at: Google Scholar
Z. H. Man, “Affine connection representation of gauge fields,” 2021, https://vixra.org/abs/2111.0054.
View at: Google Scholar
Z. H. Man, “Affine connection representation of gauge fields,” Zenodo, 2020, https://doi.org/10.5281/zenodo.4040036.
View at: Google Scholar
Z. H. Man, “Intrinsic geometry and constructivity methods for Hilbert’s 6th problem,” Zenodo, 2019, https://doi.org/10.5281/zenodo.3514523.
View at: Google Scholar
Z. H. Man, “Affine connection representation of gauge fields,” Research Square, 2021, https://www.researchsquare.com/article/rs-173799.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2022 Zhao-Hui Man. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation
Download other formats
Order printed copies
Views
298
Downloads
477
Citations