Geometric unification theory of the grand unification and gravitational interactions and new physics

TL;DR

This paper develops a geometric unification theory combining grand unification and gravity using fiber bundle theory, symmetry, and the quantitative causal principle, providing new insights into fundamental physics interactions and Noether theorem.

Contribution

It introduces a novel geometric unification framework based on QCP and symmetric principles, unifying fundamental interactions and their Lagrangians in curved spacetime.

Findings

Unified Lagrangians for four fundamental interactions derived

Overcomes previous difficulties in unifying interactions and Noether theorem

Provides a new geometric foundation for physics laws

Abstract

This paper discovers geometric unification theory of the grand unification and gravitational interactions and their new physics according to the general fiber bundle theory, symmetry and so on. Consequently, the research of this paper is based on the exact scientific bases of mathematics and physics. The Lagrangians of the grand unification and gravitational interactions are unifiedly deduced from quantitative causal principle (QCP) and satisfy the gauge invariant principle of general gauge fields interacting with Fermion and/or boson fields. The geometry and physics meanings of gauge invariant property of different physical systems are revealed, and it is discovered that all the Lagrangians of the four fundamental physics interactions are composed of the invariant quantities in corresponding spacetime structures. The difficulties that fundamental physics interactions and Noether theorem are not able to be unifiedly given and investigated are overcome, the geometric unification theory and origins of the four fundamental physics interactions and Noether theorem are shown by QCP, their two-order general Euler-Lagrange Equations and corresponding Noether conservation currents are derived in general curved spacetime. This paper further deduces QCP from symmetric principle. Consequently, geometric unification theory of the grand unification and gravitation theories and Noether theorem based on symmetric principle and the new physics are given in this paper. This paper further gives the unification of QCP and symmetric principle. Thus, this paper opens a door to both study and give new developments of geometric unification theory of physics laws, and using the new geometric unification theory, a lot of research works about different branches of physics can be anew done and expressed simpler with different symmetric characters.

Equations

$$DS-CS=0$$

$$D{S}_V={\omega }_V{S}_V$$

$$D{S}_V=d{A}_{VU}{S}_U+A_{VU}{\omega }_U{S}_U={\omega }_V{S}_V$$

$${\omega }_V=d{A}_{VU}{A}_{VU}^{-1}+A_{VU}{\omega }_U{A}_{VU}^{-1}$$

$${\Omega }_V=A_{VU}{\Omega }_U{A}_{VU}^{-1}=d{\omega }_V-{\omega }_V\Lambda {\omega }_V$$

$$\frac{1}{2}{\Omega }_{aij}^{b}d{x}^i\Lambda d{x}^j=\frac{1}{2}{A}_a^{b^{{}^{\prime }}}{\Omega }_{b^{{}^{\prime }}{i}^{{}^{\prime }}{j}^{{}^{\prime }}}^{c^{\prime }}{A}_{c^{\prime }}^{-1b}d{x}^{i^{{}^{\prime }}}\Lambda d{x}^{j^{{}^{\prime }}}$$

$${\Omega }_{ab}^{ba}(-g{)}^{\frac{1}{2}}d{x}^1\Lambda d{x}^2\Lambda d{x}^3\Lambda d{x}^4={\Omega }_{a^{{}^{\prime }}{b}^{{}^{\prime }}}^{b^{{}^{\prime }}{a}^{{}^{\prime }}}(-g^{\prime }{)}^{\frac{1}{2}}d{x}^{\prime 1}\Lambda d{x}^{\prime 2}\Lambda d{x}^{\prime 3}\Lambda d{x}^{\prime 4}$$

$$A={\int }_{U_U\cap U_V}{R}_U(-g{)}_{{}_U}^{\frac{1}{2}}d{\tau }_U={\int }_{U_U\cap U_V}{R}_V(-g{)}_{{}_V}^{\frac{1}{2}}d{\tau }_V$$

$$A=\int \frac{1}{2{\pi }^2\kappa }tr(\Omega \Lambda \Omega )=-\int \frac{1}{2{\pi }^2\kappa }{\Omega }_{ij}^a{\tilde{\Omega }}^{aij}(-g{)}^{\frac{1}{2}}d\tau$$

$$A=\int \frac{1}{2{\pi }^2\kappa }tr<<\Omega ,\Omega >>(-g{)}^{\frac{1}{2}}d\tau =\int \frac{1}{4{\pi }^2\kappa }{\Omega }_{ij}^a{\Omega }^{aij}(-g{)}^{\frac{1}{2}}d\tau$$

$$S=S_U{\Psi }_U$$

$$D^{{}^{\prime }}{S}_{Ui}={\omