aa r X i v : . [ m a t h . G M ] A p r Noname manuscript No. (will be inserted by the editor)
A generalization of intrinsic geometry and its application toHilbert’s 6th problem
Zhao-Hui Man the date of receipt and acceptance should be inserted later
Abstract
The first main work of this paper is to generalize intrinsic geometry. (1) Riemannianmanifold is generalized to geometrical manifold. (2) The expression of Erlangen program isimproved, and the concept of intrinsic geometry is generalized, so that Riemannian intrinsicgeometry which is based on the first fundamental form becomes a subgeometry of the generalizedintrinsic geometry. The Riemannian geometry is thereby incorporated into the geometricalframework of improved Erlangen program. (3) The important concept of simple connection isdiscovered, which reflects more intrinsic properties of manifold than Levi-Civita connection.The second main work of this paper is to apply the generalized intrinsic geometry to Hilbert’s6th problem at the most basic level. (1) It starts from an axiom and makes key principles, postu-lates and artificially introduced equations of fundamental physics all turned into theorems whichautomatically hold in intrinsic geometrical theory. (2) Intrinsic geometry makes gravitationalfield and gauge field unified essentially. Intrinsic geometry of external space describes gravi-tational field, and intrinsic geometry of internal space describes typical gauge field. They areunified into intrinsic geometry. (3) Intrinsic geometry makes gravitational theory and quantummechanics have the same view of time and space and unified description of evolution.
Keywords
Erlangen program · geometrical manifold · intrinsic geometry · simple connection · Riemannian geometry · reference-system · time metric · actual evolution Mathematics Subject Classification (2010)
Primary 58A05, 51P05, 70A05 · Secondary53C05, 53Z05
Contents
In 1827, Friedrich Gauss created the theory of intrinsic geometry of surface in his article
Disquisitiones Generales Circa Superficies Curves , and essentially unified the Euclidean geom-etry and various non-Euclidean geometries at that time. In 1854, Bernhard Riemann generalizedGauss’s ideas of intrinsic geometry to high dimensions in his speech
Ueber Die Hypothesen,Welche Der Geometrie Zu Grunde Liegen , and expressed the first fundamental form with met-ric tensor. In Riemannian geometry, all values of intrinsic geometrical properties are totallydetermined by metric tensor.However, in this paper it is discovered that the traditional practice that characterizing intrinsicgeometry with metric tensor is not necessarily able to cover all the intrinsic properties ofmanifold. It has been known for a long time that the coefficients of metric tensor surely remainunchanged when orthogonal transformations effect on semi-metric [14, 19]. Although therehave been many studies on semi-metric, and some non-mathematical researches have shownadvantages of semi-metric [13, 14, 53], however their mathematical meanings have not beenstudied sufficiently. Some articles treat semi-metric either as an alternative expression form ofmetric, or as an insignificant mathematical substitution [14]. And there is an article [11] whichhas noticed that semi-metric causes some indications beyond Riemannian geometry, but it arguesthat a well-done semi-metric theory should not cause such indications. We must see that themathematical significance of semi-metric has not yet been fully revealed. generalization of intrinsic geometry and its application to Hilbert’s 6th problem 3
It is noticed that a traditional intrinsic geometrical property is an invariant under identicaltransformations of metric, and we can also speak of it as an invariant under orthogonal transfor-mations of semi-metric. It indicates that two different properties under two different orthogonaltransformations of semi-metric cannot be distinguished by the traditional intrinsic geometry. Ifwe take different orthogonal transformations of semi-metric at different points of a Riemannianmanifold, the Riemannian manifold remains unchanged, and these orthogonal transformationsjust exactly constitute a local gauge transformation. That is to say, Riemannian geometry is stillnot exquisite enough. Therefore, there exists a kind of more extensive intrinsic geometry withRiemannian geometry as its subgeometry.In order to understand the generalized concept of intrinsic geometry defined later moreconveniently, the intuition of intrinsic geometry must be concisely explained here in a new waydifferent from the perspective of metric of traditional intrinsic geometry.
Fig. 1
The intuition of intrinsic geometry of curve
First, consider the case of one-dimension, that is the intuition of intrinsic geometry of curve.As shown in Fig.1, select a curve L in the plane rectangular coordinate system. Project thecoordinates of y axis onto L continuously and uniformly, and then onto x axis.In this way, the original continuous and uniform coordinate distribution becomes a continuousbut ununiform distribution via L as a medium, thus we obtain an interval S with some ununiformdistribution shown in the right figure of Fig. 1. This is actually the intuition of intrinsic geometryof curve L . It can be said that curve S is curve L in intrinsic geometry.Such an intuition of intrinsic geometry can be described strictly in the following way.Let S be a one-dimensional manifold, which is homeomorphic to an Euclidean straight line.Take two coordinate charts ( S, x ) and ( S, y ) on S such that we have a coordinate relation y = y ( x ) .As shown in the above figure, at every point of S , it shows a kind of intuition reflecting thedegree of slackness and tightness of coordinate distribution of y axis in x axis. Such a degree ofslackness and tightness can be strictly described by dydx . Then the one-dimensional manifold S given the degree of slackness and tightness dydx is the curve L defined in way of intrinsic geometry. Zhao-Hui Man
Fig. 2
The intuition of intrinsic geometry of surface
The case of two-dimensional surface is similar. The intuition of intrinsic geometry of surface z = e − ( x + y ) in the left figure of Fig.2 can be shown by the degree of slackness and tightnessof coordinate net (cid:0) u , u (cid:1) in coordinate system (cid:0) x , x (cid:1) at each point of the right figure of Fig.2.This degree of slackness and tightness can be strictly described by ∂u k ∂x i ( i, k = 1 , . It can be saidthat the degree of slackness and tightness ∂u k ∂x i of the right figure defines the surface of the leftfigure in way of intrinsic geometry. The coordinate net in this figure shows a special solution of (cid:0) u , u (cid:1) such that u (cid:0) x , x (cid:1) = x p ( x ) + ( x ) Z √ ( x ) +( x ) q ρ e − ρ dρ,u (cid:0) x , x (cid:1) = x p ( x ) + ( x ) Z √ ( x ) +( x ) q ρ e − ρ dρ. These above are intuitive descriptions of two simple cases about one and two dimensions,emphasizing the central role of the degree of slackness and tightness ∂u k ∂x i determined by twocoordinate systems in reflecting the intuition of intrinsic geometry. In this way, the generalconcept of intrinsic geometry will be defined strictly in this paper.Such a generalized intrinsic geometry is worth studying. It is discovered in this paper, thatthe geometrical properties in such a geometry can just exactly reflect the physical propertiesof elementary particles in the Standard Model, which cannot be described by the traditionalintrinsic geometry. With such a generalization, it is not only gravitational field but also gaugefield, that can be described by intrinsic geometry, and further more the whole fundamentalphysics will be unified in the intrinsic geometry. Therefore, it will naturally give a solution forHilbert’s 6th problem. (ii) Hilbert’s 6th problem. The purpose of Hilbert’s 6th problem is to axiomatize the physics. Theoretical physics atthe most basic level is an important aspect about it. The unity of the physical world has alwaysbeen a belief held by many people. The history of theoretical physics is a process that the unity generalization of intrinsic geometry and its application to Hilbert’s 6th problem 5 expands step by step. In this paper, it is essentially regarded as a process that the concept ofgeometry expands step by step.(1) From Newtonian mechanics to special relativity [16], and then to general relativity [17,18],it is a process that flat Riemannian geometry expands to general Riemannian geometry.(2) Gauge field theory actually expresses a kind of geometry that cannot be described byRiemannian geometry. It is usually described by abstract connections on abstract fibre bundles,however, that is not concrete enough.In the perspective of concrete constructivity, the generalization of intrinsic geometry in thispaper makes the concept of geometry expand further more, thereby Riemannian geometry andgauge field geometry can be uniformly described by the generalized intrinsic geometry.Besides the intrinsic geometry, the solution at the most basic level for Hilbert’s 6th problemalso depends on a constructivity method. There are two approaches to develop mathematicaltheory, one is the approach of concrete constructivity based on set theory, the other is theapproach of abstract structure based on category theory. Although the effectiveness of thesetwo research approaches is the same, without either of them, the cognition to this mathematicalintuition is not complete.For example, consider the concept of real number. From the approach of abstract structure,some conventions as the connotation of abstract structure are combined to form axiomatizeddefinition of real number, i.e. a real number is an abstract element in the complete archimedeanordered field. From the approach of concrete constructivity, natural numbers are constructedfrom empty set, then integers and rational numbers are constructed, and then irrational numbersare constructed via Dedekind cut to form the real number set. Such two concepts of real numberdefined in two approaches reflect the same mathematical intuition. And such two theories of realnumber provide a complete cognition for the intuition of real number.In the above sense, fundamental physical theories in history are too abstract and lack ofconcrete mathematical constructions. For examples:(1) In electrodynamics, the relationship between mechanics and electromagnetics can beestablished just only by Lorentz force equation
FFF = q ( EEE + vvv × BBB ) . However, various variablesin the formula are all abstract vectors and scalar, which are lack of concrete mathematicalconstructions. For example, EEE and
BBB can only be distinguished ontologically, but as two abstractvectors there is no difference of mathematical connotation between them. As a result, the Lorentzforce equation can just only be artificially introduced and regarded as a principle and cannotbecome a theorem automatically.(2) After the establishment of quantum mechanics [3–5,8,9,58–63], the quantum field theoryestablished. The suggestion of Yang-Mills theory [73] eventually led to Glashow-Weinberg-Salam’s unified theory of weak electricity [20, 29, 38, 41–43, 45, 57, 66], quantum chromody-namics [2, 6, 21, 24, 25, 27, 31–33, 56, 65] and various great unified theories [10, 23, 26, 49–51].Not long ago, Yue-Liang Wu brought gravitational field into the framework of QFT in inertialsystem [68–72].
Zhao-Hui Man
However, such fields are still abstractly defined functions and lack of concrete mathematicalconstruction. Both Hamiltonian function and Lagrangian function are abstract objects, becausefield functions composing them are abstract. On one hand, physics describes gauge field withabstract concept of connection on a fibre bundle, but without giving concrete mathematicalconstructions to the connections. On the other hand, the spinor field, which is composed ofseveral complex-valued functions, is sometimes used to refer to a charged lepton field, andsometimes a neutrino field. It is not clear in physics that how to distinguish field function ofcharged lepton and field function of neutrino by mathematical constructions.(3) Early Kaluza-Klein theory [44, 47, 48] and later string theory as well as superstringtheory [1, 12, 28, 30, 34–37, 39, 40, 55, 64, 67] attempted to provide a unified explanation of thisproblem in high dimensional space. But for Hilbert’s 6th problem, they still cannot be regardedas success.The details of the above theories will not be discussed here. What should be emphasized isthat we will supply concrete mathematical constructions for the above various physical conceptsfrom the perspective of intrinsic geometry, so that theoretical physics can be axiomatized at themost basic level.It will start from a unique basic axiom, and strictly deduce the framework of physics, andturn principles, postulates, and artificially introduced equations into theorems, so as to give akind of effective solution for Hilbert’s 6th problem at the most basic level. (iii) General definition of geometry.
In order to generalize intrinsic geometry, we have to make a few improvements to theexpression form of Erlangen program. It is mainly based on the consideration of the followingtwo issues.(1) In history, there appeared two different approaches to unify Euclidean geometry and nonEuclidean geometries. One is from Gauss and Riemann, that is to say, Gauss-Bonet theoremdistinguishes geometries with angle sum of a triangle. The other is the Erlangen program [46]proposed by Felix Klein in 1872, which distinguishes geometries with transformation group.However, Riemannian geometry was regarded as one that cannot be incorporated into theframework of Erlangen’s program, so these two approaches still fail to associate clearly.(2) Based on the idea of Erlangen program, starting from the second half of the 20th century,theoretical physics began to emphasize the notion of symmetry and research it with the conceptof group extensively. It is right, but easy to cause a kind of misunderstanding, that is, symmetryand group are regarded as equivalent things.In fact, the essential idea of symmetry is the invariance under transformations, and theessential idea of group is the relationship between transformations. The former is a geometricalproperty, and the latter is an algebraic property. Therefore, symmetry and group should not beconfused.In order to deal with the above two issues and to generalize intrinsic geometry, the expressionform of Erlangen program will be improved in this paper. The original idea of this improvement generalization of intrinsic geometry and its application to Hilbert’s 6th problem 7 has been referred to in literatures [15, 52], but they have not expressed this idea as a strictdefinition of general concept of geometry in an explicit form. Such a definition will be givenbelow.
Let C be a set and ∼ a relation of equivalence. The classification C/ ∼ is calleda geometry on C with respect to ∼ . Let t : C → C be a transformation. If ∀ [ c ] ∈ C/ ∼ , ∀ c ∈ [ c ] such that t ( c ) ∈ [ c ] , we say t is an equivalent transformation with respect to ∼ . The totality T of all equivalent transformations on C with respect to ∼ is called an equivalent transformationset on C with respect to ∼ . Evidently, ∼ and T are mutually determined, therefore, C/ ∼ canalso be denoted by C/T . ∀ [ c ] ∈ C/T is called a geometrical object . ∀ a ∈ [ c ] is called a geometrical instance of [ c ] .Denote S , S c ∈ C c , each element in S is called a point , each subset of S is called a geometricalfigure , and ( S, T ) is called a kind of geometrical theory .Let H be a set, and let a map h : C → H satisfy ∀ c , c ∈ C, c ∼ c ⇔ h ( c ) = h ( c ) . Then h induces a map ˜ h : C/ ∼→ H, [ c ] h ( c ) . Each of h and ˜ h is called a geometrical property on C . The image of h and ˜ h in H is called the value of geometrical property.Suppose there are two relations of equivalence ∼ a and ∼ b on C . If ∼ a ⊂∼ b , we say geometry C/ ∼ a is larger than C/ ∼ b , and C/ ∼ b smaller than C/ ∼ a , we also say C/ ∼ b is a subgeometry of C/ ∼ a . Remark 1.1.
The above definition is equivalent to the traditional expression of Erlangen pro-gram. It is remarkable that it characterizes geometry with a relation of equivalence, but not agroup. Why a new definition should be adopted? Because it is very inconvenient to describegeometry in the traditional form of Erlangen program in the case where a group is difficult toexpressed in an explicit form due to its complicated form or uncertain structure. But if we finda certain condition to define a relation of equivalence, according to the above new definitionwe are able to define our required geometry without specifying group structure. Thereby it willbe convenient for our study, such as what Discussion 6.1 says. In addition, Definition 2.3.2 ,Definition 2.4.2 , Definition 2.5.2 and section 2.6 are also treated in such a way.In the past, Erlangen program was used to deal with groups with simple structure. Thecorresponding geometry was confined to either local of the manifold or homogeneous manifoldsuch as constant curvature manifold. The Riemannian geometry was not incorporated into theframework of Erlangen program in traditional way. By contrast, based on the expression formof this paper, the definition of geometry can completely incorporate Riemannian geometry intothe framework of improved Erlangen program, see Discussion 2.6.2 .To say the least, if the group structure has to be emphasized, the following additional definitionis needed. The elements in equivalent transformation set T naturally imply a group structure Zhao-Hui Man with respect to composite operation of maps. The group T is called the tranformation groupof geometry C/T , and
C/T is called the geometry of group T . Therefore, the group structureexists on the equivalent transformation set naturally, and it is not necessary to make explicitrequirements in the definition of geometry as the traditional form of Erlangen program. Supposetransformation group T acting on S and transformation group T acting on S are isomorphic. ( S , T ) and ( S , T ) are called the same kind of geometrical theory . If T is a proper subgroupof T , we say T is smaller than T , and T larger than T . Evidently, the smaller the group, thelarger the geometry; conversely, the larger the group, the smaller the geometry. Definition 1.2.
On any set C , there must be a special geometry, which has only one equivalenceclass, that is C itself. This geometry is called a universal geometry . The set C is the onlygeometrical object in universal geometry, and it is called a universal geometrical object . Eachgeometrical property in universal geometry is called a universal geometrical property , andalso called a geometrical invariant on C . Each universal geometrical property with its uniquevalue is called a geometrical identity on C . Let M be a D -dimensional connected smooth real manifold. ∀ p ∈ M , take acoordinate chart ( U p , ϕ Up ) on a neighborhood U p of p . They constitute a coordinate covering { ( U p , ϕ Up ) } p ∈ M , which is called a point-by-point covering . ϕ Up is called a coordinate frame on neighborhood U p of p . For the sake of simplicity, below U p is denoted by U , and ϕ Up isdenoted by ϕ U .For any two coordinate frames ϕ U and ψ U on neighborhood U of point p , if f p , ϕ U ◦ ψ − U : ψ U ( U ) → ϕ U ( U ) is a smooth homeomorphism, f p is called a (local) reference-system onneighborhood U of point p , where ψ U is the basis coordinate frame of f p , and ϕ U is the performance coordinate frame of f p .For any local reference-systems f p and g p at p , if the coordinate frames of f p and the coordinateframes of g p are C ∞ -compatible, we say f p and g p are C ∞ -compatible . The totality of the localreference-systems that are mutually C ∞ -compatible is called a (local) reference-system space ,denoted by REF p ( U ) or REF p .The totality of all the local reference-systems with ψ U as the basis coordinate frame is denotedby REF p ( U, ψ U ) .The totality of all the local reference-systems with ϕ U as the performance coordinate frameis denoted by REF p ( ϕ U , U ) . Definition 2.1.2.
Denote
REF , S p ∈ M REF p , where ∀ p, q ∈ M all the elements in REF p and REF q are C ∞ -compatible. generalization of intrinsic geometry and its application to Hilbert’s 6th problem 9 If the map f : M → REF, p f ( p ) ∈ REF p satisfies that the slack-tights B AM and C MA indefinition 2.2.2 are all smooth real functions on M , f is called a reference-system on M . Thetotality of all reference-systems which are mutually C ∞ -compatible on M is denoted by REF M .A differential manifold M with a reference-system f is called a geometrical manifold givenshape by f , and denoted by ( M, f ) . Definition 2.1.3. ∀ p ∈ M , ∀ ρ U ◦ ψ − U ∈ REF p ( U ) induces two (local) reference-system trans-formations L ρ U ◦ ψ − U : REF p ( ψ U , U ) → REF p ( ρ U , U ) , ψ U ◦ ϕ − U ( ρ U ◦ ψ − U ) ◦ ( ψ U ◦ ϕ − U ) = ρ U ◦ ϕ − U ,R ρ U ◦ ψ − U : REF p ( U, ρ U ) → REF p ( U, ψ U ) , ϕ U ◦ ρ − U ( ϕ U ◦ ρ − U ) ◦ ( ρ U ◦ ψ − U ) = ϕ U ◦ ψ − U . ∀ f ∈ REF M , ∀ p ∈ M , suppose L f ( p ) and R f ( p ) are induced by f ( p ) . The maps L f : p L f ( p ) , L f ( p ) and R f : p R f ( p ) , R f ( p ) are called reference-system transformations onmanifold M . Remark 2.1.1.
Suppose there is a reference-system f on manifold M . Construct reference-system e in the following way: ∀ p ∈ M , on neighborhood U of point p , take the basis coordinateframe of f ( p ) as the basis coordinate frame of e ( p ) , and take the same basis coordinate frameof f ( p ) as the performance coordinate frame of e ( p ) . Thus, reference-system transformation L f sends e to f just exactly. For convenience, some index symbols have to be specified. In the absence ofa special declaration, the indices used below are valued in the following range:(1) for basis coordinate frame ( U, ξ ) , indices A, B, C, D, E = 1 , , · · · , D , such as ξ A ;(2) for performance coordinate frame ( U, x ) , indices M, N, P, Q, R = 1 , , · · · , D , such as x M . Definition 2.2.2.
Let ( M, f ) be a geometrical manifold. ∀ p ∈ M , on a neighborhood U p of point p ,let the coordinate representation of local reference-system f ( p ) be ξ A = ξ A ( x M ) , x M = x M ( ξ A ) .Their derivative functions b AM : U p → R , q b AM ( q ) , ∂ξ A ∂x M ( q ) ,c MA : U p → R , q c MA ( q ) , ∂x M ∂ξ A ( q ) , on U p are called the degrees of slackness and tightness of f ( p ) , or slack-tights for short. If itis needed to emphasize f ( p ) explicitly, b AM and c MA can be denoted by ( b f ( p ) ) AM and ( c f ( p ) ) MA .Define two kinds of smooth real functions on manifold M : B AM : M → R , p B AM ( p ) , ( b f ( p ) ) AM ( p ) C MA : M → R , p C MA ( p ) , ( c f ( p ) ) MA ( p ) , then B AM and C MA are called the slack-tights of reference-system f on manifold M . Discussion 2.2.1.
Corresponding to the two coordinate frames of f ( p ) , the tangent space T p atpoint p has two sets of natural bases ∂∂ξ A (cid:12)(cid:12)(cid:12) p , ∂∂x M (cid:12)(cid:12)(cid:12) p ∈ T p , and the cotangent space T ∗ p also has twosets of natural bases dξ A (cid:12)(cid:12) p , dx M (cid:12)(cid:12) p ∈ T ∗ p . Thus, on U p we have ( b f ( p ) ) AM = * ∂∂x M (cid:12)(cid:12)(cid:12)(cid:12) p , dξ A (cid:12)(cid:12)(cid:12) p + , ( c f ( p ) ) AM = * ∂∂ξ A (cid:12)(cid:12)(cid:12)(cid:12) p , dx M (cid:12)(cid:12)(cid:12) p + . Therefore, ∀ p ∈ M we have B AM ( p ) = * ∂∂x M (cid:12)(cid:12)(cid:12)(cid:12) p , dξ A (cid:12)(cid:12)(cid:12) p + ( p ) , C MA ( p ) = * ∂∂ξ A (cid:12)(cid:12)(cid:12)(cid:12) p , dx M (cid:12)(cid:12)(cid:12) p + ( p ) . which can concisely be denoted by B AM = (cid:28) ∂∂x M , dξ A (cid:29) , C MA = (cid:28) ∂∂ξ A , dx M (cid:29) . Definition 2.2.3.
Denote ε MN = ε MN = ε MN , , M = N , M = N , δ AB = δ AB = δ AB , , A = B , A = B .
Let ( M, f ) be a geometrical manifold. ∀ p ∈ M , let U be a neighborhood of point p .(1) The coordinate frames ( U, ξ A ) and ( U, x M ) of f ( p ) respectively inherit metric tensor fields g , δ AB dξ A ⊗ dξ B = g MN dx M ⊗ dx N h , ε MN dx M ⊗ dx N = h AB dξ A ⊗ dξ B , g MN = δ AB b AM b BN h AB = ε MN c MA c NB . If it is needed to emphasize f ( p ) explicitly, g and h can be expressed as g f ( p ) and h f ( p ) , then g MN and h AB can be expressed as ( g f ( p ) ) MN and ( h f ( p ) ) AB .(2) On manifold M , define smooth real functions G MN : M → R , p G MN ( p ) , ( g f ( p ) ) MN ( p ) H AB : M → R , p H AB ( p ) , ( h f ( p ) ) AB ( p ) , ∆ AB : M → R , p ∆ AB ( p ) , δ AB E MN : M → R , p E MN ( p ) , ε MN . Thus on the entire manifold M , two metric tensor G and H are constructed, the local restrictionsof which can be expressed as G = ∆ AB dξ A ⊗ dξ B = G MN dx M ⊗ dx N H = E MN dx M ⊗ dx N = H AB dξ A ⊗ dξ B , G MN = ∆ AB B AM B BN H AB = E MN C MA C NB . Definition 2.2.4.
Denote dξ A , h AB dξ B , dx M , g MN dx N , which induce ∂∂ξ A and ∂∂x M in tangentspace, such that D ∂∂ξ B , dξ A E = δ BA and D ∂∂x N , dx M E = ε NM . Denote c MA , ∂x M ∂ξ A b AM , ∂ξ A ∂x M , c AM , ∂x M ∂ξ A b MA , ∂ξ A ∂x M , ¯ b MA , ∂ξ A ∂x M ¯ c AM , ∂x M ∂ξ A . generalization of intrinsic geometry and its application to Hilbert’s 6th problem 11 Define the following tensors: g , δ AB dξ A ⊗ dξ B = g MN dx M ⊗ dx N = g MN dx M ⊗ dx N h , ε MN dx M ⊗ dx N = h AB dξ A ⊗ dξ B = h AB dξ A ⊗ dξ B , x , δ AB ∂∂ξ A ⊗ ∂∂ξ B = x MN ∂∂x M ⊗ ∂∂x N = x MN ∂∂x M ⊗ ∂∂x N y , ε MN ∂∂x M ⊗ ∂∂x N = y AB ∂∂ξ A ⊗ ∂∂ξ B = y AB ∂∂ξ A ⊗ ∂∂ξ B , where g MN = δ AB b AM b BN g MN = δ AB b AM b BN , h AB = ε MN c MA c NB h AB = ε MN c MA c NB , x MN = δ AB c AM c BN x MN = δ AB c MA c NB , y AB = ε MN b MA b NB y AB = ε MN b AM b BN . Proposition 2.2.1. g MN = x MN , g MN = x MN , h AB = y AB , h AB = y AB . Proof. (1) g MP b CM b DP = δ CD ⇒ g MN b CM b DN c PC c QD = δ CD c PC c QD ⇒ g P Q = x P Q . x MN , δ AB c AM c BN ⇒ g P M x MN g NQ = g P M δ AB c AM c BN g NQ = δ AB ( c AM g P M )( c BN g NQ ) = δ AB c PA c QB = g P Q ⇒ g P M x MN = ε PN ⇒ x MN = g MN .(2) h AB c PA c QB = ε P Q ⇒ h AB c PA c QB b CP b DQ = ε P Q b CP b DQ ⇒ h CD = y CD . y AB , ε MN b MA b NB ⇒ h CA y AB h BD = h CA ε MN b MA b NB h BD = ε MN ( b MA h CA )( b NB h BD ) = ε MN b CM b DN = h CD ⇒ h CA y AB = δ CB ⇒ y AB = h AB . ⊓⊔ Intrinsic geometry of reference-system. (1)
Intrinsic geometry of local reference-system. ∀ f p , g p ∈ REF p ( U ) , let slack-tights of f p and g p be ( b f ) AM and ( b g ) AM respectively.Define a relation of equivalence ∼ = , such that f p ∼ = g p if and only if ∀ q ∈ U , ( b f ) AM ( q ) =( b g ) AM ( q ) . Thus, REF p ( U ) / ∼ = is called the intrinsic geometry on REF p ( U ) , where the geomet-rical object [ f p ] is called the core of f p . (2) Intrinsic geometry of reference-system on manifold. ∀ f, g ∈ REF M , let slack-tights of f and g be ( B f ) AM and ( B g ) AM respectively.Define a relation of equivalence ≡ , such that f ≡ g if and only if ∀ p ∈ M, f ( p ) ∼ = g ( p ) . Thus,the geometry REF M / ≡ is called the strict intrinsic geometry on REF M .Define a relation of equivalence ∼ = , such that f ∼ = g if and only if ∀ p ∈ M, ( B f ) AM ( p ) =( B g ) AM ( p ) . Thus, the geometry REF M / ∼ = is called the intrinsic geometry on REF M , where thegeometrical object [ f ] is called the core of f . Due to the one to one correspondence betweenmaps [ f ] , [ p f ( p )] and p [ f ( p )] , the core of f can also be expressed as [ f ] : p [ f ( p )] . Definition 2.3.2.
Intrinsic geometry on geometrical manifold.The totality of all the geometrical manifolds on M is denoted by M ( M ) . Define a relationof equivalence ∼ = of geometrical manifolds, such that ( M, f ) ∼ = ( M, g ) if and only if f ∼ = g . Thegeometry C ( M ) , M ( M ) / ∼ = is called the intrinsic geometry on geometrical manifolds, wheregeometrical object ( M, [ f ]) is called an intrinsic geometrical manifold given shape by [ f ] .According to Definition 1.1, the value of each intrinsic geometrical property is completelydepends on the core of reference-system, and thereby depends on the slack-tights B AM or C MA . Definition 2.3.3.
Intrinsic transformation.(1) Local intrinsic transformation. [ f p ] induces equivalence classes L [ f p ] : [ h p ] [ f p ] ◦ [ h p ] and R [ f p ] : [ k p ] [ k p ] ◦ [ f p ] of L f p : h p f p ◦ h p and R f p : k p k p ◦ f p . Then we say L [ f p ] and R [ f p ] are local intrinsic transformations .(2) Intrinsic transformation on manifold. [ f ] induces equivalence classes L [ f ] : [ h ] [ f ] ◦ [ h ] and R [ f ] : [ k ] [ k ] ◦ [ f ] of L f : h f ◦ h and R f : k k ◦ f . Then we say L [ f ] and R [ f ] are intrinsic transformations of geometrical manifolds, or general gauge transformations , seeProposition 5.6.2 for reasons. Discussion 2.3.1.
Intrinsic transformation group.(1) Locally, on the neighborhood of any point p on manifold M , the slack-tights b AM or c MA of a reference-system constitute a D -order invertible square matrix. The intrinsic geometry REF p ( U ) / ∼ = is isomorphic to general linear group GL ( D , R ) .(2) On manifold M , an intrinsic transformation sends an intrinsic geometrical manifold toanother intrinsic geometrical manifold. The intrinsic geometry REF M / ∼ = is isomorphic togeneral linear group GL ( M ) , ` p ∈ M GL ( D , R ) p , where ` is disjoint union.Suppose S is a subgroup of GL ( M ) . The group structure of S may be complicated andits description may be cumbersome, it is because transformation groups at various points aredifferent from each other in general. According to the original form of Erlangen program,geometry is dependent on group, that is to say, if group structure does not described clearly,geometry could not be established. This is an important reason why in history Riemanniangeometry was not incorporated into the framework of Erlangen program.It indicates that if not adopting the concept of geometry of section 1 , we are not able tounify the viewpoints of Riemannian geometry and Erlangen program. Therefore, in order toconveniently study geometry of manifold, it is not necessary to specify detail informationsof structure of transformation group, but we can take constraints for slack-tights, and therebyconstruct a relation of equivalence about reference-systems, so that we can study geometry bythe concepts of section 1 , just like what we do in Definition 2.3.2 , Definition 2.4.2 , Definition2.5.2 and section 2.6 . generalization of intrinsic geometry and its application to Hilbert’s 6th problem 13 Definition 2.3.4.
The general linear group GL ( M ) is called intrinsic transformation group , or general gauge transformation group .Specially, for the case that the transformation groups at different points of manifold areisomorphic to each other, the general linear group GL ( D , R ) is called homogeneous intrinsictransformation group . Furthermore:Let S be a subgroup of GL ( D , R ) . If f satisfies the following two conditions: (1) ∀ p ∈ M, [ f ( p )] ∈ S ; (2) for any subgroup T of S , ∃ q ∈ M, [ f ( q )] / ∈ T ; then we say objects determinedby f , such as f , ( M, [ f ]) , L [ f ] , R [ f ] , etc., are generated by group S .As the equivalent transformation set, the totality of all intrinsic transformations generated bygroup S is used to define a relation of equivalence ∼ S , so we can define a geometry M ( M ) / ∼ S ,which is called the geometry generated by group S , also denoted by M ( M ) /S .Let us define three important subgeometries of intrinsic geometry in the following threesections, which are kernal geometry, Riemannian geometry and universal geometry. Let k be a reference-system on manifold M , and its slack-tights B AM areconstants. We say L [ k ] and R [ k ] induced by k are flat transformations of reference-systems. If det[ B AM ] = 1 is satisfied as well, we say L [ k ] and R [ k ] are unimodular flat transformations ofreference-systems, or global gauge transformations . Definition 2.4.2.
Let there be intrinsic geometrical manifolds ( M, [ f ]) and ( M, [ g ]) . Define rela-tion of equivalence ≃ , such that [ f ] ≃ [ g ] and ( M, [ f ]) ≃ ( M, [ g ]) if and only if there exists a flattransformation F [ k ] such that F [ k ] ([ f ]) = [ g ] , where F [ k ] represents L [ k ] or R [ k ] .The corresponding equivalence classes are denoted by | f | and ( M, | f | ) respectively, and | f | is called the kernal of f . The geometry C ( M ) / ≃ is called the kernal geometry on geometricalmanifolds, where the geometrical object ( M, | f | ) is called a kernal geometrical manifold .Specially, if F [ k ] is a unimodular flat transformation, C ( M ) / ≃ is called a regular kernalgeometry . Remark 2.4.1.
Kernal geometry is a subgeometry of the intrinsic geometry of Definition 2.3.2. It can be understood intuitively as follows. Consider Fig.1 of introduction section. Fix axisand scale, and rotate the entire curve L by an angle. The intrinsic geometrical curve S ′ now isdifferent from the intrinsic geometrical curve S before, but the major bending characteristicsremain unchanged under the rotation. These bending characteristics are described by variousregular kernal geometrical properties of [ S ] . Let k be a reference-system on manifold, and its slack-tights B AM satisfy ∆ AB B AM B BN = E MN . Then L [ k ] and R [ k ] induced by k are called orthogonal transformations of reference-systems. Definition 2.5.2.
Let there be intrinsic geometrical manifolds ( M, [ f ]) and ( M, [ g ]) , and theirslack-tights be ( B f ) AM and ( B g ) AM .Define relation of equivalence ≃ O , such that [ f ] ≃ O [ g ] and ( M, [ f ]) ≃ O ( M, [ g ]) if and onlyif there exists an orthogonal transformation F [ k ] such that F [ k ] ([ f ]) = [ g ] , where F [ k ] represents L [ k ] or R [ k ] . The equivalence classes are denoted by [ f ] O and ( M, [ f ] O ) respectively. We saythe geometry C ( M ) / ≃ O is Riemannian geometry , and a geometrical object ( M, [ f ] O ) is a Riemannian manifold . Proposition 2.5.1. [ f ] ≃ O [ g ] if and only if ( G f ) MN = ( G g ) MN . Proof.
We need to consider the cases of F [ k ] = R [ k ] and F [ k ] = L [ k ] , respectively.(1) Suppose R [ k ] is a transformation induced by reference-system k , such that R [ k ] ([ f ]) = [ g ] .Let the slack-tights of [ f ] be ( B f ) AM and ( C f ) MA , and the slack-tights of [ k ] be ( B k ) A ′ A and ( C k ) AA ′ .Then the slack-tights of [ g ] are ( B g ) A ′ M = ( B k ) A ′ A ( B f ) AM and ( C g ) MA ′ = ( C k ) AA ′ ( C f ) MA .According to Definition 2.2.3 , the metric tensor of [ f ] is ( G f ) MN = ∆ AB ( B f ) AM ( B f ) BN , therebythe metric tensor of [ g ] is ( G g ) MN = ∆ A ′ B ′ ( B g ) A ′ M ( B g ) B ′ N = ∆ A ′ B ′ (cid:16) ( B k ) A ′ A ( B f ) AM (cid:17) (cid:16) ( B k ) B ′ B ( B f ) BN (cid:17) = (cid:16) ∆ A ′ B ′ ( B k ) A ′ A ( B k ) B ′ B (cid:17) ( B f ) AM ( B f ) BN . Hence, ( G f ) MN = ( G g ) MN if and only if ∆ A ′ B ′ ( B k ) A ′ A ( B k ) B ′ B = ∆ AB , i.e. R [ k ] is orthogonal.(2) Suppose L [ k ] is a transformation induced by reference-system k , such that L [ k ] ([ f ]) = [ g ] .Let the slack-tights of [ f ] be ( B f ) AM and ( C f ) MA , and the slack-tights of [ k ] be ( B k ) MM ′ and ( C k ) M ′ M .Then the slack-tights of [ g ] are ( B g ) AM ′ = ( B k ) MM ′ ( B f ) AM and ( C g ) M ′ A = ( C k ) M ′ M ( C f ) MA .According to Definition 2.2.4 , we consider tensors ( H f ) AB and ( H g ) AB . ( H f ) AB = ( Y f ) AB = E MN ( B f ) AM ( B f ) BN , ( H g ) AB = E M ′ N ′ ( B g ) AM ′ ( B g ) BN ′ = E M ′ N ′ (cid:0) ( B k ) MM ′ ( B f ) AM (cid:1) (cid:0) ( B k ) NN ′ ( B f ) BN (cid:1) = (cid:16) E M ′ N ′ ( B k ) MM ′ ( B k ) NN ′ (cid:17) ( B f ) AM ( B f ) BN . Hence, ( H f ) AB = ( H g ) AB if and only if E M ′ N ′ ( B k ) MM ′ ( B k ) NN ′ = E MN , i.e. L [ k ] is orthogonal.Then due to ( H f ) AB = ( H g ) AB ⇔ ( G f ) MN = ( G g ) MN ⇔ ( G f ) MN = ( G g ) MN , the propositionis true. ⊓⊔ Discussion 2.5.1.
We notice that:(1) Definition 2.5.2 tells us that Riemannian geometry is a subgeometry of intrinsic geometryof Definition 2.3.2 .(2) Proposition 2.5.1 indicates that Definition 2.5.2 is consistent with the traditional definitionof Riemannian manifold.Hence, the intrinsic geometry of Definition 2.3.2 is larger than Riemannian geometry, andgeometrical manifold is a more fundamental concept than Riemannian manifold. According to generalization of intrinsic geometry and its application to Hilbert’s 6th problem 15 the viewpoint of Riemannian geometry, the ultimate origin of its geometrical property is metric.According to the viewpoint of geometrical manifold, the geometrical property has more basicorigin, which ultimately boils down to reference-system and its slack-tights B AM or C MA .(1) In history, the slack-tights is called a semimetric in traditional theory of Riemanniangeometry. Physicists noticed long ago [14, 53] that when researching interactions betweengravitational field and elementary particles, especially problems about spinor field, it can bedescribed only by adopting semimetric representation, and it does not work by using metricrepresentation. However, they did not realize that it means the connotation of traditional intrinsicgeometry needs to be generalized.(2) On one hand, it can be seen from Definition 2.2.3 that the slack-tights on geometricalmanifold determines the metric on Riemannian manifold. On the other hand, even when thecoefficients of metric tensors of two geometrical manifolds are completely the same, theirslack-tights are not necessarily the same. These two aspects indicate that the theory of intrinsicgeometry on geometrical manifold has richer geometrical properties than the traditional theoryof intrinsic geometry on Riemannian manifold.(3) It is important that there exists a concept of simple connection on geometrical manifold,see section 2.7 . If simple connection vanishes, Levi-Civita connection must vanish too. But ifLevi-Civita connection vanishes, the simple connection does not necessarily vanish. Therefore,simple connection reflects more intrinsic properties of manifold than Levi-Civita connection.These properties just exactly can be used to describe the characteristics of gauge fields. Let there be geometrical manifolds ( M, f ) and ( M, g ) . Define relation ofequivalence ∼ , such that ( M, f ) ∼ ( M, g ) if and only if there exists F [ k ] such that F [ k ] ([ f ]) = [ g ] ,where F [ k ] represents L [ k ] or R [ k ] .In fact such a transformation always exists, which is L [ g ◦ f − ] or R [ f − ◦ g ] . Therefore, M ( M ) becomes the only equivalence class in the geometry ˜ M ( M ) , M ( M ) / ∼ . It makes ˜ M ( M ) theuniversal geometry on geometrical manifolds. Discussion 2.6.2.
Make a summary for these subgeometries of intrinsic geometry. On geomet-rical manifold:(1) An intrinsic geometrical property is an invariant under identical intrinsic transformation of reference-systems.(2) A kernal geometrical property is an invariant under flat transformation of reference-systems.(3) A
Riemannian geometrical property is an invariant under orthogonal transformation ofreference-systems. (4) A universal geometrical property is an invariant under arbitrary transformation ofreference-systems.(5) Let e be the unit element of GL ( M ) . According to Remark 1.1, { e } as the transformationgroup of intrinsic geometry is the smallest transformation group, and GL ( M ) as the transfor-mation group of universal geometry is the largest transformation group. In other words, ongeometrical manifold, intrinsic geometry is the largest geometry, and universal geometry is thesmallest geometry. Let Γ MNP be smooth real functions on manifold M . ∀ p ∈ M , dx M and ∂∂x M arenatural basis vector fields in coordinate frame ( U, x M ) of local reference-system f ( p ) . Considerthe restriction of smooth real functions Γ MNP on U , affine connection can be expressed as: D ∂∂x N , Γ MNP dx P ⊗ ∂∂x M , Ddx N , − Γ NMP dx P ⊗ dx M . (1)In order to enable affine connection to describe intrinsic geometry, Γ MNP should be defined asthe ones dependent on slack-tights B AM or C MA , such as Levi-Civita connection Γ MNP , G MQ (cid:18) ∂G NQ ∂x P + ∂G P Q ∂x N − ∂G NP ∂x Q (cid:19) , or other forms.Levi-Civita connection is the unique torsion-free and metric-compatible connection, but it isnot fundamental enough. On one hand it is because Levi-Civita connection cannot describe theintrinsic properties determined by B AM and C MA when G MN are all constants, on the other handLevi-Civita connection is not simple enough.The torsion-free condition is very helpful to simplify theoretical form, but the metric-compatible condition restricts the further simplification of connection form. We consider thatthe metric-compatible condition D G = 0 was introduced to establish the intuition of Levi-Civitaparallel displacement, but it is not the condition that more general concept of parallel displace-ment must rely on. Therefore, in order to simplify connection furthermore, it can be imaginedthat the torsion-free condition remains and the metric-compatible condition is given up. A nicechoice is to adopt the following definition. Definition 2.7.1.
Let there be an affine connection D , which is expressed as equation (1) onperformance coordinate frame ( U, x M ) . If the connection coefficients are defined as Γ MNP , C MA ∂B AN ∂x P + ∂B AP ∂x N ! = 12 B AN ∂C MA ∂x P + B AP ∂C MA ∂x N ! , (2) D is called a simple connection . generalization of intrinsic geometry and its application to Hilbert’s 6th problem 17 Proposition 2.7.1.
The simple connection is indeed an affine connection.
Proof.
Let there be a reference-system f on manifold M . And let there be a local reference-system t p : x M ′ = x M ′ ( x M ) , which induces a reference-system transformation L t p sending f ( p ) : x M = x M ( ξ A ) to h ( p ) , t p ◦ f ( p ) : x M ′ = x M ′ ( ξ A ) , where ( U, ξ A ) is the common basiscoordinate frame of f ( p ) and h ( p ) . They can be expressed as a diagram: ( U, ξ A ) ( U, x M ′ )( U, x M ) f ( p ) h ( p ) t p Let the slack-tights of t p be b MM ′ , ∂x M ∂x M ′ , c M ′ M , ∂x M ′ ∂x M . Let the slack-tights of f be B AM and C MA . For the restriction of them on U , the slack-tights appliedtransformation L t p are B AM ′ = b MM ′ B AM , C M ′ A = c M ′ M C MA . According to Definition 2.7.1 , the original simple connection and the one applied transformation L t p are respectively ( Γ f ) MNP , C MA ∂B AN ∂x P + ∂B AP ∂x N ! , ( Γ f ) M ′ N ′ P ′ , C M ′ A ∂B AN ′ ∂x P ′ + ∂B AP ′ ∂x N ′ ! . Calculate the local transformation relation of the simple connection on U : ( Γ f ) M ′ N ′ P ′ , C M ′ A ∂B AN ′ ∂x P ′ + ∂B AP ′ ∂x N ′ ! = 12 c M ′ M C MA ∂ (cid:0) b NN ′ B AN (cid:1) ∂x P ′ + ∂ (cid:0) b PP ′ B AP (cid:1) ∂x N ′ ! = 12 c M ′ M C MA ∂b NN ′ ∂x P ′ B AN + b NN ′ ∂B AN ∂x P ′ + ∂b PP ′ ∂x N ′ B AP + b PP ′ ∂B AP ∂x N ′ ! = 12 c M ′ M C MA b NN ′ ∂B AN ∂x P ′ + b PP ′ ∂B AP ∂x N ′ ! + 12 c M ′ M C MA ∂b NN ′ ∂x P ′ B AN + ∂b PP ′ ∂x N ′ B AP ! = 12 c M ′ M C MA ∂B AN ∂x P + ∂B AP ∂x N ! b NN ′ b PP ′ + 12 c M ′ M ∂b MN ′ ∂x P ′ + ∂b MP ′ ∂x N ′ ! = ( Γ f ) MNP c M ′ M b NN ′ b PP ′ + c M ′ M ∂b MN ′ ∂x P ′ . This result is consistent with the general relation of local tansformation of affine connection.So it has been proved that the simple connection Γ MNP , C MA (cid:16) ∂B AN ∂x P + ∂B AP ∂x N (cid:17) is indeed an affineconnection, and evidently it is torsion-free. ⊓⊔ Proposition 2.7.2.
Suppose there are reference-systems g and k on manifold M . Let the slack-tights of g be B AM and C MA , the slack-tights of k be B MM ′ and C M ′ M , and the slack-tights of g ′ , L k ( g ) be B AM ′ = B AM B MM ′ and C M ′ A = C MA C M ′ M . Then let the simple connections of ( M, g ) , ( M, k ) and ( M, g ′ ) be ( Γ g ) MNP , ( Γ k ) M ′ N ′ P ′ and ( Γ g ′ ) M ′ N ′ P ′ , respectively. Thus, ( Γ g ′ ) M ′ N ′ P ′ = ( Γ g ) MNP C M ′ M B NN ′ B PP ′ + ( Γ k ) M ′ N ′ P ′ . (3) Proof.
Calculate the following smooth functions of M on an arbitrary coordinate neighborhood. ( Γ g ′ ) M ′ N ′ P ′ , C M ′ A ∂B AN ′ ∂x P ′ + ∂B AP ′ ∂x N ′ ! = 12 C M ′ M C MA ∂ (cid:0) B NN ′ B AN (cid:1) ∂x P ′ + ∂ (cid:0) B PP ′ B AP (cid:1) ∂x N ′ ! = 12 C M ′ M C MA ∂B NN ′ ∂x P ′ B AN + B NN ′ ∂B AN ∂x P ′ + ∂B PP ′ ∂x N ′ B AP + B PP ′ ∂B AP ∂x N ′ ! = 12 C M ′ M C MA B NN ′ ∂B AN ∂x P ′ + B PP ′ ∂B AP ∂x N ′ ! + 12 C M ′ M C MA ∂B NN ′ ∂x P ′ B AN + ∂B PP ′ ∂x N ′ B AP ! = 12 C M ′ M C MA ∂B AN ∂x P + ∂B AP ∂x N ! B NN ′ B PP ′ + 12 C M ′ M ∂B MN ′ ∂x P ′ + ∂B MP ′ ∂x N ′ ! = ( Γ g ) MNP C M ′ M B NN ′ B PP ′ + ( Γ k ) M ′ N ′ P ′ . ⊓⊔ Remark 2.7.1.
Simple connection proposed by this paper has two evident properties.(1) Denote Γ MNP , G MM ′ Γ M ′ NP , Λ MNP , (cid:16) ∂G NM ∂x P + ∂G PM ∂x N − ∂G NP ∂x M (cid:17) then it is easy to verify Γ MNP = 12 δ AB B BM ∂B AN ∂x P + ∂B AP ∂x N ! ,Γ MNP + Γ NP M + Γ P MN = Λ MNP + Λ NP M + Λ P MN . (2) It is evident that when G MN are all constants, Levi-Civita connection must be zero, and thecorresponding Riemannian curvature tensor also must be zero. Meanwhile, simple connectionis however not necessarily zero, and the corresponding Riemannian curvature tensor is also notnecessarily zero. It indicates that simple connection reflects more intrinsic bending-propertiesof manifold than Levi-Civita connection.
1. Riemannian manifold is generalized to geometrical manifold.2. Intrinsic geometry is generalized, so that kernal geometry, Riemannian geometry anduniversal geometry are all subgeometries of intrinsic geometry.3. The important concept of simple connection is defined, which is just exactly the key todescribe elementary particles and gauge fields from the perspective of intrinsic geometry.By applying the generalized intrinsic geometry, we can obtain an effective constructivitymethod for Hilbert’s 6th problem, which may be used to unify elementary frameworks oftheoretical physics. generalization of intrinsic geometry and its application to Hilbert’s 6th problem 19
We have to abstract and separate out physical connotations, so as to focus on mathematicalconstructions. A convenient way is to adopt the following axiom. There is only one axiom, andwe do not need any more.
The fundamental axiom.
Physical reality should be cognized by using the concept ofreference-system on geometrical manifold.
Such a concept of reference-system is strictly defined in Definition 2.1.2 . This axiom has anevident corollary as below, which can be called the principle of universal relativity . Corollary . Universal physical property should be cognized by using universal geometricalproperty of geometrical manifold.Such a concept of universal geometrical property is strictly defined in Definition 1.2 andsection 2.6 . In the following sections, universal physical properties, such as time metric, spacemetric, evolution, etc., will be strictly defined as some universal geometrical properties ofgeometrical manifold. Other main conclusions, postulates and equations of fundamental physicscan also be strictly defined, constructed and proved in pure mathematical sense.
On a neighborhood U of point p , according to Definition 2.2.3 , there are dξ and dx defined on two coordinate frames ( U, ξ A ) and ( U, x M ) of f ( p ) as ( dξ ) , δ AB dξ A dξ B = g MN dx M dx N , ( dx ) , ε MN dx M dx N = h AB dξ A dξ B . (4)We speak of dξ and dx as total space metrics of ( U, ξ A ) and ( U, x M ) , or as time metrics .On geometrical manifold ( M, f ) , dξ and dx defined as ( dξ ) , ∆ AB dξ A dξ B = G MN dx M dx N ( dx ) , E MN dx M dx N = H AB dξ A dξ B (5)are called total space metrics or time metrics of M . Remark 3.2.1.
Such a new treatment about the concept of time in this paper is very important. Itwill make the two evolution notions of gravitational theory and quantum mechanics become uni-fied and coordinated. In this way, time metric reflects the total evolution in the total-dimensionalspace, while a specific spatial metric reflects a partial evolution in a specific direction.
Definition 3.2.2.
Let P and N be closed submanifolds of manifold M = P × N . Denote r , dim P . Let s, i = 1 , · · · , r and a, m = r + 1 , · · · , D . Select some proper coordinate frames { ξ A } and { x M } such that on P there are coordinate frames { ξ s } and { x i } inherited from M , and on N there are coordinate frames { ξ a } and { x m } inherited from M . Correspondingly, twosubspace metrics can be defined on the coordinate neighborhoods on P and N respectively: ( dξ ( P ) ) , r X s =1 ( dξ s ) = δ st dξ s dξ t ( dx ( P ) ) , r X i =1 ( dx i ) = ε ij dx i dx j , ( dξ ( N ) ) , D X a = r +1 ( dξ a ) = δ ab dξ a dξ b ( dx ( N ) ) , D X m = r +1 ( dx m ) = ε mn dx m dx n . For convenience, N is called a submanifold of internal space and P is called a submanifoldof external space . dξ ( N ) and dx ( N ) are called propertime metrics . Let there be two reference-systems f and g on manifold M , if ∀ p ∈ M , f ( p ) , ϕ U ◦ ψ − U and g ( p ) , ϕ U ◦ ρ − U have the same performance coordinate frame ϕ U , namelyit can be intuitively expressed as a diagram ψ U ( U ) f ( p ) −−→ ϕ U ( U ) g ( p ) ←−− ρ U ( U ) , we say f and g motion relatively and interact mutually , and also say that f evolves in g , or f evolves on geometrical manifold ( M, g ) . Meanwhile, g evolves in f , or say g evolves on ( M, f ) . Definition 3.3.1.2.
Let there be a one-parameter group of diffeomorphisms ϕ X : M × R → M acting on M . ϕ X determines a smooth tangent vector field X on M . If X is nonzero everywhere,we say ϕ X is a set of evolution paths on M , and X is an evolution direction field on M .Let T ⊆ R be an interval. Suppose a smooth map L p : T → M constitutes a regularsubmanifold of M , and there exists a smooth tangent vector field X such that L p is on the orbit ϕ X,p ( t ) of ϕ X through p . Now the map L p is called an evolution path through p , or path forshort. The tangent vector ddt , [ L p ] = X ( p ) is called an evolution direction at p . If it does notneed to emphasize the point p , L p can be denoted by L concisely. Definition 3.3.1.3. ∀ p ∈ M , let the coordinate representations of f ( p ) , ϕ U ◦ ψ − U be x M = x M ( ξ A ) , ξ A = ξ A ( x M ) , and the coordinate representations of g ( p ) , ϕ U ◦ ρ − U be x M = x M ( ζ A ) , ζ A = ζ A ( x M ) . Then let time metrics on ( U, ϕ U ) , ( U, ψ U ) , ( U, ρ U ) be dx , dξ , dζ , respectively. generalization of intrinsic geometry and its application to Hilbert’s 6th problem 21 Each path L p is a 1-dimensional regular submanifold of M , therefore on open set U L , U ∩ L p there exist coordinate neighborhoods ( U L , ϕ UL ) , ( U L , ψ UL ) and ( U L , ρ UL ) such that the regularimbedding π : L p → M, q q (6)induces coordinate maps and parameter equations ψ U ◦ π ◦ ψ − UL : R → R D , ( ξ ) ( ξ A ) , ξ A = ξ A ( ξ ) ϕ U ◦ π ◦ ϕ − UL : R → R D , ( x ) ( x M ) , x M = x M ( x ) ρ U ◦ π ◦ ρ − UL : R → R D , ( ζ ) ( ζ A ) , ζ A = ζ A ( ζ ) which satisfy D P A =1 (cid:16) dξ A dξ (cid:17) = 1 , D P M =1 (cid:16) dx M dx (cid:17) = 1 , D P A =1 (cid:16) dζ A dζ (cid:17) = 1 . Definition 3.3.1.4. f ( p ) = ϕ U ◦ ψ − U and f − ( p ) = ψ U ◦ ϕ − U on U induce reference-systems on U L : f L ( p ) , ϕ UL ◦ ψ − UL , x = x ( ξ ) f − L ( p ) , ψ UL ◦ ϕ − UL , ξ = ξ ( x ) . Then we obtain the coordinate form of evolution of f : ξ A = ξ A ( x M ) = ξ A ( x ) ξ = ξ ( x ) , x M = x M ( ξ A ) = x M ( ξ ) x = x ( ξ ) . (7) Let L be a path on manifold M , ∀ p ∈ L . Suppose T p ( M ) and T p ( L ) are thetangent spaces at p on M and L respectively, and T ∗ p ( M ) and T ∗ p ( L ) are the cotangent spaces. ∀ p ∈ L , the regular imbedding π : L → M, q q induces tangent map and cotangent map π ∗ | p : T p ( L ) → T p ( M ) , [ γ L ] [ π ◦ γ L ] ,π ∗ | p : T ∗ p ( M ) → T ∗ p ( L ) , df d ( f ◦ π ) . (8)Evidently, restricting on L , the tangent map is an injection, and the cotangent map is a surjection. ∀ ddt L ∈ T p ( L ) , ∀ df ∈ T ∗ p ( M ) , denote ddt , π ∗ | p (cid:18) ddt L (cid:19) ∈ T p ( M ) , df L , π ∗ | p ( df ) ∈ T ∗ p ( L ) . (9)We say ddt and ddt L are equivalent , df and df L are homomorphic . They are denoted by ddt ∼ = ddt L , df ≃ df L . (10)The above locally defined concepts can also be applied to the entire manifold, and furthermore,they can be transplanted without hindrance to any-order tensor product space generated bytangent bundle and cotangent bundle, such as: (1) If ddt ∼ = ddt L , then ∀ df we say df ⊗ ddt and df ⊗ ddt L are equivalent , denoted by df ⊗ ddt ∼ = df ⊗ ddt L .(2) If df ≃ df L , then ∀ ddt we say df ⊗ ddt and df L ⊗ ddt are homomorphic , denoted by df ⊗ ddt ≃ df L ⊗ ddt . Proposition 3.3.2.1. If ddt ∼ = ddt L and df ≃ df L , then D ddt , df E = D ddt L , df L E . Proof.
The tangent vectors ddt and ddt L are respectively defined as equivalence classes [ γ ] and [ γ L ] of parameter curves, the cotengent vectors df and df L are respectively defined as equivalenceclasses [ f ] and [ f L ] of smooth functions, which satisfy γ = π ◦ γ L , f L = f ◦ π . Hence, (cid:28) ddt , df (cid:29) = (cid:28) ddt L , df L (cid:29) ⇔ h [ γ ] , [ f ] i = h [ γ L ] , [ f L ] i ⇔ d ( f ◦ γ ) dt = d ( f L ◦ γ L ) dt , where f ◦ γ = f ◦ ( π ◦ γ L ) , f L ◦ γ L = ( f ◦ π ) ◦ γ L . Evidently, f ◦ γ = f L ◦ γ L , which makes D ddt , df E = D ddt L , df L E true. ⊓⊔ Definition 3.3.2.2. ∀ p ∈ L , on coordinate neighborhood U L of point p , define b A , dξ A dx , b , dξ dx , c M , dx M dξ , c , dx dξ ,ε M , dx M dx = b c M = b A c MA , δ A , dξ A dξ = c b A = c M b AM . They determine the following smooth functions on the entire L : B A : L → R , p B A ( p ) , ( b f ( p ) ) A ( p ) C M : L → R , p C M ( p ) , ( c f ( p ) ) M ( p ) , B : L → R , p B ( p ) , ( b f ( p ) ) ( p ) C : L → R , p C ( p ) , ( c f ( p ) ) ( p ) . For convenience, if no confusion, still using notations ε and δ , we also have smooth functions: ε M , B C M = B A C MA , δ A , C B A = C M B AM . Define dξ , dx dξ dx and dx , dξ dx dξ , which induce ddξ and ddx , such that D ddξ , dξ E = 1 , D ddx , dx E = 1 . On U L we also define ¯ b A , dξ A dx , ¯ b , dξ dx , ¯ c M , dx M dξ , ¯ c , dx dξ , ¯ ε M , dx M dx = ¯ b ¯ c M = ¯ b A ¯ c AM , ¯ δ A , dξ A d ¯ ξ = ¯ c ¯ b A = ¯ c M ¯ b MA . They determine the following smooth functions on the entire L : ¯ B A : L → R , p ¯ B A ( p ) , (¯ b f ( p ) ) A ( p )¯ C M : L → R , p ¯ C M ( p ) , (¯ c f ( p ) ) M ( p ) , ¯ B : L → R , p ¯ B ( p ) , (¯ b f ( p ) ) ( p )¯ C : L → R , p ¯ C ( p ) , (¯ c f ( p ) ) ( p ) . ¯ ε M , ¯ B ¯ C M = ¯ B A ¯ C AM , ¯ δ A , ¯ C ¯ B A = ¯ C M ¯ B MA . generalization of intrinsic geometry and its application to Hilbert’s 6th problem 23 Proposition 3.3.2.2. (Evolution lemma) . Let L be a path on manifold M . Suppose there aretangent vector fields w M ∂∂x M , ¯ w M ∂∂x M and cotangent vector fields w M dx M , ¯ w M dx M on M , andthere are tangent vector fields w ddx , ¯ w ddx and cotangent vector fields ∀ w dx , ¯ w dx on L .Then the following conclusions hold: w M ∂∂x M ∼ = w ddx ⇔ w M = w ε M w M dx M ≃ w dx ⇔ ε M w M = w , ¯ w M ∂∂x M ∼ = ¯ w ddx ⇔ ¯ w M = ¯ w ¯ ε M ¯ w M dx M ≃ ¯ w dx ⇔ ¯ ε M ¯ w M = ¯ w . Proof.
The following locally discussion can also be applied to the entire manifold.1. Consider the case that basis vectors are dx M and ∂∂x M .For tangent vector, π ∗ (cid:18) ddx (cid:19) = dx M dx ∂∂x M ⇔ dx M dx ∂∂x M ∼ = ddx ⇔ ε M ∂∂x M ∼ = ddx ⇔ w ε M ∂∂x M ∼ = w ddx . Because the tangent map is an injection, then w M ∂∂x M ∼ = w ddx ⇔ w M = w ε M . For cotangent vector, dx M ≃ ε M dx ⇒ w M dx M ≃ ε M w M dx , then w M dx M ≃ w dx ⇔ ε M w M = w .2. Consider the case that basis vectors are dx M and ∂∂x M .For tangent vector, π ∗ (cid:18) ddx (cid:19) = dx M dx ∂∂x M ⇔ dx M dx ∂∂x M ∼ = ddx ⇔ ¯ ε M ∂∂x M ∼ = ddx ⇔ ¯ w ¯ ε M ∂∂x M ∼ = ¯ w ddx . Because the tangent map is an injection, then ¯ w M ∂∂x M ∼ = ¯ w ddx ⇔ ¯ w M = ¯ w ¯ ε M . For cotangent vector, dx M ≃ ¯ ε M dx ⇒ ¯ w M dx M ≃ ¯ ε M ¯ w M dx , then ¯ w M dx M ≃ ¯ w dx ⇔ ¯ ε M ¯ w M = ¯ w . ⊓⊔ On U L define g , dx dx = b b , g , dx dx = c c , h , dξ dξ = c c , h , dξ dξ = b b . They determine the following smooth functions defined on L : G , B B , G , C C , H , C C , H , B B . Proposition 3.3.3.1. On L , ddx = G ddx , ddξ = H ddξ . Proof.
At any point, tangent vector ddx can be expanded as ddx = X ddx about basis ddx , andtangent vector ddξ can be expanded as ddξ = Y ddξ about basis ddξ . (cid:28) ddx , dx (cid:29) = 1 ⇔ (cid:28) X ddx , g dx (cid:29) = 1 ⇔ Xg = 1 ⇔ X = 1 g = g ⇒ ddx = g ddx , (cid:28) ddξ , dξ (cid:29) = 1 ⇔ (cid:28) Y ddξ , h dξ (cid:29) = 1 ⇔ Y h = 1 ⇔ Y = 1 h = h ⇒ ddξ = h ddξ . This local conclusion can be applied to the entire path, so ddx = G ddx and ddξ = H ddξ holdon L . ⊓⊔ Proposition 3.3.3.2. On L , H = H AB δ A δ B , G = G MN ε M ε N . Proof.
On a neighborhood U L of any point on L , h AB δ A δ B = ε MN c MA c NB δ A δ B = ε MN dx M dξ dx N dξ = dx dξ dx dξ = h g MN ε M ε N = δ AB b AM b BN ε M ε N = δ AB dξ A dx dξ B dx = dξ dx dξ dx = g . So H = H AB δ A δ B and G = G MN ε M ε N hold on the entire L . ⊓⊔ Proposition 3.3.3.3.
The following left-hand formulas are true on L . When D > , the right-handformulas are in general false on L . G MN dx M ⊗ dx N ≃ G dx ⊗ dx ,G MN dx M ⊗ dx N ≃ G dx ⊗ dx ,H AB dξ A ⊗ dξ B ≃ H dξ ⊗ dξ ,H AB dξ A ⊗ dξ B ≃ H dξ ⊗ dξ . X MN ∂∂x M ⊗ ∂∂x N ∼ = X ddx ⊗ ddx ,X MN ∂∂x M ⊗ ∂∂x N ∼ = X ddx ⊗ ddx ,Y AB ∂∂ξ A ⊗ ∂∂ξ B ∼ = Y ddξ ⊗ ddξ ,Y AB ∂∂ξ A ⊗ ∂∂ξ B ∼ = Y ddξ ⊗ ddξ . Proof.
Due to Proposition 3.3.3.2 and evolution lemma, the left-hand formulas hold evidently.Now consider the right-hand ones.On U L when D > , x MN cannot be expressed as the form like yε M ε N . Otherwise, let x MN = yε M ε N , then the following two conclusions contradict with each other: dx M dx M = x MN dx M dx N = ( yε M ε N ) dx M dx N = y dx M dx M dx N dx N dx dx = yg dx M dx M ⇒ y = 1 g = g , D = x MN x MN = ( yε M ε N ) g MN = y ( g MN ε M ε N ) = yg ⇒ y = D g = D g . ⊓⊔ Remark 3.3.3.1.
Due to the above proposition and section 2.2 , we know that although tensors G and X have relations G MN = X MN and G MN = X MN and tensors H and Y have relations H AB = Y AB and H AB = Y AB , when considering evolution, G and H have better properties than X and Y . Let V n be the totality of n -order tensor fields generated by tangent vectorbundle and cotangent vector bundle on M . Then suppose T , t n ∂∂x ⊗ dx o ∈ V n , where smooth generalization of intrinsic geometry and its application to Hilbert’s 6th problem 25 real functions t represent the coefficients of tensor T . The regular imbedding π : L → M induces t L , t ◦ π : L → R and T L , t L n ∂∂x ⊗ dx o .(1) Let D be affine connection, and denote t L ;0 , t ; Q ε Q , then the absolute differential of T and T L can be expressed as D T , Dt ⊗ (cid:26) ∂∂x ⊗ dx (cid:27) , t ; Q dx Q ⊗ (cid:26) ∂∂x ⊗ dx (cid:27) ,D L T L , D L t L ⊗ (cid:26) ∂∂x ⊗ dx (cid:27) , t L ;0 dx ⊗ (cid:26) ∂∂x ⊗ dx (cid:27) , where Dt , t ; Q dx Q , D L t L , t L ;0 dx .(2) Gradient operator ∇ is called the actual evolution . The absolute gradient of T and T L can be expressed as ∇ T , ∇ t ⊗ (cid:26) ∂∂x ⊗ dx (cid:27) , t ; Q ∂∂x Q ⊗ (cid:26) ∂∂x ⊗ dx (cid:27) , ∇ L T L , ∇ L t L ⊗ (cid:26) ∂∂x ⊗ dx (cid:27) , t L ;0 ddx ⊗ (cid:26) ∂∂x ⊗ dx (cid:27) , where ∇ t , t ; Q ∂∂x Q , ∇ L t L , t L ;0 ddx . The gradient direction (field) ∇ t is called the actualevolution direction (field) of T . The integral curve of ∇ t is called the gradient line or actualevolution path of T . Proposition 3.3.4.1. D T ≃ D L T L if L is an arbitrary path. ∇ T ∼ = ∇ L T L if and only if L is thegradient line of T . Proof.
According to definition, the first conclusion is evident. Now consider the second conclu-sion.(1) The sufficiency. Let L be the gradient line of T . ∀ p ∈ L , the gradient direction at p is t ; Q ∂∂x Q ∈ T p ( M ) . Because tangent map is injection, there exists a unique X ddx ∈ T p ( L ) such that t ; Q ∂∂x Q ∼ = X ddx . Applying evolution lemma we obtain t ; Q = X dx Q dx (cid:12)(cid:12)(cid:12)(cid:12) L , dx Q ≃ dx Q dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L dx , thereby t ; Q dx Q ≃ X dx Q dx (cid:12)(cid:12)(cid:12)(cid:12) L dx Q dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L dx . According to Definition 3.3.1.3 , ( dξ ) = D P A =1 ( dξ A ) holds on U L , i.e. dx dx = dx Q dx Q .Substitute it into the above formula, then we obtain t ; Q dx Q ≃ Xdx . Due to evolution lemma, X = t ; Q dx Q dx = t L ;0 , and then ∇ T ∼ = ∇ L T L hold.(2) The necessity is evident. In fact, ∀ p ∈ L , suppose tangent vector t ;0 ddx on L makes ∇ T ∼ = ∇ L T L hold, so t ; Q ∂∂x Q ∼ = t ;0 ddx . And suppose ∀ p ∈ L , X Q ∂∂x Q ∼ = t ;0 ddx . Because tangentmap is injection, X Q = t ; Q , which indicates that X Q ∂∂x Q can only be the gradient direction.Additionally, p is arbitrary on L , hence L can only be the gradient line, not other paths. ⊓⊔ Definition 3.3.4.2.
According to evolution lemma, ∇ T ∼ = ∇ L T L if and only if t ; Q = t L ;0 ¯ ε Q or t ; Q = t L ;0 ε Q . These two equations are called the actual evolution equations of T . Definition 3.3.4.3.
Suppose there is a tensor product U , u Q dx Q ⊗ n ∂∂x ⊗ dx o , such that thesystem of 1-order non-homogeneous linear equations t ; Q = u Q has a unique solution t . Then ∇ t satisfies u Q dx Q ≃ u dx , u Q ∂∂x Q ∼ = u ddx . We say the actual evolution direction ∇ t = u Q ∂∂x Q is determined by u Q dx Q . Remark 3.3.4.1.
Now for any universal geometrical property defined in form of tensor producton geometrical manifold, we are able to study its actual evolution in way of absolute gradient.Then two important gradient directions will be discussed. One is the actual evolution ofpotential field of reference-system itself. The other is the case that general charge of onereference-system evolves in another reference-system. They are both attributed to geometricalproperties of geometrical manifold.
Suppose f evolves in g , that is, ∀ p ∈ M, ( U, ξ A ) f ( p ) −−→ ( U, x M ) g ( p ) ←−− ( U, ζ A ) .We will always take the following notations in coordinate frame ( U, x M ) .(1) Let the connection of geometrical manifold ( M, f ) be Λ MNP , and the connection of ( M, g ) be Γ MNP . Colon ":" is used to express the absolute derivative on ( M, f ) , and semicolon ";" is usedto express the absolute derivative on ( M, g ) , such as u Q : P = ∂u Q ∂x P + u H Λ QHP , u Q ; P = ∂u Q ∂x P + u H Γ QHP . We also call Λ MNP the potential field of f , and Γ MNP the potential field of g , or say potential forshort. In order to describe intrinsic geometry, we must adopt affine connection dependent onslack-tights. It can be either Levi-Civita connection or simple connection of Definition 2.7.1 .Because the scope of applicability of simple connection is larger than Levi-Civita connection,we suppose Λ MNP and Γ MNP are both simple connections.(2) Let the coefficients of Riemannian curvature of ( M, f ) be K MNP Q , and the coefficients ofRiemannian curvature of ( M, g ) be R MNP Q , then K MNP Q , ∂Λ MNQ ∂x P − ∂Λ MNP ∂x Q + Λ HNQ Λ MHP − Λ HNP Λ MHQ ,R MNP Q , ∂Γ MNQ ∂x P − ∂Γ MNP ∂x Q + Γ HNQ Γ MHP − Γ HNP Γ MHQ . (3) The values of indices of internal space and external space are taken according to Definition5.1.1 . generalization of intrinsic geometry and its application to Hilbert’s 6th problem 27 Discussion 3.3.5.1.
Consider evolution of f . Denote ρ MN , K MNP Q : P ε Q , then we have K MNP Q : P dx Q ≃ ρ MN dx in an arbitrary direction.Specially, we have K MNP Q : P ∂∂x Q ∼ = ρ MN ddx in the gradient direction determined by K MNP Q : P dx Q .Now due to Proposition 3.3.2.2 , we obtain the actual evolution equation K MNP Q : P = ρ MN ¯ ε Q . De-note j MNQ , ρ MN ¯ ε Q , then we obtain K MNP Q : P = j MNQ , (11)which is called the general Yang-Mills field equation of f . Then due to Proposition 3.3.4.1 wedirectly obtain the following theorem. Proposition 3.3.5.1. (Evolution theorem of general gauge field ) . ρ MN and j MNQ are bothintrinsic geometrical properties of ( M, f ) . Equation K MNP Q : P = j MNQ holds if and only if itsevolution direction field is the gradient direction field determined by K MNP Q : P dx Q . Definition 3.3.5.2.
Besides ρ MN , there are also ρ M N , G ρ MN , ρ MN , G MM ′ ρ M ′ N and ρ MN , G ρ MN . We call each of them a general charge , or charge for short. Without loss of generality, on geometrical manifold ( M, g ) we can discussthe actual evolution of charge tensor F , ρ MN dx M ⊗ dx N of f in way of absolute gradient.For the sake of simplicity, denote ρ MN by ρ MN concisely.The absolute differential of F on ( M, g ) is D F , Dρ MN ⊗ dx M ⊗ dx N , where Dρ MN , ρ MN ; R dx R , and D is the simple connection of ( M, g ) .The absolute gradient of F on ( M, g ) is ∇ F , ∇ ρ MN ⊗ dx M ⊗ dx N , where ∇ ρ MN , ρ MN ; R ∂∂x R .Due to Proposition 3.3.4.1 , we directly obtain the following theorem. Proposition 3.3.6.1. (General charge evolution theorem)
The following relations are true ongeometrical manifold ( M, g ) if and only if evolution direction field of ρ MN is the gradientdirection field ∇ ρ MN . ρ MN ; R dx R ≃ ρ MN ;0 dx ρ MN ; R ∂∂x R ∼ = ρ MN ;0 ddx , ρ MN ; R dx R ≃ ρ MN ;0 dx ρ MN ; R ∂∂x R ∼ = ρ MN ;0 ddx , (12)which are called charge evolution equations . Definition 3.3.6.1.
For more convenience, further abbreviate the notation ρ MN to ρ .(1) Call E , ρ ;0 , ρ ; R ¯ ε R and E , ρ ;0 , ρ ; R ε R the total energy or total mass of ρ .(2) Call p R , ρ ; R and p R , ρ ; R the momentum of ρ .(3) Call H , dρdx and H , dρdx the canonical energy of ρ .(4) Call P R , ∂ρ∂x R and P R , ∂ρ∂x R the canonical momentum of ρ . (5) Call V , E − H and V , E − H the potential energy of interaction.(6) Call V R , p R − P R and V R , p R − P R the momentum of interaction. Proposition 3.3.6.2.
If and only if evolution direction of ρ evolving in g is the gradient direction ∇ ρ , equation E E = p R p R holds, which is called the general energy-momentum equation of ρ . Proof.
According to Proposition 3.3.6.1 , gradient direction is equivalent to E dx ≃ p R dx R E ddx ∼ = p R ∂∂x R , E dx ≃ p R dx R E ddx ∼ = p R ∂∂x R . (13)Then we obtain the directional derivative in gradient direction: (cid:28) E ddx , E dx (cid:29) = (cid:28) p R ∂∂x R , p M dx M (cid:29) , i.e. G E E = G RM p R p M , or E E = p R p R . ⊓⊔ Proposition 3.3.6.3.
The following equations p R = E dx R dx , p R = E dx R dx (14)hold if and only if the evolution direction of ρ is the gradient direction ∇ ρ . Proof.
According to Proposition 3.3.6.1 , gradient direction is equivalent to p R ∂∂x R ∼ = E ddx and p R ∂∂x R ∼ = E ddx . Then due to evolution lemma we immediately obtain p R = E dx R dx and p R = E dx R dx . ⊓⊔ Remark 3.3.6.1.
In the gradient direction, the conclusion of this proposition is consistent withthe traditional p = mv . Definition 3.3.6.2.
Let L be the totality of paths from a to b . And suppose L ρ ∈ L , and theevolution parameter x satisfies t a , x ( a ) < x ( b ) , t b . The functional s ρW (cid:0) L ρ (cid:1) , Z L ρ Dρ = Z t b t a E dx = Z t b t a p R dx R is called general evolution quantity of ρ . Proposition 3.3.6.4. (Extreme value theorem of general evolution quantity) L ρ is the gradientline of ρ if and only if δs ρW (cid:0) L ρ (cid:1) = 0 . Proof.
Let the parameter equation of L ρ be x R = x R ( x ) , t a x t b , and let the parameter equation of L ρ + δL ρ be x R = x R ( x ) + δx R ( x ) , t a x t b , δx R ( t a ) = δx R ( t b ) = 0 . generalization of intrinsic geometry and its application to Hilbert’s 6th problem 29 Let the unit tangent vector on L ρ at any x be X , π ∗ (cid:18) ddx (cid:19) , dx R dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x ∂∂x R = ε R (cid:0) x (cid:1) ∂∂x R , and let the unit tangent vector on L ρ + δL ρ be X + δX , d (cid:0) x R + δx R (cid:1) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x ∂∂x R = dx R dx + δ dx R dx !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x ∂∂x R = (cid:16) ε R (cid:0) x (cid:1) + δε R (cid:0) x (cid:1)(cid:17) ∂∂x R . Then consider the variation of s ρW ( L ρ ) . ∆s ρW ( L ρ ) = ∆ Z L ρ p R ε R dx = Z L ρ + δL ρ p R ε R dx − Z L ρ p R ε R dx = Z L ρ + δL ρ ρ ; R ε R dx − Z L ρ ρ ; R ε R dx = Z L ρ + δL ρ h X, Dρ i dx − Z L ρ h X, Dρ i dx = Z t b t a D X + δX, Dρ (cid:16) x R + δx R (cid:17)E dx − Z t b t a D X, Dρ (cid:16) x R (cid:17)E dx = Z t b t a * X + δX, Dρ ( x R ) + ∂Dρ ( x R ) ∂x M δx M + o ( δx ) + dx − Z t b t a D X, Dρ ( x R ) E dx = Z t b t a (cid:18) h X + δX, Dρ i + (cid:28) X + δX, ∂Dρ∂x M δx M (cid:29)(cid:19) dx − Z t b t a h X, Dρ i dx + o ( δx )= Z t b t a (cid:18) h δX, Dρ i + (cid:28) X, ∂Dρ∂x M δx M (cid:29)(cid:19) dx + o ( δx )= Z t b t a ( h δX, Dρ i + h X, δDρ i ) dx + o ( δx ) = Z t b t a h δX, Dρ i dx + Z t b t a δDρ + o ( δx )= Z t b t a h δX, Dρ i dx + o ( δx ) . Thus we obtain δs ρW = Z t b t a h δX, Dρ i dx . When b → a , δds ρW = h δX, Dρ i dx . The directional derivative h X, Dρ i = ρ ;0 cos θ , where θ is the included angle between evolution direction X and the gradient direction. Then h δX, Dρ i = ρ ;0 δ cos θ = − ρ ;0 sin θδθ . Now δds ρW = − ρ ;0 sin θδθdx . Hence, for general ρ , δds ρW = 0 if and only if sin θ = 0 , namely evolution direction at this pointis exactly the gradient direction (take the positive direction without loss of generality).Take integration from a to b , then δ R t b t a ds ρW = 0 if and only if evolution direction at eachpoint of L ρ is the gradient direction of ρ . In other words, δs ρW = 0 if and only if L ρ is the gradientline of ρ . ⊓⊔ Remark 3.3.6.2.
In the Minkowski coordinate frame defined later, evolution parameter x willbe changed to ˜ x τ , then there still exists a concept of gradient direction. Correspondingly, theevolution quantity R t b t a E dx will present as R τ b τ a ˜ m τ dτ , where ˜ m τ is rest-mass. Thus, the principleof least action will become a theorem, no longer as a principle. The most general abstract theory about conserved quantity is the Neother’s theorem. However,it is not enough to just only content with abstract point of view. In consideration of that aconserved quantity is a geometrical property, we need to study its concrete construction in wayof intrinsic geometry.
Definition 3.3.7.1.
Denote [ ρΓ G ] , ∂ρ∂x G − ρ ; G , ∂ρ MN ∂x G − ρ MN ; G = ρ MH Γ HNG + ρ HN Γ HMG , [ ρΓ ] , dρdx − ρ ;0 , dρ MN dx − ρ MN ;0 = ρ MH Γ HN + ρ HN Γ HM . [ ρΓ Q ] , G GQ [ ρΓ G ] , [ ρΓ ] , G [ ρΓ ] . [ ρB P Q ] , ρ MH ∂Γ HNQ ∂x P − ∂Γ HNP ∂x Q ! + ρ HN ∂Γ HMQ ∂x P − ∂Γ HMP ∂x Q ! , [ ρR P Q ] , ρ MH R HNP Q + ρ HN R HMP Q . [ ρF P Q ] , ∂ [ ρΓ Q ] ∂x P − ∂ [ ρΓ P ] ∂x Q , [ ρE P Q ] , [ ρΓ Q ] ; P − [ ρΓ P ] ; Q . Proposition 3.3.7.1.
The following two equations hold: (1)[ ρF P Q ] = [ ρE P Q ];(2)[ ρF P Q ] − [ ρB P Q ] = (cid:16) ρ MH,P Γ HNQ − ρ MH,Q Γ HNP (cid:17) + (cid:16) ρ HN,P Γ HMQ − ρ HN,Q Γ HMP (cid:17) . Proof. [ ρE P Q ] = [ ρΓ Q ] ; P − [ ρΓ P ] ; Q = (cid:16) ρ MH Γ HNQ + ρ HN Γ HMQ (cid:17) ; P − (cid:16) ρ MH Γ HNP + ρ HN Γ HMP (cid:17) ; Q = ρ MH (cid:16) Γ HNQ ; P − Γ HNP ; Q (cid:17) + ρ HN (cid:16) Γ HMQ ; P − Γ HMP ; Q (cid:17) + (cid:16) ρ MH ; P Γ HNQ − ρ MH ; Q Γ HNP (cid:17) + (cid:16) ρ HN ; P Γ HMQ − ρ HN ; Q Γ HMP (cid:17) = ρ MH (cid:16) Γ HNQ ; P − Γ HNP ; Q (cid:17) + ρ HN (cid:16) Γ HMQ ; P − Γ HMP ; Q (cid:17) + (cid:16) ρ MH,P Γ HNQ + ρ HN,P Γ HMQ (cid:17) − (cid:16) ρ MH,Q Γ HNP + ρ HN,Q Γ HMP (cid:17) − (cid:16) ρ MG Γ GHP Γ HNQ − ρ MG Γ GHQ Γ HNP (cid:17) − (cid:16) ρ GN Γ GHP Γ HMQ − ρ GN Γ GHQ Γ HMP (cid:17) = ρ MH (cid:16) Γ HNQ ; P − Γ HNP ; Q (cid:17) + ρ HN (cid:16) Γ HMQ ; P − Γ HMP ; Q (cid:17) + (cid:16) ρ MH,P Γ HNQ + ρ HN,P Γ HMQ (cid:17) − (cid:16) ρ MH,Q Γ HNP + ρ HN,Q Γ HMP (cid:17) − (cid:16) ρ MH Γ HGP Γ GNQ − ρ MH Γ HGQ Γ GNP (cid:17) − (cid:16) ρ HN Γ HGP Γ GMQ − ρ HN Γ HGQ Γ GMP (cid:17) = (cid:16) ρ MH,P Γ HNQ + ρ HN,P Γ HMQ (cid:17) − (cid:16) ρ MH,Q Γ HNP + ρ HN,Q Γ HMP (cid:17) + ρ MH (cid:16) Γ HNQ ; P − Γ HNP ; Q + Γ HGQ Γ GNP − Γ HGP Γ GNQ (cid:17) + ρ HN (cid:16) Γ HMQ ; P − Γ HMP ; Q + Γ HGQ Γ GMP − Γ HGP Γ GMQ (cid:17) = (cid:16) ρ MH,P Γ HNQ − ρ MH,Q Γ HNP (cid:17) + (cid:16) ρ HN,P Γ HMQ − ρ HN,Q Γ HMP (cid:17) + ρ MH ∂Γ HNQ ∂x P − ∂Γ HNP ∂x Q ! + ρ HN ∂Γ HMQ ∂x P − ∂Γ HMP ∂x Q ! generalization of intrinsic geometry and its application to Hilbert’s 6th problem 31 = (cid:16) ρ MH,P Γ HNQ − ρ MH,Q Γ HNP (cid:17) + (cid:16) ρ HN,P Γ HMQ − ρ HN,Q Γ HMP (cid:17) + [ ρB P Q ]= ∂∂x P (cid:16) ρ MH Γ HNQ + ρ HN Γ HMQ (cid:17) − ∂∂x Q (cid:16) ρ MH Γ HNP + ρ HN Γ HMP (cid:17) = ∂ [ ρΓ Q ] ∂x P − ∂ [ ρΓ P ] ∂x Q = [ ρF P Q ] . ⊓⊔ Proposition 3.3.7.2.
The following two equations hold: (1) ∂p P ∂x Q − ∂p Q ∂x P − [ ρF P Q ] = 0;(2) dp P dx − ∂E ∂x P + p Q ∂ε Q ∂x P − [ ρF P Q ] ε Q = 0 . Proof.
According to Definition 3.3.6.1 , ∂P P ∂x Q − ∂P Q ∂x P = 0 ⇔ ∂p P ∂x Q − ∂p Q ∂x P + ∂ [ ρΓ P ] ∂x Q − ∂ [ ρΓ Q ] ∂x P = 0 ⇔ ∂p P ∂x Q − ∂p Q ∂x P − [ ρF P Q ] = 0 . Consider π ∗ (cid:16) ∂p P ∂x Q dx Q − ∂p Q ∂x P dx Q − [ ρF P Q ] dx Q (cid:17) . π ∗ : ∂p P ∂x Q dx Q ∂p P ∂x Q dx Q dx dx = dp P dx dx ,π ∗ : ∂p Q ∂x P dx Q ∂p Q ∂x P dx Q dx dx = ∂ (cid:16) p Q dx Q dx (cid:17) ∂x P dx − p Q ∂∂x P dx Q dx ! dx = ∂E ∂x P dx − p Q ∂ε Q ∂x P dx ,π ∗ : [ ρF P Q ] dx Q [ ρF P Q ] dx Q dx dx = [ ρF P Q ] ε Q dx . Then dp P dx dx − ∂E ∂x P dx + p Q ∂ε Q ∂x P dx − [ ρF P Q ] ε Q dx = 0 , finally dp P dx − ∂E ∂x P + p Q ∂ε Q ∂x P − [ ρF P Q ] ε Q = 0 . ⊓⊔ Proposition 3.3.7.3.
With torsion-free connection, the following two equations hold: (1) p P ; Q − p Q ; P − [ ρB P Q ] = 0;(2) p P ;0 − E P + p Q ε Q P − [ ρB P Q ] ε Q = 0 . Proof.
According to Proposition 3.3.7.2 , ∂p P ∂x Q − ∂p Q ∂x P − [ ρF P Q ] = 0 . Substitute equation (2) ofProposition 3.3.7.1 into this one, then we obtain ∂p P ∂x Q − ∂p Q ∂x P − (cid:16) ρ MH,P Γ HNQ − ρ MH,Q Γ HNP (cid:17) − (cid:16) ρ HN,P Γ HMQ − ρ HN,Q Γ HMP (cid:17) = [ ρB P Q ] ⇔ ∂ρ MN ; P ∂x Q − ∂ρ MN ; Q ∂x P − (cid:16) ρ MH,P Γ HNQ − ρ MH,Q Γ HNP (cid:17) − (cid:16) ρ HN,P Γ HMQ − ρ HN,Q Γ HMP (cid:17) = [ ρB P Q ] ⇔ (cid:18) ∂ρ MN ; P ∂x Q − ρ MH,P Γ HNQ − ρ HN,P Γ HMQ (cid:19) − (cid:18) ∂ρ MN ; Q ∂x P − ρ MH,Q Γ HNP − ρ HN,Q Γ HMP (cid:19) = [ ρB P Q ] ⇔ (cid:18) ∂ρ MN ; P ∂x Q − ρ MH,P Γ HNQ − ρ HN,P Γ HMQ − ρ MN ; H Γ HP Q (cid:19) − (cid:18) ∂ρ MN ; Q ∂x P − ρ MH,Q Γ HNP − ρ HN,Q Γ HMP − ρ MN ; H Γ HQP (cid:19) + ρ MN ; H (cid:16) Γ HP Q − Γ HQP (cid:17) = [ ρB P Q ] ⇔ ρ MN ; P ; Q − ρ MN ; Q ; P = [ ρB P Q ] ⇔ p P ; Q − p Q ; P − [ ρB P Q ] = 0 . Consider π ∗ (cid:0) p P ; Q dx Q − p Q ; P dx Q − [ ρB P Q ] dx Q (cid:1) . π ∗ : p P ; Q dx Q p P ; Q dx Q dx dx = p P ;0 dx ,π ∗ : p Q ; P dx Q p Q ; P dx Q dx dx = p Q dx Q dx ! ; P − p Q dx Q dx ! ; P dx = E P dx − p Q ε Q P dx ,π ∗ : [ ρB P Q ] dx Q [ ρB P Q ] dx Q dx dx = [ ρB P Q ] ε Q dx . Then p P ;0 dx − E P dx + p Q ε Q P dx − [ ρB P Q ] ε Q dx = 0 , finally p P ;0 − E P + p Q ε Q P − [ ρB P Q ] ε Q = 0 . ⊓⊔ Proposition 3.3.7.4.
With torsion-free connection, the following three equations hold: (1) p P ; Q − p Q ; P − [ ρR P Q ] = 0;(2) p P ;0 − E P + p Q ε Q P − [ ρR P Q ] ε Q = 0; . (3)[ ρB P Q ] = [ ρR P Q ] . Proof.
The covariant derivatives of p P , ρ ; P , ρ MN ; P = ρ MN,P − ρ MH Γ HNP − ρ HN Γ HMP are p P ; Q = ρ MN ; P ; Q = ρ MN ; P,Q − ρ MH ; P Γ HNQ − ρ HN ; P Γ HMQ − ρ MN ; H Γ HP Q ,p Q ; P = ρ MN ; Q ; P = ρ MN ; Q,P − ρ MH ; Q Γ HNP − ρ HN ; Q Γ HMP − ρ MN ; H Γ HQP . Substract them: p P ; Q − p Q ; P = (cid:0) ρ MN ; P,Q − ρ MN ; Q,P (cid:1) + (cid:16) ρ MH ; Q Γ HNP − ρ MH ; P Γ HNQ (cid:17) + (cid:16) ρ HN ; Q Γ HMP − ρ HN ; P Γ HMQ (cid:17) + (cid:16) ρ MN ; H Γ HQP − ρ MN ; H Γ HP Q (cid:17) = (cid:0) ρ MN ; P,Q − ρ MN ; Q,P (cid:1) + (cid:16) ρ MH ; Q Γ HNP − ρ MH ; P Γ HNQ (cid:17) + (cid:16) ρ HN ; Q Γ HMP − ρ HN ; P Γ HMQ (cid:17) generalization of intrinsic geometry and its application to Hilbert’s 6th problem 33 = (cid:16) ρ MN,P − ρ MH Γ HNP − ρ HN Γ HMP (cid:17) ,Q − (cid:16) ρ MN,Q − ρ MH Γ HNQ − ρ HN Γ HMQ (cid:17) ,P + (cid:16) ρ MH,Q − ρ MG Γ GHQ − ρ GH Γ GMQ (cid:17) Γ HNP − (cid:16) ρ MH,P − ρ MG Γ GHP − ρ GH Γ GMP (cid:17) Γ HNQ + (cid:16) ρ HN,Q − ρ HG Γ GNQ − ρ GN Γ GHQ (cid:17) Γ HMP − (cid:16) ρ HN,P − ρ HG Γ GNP − ρ GN Γ GHP (cid:17) Γ HMQ = (cid:18)(cid:16) ρ MH Γ HNQ (cid:17) ,P + (cid:16) ρ HN Γ HMQ (cid:17) ,P (cid:19) − (cid:18)(cid:16) ρ MH Γ HNP (cid:17) ,Q + (cid:16) ρ HN Γ HMP (cid:17) ,Q (cid:19) + (cid:16) ρ MH,Q − ρ MG Γ GHQ − ρ GH Γ GMQ (cid:17) Γ HNP − (cid:16) ρ MH,P − ρ MG Γ GHP − ρ GH Γ GMP (cid:17) Γ HNQ + (cid:16) ρ HN,Q − ρ HG Γ GNQ − ρ GN Γ GHQ (cid:17) Γ HMP − (cid:16) ρ HN,P − ρ HG Γ GNP − ρ GN Γ GHP (cid:17) Γ HMQ = (cid:16) ρ MH Γ HNQ,P + ρ HN Γ HMQ,P (cid:17) − (cid:16) ρ MH Γ HNP,Q + ρ HN Γ HMP,Q (cid:17) + (cid:16) − ρ MG Γ GHQ Γ HNP − ρ GH Γ GMQ Γ HNP (cid:17) − (cid:16) − ρ MG Γ GHP Γ HNQ − ρ GH Γ GMP Γ HNQ (cid:17) + (cid:16) − ρ HG Γ GNQ Γ HMP − ρ GN Γ GHQ Γ HMP (cid:17) − (cid:16) − ρ HG Γ GNP Γ HMQ − ρ GN Γ GHP Γ HMQ (cid:17) = ρ MH ( Γ HNQ,P − Γ HNP,Q + Γ HGP Γ GNQ − Γ HGQ Γ GNP ) + ρ HN ( Γ HMQ,P − Γ HMP,Q + Γ HGP Γ GMQ − Γ HGQ Γ GMP )= ρ MH R HNP Q + ρ HN R HMP Q = [ ρR P Q ] . That is p P ; Q − p Q ; P − [ ρR P Q ] = 0 . And compare it with equation (1) of Proposition 3.3.7.3, then [ ρB P Q ] = [ ρR P Q ] is obtained. Finally, due to equation (2) of Proposition 3.3.7.3 , p P ;0 − E P + p Q ε Q P − [ ρR P Q ] ε Q = 0 holds. ⊓⊔ Definition 3.3.7.2.
According to the above propositions, equations F P , dp P dx = ∂E ∂x P − p Q ∂ε Q ∂x P + [ ρF P Q ] ε Q ,f P , p P ;0 = E P − p Q ε Q P + [ ρR P Q ] ε Q . are called the general Lorentz force equations , and the intrinsic geometrical properties F P and f P are called general force on ρ . Proposition 3.3.7.5.
Suppose there is a tensor Y MN satisfies Y MN ; M = − G ( E N − p Q ε Q N +[ ρR NQ ] ε Q ) − ¯ ε MM p N , and let W MN , E ¯ ε M ¯ ε N , then in gradient direction of ρ , intrinsic geo-metrical property T MN , W MN + Y MN satisfies T MN ; M = 0 , which can be called the conservation of energy-momentum of ρ . Proof.
According to Proposition 3.3.6.3 , E ¯ ε N = p N in gradient direction of ρ . Then W MN ; M = (cid:0) ¯ ε M p N (cid:1) ; M = p N ; M ¯ ε M + ¯ ε MM p N = p N ;0 + ¯ ε MM p N = − Y MN ; M , Thus, ( W MN + Y MN ) ; M = 0 , that is T MN ; M = 0 . ⊓⊔ For any two non-vanishing smooth tangent vector fields X and Y on manifold M , let L Y be the Lie derivative operator induced by the one-parameter group of diffeomorphisms ϕ Y corresponding to Y . According to a well-known theorem [7], Lie derivative equation [ X, Y ] = L Y X holds.On one hand, suppose H is the field of unit tangent vector in gradient directions of ρ , and ϕ H is the one-parameter group of diffeomorphisms corresponding to H , and the the parameter of ϕ H is x . The Lie derivative equation induced by ϕ H is [ X, H ] = L H X . Lie derivative operator L H and tangent vector field ddx are both uniquely determined by H , so it can be denoted that ddx X , L H X . Thus, [ X, H ] = L H X becomes [ X, H ] = ddx X .On the other hand, the regular imbedding of path induces H ∼ = H L . Then according toProposition 3.3.2.1 , for any smooth function f , equation h H, df i = h H L , df L i holds. Notice that H L and ddx are the same, hence we have Hf = ddx f L .In a word, L H and H L are both uniquely determined by the gradient direction H . Due to theabove discussion, we immediately obtain the following proposition. Proposition 3.3.8.1.
Let H be the field of unit tangent vector in gradient directions of ρ , for any X and any f , equations [ X, H ] = ddx X, Hf = ddx f L (15)hold if and only if ddx is the field of unit tangent vector in gradient directions of ρ . Definition 3.3.8.1.
Equation [ X, H ] = ddx X is called the general Heisenberg equation . Equa-tion Hf = ddx f L is called the general Schrödinger equation . Discussion 3.3.8.2.
Both the two equations describe gradient direction field. H not only can betaken as the gradient direction determined by Dρ , but can also taken as the gradient direction ofany other geometrical property.According to Definition 3.3.4.1 , gradient operator is a universal geometrical property ofgeometrical manifold, so these two equations remain unchanged under arbitrary tranformationof reference-systems. These two equations reflect two descriptions of the same geometricalproperty by two mutually dual linear spaces, which are tangent bundle and cotangent bundle.From real-valued evolution equations on tangent bundle and cotangent bundle, we can surelydeduce complex-valued evolution equations on operator space and state space, such as section5.6 . What they describe is none other than gradient direction. It is true both for wave functionand field function.We can say that the value of gradient direction is determined by intrinsic geometry, andit is independent of either real-valued form or complex-valued form. The effectivenesses ofdescribing intrinsic geometry with complex-valued form and real-valued form are the same. In generalization of intrinsic geometry and its application to Hilbert’s 6th problem 35 a word, there is no need to be constrained on theoretical forms. Intrinsic geometry and gradientdirection are the very essences which should be grasped.The only necessity of using complex form is that it is the most convenient for describingthe coherent superposition of propagator. However, it is a different problem from that of thissection and it will be specifically discussed in the next section. In order to achieve the purposeof clarifying concepts, it is beneficial to separate the two equations here from the constructionof coherent superposition of propagators of the next section. Intrinsic geometry relies on reference-system on manifold, hence:(1) It is impossible for the coordinate of a single point to reflect the full picture of intrinsicgeometrical shape of manifold.(2) It is also impossible for a single gradient direction to reflect the full picture of intrinsicgeometrical shape of manifold.In order to describe the intrinsic geometrical shape of ( M, g ) , it is meaningful to studydistribution of gradient directions of ρ on ( M, g ) . Intuitively:(1) If g is trivial, the gradient direction field of ρ distributes uniformly on the flat ( M, g ) .(2) If g is non-trivial, the intrinsic geometrical shape of ( M, g ) effects the distribution ofgradient directions of ρ . Definition 3.4.1.
Let ρ be a geometrical property determined by f , then ρ is a universal geomet-rical property on ( M, g ) , and H , ∇ ρ is a gradient direction field of ρ on ( M, g ) .Let T be the totality of flat transformations defined in section 2.4 . ∀ T ∈ T , the flattransformation T : f T f induces a transformation T ∗ : ρ T ∗ ρ . Denote H T , ∇ ( T ∗ ρ ) , | ρ | , { ρ T , T ∗ ρ | T ∈ T } and | H | , { H T | T ∈ T } .Let ϕ H be the one-parameter group of diffeomorphisms corresponding to H , the parameterof which is x . ∀ a ∈ M , suppose ϕ H,a is the orbit through point a . Without loss of generality,let a = ϕ H,a (0) . We say ϕ | H | ,a , { ϕ X,a | X ∈ | H |} is a system of gradient lines of | ρ | through a . ∀ t ∈ R + , we say ϕ | H | ,a ( t ) , { ϕ X,a ( t ) | X ∈ | H |} is the evolution image of a at x = t . ∀ Ω ⊆ T , | H Ω | , { H T | T ∈ Ω } is a subset of | H | , and ϕ | H Ω | ,a , { ϕ X,a | X ∈ | H Ω |} is a subsetof ϕ | H | ,a . Correspondingly, ∀ t ∈ R + , ϕ | H Ω | ,a ( t ) , { ϕ H,a ( t ) | X ∈ | X Ω |} is a subset of ϕ | H | ,a ( t ) . ∀ a ∈ M , the restrictions of | H | and | H Ω | at a are denoted by | H ( a ) | , { H T ( a ) | T ∈ T } and | H Ω ( a ) | , { H T ( a ) | T ∈ Ω } , respectively. Remark 3.4.1.
When t = 0 , intuitively, the gradient directions | H ( a ) | of | ρ | start from a and pointto all directions around a uniformly. If ( M, g ) is not flat, when evolving to a certain time t > ,the distribution of gradient directions on ϕ | H | ,a ( t ) is no longer as uniform as they are around a . The following definition precisely characterizes such a distribution, which thereby reflect theintrinsic geometrical shape in a new approach.
Definition 3.4.2. (Evolution distribution) . Let transformation L g − act on g , then we obtain thetrivial e , L g − ( g ) . Now ( M, g ) is sent to a flat ( M, e ) , and the gradient direction field | H | of | ρ | on ( M, g ) is sent to a gradient direction field | O | of | ρ | on ( M, e ) . Correspondingly, ϕ | H | ,a ( t ) issent to ϕ | O | ,a ( t ) . In a word, L g − induces the following two maps: g − ∗ : | H | → | O | , g − ∗∗ : ϕ | H | ,a → ϕ | O | ,a . ∀| H Ω | ⊆ | H | , denote | O Ω | , g − ∗ ( | H Ω | ) ⊆ | O | and ϕ | O Ω | ,a , g − ∗∗ (cid:0) ϕ | H Ω | ,a (cid:1) ⊆ ϕ | O | ,a . Further-more, ∀ t ∈ R + , we say the measure P (cid:0) ϕ | O Ω | ,a ( t ) (cid:1) = P (cid:0) g − ∗∗ (cid:0) ϕ | H Ω | ,a ( t ) (cid:1)(cid:1) is the distribution ofgradient directions of | ρ | on ϕ | H Ω | ,a ( t ) , or the evolution distribution of | ρ | at time x = t afterstarting from a in directions | H Ω | .Due to T ∼ = GL ( D , R ) , let Ω be a certain neighborhood of T , with respect to the topologyof GL ( D , R ) . Now at the starting point a , we say | H Ω ( a ) | is a neighborhood of H T ( a ) , and | O Ω ( a ) | , g − ∗ ( | H Ω ( a ) | ) is a neighborhood of O T ( a ) , g − ∗ ( H T ( a )) .When Ω is sufficiently small, | H Ω ( a ) | and | O Ω ( a ) | are both sufficiently small, and ∀ t ∈ R + , ϕ | H Ω | ,a ( t ) and ϕ | O Ω | ,a ( t ) are also sufficiently small. Concretely, when Ω → T , H T = lim Ω → T | H Ω | , H T ( a ) = lim Ω → T | H Ω ( a ) | , and the evolution image ϕ | H Ω | ,a ( t ) of a at t will approach to a point b T , ϕ H T ,a ( t ) = lim | H Ω ( a ) |→ H T ( a ) ϕ | H Ω | ,a ( t ) .The limit w a ( b T ) , dV O T dV H T , lim Ω → T P (cid:0) ϕ | O Ω | ,a ( t ) (cid:1) P (cid:0) ϕ | H Ω | ,a ( t ) (cid:1) = lim Ω → T P (cid:0) g − ∗∗ (cid:0) ϕ | H Ω | ,a ( t ) (cid:1)(cid:1) P (cid:0) ϕ | H Ω | ,a ( t ) (cid:1) (16)is called the distribution density of gradient directions at time x = t after starting from a indirection H T , or distribution density of evolution . Remark 3.4.2.
Radon-Nikodym theorem [54] guarantees the existence of such a limit. Andevidently such a distribution density is an intrinsic geometrical property of ( M, g ) .For any two points a and b on manifold M , it anyway makes sense to discuss the gradient lineof | ρ | from a to b . It is because even if the gradient line of ρ starting from a does not pass through b , it just only needs to carry out a certain flat transformation T defined in section 2.4 to obtain a ρ ′ , T ∗ ρ so that the gradient line of ρ ′ starting from a can just exactly pass through b . Intuitively,when | ρ | takes two different initial directions of motion, | ρ | presents as ρ and ρ ′ , respectively. Definition 3.4.3. (Evolutor) ∀ a, b ∈ M , if ∃ ρ ′ ∈ | ρ | such that a and b are both on the gradientline L ( b, a ) of ρ ′ , then we say L ( b, a ) is a gradient line of | ρ | .According to Definition 3.3.6.2 , let s L ( b, a ) be the evolution quantity on L ( b, a ) . Denote r L ( b, a ) , p w a ( b ) , R L ( b, a ) , r L ( b, a ) e is L ( b,a ) , which are both intrinsic geometrical properties of ( M, g ) . We say R L ( b, a ) is the evolutor of | ρ | on L ( b, a ) . generalization of intrinsic geometry and its application to Hilbert’s 6th problem 37 Definition 3.4.4. (Propagator)
Let L ( b, a ) be the totality of gradient lines of | ρ | from a to b . ∀ L ( b, a ) ∈ L ( b, a ) , let R L ( b, a ) be the evolutor of | ρ | on L ( b, a ) . The intrinsic geometrical property K ( b, a ) , X L ∈ L ( b,a ) R L ( b, a ) (17)is called the propagator of | ρ | from a to b . Remark 3.4.3.
Abstractly, propagator is defined as the Green function of evolution equation.Concretely, propagator still needs a constructive definition. One method is to construct withFeynman path integral [22] R x b x a e iS D x ( t ) , which is expressed in form of functional integral.However, until now the functional integral has strict definition just in some special cases, butthe strict definition in general case is still an unsolved problem.Definition 3.4.4 reduces the scope of summation to the totality of gradient lines from a to b .If ( M, g ) is flat, | ρ | has a unique gradient line from a to b , but it is not unique for general ( M, g ) . Remark 3.4.4.
As the simplest example, consider the propagator of free particle.In this case ( M, g ) is flat. In sense of Remark 3.4.1 , the system of gradient lines of | ρ | starting from a spreads uniformly in all directions around a . No matter where b is, w a ( b ) isidentically equal to . Then for a fixed b , the evolutor on gradient line L ( b, a ) is R L ( b, a ) = r L ( b, a ) e is L ( b,a ) = p w a ( b ) e is L ( b,a ) = e is L ( b,a ) . Because there is only one element in L ( b, a ) , thepropagator is K ( b, a ) = R L ( b, a ) = e is L ( b,a ) . Of course, if normalizing on wavefront, there wouldbe a coefficient of normalization. Definition 3.4.5. (Wave function) . Let a be a point on geometrical manifold ( M, g ) . ∀ a ∈ M , d ( a, a ) is the geodesic distance between a and a . Denote Σ a ( a ) , { q ∈ M | d ( a, q ) = d ( a, a ) } .If function ψ : M → C satisfies both the following two conditions, then ψ is called a wavefunction of | ρ | on ( M, g ) .(1) ∃ r > such that lim d ( a,a ) → r ψ ( a ) = 0 , or lim d ( a,a ) →∞ ψ ( a ) = 0 .(2) ∀ a , a ∈ M , ψ ( a ) = Z Σ a ( a ) K ( a, q ) ψ ( q ) dσ q . Remark 3.4.5.
The propagator K ( a, q ) is an intrinsic geometrical property of ( M, g ) , so theabove defined wave function ψ is also an intrinsic geometrical property of ( M, g ) .
1. This section proposes an intrinsic geometrical solution for Hilbert’s 6th problem at themost basic level, that is, starting from an axiom, based on intrinsic geometry, the theoreticalframework at the most basic level of physics is deduced in sense of pure mathematics.2. Postulates that have a status as principle in physics are theorems that hold automatically inintrinsic geometry, such as Yang-Mills field equation, Lorentz force equation, energy-momentum equation, conservation law of energy-momentum, gravitational field equation(see Discussion5.4.4 ), least action principle, Schrödinger equation, Heisenberg equation, Dirac equation(seesection 5.6 ), etc., which in intrinsic geometry are no longer necessary to be regarded as principlesand postulates.In addition, this section adopts general coordinate form, the evolution parameter of which is x . In section 5.2 , the Minkowski coordinate will be constructed, the evolution parameter ofwhich is x τ . It has to be emphasized that no matter what coordinate forms are adopted, theirgeometrical essences are the same. Intrinsic geometrical properties in some special cases willbe discussed in the following sections. In this section, indices of internal and external space are taken values according to Definition5.1.1 .
Definition 4.1.
Let the local coordinate representation of reference-system k be x ′ j = − δ ji x i , x ′ n = δ nm x m , then we say P , L [ k ] : x i → − x i , x m → x m is parity inversion . Let the localcoordinate representation of reference-system h be x ′ j = δ ji x i , x ′ n = − δ nm x m , then we say C , L [ h ] : x i → x i , x m → − x m is charge conjugate inversion . In addition, denote T : x → − x ,which is called time coordinate invesion .Their composition CP T : x R → − x R , x → − x is called total inversion of coordinates . Definition 4.2.
Reviewing Definition 3.2.1 , without loss of generality, positive or negativesign of metric, which marks two opposite directions of evolution, is independent of positiveor negative sign of coordinate. Let N be a closed submanifold of M , and its metric be dx ( N ) .The transformation T ( N )0 : dx ( N ) → − dx ( N ) is called a single inversion of space metric on N .Specially, when N = M , T ( M )0 : dx → − dx is called a single inversion of time metric .Denote the totality of closed submanifolds of M by B ( M ) , and denote T ( M ) , Q B ∈ B ( M ) T ( B )0 .We say T ( M ) is total inversion of metrics . Definition 4.3. T , T ( M ) T is called time inversion . The joint transformation of total inversionof coordinates CP T and total inversion of metrics T ( M ) is called space-time inversion , that is CP T T ( M ) = CP T . Remark 4.1.
Summerize the above definitions, then we have:
CP T : x R → − x R , x → − x , dx R → dx R , dx → dx ,T ( M ) : x R → x R , x → x , dx R → − dx R , dx → − dx ,CP T : x R → − x R , x → − x , dx R → − dx R , dx → − dx . generalization of intrinsic geometry and its application to Hilbert’s 6th problem 39 Proposition 4.1.
Consider
CP T on g . Denote s , Z L Dρ , and D P e is , (cid:16) ∂∂x P + i [ ρΓ P ] (cid:17) e is , then:(1) CP T : Dρ → Dρ , (2) CP T : D P e is → − D P e is . Proof. (1) On one hand, for CP : x R → − x R we have CP : ∂∂x R → − ∂∂x R . According to thedefinition of simple connection we obtain CP : Γ MNP → − Γ MNP . In consideration of that ρ isdetermined by f , so ρ is required to remain unchanged under transformations of g .Due to p R , ∂ρ∂x R + [ ρΓ P ] we know CP : p R → − p R . On the other hand, according to thedefinition of T we immediately obtain T : dx R → − dx R . In summary, we have CP T : p R dx R → ( − p R )( − dx R ) , that is CP T : Dρ → Dρ .(2) On one hand, in consideration of CP : ∂∂x P + i [ ρΓ P ] → − ∂∂x P − i [ ρΓ P ] , and dueto s , Z L Dρ = Z L p R dx R we know CP : s → − s , hence CP : (cid:16) ∂∂x P + i [ ρΓ P ] (cid:17) e is → (cid:16) − ∂∂x P − i [ ρΓ P ] (cid:17) e − is , that is CP : D P e is → − D P e − is . On the other hand, ∂∂x P + i [ ρΓ P ] and p R are both independent of metrics, therefore ∂∂x P + i [ ρΓ P ] and p R both remain unchangedunder transformation T . Due to T : dx R → − dx R , then T : p R dx R → − p R dx R , that is T : Dρ → − Dρ , hence T : s → − s . Moreover, T : D P e is → D P e − is = D P ( e is ) ∗ . In sum-mary, we have CP T : D P e is → − D P ( e − is ) ∗ = − D P e is . ⊓⊔ Remark 4.2.
The above proposition gives the intrinsic geometrical origin of
CP T invariance.In addition, in physics there is a complex conjugation in time inversion about wave function T : ψ ( x, t ) → ψ ∗ ( x, − t ) , the mathematical orgin of which can essentially be attributed to Definition4.1 , Definition 4.2 and Definition 4.3 . In fact we have the following proposition, where thecoordinate ˜ x µ is Minkowski coordinate defined in section 5.2 such that ˜ x i = x i , ˜ x = x , andMinkowski metric satisfies ( d ˜ x τ ) = ( d ˜ x ) − P i ( d ˜ x i ) , ˜ m τ , ˜ ρ ; τ . Proposition 4.2.
Denote S , Z L ˜ m τ d ˜ x τ and ψ (˜ x i , ˜ x ) , f (˜ x i , ˜ x ) e iS , then T : ψ (˜ x i , ˜ x ) → ψ ∗ (˜ x i , − ˜ x ) . Proof.
According to Definition 4.3 we know T : ˜ m τ → ˜ m τ , d ˜ x τ → − d ˜ x τ , ˜ x → − ˜ x ,therefore T : ˜ m τ d ˜ x τ → − ˜ m τ d ˜ x τ , then T : S → − S, f (˜ x i , ˜ x ) → f (˜ x i , − ˜ x ) , and furthermore T : f (˜ x i , ˜ x ) e iS → f (˜ x i , − ˜ x ) e − iS , that is T : ψ (˜ x i , ˜ x ) → ψ ∗ (˜ x i , − ˜ x ) . ⊓⊔ Suppose M = P × N , D , dimM and r , dimP = 3 . According to Definition3.2.2 , we have a submanifold of external space P and a submanifold of internal space N . P inherits coordinate { ξ s }{ x i } from M , and N inherits coordinate { ξ a }{ x m } from M . The valuesof indices are specified as follows.(1) Total indices of basis coordinate frame ξ are A, B, C, D = 1 , , · · · , D . Total indices ofperformance coordinate frame x are M, N, P, Q = 1 , , · · · , D . (2) External indices of ξ are s, t, u, v = 1 , , · · · , r . Internal indices of ξ are a, b, c, d = r + 1 , r + 2 , · · · , D .(3) External indices of x are i, j, k, l = 1 , , · · · , r . Internal indices of x are m, n, p, q = r + 1 , r + 2 , · · · , D .(4) Regular indices of ξ are S, T, U, V = 1 , , · · · , r, τ . Minkowski indices of ξ are α, β, γ, δ =0 , , , · · · , r .(5) Regular indices of x are I, J, K, L = 1 , , · · · , r, τ . Minkowski indices of x are µ, ν, ρ, σ =0 , , , · · · , r . Definition 5.1.2.
Let there be a smooth tangent vector field X on ( M, f ) . If ∀ p ∈ M , X ( p ) = b A ∂∂ξ A (cid:12)(cid:12)(cid:12) p = c M ∂∂x M (cid:12)(cid:12)(cid:12) p satisfies that b a are not all zero and c m are not all zero, then we say X is internal-directed . Proposition 5.1.1.
Suppose M = P × N and X is a smooth tangent vector field on M . Fix apoint o ∈ M . If X is internal-directed, then:(1) There exist a unique ( r + 1) -dimensional regular submanifold γ : ˜ M → M, p p and aunique smooth tangent vector field ˜ X on ˜ M such that: (i) P × { o } is a closed submanifold of ˜ M , (ii) tangent map γ ∗ : T ( ˜ M ) → T ( M ) satisfies that ∀ q ∈ ˜ M , γ ∗ : ˜ X ( q ) X ( q ) . Such an ˜ M iscalled a submanifold with classical spacetime determined by X through o .(2) Let ϕ X be the one-parameter group of diffeomorphisms on M corresponding to X , and ϕ ˜ X be the one-parameter group of diffeomorphisms on ˜ M corresponding to ˜ X . Thus, we have ϕ ˜ X = ϕ X | ˜ M . Proof.
Step 1: construction of ˜ M . We can define a closed submanifold P × { o } on M through o via parameter equation x m = x mo . Then let ϕ X : M × R → M be the one-parameter group ofdiffeomorphisms corresponding to X . Restrict ϕ X to P × { o } and we obtain ϕ X | P ×{ o } : P × { o } × { t } 7→ P ′ × { o ′ } , where points o and o ′ are on the same orbit L o , ϕ X,o . P × { o } and P ′ × { o ′ } are bothhomeomorphic to P . If we do not distinguish P and P ′ , we have ϕ X | P ×{ o } : P × { o } × R → P × L o . Then consider all of such { o } on the entire orbit L o , and we obtain a map ϕ X | P × L o : P × L o × R → P × L o . Denote ˜ M , P × L o , then ϕ X | ˜ M : ˜ M × R → ˜ M constitutes a one-parameter group of diffeomorphisms on ˜ M .Step 2: constructions of γ : ˜ M → M and ˜ X . Because X is internal-directed, the restrictionof X to L o , ϕ X,o : R → M is non-vanishing everywhere and L o is an injection. The image setof L o can also be denoted by L o . ∀ t ∈ R , q , ϕ X,o ( t ) ∈ L o , we can define a closed submanifold generalization of intrinsic geometry and its application to Hilbert’s 6th problem 41 N q on M through q via parameter equation x i = x iq , and N q is homeomorphic to N . Due to theone-to-one correspondence between q and N q , L o → N is a regular imbedding. Furthermore:(1) γ : P × L o → P × N is a regular imbedding, that is, γ : ˜ M → M . Hence the tangentmap γ ∗ : T ( ˜ M ) → T ( M ) is an injection. Therefore, the smooth tangent vector ˜ X which satisfies ∀ q ∈ ˜ M , γ ∗ : ˜ X ( q ) X ( q ) is uniquely defined by X via γ − ∗ .(2) We notice that o ∈ ˜ M , hence L o , ϕ X,o is an orbit of ϕ X | ˜ M . In consideration of that L o uniquely determines γ , and γ uniquely determines γ ∗ , and γ − ∗ uniquely determines ˜ X , so finally ˜ X is uniquely determined by ϕ X | ˜ M . Thus, we have ϕ ˜ X = ϕ X | ˜ M .In summary of (1) and (2), it also indicates that ˜ M is determined by X , therefore it is unique. ⊓⊔ Remark 5.1.1. (1) ˜ M is not independent of M , but determined by smooth tangent vector field X on M .(2) ˜ M is a regular submanifold of M , so not all intrinsic geometrical properties of M can beinherited by ˜ M .(3) The correspondence between ˜ X and the restriction of X to ˜ M is one-to-one. For conve-nience, we later will not distinguish the notations X and ˜ X on ˜ M , but uniformly denote them by X . (4) An arbitrary path ˜ L : T → ˜ M , t p on ˜ M uniquely corresponds to a path L , γ ◦ ˜ L : T → M, t p on M . Evidently the image sets of L and ˜ L are the same, that is, L ( T ) = ˜ L ( T ) .For convenience, we later will not distinguish the notations L and ˜ L on ˜ M , but uniformly denotethem by L . Let ˜ M be the submanifold with classical spacetime determined by X on ( M, f ) , and L be a path on an orbit of ϕ ˜ X of ˜ M . ∀ p ∈ L , suppose on neighborhood U that f ( p ) : ξ A = ξ A ( x M ) = ξ A ( x ) , ξ = ξ ( x ) . Thus: (1) There exists a unique local reference-system ˜ f ( p ) on ˜ U , U ∩ ˜ M such that ˜ f ( p ) : ξ U = ξ U ( x K ) = ξ U ( x ) , ξ = ξ ( x ) and ( dx τ ) = D P m = r +1 ( dx m ) , ( dξ τ ) = D P a = r +1 ( dξ a ) .(2) The above coordinate frames ( ˜ U , ξ U ) and ( ˜ U , x K ) of ˜ f ( p ) uniquely determine coordinateframes ( ˜ U , ˜ ξ α ) and ( ˜ U , ˜ x µ ) , such that ˜ f ( p ) : ˜ ξ α = ˜ ξ α (˜ x µ ) = ˜ ξ α (˜ x τ ) , ˜ ξ τ = ˜ ξ τ (˜ x τ ) and coordinates ˜ ξ s = ξ s , ˜ ξ τ = ξ τ , ˜ ξ = ξ , ˜ x i = x i , ˜ x τ = x τ , ˜ x = x . Proof. (1) According to the proof of previous proposition, if L is on an orbit of ϕ ˜ X , L is a regularsubmanifold of N . Let the metrics on N be ( dx τ ) = D P m = r +1 ( dx m ) and ( dξ τ ) = D P a = r +1 ( dξ a ) .Review Definition 3.3.1.3 and we know the regular imbedding π : L → N, q q inducesparameter equations x m = x mτ ( x τ ) and ξ a = ξ aτ ( ξ τ ) of L . Then substitute them into f ( p ) and weobtain ξ A = ξ A (cid:16) x M (cid:17) = ξ A (cid:0) x (cid:1) ⇔ ξ u = ξ u (cid:16) x k , x mτ ( x τ ) (cid:17) = ξ uL (cid:0) x (cid:1) ξ aτ ( ξ τ ) = ξ a (cid:16) x k , x mτ ( x τ ) (cid:17) = ξ aL (cid:0) x (cid:1) ⇔ ξ u = ξ u (cid:16) x k , x mτ ( x τ ) (cid:17) = ξ uL (cid:0) x (cid:1) ξ τ = ( ξ aτ ) − ◦ ξ a (cid:16) x k , x mτ ( x τ ) (cid:17) = ( ξ aτ ) − ◦ ξ aL (cid:0) x (cid:1) ⇔ ξ u = ξ uτ (cid:16) x k , x τ (cid:17) = ξ uL (cid:0) x (cid:1) ξ τ = ξ τ (cid:16) x k , x τ (cid:17) = ξ τL (cid:0) x (cid:1) ⇔ ξ U = ξ U ( x K ) = ξ UL ( x ) , abbreviated to ξ U = ξ U ( x K ) = ξ U ( x ) . (2) As same as the above, we also obtain x K = x K ( ξ U ) = x K ( ξ ) . The relation between twoparameters x and x τ of L can be expressed as x τ = x τL ( ξ ( x )) , and the relation between twoparameters ξ and ξ τ can be expressed as ξ τ = ξ τL ( x ( ξ )) . Substitute them into ˜ f ( p ) , then ξ U = ξ U ( x K ) = ξ U ( x ) ⇔ ξ u = ξ u (cid:16) x k , x τL (cid:0) ξ (cid:0) x (cid:1)(cid:1)(cid:17) = ξ uL (cid:16) x (cid:16) ( x τL ) − ( x τ ) (cid:17)(cid:17) ξ τL (cid:0) x (cid:0) ξ (cid:1)(cid:1) = ξ τ (cid:16) x k , x τL (cid:0) ξ (cid:0) x (cid:1)(cid:1)(cid:17) = ξ τL (cid:16) x (cid:16) ( x τL ) − ( x τ ) (cid:17)(cid:17) ξ (cid:16) ξ τL − ( ξ τ ) (cid:17) = ( x τL ) − ( x τ ) (18) ⇔ ξ u = ξ u (cid:16) x k , x τL (cid:0) ξ (cid:0) x (cid:1)(cid:1)(cid:17) = ξ uL (cid:16) x (cid:16) ( x τL ) − ( x τ ) (cid:17)(cid:17) ξ = ξ (cid:16) ( ξ τL ) − (cid:16) ξ τ (cid:16) x k , x τL (cid:0) ξ (cid:0) x (cid:1)(cid:1)(cid:17)(cid:17)(cid:17) = ( x τL ) − ( x τ ) ξ τ = ξ τL (cid:16) x (cid:16) ( x τL ) − ( x τ ) (cid:17)(cid:17) , abbreviated to ξ u = ˜ ξ u ( x k , x ) = ˜ ξ uL ( x τ ) ξ = ˜ ξ ( x k , x ) = ˜ ξ L ( x τ ) ξ τ = ˜ ξ τ ( x τ ) . Denote ˜ ξ s , ξ s , ˜ ξ τ , ξ τ , ˜ ξ , ξ , ˜ x i , x i , ˜ x τ , x τ , ˜ x , x , hence we have ˜ ξ α = ˜ ξ α (˜ x µ ) =˜ ξ αL (˜ x τ ) , ˜ ξ τ = ˜ ξ τ (˜ x τ ) , abbreviated to ˜ ξ α = ˜ ξ α (˜ x µ ) = ˜ ξ α (˜ x τ ) , ˜ ξ τ = ˜ ξ τ (˜ x τ ) . ⊓⊔ Definition 5.2.1. ˜ f is called a classical spacetime reference-system on ˜ M , and ( ˜ M , ˜ f ) is calleda gravitational manifold . ( ˜ U , ξ U ) and ( ˜ U , x K ) are called regular coordinate frames on ( ˜ M , ˜ f ) ,meanwhile ( ˜ U , ˜ ξ α ) and ( ˜ U , ˜ x µ ) are called Minkowski coordinate frames on ( ˜ M , ˜ f ) . Remark 5.2.1. ˜ f is uniquely determined by f , and ˜ f encapsulates the internal space of f .Although ( ˜ M , ˜ f ) can reflect intrinsic geometrical properties of external space of ( M, f ) , itcannot totally reflect intrinsic geometrical properties of internal space of ( M, f ) . There is afurther illustration in Discussion 5.4.2 .In addition, ˜ f presents locally as ˜ f ( p ) , so the mathematical origin of the principle of equiv-alence has been able to be attributed to Definition 2.1.2 , Definition 2.2.2 and Definition 5.2.1. Of course its physical connotation can only be endowed by the fundamental axiom of section3.1 . Besides, with respect to the gravitational field equation, see Discussion 5.4.4 . generalization of intrinsic geometry and its application to Hilbert’s 6th problem 43 It is well-known that there is no logically strict definition of inertial system in physics.However, now we can give a strict definition to inertial system from the perspective of intrinsicgeometry. It is a reference-system, not a coordinate frame.
Definition 5.2.2.
Suppose we have a geometrical manifold ( ˜
M , ˜ g ) . F ˜ g is a transformation inducedby ˜ g .(1) According to Discussion 5.3.2 , if d ˜ ζ τ = d ˜ x τ , then we must have ˜ G µν = ˜ ∆ αβ ˜ B αµ ˜ B βν = ˜ E µν .Thus, we say ˜ g is orthogonal , F ˜ g is an orthogonal transformation , and ( ˜ M , ˜ g ) is an isotropicspacetime . For convenience, d ˜ ζ τ and d ˜ x τ are uniformly denoted by dτ .(2) If the slack-tights ˜ B αµ and ˜ C µα of ˜ g are constants on ˜ M , then we say ˜ g is flat , F ˜ g is a flattransformations , and ( ˜ M , ˜ g ) is a flat spacetime .(3) If ˜ g is both orthogonal and flat, then we say ˜ g is an inertial-system , F ˜ g is a Lorentztransformations , and the isotropic and flat ( ˜
M , ˜ g ) is Minkowski spacetime . Proposition 5.2.2.
Let L be a path on ˜ M , and ˜ U be a coordinate neighborhood. Denote ˜ U L , ˜ U ∩ L , and c , (cid:12)(cid:12) dx i /dx (cid:12)(cid:12) . If ∀ q ∈ ˜ U L , tangent vector [ L ] ∈ T q ( M ) is not internal-directed, then:(i) we have c = 1 on path ˜ U L . (ii) c = 1 remains unchanged under orthogonal transformation.(iii) c = 1 remains unchanged under Lorentz transformation. Proof.
In regular coordinate frame ( ˜
U , x K ) , ˜ U L can be described by equations x i = x i ( x ) and x τ = const about parameter x . We notice that x τ = const , so there does not exist an equation of ˜ U L with respect to parameter ˜ x τ in Minkowski coordinate frame ( ˜ U , ˜ x µ ) . Therefore, we alwayshave d ˜ x τ = dx τ = 0 on ˜ U L . Thus, c = (cid:12)(cid:12) dx i /dx (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) ± dx i / p ( dx i ) + ( dx τ ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12) ± dx i /dx i (cid:12)(cid:12) = 1 .According to Definition 5.2.2 , an orthogonal transformation satisfies d ˜ ζ τ = d ˜ x τ = 0 , hence c ′ = (cid:12)(cid:12) dζ s /dζ (cid:12)(cid:12) = |± dζ s /dζ s | = 1 . It is naturally true for Lorentz transformation as an orthogonalone. ⊓⊔ Remark 5.2.2.
The above proposition indicates the limitation of Minkowski coordinate, andalso turns the principle of constancy of light velocity to a theorem, i.e. conclusion (iii). Itsmathematical origin can be attributed to Definition 3.2.1 .
Discussion 5.2.1.
Let ˜ ρ be a charge of ˜ f , and L be a gradient line of ˜ ρ on ( ˜ M , ˜ g ) .(1) In the basis coordinate frame { ˜ ζ α } of ˜ g , suppose the parameter equation of L satisfies ˜ ζ s = const . Thus on L we have velocity d ˜ ζ s dτ = 0 and proper time d ˜ ζ = dτ . This is the intrinsicgeometrical treatment of the inertial relative rest state of physics, which is not a stopped evolution,but just an evolution in a special direction.(2) In the performance coordinate frame { ˜ x µ } of ˜ g , on L we have velocity d ˜ x i dτ = ˜ C i andcoordinate time d ˜ x = ˜ C τ dτ , where the slack-tights ˜ C µα are constants. This is the intrinsicgeometrical treatment of the inertial relative motion of physics. According to Definition 3.3.2.2 , suppose the slack-tights of ( ˜
M , ˜ f ) aboutregular coordinate x K are B SI and C IS , such that dξ S = B SI dx I ≃ B S dx C I ∂∂x I ∼ = C ddx = ddξ , dx I = C IS dξ S ≃ C I dξ B S ∂∂ξ S ∼ = B ddξ = ddx . (19)The slack-tights of ( ˜ M , ˜ f ) about Minkowski coordinate ˜ x µ are ˜ B αµ and ˜ C µα , such that d ˜ ξ α = ˜ B αµ d ˜ x µ ≃ ˜ B ατ d ˜ x τ ˜ C µτ ∂∂ ˜ x µ ∼ = ˜ C ττ dd ˜ x τ = dd ˜ ξ τ , d ˜ x µ = ˜ C µα d ˜ ξ α ≃ ˜ C µτ d ˜ ξ τ ˜ B ατ ∂∂ ˜ ξ α ∼ = ˜ B ττ dd ˜ ξ τ = dd ˜ x τ . (20)Applying the chain rule of differentiation, it is not difficult to obtain the following relationsbetween regular slack-tights and Minkowski slack-tights via formula (18). ˜ B si = − B si ˜ B i = − B τi δ τ ˜ C iτ = C i δ τ , ˜ B sτ = B sτ ˜ B τ = B ττ δ τ ˜ C ττ = C τ δ τ , ˜ B s = B s ˜ B = B τ δ τ ˜ C τ = C δ τ ; ˜ C is = − C is ˜ C s = − C τs ε τ ˜ B sτ = B s ε τ , ˜ C iτ = C iτ ˜ C τ = C ττ ε τ ˜ B ττ = B τ ε τ , ˜ C i = C i ˜ C = C τ ε τ ˜ B τ = B ε τ , B si = − ˜ B si B τi = − ˜ B i ˜ δ τ C i = − ˜ C iτ ˜ δ τ , B s = ˜ B s B τ = ˜ B ˜ δ τ C = ˜ C τ ˜ δ τ , B sτ = ˜ B sτ B ττ = ˜ B τ ˜ δ τ C τ = ˜ C ττ ˜ δ τ ; C is = − ˜ C is C τs = − ˜ C s ˜ ε τ B s = ˜ B sτ ˜ ε τ , C i = ˜ C i C τ = ˜ C ˜ ε τ B = ˜ B τ ˜ ε τ , C iτ = ˜ C iτ C ττ = ˜ C τ ˜ ε τ B τ = ˜ B ττ ˜ ε τ , where ε IJ , C IS B SJ , δ ST = B SI C IT , ε I , B C I = B S C IS , δ S , C B S = C I B SI . ˜ ε µν , ˜ C µα ˜ B αν , ˜ δ αβ = ˜ B αµ ˜ C µβ , ˜ ε µτ , ˜ B ττ ˜ C µτ = ˜ B ατ ˜ C µα , ˜ δ ατ , ˜ C ττ ˜ B ατ = ˜ C µτ ˜ B αµ . The evolution lemma of Proposition 3.3.2.2 can be expressed in Minkowski coordinate as w µ ∂∂ ˜ x µ ∼ = w τ dd ˜ x τ ⇔ w µ = w τ ˜ ε µτ w µ d ˜ x µ ≃ w τ d ˜ x τ ⇔ ˜ ε µτ w µ = w τ , ¯ w µ ∂∂ ˜ x µ ∼ = ¯ w τ dd ˜ x τ ⇔ ¯ w µ = ¯ w τ ˜¯ ε τµ ¯ w µ d ˜ x µ ≃ ¯ w τ d ˜ x τ ⇔ ˜¯ ε τµ ¯ w µ = ¯ w τ , where ˜¯ ε µν , ˜¯ B µα ˜¯ C αν = ε µν , ˜¯ δ αβ , ˜¯ C αµ ˜¯ B µβ = δ αβ , ˜¯ ε τµ , ˜¯ B ττ ˜¯ C τµ = ˜¯ B τα ˜¯ C αµ , ˜¯ δ τα , ˜¯ C ττ ˜¯ B τα = ˜¯ C τµ ˜¯ B µα . generalization of intrinsic geometry and its application to Hilbert’s 6th problem 45 Discussion 5.3.2.
According to Definition 3.2.1 , on the neighborhood ˜ U of p on ( ˜ M , f ) , timemetrics dξ and dx of ˜ f ( p ) in coordinate frames ( ˜ U , ξ S ) and ( ˜ U , x I ) satisfy ( dξ ) , r X s =1 ( dξ s ) + ( dξ τ ) = δ ST dξ S dξ T = δ ST b SI b TJ dx I dx J = g IJ dx I dx J , ( dx ) , r X i =1 ( dx i ) + ( dx τ ) = ε IJ dx I dx J = ε IJ c IS c JT dξ S dξ T = h ST dξ S dξ T . where ( dξ τ ) , D P a = r +1 ( dξ a ) and ( dx τ ) , D P m = r +1 ( dx m ) . As differential forms defined on mani-fold, time metrics of ( ˜ M , ˜ f ) satisfy ( dξ ) , ∆ ST dξ S dξ T = G IJ dx I dx J , ( dx ) , E IJ dx I dx J = H ST dξ S dξ T . G IJ , ∆ ST B SI B TJ ,H ST , E IJ C IS C JT . Denote ˜ ε µν = ˜ ε µν , µ = ν = 0 − , µ = ν = 00 , µ = ν , ˜ δ αβ = ˜ δ αβ , α = β = 0 − , α = β = 00 , α = β . Thus, we can define proper-times d ˜ ξ τ and d ˜ x τ in Minkowski coordinate frames ( ˜ U , ˜ ξ α ) and ( ˜ U , ˜ x µ ) as below: ( d ˜ ξ τ ) = ( dξ ) − r X s =1 ( dξ s ) = ˜ δ αβ d ˜ ξ α d ˜ ξ β = ˜ δ αβ ˜ b αµ ˜ b βν d ˜ x µ d ˜ x ν = ˜ g µν d ˜ x µ d ˜ x ν , ( d ˜ x τ ) = ( dx ) − r X i =1 ( dx i ) = ˜ ε µν d ˜ x µ d ˜ x ν = ˜ ε µν ˜ c µα ˜ c νβ d ˜ ξ α d ˜ ξ β = ˜ h αβ d ˜ ξ α d ˜ ξ β . And there are differential forms on ( ˜
M , ˜ f ) as below: ( d ˜ ξ τ ) , ˜ ∆ αβ d ˜ ξ α d ˜ ξ β = ˜ G µν d ˜ x µ d ˜ x ν , ( d ˜ x τ ) , ˜ E µν d ˜ x µ d ˜ x ν = ˜ H αβ d ˜ ξ α d ˜ ξ β . ˜ G µν , ˜ ∆ αβ ˜ B αµ ˜ B βν , ˜ H αβ , ˜ E µν ˜ C µα ˜ C νβ . It is easy to know there are relations between regular metric G IJ and Minkowski metric ˜ G µν asbelow: G ττ = ˜ G ττ ˜ G G = ˜ δ τ ˜ δ τ ˜ ε τ ˜ ε τ ˜ G ττ ˜ G ˜ G ττ G iτ = − ˜ G i ˜ G ˜ ε τ G = − ˜ δ τ ˜ δ τ ˜ ε τ ˜ G ττ ˜ G ˜ G i G τj = − ˜ G j ˜ G ˜ ε τ G = − ˜ δ τ ˜ δ τ ˜ ε τ ˜ G ττ ˜ G ˜ G j G ij = − ˜ G ij ˜ G G = − ˜ δ τ ˜ δ τ ˜ ε τ ˜ ε τ ˜ G ττ ˜ G ˜ G ij , ˜ G = G G ττ ˜ G ττ = δ τ δ τ ε τ ε τ G G ττ G ˜ G i = − G iτ G ττ ε τ ˜ G ττ = − δ τ δ τ ε τ G G ττ G iτ ˜ G j = − G τj G ττ ε τ ˜ G ττ = − δ τ δ τ ε τ G G ττ G τj ˜ G ij = − G ij G ττ ˜ G ττ = − δ τ δ τ ε τ ε τ G G ττ G ij . Denote ˜ G ττ , ˜ B ττ ˜ B ττ , ˜ G ττ , ˜ C ττ ˜ C ττ , then it is easy to obtain relations: ˜ G ττ = δ τ δ τ ε τ ε τ G , G = ˜ δ τ ˜ δ τ ˜ ε τ ˜ ε τ ˜ G ττ . The absolute differential and absolute gradient of section 3.3.4 can be ex-pressed on ˜ M in Minkowski coordinate as:(1) Let ˜ D be affine connection on ˜ M , and denote ˜ t L ; τ , ˜ t ; σ ˜ ε στ , then the absolute differentialof ˜ T and ˜ T L are ˜ D ˜ T , ˜ D ˜ t ⊗ (cid:26) ∂∂ ˜ x ⊗ d ˜ x (cid:27) , ˜ t ; σ d ˜ x σ ⊗ (cid:26) ∂∂ ˜ x ⊗ d ˜ x (cid:27) , ˜ D L ˜ T L , ˜ D L ˜ t L ⊗ (cid:26) ∂∂ ˜ x ⊗ d ˜ x (cid:27) , ˜ t L ; τ d ˜ x τ ⊗ (cid:26) ∂∂ ˜ x ⊗ d ˜ x (cid:27) , where ˜ D ˜ t , ˜ t ; σ d ˜ x σ , ˜ D L ˜ t L , ˜ t L ; τ d ˜ x τ .(2) The gradient operator ˜ ∇ is the actual evolution on ˜ M . Thus, the absolute gradient of ˜ T and ˜ T L are ˜ ∇ ˜ T , ˜ ∇ ˜ t ⊗ (cid:26) ∂∂ ˜ x ⊗ d ˜ x (cid:27) , ˜ t ; σ ∂∂ ˜ x σ ⊗ (cid:26) ∂∂ ˜ x ⊗ d ˜ x (cid:27) , ˜ ∇ L ˜ T L , ˜ ∇ L ˜ t L ⊗ (cid:26) ∂∂ ˜ x ⊗ d ˜ x (cid:27) , ˜ t L ; τ dd ˜ x τ ⊗ (cid:26) ∂∂ ˜ x ⊗ d ˜ x (cid:27) , where ˜ ∇ ˜ t , ˜ t ; σ ∂∂ ˜ x σ , ˜ ∇ L ˜ t L , ˜ t L ; τ dd ˜ x τ .Now Proposition 3.3.4.1 can be expressed as: ˜ D ˜ T ≃ ˜ D L ˜ T L if L is an arbitrary path. ˜ ∇ ˜ T ∼ = ˜ ∇ L ˜ T L if and only if L is the gradient line of ˜ T . The actual evolution equation of ˜ T is ˜ t ; σ = ˜ t L ; τ ˜¯ ε τσ or ˜ t ; σ = ˜ t ; τL ˜ ε στ . Discussion 5.4.2.
Similar to Discussion 3.3.5.1 we have ˜ K µνρσ : ρ = ˜ j µνσ , where ˜ ρ µντ , ˜ K µνρσ : ρ ˜ ε στ , ˜ j µνσ , ˜ ρ µντ ˜¯ ε τσ . Consider the case where external space is flat, then just only the internal component ˜ ρ τ does not vanish. Thus we have Minkowski Yang-Mills field equation ˜ K ρσ : ρ = ˜ j σ , where ˜ j σ , ˜ ρ τ ˜¯ ε τσ .Now there is a problem. Several internal dimensions of ( M, f ) become just one dimension of ( ˜ M , ˜ f ) via the encapsulation of classical spacetime. ( M, f ) has several internal charges ρ mn , but ( ˜ M , ˜ f ) has just one, which is ρ ττ in regular form, or ˜ ρ τ in Minkowski form. As what Remark5.2.1 says, ( ˜ M , ˜ f ) cannot totally reflect all the intrinsic geometrical properties of internal spaceof ( M, f ) .On the premise of not abandoning the four-dimensional spacetime, if we want to describegauge fields, the only way is to put those degrees of freedom of internal space to the phase ofcomplex-valued field function. This way is effective, but not natural at all.The logically more natural way is to abandon the framework of four-dimensional spacetime.We should put internal space and external space together to describe their unified intrinsicgeometry, rather than based on the rigid intuition of four-dimensional spacetime, artificiallysetting up several abstract degrees of freedom which are irrelevant to the concept of time andspace to describe the so called gauge fields. generalization of intrinsic geometry and its application to Hilbert’s 6th problem 47 Both for gauge fields and for gravitational fields, their concepts of time and space should beunified. The gravitational fields are described by the intrinsic geometry of external space, andthe gauge fields are described by the intrinsic geometry of internal space. They are unified inintrinsic geometry.Therefore, the complex-valued expression form of traditional gauge field theory is a historicalnecessity, but not a logical necessity. It can be seen later that as long as expanding thoseencapsulated dimensions, we can clearly illustrate the geometrical properties of internal space.Especially, if understanding in way of intrinsic geometry, some man-made postulates of StandardModel of particle physics will be unnecessary, because they will appear automatically.
Definition 5.4.1.
Denote ˜ ρ µν τ of ˜ f concisely by ˜ ρ . According to section 3.3.6 , we have thefollowing definitions in Minkowski coordinate on ( ˜ M , ˜ g ) .(1) Call ˜ m τ , ˜ ρ ; τ and ˜ m τ , ˜ ρ ; τ the rest mass of ˜ ρ .(2) Call ˜ p µ , ˜ ρ ; µ and ˜ p µ , ˜ ρ ; µ the energy-momentum of ˜ ρ , and ˜ E , ˜ p , ˜ E , ˜ p the energy of ˜ ρ .(3) Call ˜ M τ , d ˜ ρd ˜ x τ and ˜ M τ , d ˜ ρd ˜ x τ the canonical rest mass of ˜ ρ .(4) Call ˜ P µ , ∂ ˜ ρ∂ ˜ x µ and ˜ P µ , ∂ ˜ ρ∂ ˜ x µ the canonical energy-momentum of ˜ ρ , and ˜ H , − ˜ P , ˜ H , − ˜ P the canonical energy of ˜ ρ . Discussion 5.4.3.
Similar to Proposition 3.3.6.2 , if and only if the evolution direction of ˜ ρ is thegradient direction ˜ ∇ ˜ ρ on ( ˜ M , ˜ g ) , the directional derivative is D ˜ m τ dd ˜ x τ , ˜ m τ d ˜ x τ E = D ˜ p µ ∂∂ ˜ x µ , ˜ p µ d ˜ x µ E ,that is ˜ G ττ ˜ m τ ˜ m τ = ˜ G µν ˜ p µ ˜ p ν , or ˜ m τ ˜ m τ = ˜ p µ ˜ p µ , which is the mathematical origin of energy-momentum equation of physics.In addition, as what Proposition 3.3.6.3 says, according to evolution lemma, if and only if wetake the gradient direction ˜ ∇ ˜ ρ , we have ˜ p µ = ˜ m τ d ˜ x µ d ˜ x τ and ˜ p µ = ˜ m τ d ˜ x µ d ˜ x τ . This is the mathematicalorigin of traditional definition of momentum.Similar to discussions of section 3.3.7 , denote [ ˜ ρ ˜ Γ ω ] , [ ˜ ρ µν ˜ Γ ω ] , ∂ ˜ ρ∂ ˜ x ω − ˜ ρ ; ω , ∂ ˜ ρ µν ∂ ˜ x ω − ˜ ρ µν ; ω = ˜ ρ µχ ˜ Γ χνω + ˜ ρ χν ˜ Γ χµω , [ ˜ ρ ˜ Γ τ ] , [ ˜ ρ µν ˜ Γ τ ] , d ˜ ρd ˜ x τ − ˜ ρ ; τ , d ˜ ρ µν d ˜ x τ − ˜ ρ µν ; τ = ˜ ρ µχ ˜ Γ χντ + ˜ ρ χν ˜ Γ χµτ . [ ˜ ρ ˜ Γ ω ] , g χω [ ˜ ρ ˜ Γ χ ] , [ ˜ ρ ˜ Γ τ ] , g ττ [ ˜ ρ ˜ Γ τ ] . [ ˜ ρ ˜ B ρσ ] , ˜ ρ µχ ∂ ˜ Γ χνσ ∂ ˜ x ρ − ∂ ˜ Γ χνρ ∂ ˜ x σ ! + ˜ ρ χν ∂ ˜ Γ χµσ ∂ ˜ x ρ − ∂ ˜ Γ χµρ ∂ ˜ x σ ! , [ ˜ ρ ˜ R ρσ ] , ˜ ρ µχ ˜ R χνρσ + ˜ ρ χν ˜ R χµρσ , [ ˜ ρ ˜ F ρσ ] , ∂ [ ˜ ρ ˜ Γ σ ] ∂ ˜ x ρ − ∂ [ ˜ ρ ˜ Γ ρ ] ∂ ˜ x σ , [ ˜ ρ ˜ E ρσ ] , [ ˜ ρ ˜ Γ σ ] ; ρ − [ ˜ ρ ˜ Γ ρ ] ; σ . Then for the same reason as Proposition 3.3.7.2 , we can strictly prove the
Lorentz force equationof ˜ ρ , which is ˜ F ρ , d ˜ p ρ d ˜ x τ = ∂ ˜ m τ ∂ ˜ x ρ − ˜ p σ ∂ ˜ ε στ ∂ ˜ x ρ + [ ˜ ρ ˜ F ρσ ]˜ ε στ . And for the same reason as Proposition 3.3.7.5, we have conservation of energy-momentum ˜ T µν ; µ = 0 . Definition 5.4.2.
The following three conditions are uniformly called traditional standardconditions :(1) Rest mass condition: ∂ µ ˜ m τ = 0 .(2) Canonical mass condition: ˜ Γ µντ , ˜ Γ µνρ ˜ ε ρτ = 0 and ˜ Γ µνρ γ ρ = 0 , where γ ρ are generators ofDirac algebra.(3) Simple perspective condition: ∂ ν ˜ ε µτ = 0 . Remark 5.4.1.
Conditions (1) and (3) make the above Lorentz force simplify to ˜ F ρ = [ ˜ ρ ˜ F ρσ ]˜ ε στ ,which is the general essence of interaction force of physics, and also is the mathematical originof Lorentz force FFF = q ( EEE + vvv × BBB ) or F ρ = j σ F ρσ of electrodynamics. In this sense, we canargue that Lorentz force equation has become a theorem, and no longer as a principle.Condition (2) is the general mathematical expression of the following example. Take electro-dynamics with natural units for example. The canonical energy-momentum of electric chargedparticle is H = E + qϕ, PPP = ppp + qAAA. We notice that there is no concept of canonical mass ˜ M τ in physics. It is because that there is H = qϕ + p ( PPP − qAAA ) + m in electrodynamics, which indicates that electromagnetic potentialfield ( ϕ, AAA ) contributes qϕ to energy and qAAA to momentum, but it contributes nothing to restmass of q . This implies that if we define ˜ M τ , ˜ m τ + q ˜ A τ , ˜ A τ , ϕγ + AAA · uuu = A ρ dx ρ dτ , then electrodynamics actually requires ˜ M τ = ˜ m τ , ˜ A τ = A ρ dx ρ dτ = 0 by default, the generalmathematical expression of which is ˜ Γ µντ , ˜ Γ µνρ ˜ ε ρτ = 0 .In a word, the above three conditions are necessary conditions for transitioning in puremathematical sense to traditional theory of physics. Discussion 5.4.4.
Suppose ˜ C (˜ x ) µν is a zero-order or two-order tensor, such as ˜ R µν − ˜ G µν ˜ R ,which just only depends on intrinsic geometrical properties of ( ˜ M , ˜ g ) , such that ˜ C (˜ x ) µν ; µ = 0 .We distinguish them with index (˜ x ) . And suppose T (˜ ρ ) µν ; µ = 0 is the conservation of energy-momentum of ˜ ρ . We distinguish various T (˜ ρ ) µν with index ( ˜ ρ ) . Hence, ∀ c (˜ x ) , c (˜ ρ ) ∈ R , we have X ˜ x c (˜ x ) C (˜ x ) µν + X ˜ ρ c (˜ ρ ) ˜ T (˜ ρ ) µν ; µ = 0 . If the ergodic ranges of the summations are sufficiently large, we immediately obtain X ˜ x c (˜ x ) C (˜ x ) µν + X ˜ ρ c (˜ ρ ) ˜ T (˜ ρ ) µν = 0 , which is the general gravitational field equation, where the dimensions among various terms areharmonized by constants c (˜ x ) , c (˜ ρ ) . It is the mathematical origin of Einstein’s gravitational fieldequation. generalization of intrinsic geometry and its application to Hilbert’s 6th problem 49 Definition 5.4.3.
It is similar to Definition 3.3.6 . Let ˜ L be the totality of paths on ˜ M from point a to point b . And let L ˜ ρ ∈ ˜ L , and parameter ˜ x τ satisfy τ a , ˜ x τ ( a ) < ˜ x τ ( b ) , τ b . We say thefunctional ˜ s ˜ ρ ˜ W ( L ˜ ρ ) , Z L ˜ ρ ˜ D ˜ ρ = Z τ b τ a ˜ m τ d ˜ x τ = Z τ b τ a ˜ p µ d ˜ x µ = Z τ b τ a d ˜ x τ d ˜ x (cid:16) ˜ M τ − [ ˜ ρ ˜ Γ σ ]˜ ε στ (cid:17) d ˜ x is the action of ˜ ρ . For the same reason as the proof of Proposition 3.3.6.4 , we have the followingtheorem, which is the mathematical origin of the principle of least action of physics. Proposition 5.4.1. (Theorem of least action) L ˜ ρ is the gradient line of ˜ ρ if and only if δ ˜ s ˜ ρ ˜ W (cid:0) L ˜ ρ (cid:1) = 0 . This section does not discuss the general abstract theory of Legendre transformation, butdiscusses the relationship between energy-momentum equation and the concrete construction ofLegendre transformation.
Definition 5.5.1.
Denote ˜ L , ˜ m τ d ˜ x τ d ˜ x = d ˜ x τ d ˜ x (cid:16) ˜ M τ − [ ˜ ρ ˜ Γ σ ]˜ ε στ (cid:17) , ˜ L , ˜ M τ d ˜ x τ d ˜ x = d ˜ x τ d ˜ x (cid:16) ˜ m τ + [ ˜ ρ ˜ Γ σ ]˜ ε στ (cid:17) . Evidently we have ˜ L = ˜ L on canonical mass condition. We say ˜ L is Lagrangian densigy of ˜ ρ . According to Definition 5.4.1 , we have ˜ M τ d ˜ x τ = ˜ P k d ˜ x k − ˜ H d ˜ x , therefore ˜ H = ˜ P k d ˜ x k d ˜ x − ˜ M τ d ˜ x τ d ˜ x , that is ˜ H = ˜ P k d ˜ x k d ˜ x − ˜ L . We say ˜ H is Hamiltonian density of ˜ ρ . This is the intrinsicgeometrical origin of Legendre transformation of physics. Proposition 5.5.1.
Denote ˜ v k , d ˜ x k d ˜ x , and regard ˜ L = ˜ P k ˜ v k − ˜ H as function ˜ L (˜ x k , ˜ v k ) . Thus,on traditional standard conditions we have dd ˜ x ∂ ˜ L ∂ ˜ v k ! − ∂ ˜ L ∂ ˜ x k = 0 . Proof.
On traditional standard conditions, we can obtain Euler-Lagrange equation from thedefinition of Lagrangian density. Concretely: ˜ L , ˜ M τ d ˜ x τ d ˜ x = d ˜ ρd ˜ x τ d ˜ x τ d ˜ x = ∂ ˜ ρ∂ ˜ x µ d ˜ x µ d ˜ x τ d ˜ x τ d ˜ x . Accordingly, ∂ ˜ L ∂ ˜ x σ = ∂∂ ˜ x σ (cid:18) ∂ ˜ ρ∂ ˜ x µ d ˜ x µ d ˜ x τ d ˜ x τ d ˜ x (cid:19) . On traditional standard conditions, ∂ ˜ L ∂ ˜ x σ = d ˜ x µ d ˜ x τ d ˜ x τ d ˜ x ∂∂ ˜ x σ (cid:18) ∂ ˜ ρ∂ ˜ x µ (cid:19) = d ˜ x µ d ˜ x τ d ˜ x τ d ˜ x ∂∂ ˜ x µ (cid:18) ∂ ˜ ρ∂ ˜ x σ (cid:19) = d ˜ x µ d ˜ x τ d ˜ x τ d ˜ x ∂ ˜ P σ ∂ ˜ x µ = d ˜ P σ d ˜ x . Thus we obtain Euler-Lagrange equation d ˜ P k d ˜ x − ∂ ˜ L ∂ ˜ x k = 0 . In consideration of ˜ P k = ∂ ˜ L ∂ ˜ v k , then dd ˜ x ∂ ˜ L ∂ ˜ v k ! − ∂ ˜ L ∂ ˜ x k = 0 . ⊓⊔ Remark 5.5.1.
The above proposition is the mathematical origin of Euler-Lagrange equationof motion of physics. It should be clarified that the above Euler-Lagrange equation holds inarbitrary directions, while the Euler-Lagrange equation of traditional theory holds just only ingradient direction.The reason why such a situation happens is that their definitions of momentum are different.Definition 3.3.6.1 defines the momentum in arbitrary directions. Due to Proposition 3.3.6.3 andRemark 3.3.6.1 , p R = E dx R dx and ˜ p µ = ˜ m τ d ˜ x µ d ˜ x τ just hold in gradient direction. But traditionaltheory denotes p , mv in arbitrary directions. Such two different ways of defining momentummake the conditions of the holding of Euler-Lagrange equation different.(1) When ˜ p µ = ˜ m τ d ˜ x µ d ˜ x τ holds just only in gradient direction, Euler-Lagrange equation holdsin arbitrary directions. At this time, what we can obtain from Proposition 5.4.1 is just only theformer.(2) When we denote p , mv in arbitrary directions, Euler-Lagrange equation holds just onlyin gradient direction. At this time, what we can obtain from Proposition 5.4.1 is just only thelatter.No matter we use which way of definition, there is always a formula that can describe gradientdirection, which is either ˜ p µ = ˜ m τ d ˜ x µ d ˜ x τ or Euler-Lagrange equation. This section proves two propositions. The first one illustrates the intrinsic geometrical originof Dirac equation, and makes it no longer a principle but a theorem. The second one clarifiesthe intrinsic geometrical origin of gauge transformation and gauge invariance, and shows howthe transformation of reference-systems characterizes the general gauge transformation.
Proposition 5.6.1.
On traditional standard conditions, suppose there is a smooth real function f (˜ x µ ) on an isotropic ( ˜ M , ˜ g ) , and define Dirac algebras γ µ and γ α such that γ µ = ˜ C µα γ α , γ α γ β + γ β γ α = 2˜ δ αβ , γ µ γ ν + γ ν γ µ = 2 ˜ G µν , ˜ Γ µνσ γ σ = 0 , γ µ ∂f∂ ˜ x µ = 0 , Z f dV = 1 . Let ˜ ρ , ˜ ρ ων bea charge of reference-system ˜ f , and denote ˜ S , Z ˜ sdV , ˜ s = Z ˜ L d ˜ x = f ˜ S, ˜ M τ , Z ˜ m τ dV , ˜ m τ = f ˜ M τ , ˜P , Z ˜ ρdV , ˜ ρ = f ˜P . Then denote ψ (˜ x µ ) , f (˜ x µ ) e i ˜ S , ˜ D µ , ∂∂ ˜ x µ − i [˜P ˜ Γ µ ] . If and only if we take the gradient direction ˜ ∇ ˜ ρ , equation iγ µ ˜ D µ ψ = ˜ M τ ψ holds. generalization of intrinsic geometry and its application to Hilbert’s 6th problem 51 Proof.
Due to Discussion 5.4.3 , if and only if we take the gradient direction of ˜ ρ thereexists energy-momentum equation ˜ ρ ; µ ˜ ρ ; µ = ˜ ρ ; τ ˜ ρ ; τ , that is ˜ G µν ˜ ρ ; µ ˜ ρ ; ν = ˜ m τ due to isotropy. Then, ( γ µ ˜ ρ ; µ )( γ ν ˜ ρ ; ν )+( γ ν ˜ ρ ; ν )( γ µ ˜ ρ ; µ ) = 2 ˜ m τ , hence ( γ µ ˜ ρ ; µ )( γ ν ˜ ρ ; ν ) = ˜ m τ , that is ( γ µ ˜ ρ ; µ ) = ˜ m τ . Withoutloss of generality, we take γ µ ˜ ρ ; µ = − ˜ m τ . Due to canonical mass condition ˜ Γ µνσ γ σ = 0 we have γ µ [˜P ˜ Γ µ ] = 0 . And with condition γ µ ∂f∂ ˜ x µ = 0 it can be obtained that γ µ ˜ ρ ; µ = − ˜ m τ ⇔ γ µ ∂ ˜ s∂x µ = − ˜ m τ ⇔ γ µ ∂ ˜ S∂x µ = − ˜ M τ ⇔ γ µ ∂ ˜ S∂x µ − [˜P ˜ Γ µ ] ! = − ˜ M τ ⇔ iγ µ i ∂ ˜ S∂x µ f e i ˜ S − i [˜P ˜ Γ µ ] f e i ˜ S ! = ˜ M τ f e i ˜ S ⇔ iγ µ ∂f∂ ˜ x µ e i ˜ S + i ∂ ˜ S∂x µ f e i ˜ S − i [˜P ˜ Γ µ ] f e i ˜ S ! = ˜ M τ f e i ˜ S ⇔ iγ µ ∂ (cid:16) f e i ˜ S (cid:17) ∂ ˜ x µ − i [˜P ˜ Γ µ ] f e i ˜ S = ˜ M τ f e i ˜ S ⇔ iγ µ (cid:18) ∂ψ∂ ˜ x µ − i [˜P ˜ Γ µ ] ψ (cid:19) = ˜ M τ ψ ⇔ iγ µ (cid:18) ∂∂ ˜ x µ − i [˜P ˜ Γ µ ] (cid:19) ψ = ˜ M τ ψ ⇔ iγ µ ˜ D µ ψ = ˜ M τ ψ. ⊓⊔ Remark 5.6.1.
Until now, Dirac equation has become a theorem. According to this theorem,Dirac equation also reflects the notion of gradient direction, it thereby describes the effects ofintrinsic geometry of manifold on gradient direction. This is the mathematical origin of theeffectiveness of Dirac equation of physics.
Proposition 5.6.2. On ˜ M let the slack-tights of ˜ g be ˜ B αµ and ˜ C µα , and the slack-tights of ˜ k be ˜ B µµ ′ and ˜ C µ ′ µ . Let ˜ g ′ , L [˜ k ] (˜ g ) , then the slack-tights of ˜ g ′ are ˜ B αµ ′ = B αµ ˜ B µµ ′ and ˜ C µ ′ α = ˜ C µα ˜ C µ ′ µ .Define Dirac algebras γ α , γ µ = ˜ C µα γ α and γ µ ′ = ˜ C µ ′ α γ α , such that γ α γ β + γ β γ α = 2˜ δ αβ , γ µ γ ν + γ ν γ µ = 2 ˜ G µν , γ µ ′ γ ν ′ + γ ν ′ γ µ ′ = 2 ˜ G µ ′ ν ′ , where ˜ G µν = ˜ δ αβ ˜ C µα ˜ C νβ is the metric tensor of ˜ g ,and ˜ G µ ′ ν ′ = ˜ δ αβ ˜ C µ ′ α ˜ C ν ′ β is the metric tensor of ˜ g ′ .Let ˜ ρ be a charge of ˜ f , and ˜ D be the simple connection on ( ˜ M , ˜ g ) and ( ˜ M , ˜ g ′ ) . According toProposition 5.6.1 , suppose ( ˜ Γ ˜ g ) µνσ γ σ = 0 , the Dirac equation of ˜ ρ on ( ˜ M , ˜ g ) is iγ µ ˜ D µ ψ = ˜ M τ ψ ,and suppose ( ˜ Γ ˜ g ′ ) µνσ γ σ = 0 , the Dirac equation of ˜ ρ on ( ˜ M , ˜ g ′ ) is iγ µ ′ ˜ D µ ′ ψ ′ = ˜ M τ ′ ψ ′ .Then we have the following conclusions under transformation L [˜ k ] : ˜ g ˜ g ′ .(1) L [˜ k ] : ˜ D µ ˜ D µ ′ = ˜ B µµ ′ ( ˜ D µ − i∂ µ θ ) .(2) L [˜ k ] : ψ ψ ′ = ψe iθ .(3) For a selected ˜ ρ , the smooth real function θ is deternimed by L [ k ] ∈ GL ( M ) . See Discussion2.3.1 for the definition of general linear group GL ( M ) .(4) Suppose ˜ g satisfies the condition of Proposition 5.6.1 , then | ψ | and (cid:12)(cid:12)(cid:12) iγ µ ˜ D µ ψ (cid:12)(cid:12)(cid:12) are bothuniversal geometrical properties of ˜ M .(5) The necessary condition of simultaneous ( ˜ Γ ˜ g ) µνσ γ σ = 0 and ( ˜ Γ ˜ g ′ ) µνσ γ σ = 0 is that L [˜ k ] is anorthogonal transformation. Proof.
Under the transformation L [˜ k ] : ˜ g ˜ g ′ we have L [˜ k ] : ˜ B αµ ˜ B αµ ′ = B αµ ˜ B µµ ′ , ˜ C µα ˜ C µ ′ α = ˜ C µα ˜ C µ ′ µ , ˜ B ττ ˜ B ττ ′ = B ττ ˜ B ττ ′ , ˜ C ττ ˜ C τ ′ τ = ˜ C ττ ˜ C τ ′ τ .L [˜ k ] : ∂∂ ˜ x µ ∂∂ ˜ x µ ′ = ˜ B µµ ′ ∂∂ ˜ x µ , dd ˜ x τ dd ˜ x τ ′ = ˜ B ττ ′ dd ˜ x τ .L [˜ k ] : ρ ; µ ρ ; µ ′ = ρ ; µ ˜ B µµ ′ , ˜ M τ ˜ M τ ′ = ˜ M τ ˜ B ττ ′ .L [˜ k ] : γ µ γ µ ′ = ˜ C µ ′ µ γ µ . According to equation(21) below, it is obtained that L [˜ k ] : [˜P ˜ Γ µ ] [˜P ˜ Γ µ ′ ] = ˜ B µµ ′ ([˜P ˜ Γ µ ] + r µ ) . Correspondingly, the transformation of ˜ D µ is ˜ D µ , ∂∂ ˜ x µ − i [˜P ˜ Γ µ ] ˜ D µ ′ , ∂∂ ˜ x µ ′ − i [˜P ˜ Γ µ ′ ] = ˜ B µµ ′ ∂∂ ˜ x µ − i ˜ B µµ ′ ([˜P ˜ Γ µ ] + r µ ) = ˜ B µµ ′ ( ˜ D µ − ir µ ) . The transformation of ψ is ψ = f ei Z (cid:16) ˜P ; µ + [˜P ˜ Γ µ ] (cid:17) d ˜ x µ ψ ′ = f ei Z (cid:16) ˜P ; µ ′ + [˜P ˜ Γ µ ′ ] (cid:17) d ˜ x µ ′ = f ei Z (cid:16) ˜P ; µ ′ + ˜ B µµ ′ ([˜P ˜ Γ µ ] + r µ ) (cid:17) d ˜ x µ ′ , that is ψ ψ ′ = ψei Z r µ d ˜ x µ . Denote θ , Z r µ d ˜ x µ , r µ = ∂ µ θ , thus we obtain ψ ψ ′ = ψe iθ .We have proved (1) and (2) as the above. Now in order to prove (4), we can substitute theabove transformations into iγ µ ˜ D µ ψ = ˜ M τ ψ of ˜ g . iγ µ ˜ D µ ψ = ˜ M τ ψ ⇔ i (cid:16) ˜ B µµ ′ γ µ ′ (cid:17) (cid:16) ˜ C ν ′ µ ( ˜ D ν ′ + i∂ ν ′ θ ) (cid:17) (cid:16) ψ ′ e − iθ (cid:17) = ˜ M τ (cid:16) ψ ′ e − iθ (cid:17) ⇔ iγ µ ′ ( ˜ D µ ′ + i∂ µ ′ θ ) (cid:16) ψ ′ e − iθ (cid:17) = ˜ M τ ψ ′ e − iθ ⇔ iγ µ ′ (cid:16) ∂ µ ′ − i [˜P ˜ Γ µ ′ ] + i∂ µ ′ θ (cid:17) (cid:16) ψ ′ e − iθ (cid:17) = ˜ M τ ψ ′ e − iθ ⇔ iγ µ ′ ∂ µ ′ ψ ′ e − iθ + iγ µ ′ ψ ′ ∂ µ e − iθ + γ µ ′ [˜P ˜ Γ µ ′ ] ψ ′ e − iθ − γ µ ′ ψ ′ e − iθ ∂ µ ′ θ = ˜ M τ ψ ′ e − iθ ⇔ iγ µ ′ ∂ µ ′ ψ ′ + γ µ ′ ψ ′ ∂ µ ′ θ + γ µ ′ [˜P ˜ Γ µ ′ ] ψ ′ − γ µ ′ ψ ′ ∂ µ ′ θ = ˜ M τ ψ ′ ⇔ iγ µ ′ ∂ µ ′ ψ ′ + γ µ ′ [˜P ˜ Γ µ ′ ] ψ ′ = ˜ M τ ψ ′ ⇔ iγ µ ′ ˜ D µ ′ ψ ′ = ˜ M τ ψ ′ . Consequently, iγ µ ′ ˜ D µ ′ ψ ′ = ˜ M τ ψ ′ = ˜ M τ ψe iθ = (cid:16) iγ µ ˜ D µ ψ (cid:17) e iθ , hence (cid:12)(cid:12)(cid:12) iγ µ ′ ˜ D µ ′ ψ ′ (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) iγ µ ˜ D µ ψ (cid:12)(cid:12)(cid:12) .In addition, it is evident that | ψ ′ | = | ψ | . It indicates that (cid:12)(cid:12)(cid:12) iγ µ ˜ D µ ψ (cid:12)(cid:12)(cid:12) and | ψ | remain unchangedunder transformation L [˜ k ] . Due to the arbitrariness of [˜ k ] , according to section 2.6 , (cid:12)(cid:12)(cid:12) iγ µ ˜ D µ ψ (cid:12)(cid:12)(cid:12) isa universal geometrical property of ˜ M .Next we consider (5). The above iγ µ ′ ˜ D µ ′ ψ ′ = ˜ M τ ψ ′ is obtained in the case where ˜ g satisfiesthe condition ( ˜ Γ ˜ g ) µνσ γ σ = 0 of Proposition 5.6.1 . When ˜ g ′ satisfies condition ( ˜ Γ ˜ g ′ ) µ ′ ν ′ σ ′ γ σ ′ = 0 ,according to Proposition 5.6.1 we have iγ µ ′ ˜ D µ ′ ψ ′ = ˜ M τ ′ ψ ′ . Compare such two equations weobtain ˜ M τ = ˜ M τ ′ , that is ˜ M τ = ˜ B ττ ′ ˜ M τ , or ˜ B ττ ′ = 1 . According to Definition 5.2.2 , L [˜ k ] is anorthogonal transformation. generalization of intrinsic geometry and its application to Hilbert’s 6th problem 53 In order to prove (3), we need to calculate r µ . According to Proposition 2.7.2 , we have ( ˜ Γ ˜ g ′ ) µ ′ ν ′ σ ′ = ( ˜ Γ ˜ g ) µνσ ˜ C µ ′ µ ˜ B νν ′ ˜ B σσ ′ + ( ˜ Γ ˜ k ) µ ′ ν ′ σ ′ . Due to Discussion 5.4.3 we know [ ˜ ρ ˜ Γ µ ] , [ ˜ ρ ων ˜ Γ µ ] , ˜ ρ ωχ ( ˜ Γ ˜ g ) χνµ + ˜ ρ χν ( ˜ Γ ˜ g ) χωµ and [ ˜ ρ ′ ˜ Γ µ ′ ] , [ ˜ ρ ω ′ ν ′ ˜ Γ µ ′ ] = ˜ ρ ω ′ χ ′ ( ˜ Γ ˜ g ′ ) χ ′ ν ′ µ ′ + ˜ ρ χ ′ ν ′ ( ˜ Γ ˜ g ′ ) χ ′ ω ′ µ ′ . [ ˜ ρ ˜ Γ µ ′ ] , [ ˜ ρ ων ˜ Γ µ ′ ] , ˜ C ω ′ ω ˜ C ν ′ ν [ ˜ ρ ω ′ ν ′ ˜ Γ µ ′ ] = ˜ C ω ′ ω ˜ C ν ′ ν (cid:16) ˜ ρ ω ′ χ ′ ( ˜ Γ ˜ g ′ ) χ ′ ν ′ µ ′ + ˜ ρ χ ′ ν ′ ( ˜ Γ ˜ g ′ ) χ ′ ω ′ µ ′ (cid:17) = ˜ C ω ′ ω ˜ C ν ′ ν (cid:16) ˜ ρ ω ′ χ ′ (cid:16) ( ˜ Γ ˜ g ) χρµ ˜ C χ ′ χ ˜ B ρν ′ ˜ B µµ ′ + ( ˜ Γ ˜ k ) χ ′ ν ′ µ ′ (cid:17) + ˜ ρ χ ′ ν ′ (cid:16) ( ˜ Γ ˜ g ) χσµ ˜ C χ ′ χ ˜ B σω ′ ˜ B µµ ′ + ( ˜ Γ ˜ k ) χ ′ ω ′ µ ′ (cid:17)(cid:17) = ˜ ρ ωχ ( ˜ Γ ˜ g ) χνµ ˜ B µµ ′ + ˜ ρ χν ( ˜ Γ ˜ g ) χωµ ˜ B µµ ′ + ˜ ρ ωχ ′ ( ˜ Γ ˜ k ) χ ′ ν ′ µ ′ ˜ C ν ′ ν + ˜ ρ χ ′ ν ′ ( ˜ Γ ˜ k ) χ ′ ω ′ µ ′ ˜ C ω ′ ω = ˜ ρ ωχ ( ˜ Γ ˜ g ) χνµ ˜ B µµ ′ + ˜ ρ χν ( ˜ Γ ˜ g ) χωµ ˜ B µµ ′ + ˜ ρ ωχ ( ˜ Γ ˜ k ) χ ′ ν ′ σ ′ ˜ C σ ′ µ ˜ B χχ ′ ˜ C ν ′ ν ˜ B µµ ′ + ˜ ρ χν ( ˜ Γ ˜ k ) χ ′ ω ′ σ ′ ˜ C σ ′ µ ˜ B χχ ′ ˜ C ω ′ ω ˜ B µµ ′ = (cid:16) [ ˜ ρ ˜ Γ µ ] + ˜ ρ ωχ ( ˜ Γ ˜ k ) χ ′ ν ′ σ ′ ˜ B χχ ′ ˜ C ν ′ ν ˜ C σ ′ µ + ˜ ρ χν ( ˜ Γ ˜ k ) χ ′ ω ′ σ ′ ˜ B χχ ′ ˜ C ω ′ ω ˜ C σ ′ µ (cid:17) ˜ B µµ ′ , hence [˜P ˜ Γ µ ′ ] = (cid:16) [˜P ˜ Γ µ ] + ˜P ωχ ( ˜ Γ ˜ k ) χ ′ ν ′ σ ′ ˜ B χχ ′ ˜ C ν ′ ν ˜ C σ ′ µ + ˜P χν ( ˜ Γ ˜ k ) χ ′ ω ′ σ ′ ˜ B χχ ′ ˜ C ω ′ ω ˜ C σ ′ µ (cid:17) ˜ B µµ ′ = (cid:16) [˜P ˜ Γ µ ] + r ωνµ (cid:17) ˜ B µµ ′ = (cid:16) [˜P ˜ Γ µ ] + r µ (cid:17) ˜ B µµ ′ , (21)where r µ , r ωνµ , ˜P ωχ ( ˜ Γ ˜ k ) χ ′ ν ′ σ ′ ˜ B χχ ′ ˜ C ν ′ ν ˜ C σ ′ µ + ˜P χν ( ˜ Γ ˜ k ) χ ′ ω ′ σ ′ ˜ B χχ ′ ˜ C ω ′ ω ˜ C σ ′ µ .We notice that ˜ C and ˜ B in the above r µ are slack-tights of [˜ k ] , and ( ˜ Γ ˜ k ) µ ′ ν ′ σ ′ = 12 ˜ C µ ′ µ ∂ ˜ B µν ′ ∂ ˜ x σ ′ + ∂ ˜ B µσ ′ ∂x ν ′ ! is simple connection of [ k ] , therefore for a selected ˜ ρ we know r µ is uniquely determined by L [˜ k ] ,and furthermore θ , Z r µ d ˜ x µ is uniquely determined by L [˜ k ] . In consideration of that L [˜ k ] of ˜ M is determined by L [ k ] of M , so r µ and θ are finally determined by L [ k ] . ⊓⊔ Remark 5.6.2.
Conclusions (1)(2)(3) are the reasons why Definition 2.3.3 calls L [ k ] a generalgauge transformation. Conclusion (4) can also be called the gauge invariance , which utill nowhas become a theorem, no longer been as a principle. This proposition indicates that ψ ψ ′ and ˜ D µ ˜ D µ ′ and gauge invariance such three things have the same mathematical origin, which isthe intrinsic transformation L [ k ] , and their physical connotations just only come from the axiomand corollary of section 3.1 . L [ k ] is an element of general linear group, therefore gauge fieldsand gauge transformations defined by whatever subgroup of general linear group can always becharacterized by [ k ] and L [ k ] . This is the intrinsic geometrical origin of gauge field and gaugetransformation.In summary, without the viewpoints of intrinsic geometry, it is impossible to clarify thatthere exists a more fundamental mathematical essence than gauge transformations ψ ψ ′ and ˜ D µ ˜ D µ ′ . In consideration of Discussion 5.4.2 , total superiority of viewpoints of intrinsicgeometry can be brought into full play just only on manifold M rather than ˜ M , complete detailsof various intrinsic geometrical properties of gauge field can thereby be presented on M . Hence,next we are going to stop the discussions about intrinsic geometry of ˜ M , but to focus on intrinsicgeometry of M . The intrinsic geometrical properties of the following sections still can only bedescribed by simple connection, but not Levi-Civita connection. Definition 5.6.1.
Let there be a geometrical manifold ( M, k ) , such that M = P × N , r , dimP = 3 and D , dimM = 5 or or .(1) Suppose the local coordinate representation of k is x m ′ = x m ′ ( x m ) and x i ′ = δ i ′ i x i , whichsatisfies internal standard conditions : (i) G mn = const , (ii) G mn = 0 when m = n . We say k isa typical gauge field , and L [ k ] is a typical gauge transformation .(2) Suppose the local coordinate representation of k is x m ′ = x m ′ ( x M ) and x i ′ = x i ′ ( x i ) ,we say k is a typical gauge field with gravitation , and L [ k ] is a typical gravitational gaugetransformation . Suppose f satisfies the (1) of Definition 5.6.1 , and geometrical manifold ( M, f ) satisfies D = r + 2 = 5 . On a neighborhood U of any point p the coordinate representation of f ( p ) is ξ a = ξ a ( x m ) and ξ s = δ si x i , such that G ( D − D − = G DD . We say f is a weak andelectromagnetic unified field . The reason for such naming lies in the following proposition. Proposition 6.1.
Let the simple connection of the above ( M, f ) be Λ MNP and Λ MNP . And let thecoefficients of curvature of ( M, f ) be K MNP Q and K MNP Q . Denote B P , √ (cid:0) Λ DD P + Λ ( D − D − P (cid:1) A P , √ (cid:0) Λ DD P − Λ ( D − D − P (cid:1) , A P , √ (cid:0) Λ ( D − D P + Λ D ( D − P (cid:1) A P , √ (cid:0) Λ ( D − D P − Λ D ( D − P (cid:1) . B P Q , √ (cid:0) K DD P Q + K ( D − D − P Q (cid:1) F P Q , √ (cid:0) K DD P Q − K ( D − D − P Q (cid:1) , F P Q , √ (cid:0) K ( D − D P Q + K D ( D − P Q (cid:1) F P Q , √ (cid:0) K ( D − D P Q − K D ( D − P Q (cid:1) . And denote g , q(cid:0) G ( D − D − (cid:1) + (cid:0) G DD (cid:1) . Thus the following equations hold, and they are allintrinsic geometrical properties of ( M, f ) . B P Q = ∂B Q ∂x P − ∂B P ∂x Q ,F P Q = ∂A Q ∂x P − ∂A P ∂x Q + g (cid:16) A P A Q − A P A Q (cid:17) ,F P Q = ∂A Q ∂x P − ∂A P ∂x Q + g (cid:16) A P A Q − A P A Q (cid:17) ,F P Q = ∂A Q ∂x P − ∂A P ∂x Q − g (cid:16) A P A Q − A P A Q (cid:17) . Proof.
According to the definition, the slack-tights of f satisfy that B sm = 0 , C ia = 0 . B si = δ si , B ai = 0 , C is = δ is , C ms = 0 . The metric of f satisfies that G mn = 0( m = n ) , G mn = const , G MN , δ AB B AM B BN and G MN = δ AB C MA C NB . Concretely: generalization of intrinsic geometry and its application to Hilbert’s 6th problem 55 G ij = δ st B si B tj + δ ab B ai B bj = δ st δ si δ tj = δ ij G in = δ st B si B tn + δ ab B ai B bn = 0 G mj = δ st B sm B tj + δ ab B am B bj = 0 G mn = B D − m B D − n + B D m B D n , G ij = δ st C is C jt = δ st δ is δ jt = δ ij G in = δ st C is C nt = 0 G mj = δ st C ms C jt = 0 G mn = C m D − C n D − + C m D C n D . Calculate the simeple connection of f , that is Λ MNP , C MA (cid:16) ∂B AN ∂x P + ∂B AP ∂x N (cid:17) and Λ MNP , G MM ′ Λ M ′ NP , then we obtain Λ iNP = 0 Λ mjk = 0 Λ mnP = 12 C ma (cid:18) ∂B an ∂x P + ∂B aP ∂x n (cid:19) Λ mNp = 12 C ma (cid:18) ∂B aN ∂x p + ∂B ap ∂x N (cid:19) , Λ iNP = G iM ′ Λ M ′ NP = G ii ′ Λ i ′ NP = 0 Λ mjk = G mM ′ Λ M ′ jk = G mm ′ Λ m ′ jk = 0 Λ mnP = 12 δ ab B bm (cid:18) ∂B an ∂x P + ∂B aP ∂x n (cid:19) Λ mNp = 12 δ ab B bm (cid:18) ∂B aN ∂x p + ∂B ap ∂x N (cid:19) . Calculate the coefficients of curvature of f , that is K mnP Q , ∂Λ mnQ ∂x P − ∂Λ mnP ∂x Q + Λ mHP Λ HnQ − Λ HnP Λ mHQ and K mnP Q , G mM ′ K M ′ nP Q = G mm ′ K m ′ nP Q , then we obtain K D − D − P Q = ∂Λ D − D − Q ∂x P − ∂Λ D − D − P ∂x Q + Λ D − D P Λ D ( D − Q − Λ D ( D − P Λ D − D Q K D − D P Q = ∂Λ D − D Q ∂x P − ∂Λ D − D P ∂x Q + Λ D − D − P Λ D − D Q + Λ D − D P Λ DD Q − Λ D − D P Λ D − D − Q − Λ DD P Λ D − D Q K D ( D − P Q = ∂Λ D ( D − Q ∂x P − ∂Λ D ( D − P ∂x Q + Λ D ( D − P Λ D − D − Q + Λ DD P Λ D ( D − Q − Λ D − D − P Λ D ( D − Q − Λ D ( D − P Λ DD Q K DD P Q = ∂Λ DD Q ∂x P − ∂Λ DD P ∂x Q + Λ D ( D − P Λ D − D Q − Λ D − D P Λ D ( D − Q . K ( D − D − P Q = ∂Λ ( D − D − Q ∂x P − ∂Λ ( D − D − P ∂x Q + G DD (cid:0) Λ ( D − D P Λ D ( D − Q − Λ D ( D − P Λ ( D − D Q (cid:1) K D ( D − P Q = ∂Λ D ( D − Q ∂x P − ∂Λ D ( D − P ∂x Q + G DD (cid:0) Λ DD P Λ D ( D − Q − Λ D ( D − P Λ DD Q (cid:1) + G ( D − D − (cid:0) Λ D ( D − P Λ ( D − D − Q − Λ ( D − D − P Λ D ( D − Q (cid:1) K ( D − D P Q = ∂Λ ( D − D Q ∂x P − ∂Λ ( D − D P ∂x Q + G DD (cid:0) Λ ( D − D P Λ DD Q − Λ DD P Λ ( D − D Q (cid:1) + G ( D − D − (cid:0) Λ ( D − D − P Λ ( D − D Q − Λ ( D − D P Λ ( D − D − Q (cid:1) K DD P Q = ∂Λ DD Q ∂x P − ∂Λ DD P ∂x Q + G ( D − D − (cid:0) Λ D ( D − P Λ ( D − D Q − Λ ( D − D P Λ D ( D − Q (cid:1) . Hence, B P Q , √ (cid:0) K DD P Q + K ( D − D − P Q (cid:1) = 1 √ ∂ (cid:0) Λ DD Q + Λ ( D − D − Q (cid:1) ∂x P − √ ∂ (cid:0) Λ DD P + Λ ( D − D − P (cid:1) ∂x Q = ∂B Q ∂x P − ∂B P ∂x Q .F P Q , √ (cid:0) K DD P Q − K ( D − D − P Q (cid:1) = 1 √ (cid:18) ∂Λ DD Q ∂x P − ∂Λ DD P ∂x Q + G ( D − D − (cid:0) Λ D ( D − P Λ ( D − D Q − Λ ( D − D P Λ D ( D − Q (cid:1)(cid:19) − √ (cid:18) ∂Λ ( D − D − Q ∂x P − ∂Λ ( D − D − P ∂x Q + G DD (cid:0) Λ ( D − D P Λ D ( D − Q − Λ D ( D − P Λ ( D − D Q (cid:1)(cid:19) = ∂A Q ∂x P − ∂A P ∂x Q + g (cid:0) Λ D ( D − P Λ ( D − D Q − Λ ( D − D P Λ D ( D − Q (cid:1) = ∂A Q ∂x P − ∂A P ∂x Q + g (cid:16) A P A Q − A P A Q (cid:17) . Similarly we also obtain F P Q = ∂A Q ∂x P − ∂A P ∂x Q + g (cid:16) A P A Q − A P A Q (cid:17) and F P Q = ∂A Q ∂x P − ∂A P ∂x Q − g (cid:16) A P A Q − A P A Q (cid:17) . ⊓⊔ Remark 6.1.
Comparing the conclusions of the above proposition and the Glashow-Weinberg-Salam theory of physics, we know that this proposition gives another mathematical treatmentof weak and electromagnetic field of physics. It does not abstractly define gauge potentialsas the adjoint representation of group U (1) × SU (2) , but gives concrete intrinsic geometricalconstructions.In addition, the following proposition will prove the characteristics of chirality of leptonsfrom the perspective of intrinsic geometry. This is beyond the reach of the traditional methodthat abstractly regards leptons as the fundamental representation of group U (1) × SU (2) . Definition 6.2.
Suppose f and g both satisfy Definition 6.1 . According to Definition 3.3.1.1and Definition 3.3.5.2 , let ρ mn of f evolve on ( M, g ) . Then l , ( ρ ( D − D − , ρ DD ) is called electric charged lepton , and ν , ( ρ D ( D − , ρ ( D − D ) is called neutrino . l and ν are uniformlycalled leptons , denoted by L . And L √ (cid:0) (cid:1) is called left-handed lepton , L √ (cid:0) − (cid:1) is called right-handed lepton , denoted by l L , √ (cid:0) ρ ( D − D − + ρ DD (cid:1) ,l R , √ (cid:0) ρ ( D − D − − ρ DD (cid:1) , ν L , √ (cid:0) ρ D ( D − + ρ ( D − D (cid:1) ,ν R , √ (cid:0) ρ D ( D − − ρ ( D − D (cid:1) . On ( M, g ) we define W P , √ (cid:0) Γ ( D − D P + Γ D ( D − P (cid:1) ,W P , √ (cid:0) Γ ( D − D P − Γ D ( D − P (cid:1) , Z P , √ (cid:0) Γ ( D − D − P + Γ DD P (cid:1) ,A P , √ (cid:0) Γ ( D − D − P − Γ DD P (cid:1) , generalization of intrinsic geometry and its application to Hilbert’s 6th problem 57 and say the intrinsic geometrical property A p is electromagnetic potential , Z p is Z potential , W P and W P are W potential . Then denote W + P , √ (cid:0) W P − iW P (cid:1) W − P , √ (cid:0) W P + iW P (cid:1) , l + L = 1 √ l L − il R ) l − L = 1 √ l L + il R ) , l + R , √ l R − il L ) l − R , √ l R + il L ) . Proposition 6.2. If ( M, g ) satisfies symmetry condition Γ ( D − D P = Γ D ( D − P , then intrinsicgeometrical properties l and ν of f satisfy the following conclusions on ( M, g ) . l L ; P = ∂ P l L − gl L Z P − gl R A P − gν L W P ,l R ; P = ∂ P l R − gl R Z P − gl L A P ,ν L ; P = ∂ P ν L − gν L Z P − gl L W P ,ν R ; P = ∂ P ν R − gν R Z P . l − L ; P = ∂ P l − L − gl − L Z P − gl − R A P − gν L W − P ,l + L ; P = ∂ P l + L − gl + L Z P − gl + R A P − gν L W + P ,l − R ; P = ∂ P l − R − gl − R Z P − gl − L A P − igν L W − P ,l + R ; P = ∂ P l + R − gl + R Z P − gl + L A P + igν L W + P ,ν L ; P = ∂ P ν L − gν L Z P − gl + L W − P − gl − L W + P ,ν R ; P = ∂ P ν R − gν R Z P . Proof.
Due to ρ mn ; P = ∂ P ρ mn − ρ Hn Γ HmP − ρ mH Γ HnP = ∂ P ρ mn − ρ hn Γ hmP − ρ mh Γ hnP , we have ρ ( D − D − P = ∂ P ρ ( D − D − − ρ ( D − D − Γ D − D − P − (cid:0) ρ D ( D − + ρ ( D − D (cid:1) Γ D ( D − P ,ρ DD ; P = ∂ P ρ DD − ρ DD Γ DD P − (cid:0) ρ D ( D − + ρ ( D − D (cid:1) Γ D − D P ,ρ D ( D − P = ∂ P ρ D ( D − − (cid:16) ρ ( D − D − Γ D − D P + ρ DD Γ D ( D − P (cid:17) − ρ D ( D − (cid:16) Γ D − D − P + Γ DD P (cid:17) ,ρ ( D − D ; P = ∂ P ρ ( D − D − (cid:16) ρ ( D − D − Γ D − D P + ρ DD Γ D ( D − P (cid:17) − ρ ( D − D (cid:16) Γ D − D − P + Γ DD P (cid:17) . ⇒ l L ; P = ∂ P l L − √ ρ ( D − D − Γ D − D − P − √ ρ DD Γ DD P − gν L W P ,l R ; P = ∂ P l R − √ ρ ( D − D − Γ D − D − P + √ ρ DD Γ DD P ,ν L ; P = ∂ P ν L − gl L W P − ν L (cid:16) Γ D − D − P + Γ DD P (cid:17) ,ν R ; P = ∂ P ν R − ν R (cid:16) Γ D − D − P + Γ DD P (cid:17) . ⇒ l L ; P = ∂ P l L − gl L Z P − gl R A P − gν L W P ,l R ; P = ∂ P l R − gl R Z P − gl L A P ,ν L ; P = ∂ P ν L − gν L Z P − gl L W P ,ν R ; P = ∂ P ν R − gν R Z P . ⇔ l − L ; P = ∂ P l − L − gl − L Z P − gl − R A P − gν L W − P ,l + L ; P = ∂ P l + L − gl + L Z P − gl + R A P − gν L W + P , l − R ; P = ∂ P l − R − gl − R Z P − gl − L A P − igν L W − P ,l + R ; P = ∂ P l + R − gl + R Z P − gl + L A P + igν L W + P , ν L ; P = ∂ P ν L − gν L Z P − gl + L W − P − gl − L W + P ,ν R ; P = ∂ P ν R − gν R Z P . ⊓⊔ Discussion 6.1.
According to Discussion 2.6.2 , the transformation group of intrinsic geometryis the subgroup { e } which is uniquely made up of the unit element e of general linear group.Its symmetry is the smallest and never breaks, because it is too small to break. In other words,intrinsic geometry is the largest geometry of geometrical manifold, and its geometrical propertiesare the richest, so that any irregular smooth shape is an intrinsic geometrical property, it therebycan anyway be precisely characterized by slack-tights.We can obtain whatever subgeometry of intrinsic geomtry by way of restricting slack-tightsvia some symmetry conditions , like what section 2.4 and section 2.5 do. Γ ( D − D P = Γ D ( D − P in Proposition 6.2 is just exactly such a symmetry condition, which can be regarded as anequivalent condition to define a kind of subgeometry of intrinsic geometry to describe the weakand eletromagnetic unified field. In summary:(1) The traditional physics starts from a very large symmetry group, and reduces symmetriesin way of symmetry breaking to approach the target geometry.(2) The viewpoint of intrinsic geometry starts from the smallest symmetry group { e } , andadds symmetries in way of symmetry condition to approach the target geometry.Such two ways must lead to the same destination. They both go towards the same specificgeometry.From the perspective of unification of time and space, it is better to focus on concretegeometrical construction than to focus on abstract algebraic structure, and it is better to studyhow to add symmetry conditions than to introduce symmetry breaking. Remark 6.2.
If we do not consider from the perspective of intrinsic geometry, the Hilbert’s 6thproblem can never be solved at the most basic level. In addition:(1) Proposition 6.2 gives the mathematical essence of Glashow-Weinberg-Salam theoryfrom the perspective of intrinsic geometry, even the coupling constant g becomes an intrinsicgeometrical property. Furthermore, we are able to study Dirac equations of l L , l R , ν L , ν R in wayof section 5.6 , and moreover to construct complex-valued Lagrangian density, and to treat QFTfrom the perspective of intrinsic geometry in sense of section 3.4 . However, such topics arebeyond the subject of this paper, we will not discuss them.(2) We notice that the coupling constants of Z P and A P in Proposition 6.2 satisfy g Z = g A = g ,which is comprehensible, because it is just a conclusion at the most basic level. Only when weconsider a kind of medium that is called Higgs field, there appears a Weinberg angle and therebywe have g Z = g A . When we consider Higgs boson as a zero-spin pair of neutrinos, the Higgsboson will lose its fundamentality and it thereby does not have enough importance in theory ofthe most basic level. This paper does not concern such non-fundamental objects and properties.(3) The mixing of leptons of three generations will automatically appear as an intrinsicgeometrical property on the geometrical manifold of Definition 8.1 . generalization of intrinsic geometry and its application to Hilbert’s 6th problem 59 Suppose f satisfies the (1) of Definition 5.6.1 , and geometrical manifold ( M, f ) satisfies D = r + 3 = 6 . On a neighborhood U of any point p the coordinate representation of f ( p ) is ξ a = ξ a ( x m ) and ξ s = δ si x i , such that G ( D − D − = G ( D − D − = G DD . We say f is a strong interaction field . Definition 7.2.
Suppose f and g both satisfy Definition 7.1 . According to Definition 3.3.1.1and Definition 3.3.5.2 , let ρ mn of f evolve on ( M, g ) . Define d , ( ρ ( D − D − , ρ ( D − D − ) d , ( ρ ( D − D − , ρ DD ) d , ( ρ DD , ρ ( D − D − ) , u , ( ρ ( D − D − , ρ ( D − D − ) u , ( ρ ( D − D , ρ D ( D − ) u , ( ρ D ( D − , ρ ( D − D ) . We say d and u are red color charge , d and u are blue color charge , d and u are greencolor charge . Then d , d , d are called down-type color charge , uniformly denoted by d , and u , u , u are called up-type color charge , uniformly denoted by u . d and u are uniformly called color charge , denoted by q . We say q √ (cid:0) (cid:1) is left-handed color charge , and q √ (cid:0) − (cid:1) arecalled right-handed color charge . They are d L , √ (cid:0) ρ ( D − D − + ρ ( D − D − (cid:1) d L , √ (cid:0) ρ ( D − D − + ρ DD (cid:1) d L , √ (cid:0) ρ DD + ρ ( D − D − (cid:1) , d R , √ (cid:0) ρ ( D − D − − ρ ( D − D − (cid:1) d R , √ (cid:0) ρ ( D − D − − ρ DD (cid:1) d R , √ (cid:0) ρ DD − ρ ( D − D − (cid:1) . u L , √ (cid:0) ρ ( D − D − + ρ ( D − D − (cid:1) u L , √ (cid:0) ρ ( D − D + ρ D ( D − (cid:1) u L , √ (cid:0) ρ D ( D − + ρ ( D − D (cid:1) , u R , √ (cid:0) ρ ( D − D − − ρ ( D − D − (cid:1) u R , √ (cid:0) ρ ( D − D − ρ D ( D − (cid:1) u R , √ (cid:0) ρ D ( D − − ρ ( D − D (cid:1) . On ( M, g ) we denote g s , q(cid:0) G ( D − D − (cid:1) + (cid:0) G DD (cid:1) = q(cid:0) G ( D − D − (cid:1) + (cid:0) G ( D − D − (cid:1) = q(cid:0) G ( D − D − (cid:1) + (cid:0) G DD (cid:1) . U P , √ (cid:0) Γ ( D − D − P + Γ ( D − D − P (cid:1) V P , √ (cid:0) Γ ( D − D − P − Γ ( D − D − P (cid:1) , X P , √ (cid:0) Γ ( D − D − P + Γ ( D − D − P (cid:1) Y P , √ (cid:0) Γ ( D − D − P − Γ ( D − D − P (cid:1) , U P , √ (cid:0) Γ ( D − D − P + Γ DD P (cid:1) V P , √ (cid:0) Γ ( D − D − P − Γ DD P (cid:1) , X P , √ (cid:0) Γ ( D − D P + Γ D ( D − P (cid:1) Y P , √ (cid:0) Γ ( D − D P − Γ D ( D − P (cid:1) , U P , √ (cid:0) Γ DD P + Γ ( D − D − P (cid:1) V P , √ (cid:0) Γ DD P − Γ ( D − D − P (cid:1) , X P , √ (cid:0) Γ D ( D − P + Γ ( D − D P (cid:1) Y P , √ (cid:0) Γ D ( D − P − Γ ( D − D P (cid:1) . We notice that there are just only three independent ones in U P , U P , U P , V P , V P and V P . Withoutloss of generality, let R P , a R U P + b R U P + c R U P S P , a S U P + b S U P + c S U P T P , a T U P + b T U P + c T U P , U P , α R R P + α S S P + α T T P U P , β R R P + β S S P + β T T P U P , γ R R P + γ S S P + γ T T P , where the coefficients matrix are non-singular. Proposition 7.1.
Let λ a ( a = 1 , , · · · , be the Gell-Mann matrices, and T a , λ a be the gener-ators of SU (3) group. When ( M, g ) satisfies symmetry condition Γ ( D − D − P + Γ ( D − D − P + Γ DD P = 0 , denote A P , A P A P A P A P A P A P A P A P A P , A P , X P + iY P A P , X P − iY P A P , S P + 1 √ T P , A P , X P + iY P A P , X P − iY P A P , − S P + 1 √ T P , A P , X P + iY P A P , X P − iY P A P , − √ T P . Thus, A P = T a A aP if and only if A P , X P , A P , Y P , A P , S P , A P , X P , A P , Y P , A P , X P , A P , Y P and A P , T P . Proof.
We just need to substitute the Gell-Mann matrices λ , , λ , − i i , λ , − , λ , ,λ , − i i , λ , , λ , − i i , λ , √ − . into A P = T a A aP and directly verify them. ⊓⊔ Remark 7.1.
On one hand, the above proposition indicates that Definition 7.1 is another math-ematical treatment of strong interaction field of physics. It does not anymore define the gaugepotentials abstractly like QCD based on SU (3) theory, but gives concrete intrinsic geometricalconstructions. On the other hand, the above proposition implies that if we take appropriatesymmetry conditions, the algebraic properties of SU (3) group can be described by the trans-formation group GL (3 , R ) of internal space of g . In other words, there exists a homomorphismfrom GL (3 , R ) to SU (3) . generalization of intrinsic geometry and its application to Hilbert’s 6th problem 61 Suppose f satisfies the (1) of Definition 5.6.1 , and geometrical manifold ( M, f ) satisfies D = r + 5 = 8 . On a neighborhood U of any point p the coordinate representation of f ( p ) is ξ a = ξ a ( x m ) and ξ s = δ si x i , such that G ( D − D − = G ( D − D − and G ( D − D − = G ( D − D − = G DD . We say f is a typical unified gauge field . Definition 8.2.
Suppose f and g both satisfy Definition 8.1 . According to Definition 3.3.1.1and Definition 3.3.5.2 , let ρ mn of f evolve on ( M, g ) . Define l , (cid:0) ρ ( D − D − , ρ ( D − D − (cid:1) d , ( ρ ( D − D − , ρ ( D − D − ) d , ( ρ ( D − D − , ρ DD ) d , ( ρ DD , ρ ( D − D − ) , ν , (cid:0) ρ ( D − D − , ρ ( D − D − (cid:1) u , ( ρ ( D − D − , ρ ( D − D − ) u , ( ρ ( D − D , ρ D ( D − ) u , ( ρ D ( D − , ρ ( D − D ) . And Denote l L , √ (cid:0) ρ ( D − D − + ρ ( D − D − (cid:1) l R , √ (cid:0) ρ ( D − D − − ρ ( D − D − (cid:1) , ν L , √ (cid:0) ρ ( D − D − + ρ ( D − D − (cid:1) ν R , √ (cid:0) ρ ( D − D − − ρ ( D − D − (cid:1) , d L , √ (cid:0) ρ ( D − D − + ρ ( D − D − (cid:1) d L , √ (cid:0) ρ ( D − D − + ρ DD (cid:1) d L , √ (cid:0) ρ DD + ρ ( D − D − (cid:1) , d R , √ (cid:0) ρ ( D − D − − ρ ( D − D − (cid:1) d R , √ (cid:0) ρ ( D − D − − ρ DD (cid:1) d R , √ (cid:0) ρ DD − ρ ( D − D − (cid:1) , u L , √ (cid:0) ρ ( D − D − + ρ ( D − D − (cid:1) u L , √ (cid:0) ρ ( D − D + ρ D ( D − (cid:1) u L , √ (cid:0) ρ D ( D − + ρ ( D − D (cid:1) , u R , √ (cid:0) ρ ( D − D − − ρ ( D − D − (cid:1) u R , √ (cid:0) ρ ( D − D − ρ D ( D − (cid:1) u R , √ (cid:0) ρ D ( D − − ρ ( D − D (cid:1) . On ( M, g ) we denote g , q(cid:0) G ( D − D − (cid:1) + (cid:0) G ( D − D − (cid:1) ,g s , q(cid:0) G ( D − D − (cid:1) + (cid:0) G DD (cid:1) = q(cid:0) G ( D − D − (cid:1) + (cid:0) G ( D − D − (cid:1) = q(cid:0) G ( D − D − (cid:1) + (cid:0) G DD (cid:1) , Z P , √ Γ ( D − D − P + Γ ( D − D − P ) A P , √ Γ ( D − D − P − Γ ( D − D − P ) , W P , √ Γ ( D − D − P + Γ ( D − D − P ) W P , √ Γ ( D − D − P − Γ ( D − D − P ) , U P , √ (cid:0) Γ ( D − D − P + Γ ( D − D − P (cid:1) V P , √ (cid:0) Γ ( D − D − P − Γ ( D − D − P (cid:1) , X P , √ (cid:0) Γ ( D − D − P + Γ ( D − D − P (cid:1) Y P , √ (cid:0) Γ ( D − D − P − Γ ( D − D − P (cid:1) , U P , √ (cid:0) Γ ( D − D − P + Γ DD P (cid:1) V P , √ (cid:0) Γ ( D − D − P − Γ DD P (cid:1) , X P , √ (cid:0) Γ ( D − D P + Γ D ( D − P (cid:1) Y P , √ (cid:0) Γ ( D − D P − Γ D ( D − P (cid:1) , U P , √ (cid:0) Γ DD P + Γ ( D − D − P (cid:1) V P , √ (cid:0) Γ DD P − Γ ( D − D − P (cid:1) , X P , √ (cid:0) Γ D ( D − P + Γ ( D − D P (cid:1) Y P , √ (cid:0) Γ D ( D − P − Γ ( D − D P (cid:1) . Definition 8.3.
Define the symmetry condition of unification:(1) Basic conditions, No.1: G ( D − D − = G ( D − D − ,G ( D − D − = G ( D − D − = G DD , (2) Basic conditions, No.2: Γ ( D − D − P = Γ ( D − D − P ,Γ ( D − D − P + Γ ( D − D − P + Γ DD P = 0 , (3) MNS mixing conditions of weak interaction, No.1: Γ D − D − P = c D − D − Γ D − D − P ,Γ D − D − P = c D − D − Γ D − D − P ,Γ D ( D − P = c DD − Γ D − D − P , Γ D − D − P = c D − D − Γ D − D − P ,Γ D − D − P = c D − D − Γ D − D − P ,Γ D ( D − P = c DD − Γ D − D − P , c D − D − = c D − D − ,c D − D − = c D − D − ,c DD − = c DD − , (4) MNS mixing conditions of weak interaction, No.2: ρ ( D − D − = ρ ( D − D − ,ρ ( D − D − = ρ ( D − D − ,ρ D ( D − = ρ D ( D − , ρ ( D − D − = ρ ( D − D − ,ρ ( D − D − = ρ ( D − D − ,ρ ( D − D = ρ ( D − D , (5) CKM mixing conditions of strong interaction, No.1: Γ D − D − P = c D − D − Γ D − D − P ,Γ D − D − P = c D − D − Γ D − D − P ,Γ D − D P = c D − D Γ D − D − P , Γ D − D − P = c D − D − Γ D − D − P ,Γ D − D − P = c D − D − Γ D − D − P ,Γ D − D P = c D − D Γ D − D − P , c D − D − = c D − D − = c D − D ,c D − D − = c D − D − = c D − D , (6) CKM mixing conditions of strong interaction, No.2: ρ ( D − D − = ρ ( D − D − = ρ D ( D − ,ρ ( D − D − = ρ ( D − D − = ρ D ( D − , ρ ( D − D − = ρ ( D − D − = ρ ( D − D ,ρ ( D − D − = ρ ( D − D − = ρ ( D − D , where c mn are constants. generalization of intrinsic geometry and its application to Hilbert’s 6th problem 63 Proposition 8.1.
When ( M, g ) satisfies the symmetry conditions of Definition 8.3 , denote l ′ , ρ ( D − D − + c D − D − (cid:0) ρ ( D − D − + ρ ( D − D − (cid:1) + c D − D − (cid:0) ρ ( D − D − + ρ ( D − D − (cid:1) + c DD − (cid:0) ρ D ( D − + ρ ( D − D (cid:1) ,ρ ( D − D − + c D − D − (cid:0) ρ ( D − D − + ρ ( D − D − (cid:1) + c D − D − (cid:0) ρ ( D − D − + ρ ( D − D − (cid:1) + c DD − (cid:0) ρ D ( D − + ρ ( D − D (cid:1)! .ν ′ , ρ ( D − D − + c D − D − (cid:0) ρ ( D − D − + ρ ( D − D − (cid:1) + c D − D − (cid:0) ρ ( D − D − + ρ ( D − D − (cid:1) + c DD − (cid:0) ρ D ( D − + ρ ( D − D (cid:1) ,ρ ( D − D − + c D − D − (cid:0) ρ ( D − D − + ρ ( D − D − (cid:1) + c D − D − (cid:0) ρ ( D − D − + ρ ( D − D − (cid:1) + c DD − (cid:0) ρ D ( D − + ρ ( D − D (cid:1)! . Thus, the intrinsic geometrical properties l and ν of f satisfy the following conclusions on ( M, g ) . l L ; P = ∂ P l L − gl L Z P − gl R A P − gν ′ L W P ,l R ; P = ∂ P l R − gl R Z P − gl L A P ,ν L ; P = ∂ P ν L − gν L Z P − gl ′ L W P ,ν R ; P = ∂ P ν R − gν R Z P . (22) Proof.
First, ρ mn of f can be calculated as below: ρ mn ; P = ∂ P ρ mn − ρ Hn Γ HmP − ρ mH Γ HnP = ∂ P ρ mn − ρ ( D − n Γ D − mP − ρ ( D − n Γ D − mP − ρ ( D − n Γ D − mP − ρ ( D − n Γ D − mP − ρ D n Γ D mP − ρ m ( D − Γ D − nP − ρ m ( D − Γ D − nP − ρ m ( D − Γ D − nP − ρ m ( D − Γ D − nP − ρ m D Γ D nP . According to Definition 8.2 and Definition 8.3 , by calculation we obtain that l L ; P = ∂ P l L − gl L Z P − gl R A P − gν L W P − h c D − D − (cid:0) ρ ( D − D − + ρ ( D − D − (cid:1) + c D − D − (cid:0) ρ ( D − D − + ρ ( D − D − (cid:1)i g √ W P − h c D − D − (cid:0) ρ ( D − D − + ρ ( D − D − (cid:1) + c D − D − (cid:0) ρ ( D − D − + ρ ( D − D − (cid:1)i g √ W P − h c DD − (cid:0) ρ D ( D − + ρ ( D − D (cid:1) + c DD − (cid:0) ρ D ( D − + ρ ( D − D (cid:1)i g √ W P ,l R ; P = ∂ P l R − gl R Z P − gl L A P , ν L ; P = ∂ P ν L − gν L Z P − gl L W P − h c D − D − (cid:0) ρ ( D − D − + ρ ( D − D − (cid:1) + c D − D − (cid:0) ρ ( D − D − + ρ ( D − D − (cid:1)i g √ W P − h c D − D − (cid:0) ρ ( D − D − + ρ ( D − D − (cid:1) + c D − D − (cid:0) ρ ( D − D − + ρ ( D − D − (cid:1)i g √ W P − h c DD − (cid:0) ρ D ( D − + ρ ( D − D (cid:1) + c DD − (cid:0) ρ ( D − D + ρ D ( D − (cid:1)i g √ W P ,ν R ; P = ∂ P ν R − gν R Z P . Then, according to definitions of l ′ and ν ′ , we obtain that l ′ L = l L + c D − D − √ (cid:0) ρ ( D − D − + ρ ( D − D − (cid:1) + c D − D − √ (cid:0) ρ ( D − D − + ρ ( D − D − (cid:1) + c DD − √ (cid:0) ρ D ( D − + ρ ( D − D (cid:1) + c D − D − √ (cid:0) ρ ( D − D − + ρ ( D − D − (cid:1) + c D − D − √ (cid:0) ρ ( D − D − + ρ ( D − D − (cid:1) + c DD − √ (cid:0) ρ D ( D − + ρ ( D − D (cid:1) ν ′ L = ν L + c D − D − √ (cid:0) ρ ( D − D − + ρ ( D − D − (cid:1) + c D − D − √ (cid:0) ρ ( D − D − + ρ ( D − D − (cid:1) + c DD − √ (cid:0) ρ D ( D − + ρ ( D − D (cid:1) + c D − D − √ (cid:0) ρ ( D − D − + ρ ( D − D − (cid:1) + c D − D − √ (cid:0) ρ ( D − D − + ρ ( D − D − (cid:1) + c DD − √ (cid:0) ρ D ( D − + ρ ( D − D (cid:1) . Substitute them into the previous equations, and we obtain that l L ; P = ∂ P l L − gl L Z P − gl R A P − gν ′ L W P ,l R ; P = ∂ P l R − gl R Z P − gl L A P , , ν L ; P = ∂ P ν L − gν L Z P − gl ′ L W P ,ν R ; P = ∂ P ν R − gν R Z P . ⊓⊔ Remark 8.1.
Reviewing Discussion 6.1 , we know the above proposition gives the mathematicalessence of MNS mixing of weak interaction from the perspective of intrinsic geometry. In math-ematics, the MNS mixing automatically appears as an intrinsic geometrical property, thereforeit is not necessary to postulate artificially like that in physics.In physics, e , µ and τ have just only ontological differences, but they have no difference inmathematical connotation. By contrast, Proposition 8.1 tells us that leptons of three generationsshould be constructed by different linear combinations of ρ ( D − D − , ρ ( D − D − , ρ ( D − D − , ρ ( D − D − , ρ D ( D − , ρ ( D − D , ρ ( D − D − , ρ ( D − D − , ρ ( D − D − , ρ ( D − D − , ρ D ( D − and ρ ( D − D . Thus, e , µ and τ may have concrete and distinguishable mathematical connotations.For example, suppose a τ , b τ , a τ mn , b τ mn are constants, then we can image that e , l = ( ρ ( D − D − , ρ ( D − D − ) ,ν e , ν = ( ρ ( D − D − , ρ ( D − D − ) . generalization of intrinsic geometry and its application to Hilbert’s 6th problem 65 µ , a µ e + 12 (cid:16) a µ D − D − ρ ( D − D − + a µ D − D − ρ ( D − D − + a µ DD − ρ D ( D − ,a µ D − D − ρ ( D − D − + a µ D − D − ρ ( D − D − + a µ DD − ρ D ( D − (cid:17) .ν µ , b µ ν e + 12 (cid:16) b µ D − D − ρ ( D − D − + b µ D − D − ρ ( D − D − + b µ DD − ρ D ( D − ,b µ D − D − ρ ( D − D − + b µ D − D − ρ ( D − D − + b µ DD − ρ D ( D − (cid:17) . τ , a τ µ + 12 (cid:16) a τ D − D − ρ ( D − D − + a τ D − D − ρ ( D − D − + a τ DD − ρ ( D − D ,a τ D − D − ρ ( D − D − + a τ D − D − ρ ( D − D − + a τ DD − ρ ( D − D (cid:17) .ν τ , b τ ν µ + 12 (cid:16) b τ D − D − ρ ( D − D − + b τ D − D − ρ ( D − D − + b τ DD − ρ ( D − D ,b τ D − D − ρ ( D − D − + b τ D − D − ρ ( D − D − + b τ DD − ρ ( D − D (cid:17) . Proposition 8.2.
When ( M, g ) satisfies the symmetry conditions of Definition 8.3 , denote d ′ L , √ c D − D − ( ρ ( D − D − + ρ ( D − D − ) + 12 √ c D − D − ( ρ ( D − D − + ρ ( D − D − )+ 12 √ c D − D − ( ρ ( D − D − + ρ ( D − D − ) + 12 √ c D − D − ( ρ ( D − D − + ρ ( D − D − ) d ′ L , √ c D − D ( ρ ( D − D − + ρ ( D − D − ) + 12 √ c D − D − ( ρ ( D − D + ρ D ( D − )+ 12 √ c D − D ( ρ ( D − D − + ρ ( D − D − ) + 12 √ c D − D − ( ρ ( D − D + ρ D ( D − ) d ′ L , √ c D − D − ( ρ ( D − D + ρ D ( D − ) + 12 √ c D − D ( ρ ( D − D − + ρ ( D − D − )+ 12 √ c D − D − ( ρ ( D − D + ρ D ( D − ) + 12 √ c D − D ( ρ ( D − D − + ρ ( D − D − ) u ′ L , √ c D − D − ( ρ ( D − D − + ρ ( D − D − ) + 12 √ c D − D − ( ρ ( D − D − + ρ ( D − D − )+ 12 √ c D − D − ( ρ ( D − D − + ρ ( D − D − ) + 12 √ c D − D − ( ρ ( D − D − + ρ ( D − D − ) u ′ L , √ c D − D − ( ρ ( D − D − + ρ ( D − D − ) + 12 √ c D − D − ( ρ ( D − D − + ρ ( D − D − )+ 12 √ c D − D ( ρ ( D − D + ρ D ( D − ) + 12 √ c D − D ( ρ ( D − D + ρ D ( D − ) u ′ L , √ c D − D ( ρ ( D − D + ρ D ( D − ) + 12 √ c D − D ( ρ ( D − D + ρ D ( D − )+ 12 √ c D − D − ( ρ ( D − D − + ρ ( D − D − ) + 12 √ c D − D − ( ρ ( D − D − + ρ ( D − D − ) . Then the intrinsic geometrical properties d , d , d , u , u and u of f satisfy the followingconclusions on ( M, g ) . d L ; P = ∂ P d L − g s d L U P + g s d L V P − g s d L V P − g s u L X P − g s u L X P + g s u L Y P − g s u L X P − g s u L Y P − gu ′ L W P d L ; P = ∂ P d L − g s d L U P + g s d L V P − g s d L V P − g s u L X P − g s u L X P + g s u L Y P − g s u L X P − g s u L Y P − gu ′ L W P d L ; P = ∂ P d L − g s d L U P + g s d L V P − g s d L V P − g s u L X P − g s u L X P + g s u L Y P − g s u L X P − g s u L Y P − gu ′ L W P d R ; P = ∂ P d R − g s d L V P + g s d L U P − g s d L U P + g s u L Y P + g s u L X P − g s u L Y P − g s u L X P − g s u L Y P d R ; P = ∂ P d R − g s d L V P + g s d L U P − g s d L U P + g s u L Y P + g s u L X P − g s u L Y P − g s u L X P − g s u L Y P d R ; P = ∂ P d R − g s d L V P + g s d L U P − g s d L U P + g s u L Y P + g s u L X P − g s u L Y P − g s u L X P − g s u L Y P u L ; P = ∂ P u L − g s u L U P − g s u L X P − g s u L Y P − g s u L X P + g s u L Y P − g s d L X P + g s d L Y P − g s d L Y P − gd ′ L W P u L ; P = ∂ P u L − g s u L U P − g s u L X P − g s u L Y P − g s u L X P + g s u L Y P − g s d L X P + g s d L Y P − g s d L Y P − gd ′ L W P u L ; P = ∂ P u L − g s u L U P − g s u L X P − g s u L Y P − g s u L X P + g s u L Y P − g s d L X P + g s d L Y P − g s d L Y P − gd ′ L W P u R ; P = ∂ P u R − g s u R U P + g s u R X P + g s u R Y P + g s u R X P − g s u R Y P u R ; P = ∂ P u R − g s u R U P + g s u R X P + g s u R Y P + g s u R X P − g s u R Y P u R ; P = ∂ P u R − g s u R U P + g s u R X P + g s u R Y P + g s u R X P − g s u R Y P . Proof.
Substitute Definition 8.2 into ρ mn and consider Definition 8.3 , then by calculation wefinally obtain that d L ; P = ∂ P d L − g s d L U P + g s d L V P − g s d L V P − g s u L X P − g s u L X P + g s u L Y P − g s u L X P − g s u L Y P − c D − D − (cid:0) ρ ( D − D − + ρ ( D − D − (cid:1) g √ W P − c D − D − (cid:0) ρ ( D − D − + ρ ( D − D − (cid:1) g √ W P − c D − D − (cid:0) ρ ( D − D − + ρ ( D − D − (cid:1) g √ W P − c D − D − (cid:0) ρ ( D − D − + ρ ( D − D − (cid:1) g √ W P d L ; P = ∂ P d L − g s d L U P + g s d L V P − g s d L V P − g s u L X P − g s u L X P + g s u L Y P − g s u L X P − g s u L Y P − c D − D − (cid:0) ρ ( D − D − + ρ ( D − D − (cid:1) g √ W P − c D − D − (cid:0) ρ ( D − D − + ρ ( D − D − (cid:1) g √ W P − c D − D (cid:0) ρ ( D − D + ρ D ( D − (cid:1) g √ W P − c D − D (cid:0) ρ ( D − D + ρ D ( D − (cid:1) g √ W P d L ; P = ∂ P d L − g s d L U P + g s d L V P − g s d L V P generalization of intrinsic geometry and its application to Hilbert’s 6th problem 67 − g s u L X P − g s u L X P + g s u L Y P − g s u L X P − g s u L Y P − c D − D (cid:0) ρ ( D − D + ρ D ( D − (cid:1) g √ W P − c D − D (cid:0) ρ ( D − D + ρ D ( D − (cid:1) g √ W P − c D − D − (cid:0) ρ ( D − D − + ρ ( D − D − (cid:1) g √ W P − c D − D − (cid:0) ρ ( D − D − + ρ ( D − D − (cid:1) g √ W P d R ; P = ∂ P d R − g s d L V P + g s d L U P − g s d L U P + g s u L Y P + g s u L X P − g s u L Y P − g s u L X P − g s u L Y P d R ; P = ∂ P d R − g s d L V P + g s d L U P − g s d L U P + g s u L Y P + g s u L X P − g s u L Y P − g s u L X P − g s u L Y P d R ; P = ∂ P d R − g s d L V P + g s d L U P − g s d L U P + g s u L Y P + g s u L X P − g s u L Y P − g s u L X P − g s u L Y P u L ; P = ∂ P u L − g s u L U P − g s u L X P − g s u L Y P − g s u L X P + g s u L Y P − g s d L X P + g s d L Y P − g s d L Y P − c D − D − (cid:0) ρ ( D − D − + ρ ( D − D − (cid:1) g √ W P − c D − D − (cid:0) ρ ( D − D − + ρ ( D − D − (cid:1) g √ W P − c D − D − (cid:0) ρ ( D − D − + ρ ( D − D − (cid:1) g √ W P − c D − D − (cid:0) ρ ( D − D − + ρ ( D − D − (cid:1) g √ W P u L ; P = ∂ P u L − g s u L U P − g s u L X P − g s u L Y P − g s u L X P + g s u L Y P − g s d L X P + g s d L Y P − g s d L Y P − c D − D (cid:0) ρ ( D − D − + ρ ( D − D − (cid:1) g √ W P − c D − D − (cid:0) ρ ( D − D + ρ D ( D − (cid:1) g √ W P − c D − D (cid:0) ρ ( D − D − + ρ ( D − D − (cid:1) g √ W P − c D − D − (cid:0) ρ ( D − D + ρ D ( D − (cid:1) g √ W P u L ; P = ∂ P u L − g s u L U P − g s u L X P − g s u L Y P − g s u L X P + g s u L Y P − g s d L X P + g s d L Y P − g s d L Y P − c D − D − (cid:0) ρ ( D − D + ρ D ( D − (cid:1) g √ W P − c D − D (cid:0) ρ ( D − D − + ρ ( D − D − (cid:1) g √ W P − c D − D − (cid:0) ρ ( D − D + ρ D ( D − (cid:1) g √ W P − c D − D (cid:0) ρ ( D − D − + ρ ( D − D − (cid:1) g √ W P u R ; P = ∂ P u R − g s u R U P + g s u R X P + g s u R Y P + g s u R X P − g s u R Y P u R ; P = ∂ P u R − g s u R U P + g s u R X P + g s u R Y P + g s u R X P − g s u R Y P u R ; P = ∂ P u R − g s u R U P + g s u R X P + g s u R Y P + g s u R X P − g s u R Y P . Then we just need to substitute the definitions of d ′ L , d ′ L , d ′ L , u ′ L , u ′ L , u ′ L into the aboveequations. ⊓⊔ Remark 8.2.
The above proposition gives the mathematical essence of CKM mixing of strong in-teraction from the perspective of intrinsic geometry. In mathematics, d ′ L , d ′ L , d ′ L , u ′ L , u ′ L , u ′ L automatically appear as intrinsic geometrical properties, which are thereby not necessary to bepostulated artificially like that in physics. Definition 8.4.
If reference-system f satisfies ρ ( D − D − = ρ ( D − D − = ρ DD = ρ ( D − D − = ρ ( D − D − = ρ ( D − D = ρ D ( D − = ρ D ( D − = ρ ( D − D = 0 , we say f is a lepton field , otherwise f is a hadron field . Suppose f is a hadron field. For d , d , d , u , u , u , if f satisfies that fiveof them are zero and the other one is non-zero, we say f is a single quark . Proposition 8.3.
There does not exist a single quark. In other words, if five of d , d , d , u , u , u are zero, then d = d = d = u = u = u = 0 . Remark 8.3.
For a single down-type quark, the above proposition is evident. Without loss ofgenerality let u = u = u = 0 and d = d = 0 , thus ρ ( D − D − = ρ ( D − D − = ρ DD = 0 ,hence we must have d = 0 .For a single up-type quark, this paper has not made progress on the proof yet. Anyway, inconsideration of that in physics the color confinement has no clear mathematical connotation,by contrast, Proposition 8.3 explicitly gives the mathematical connotation of color confinementfrom the perspective of intrinsic geometry, which itself is very significant.
1. The fundamental theory of intrinsic geometry.(1) Riemannian manifold is generalized to geometrical manifold.(2) The expression form of Erlangen program is improved. The concept of intrinsic geometryis generalized, so that the Riemannian intrinsic geometry based on the first fundamental formbecomes a subgeometry of the generalized intrinsic geometry, and thereby Riemannian geometrycan be incorporated into the geometrical framework of improved Erlangen program.(3) The concept of simple connection is discovered, which reflects more intrinsic propertiesof manifold than Levi-Civita connection.2. The application of intrinsic geometry to Hilbert’s 6th problem.(1) This paper applies the generalized intrinsic geometry to Hilbert’s 6th problem at the mostbasic level, so that the kernel concepts, conclusions, postulates and equations of fundamentalphysics can be strictly defined, constructed and proved in pure mathematical sense.(2) Gravitational field and gauge field have been unified essentially via the concepts ofreference-system and simple connection. Intrinsic geometry of external space describes gravi-tational field, and intrinsic geometry of internal space describes typical gauge field. They havebeen unified into intrinsic geometry.(3) The viewpoints of section 3.4 make gravitational theory and quantum mechanics havethe same view of time and space and unified description of evolution, so we say they are unifiedinto intrinsic geometry. generalization of intrinsic geometry and its application to Hilbert’s 6th problem 69
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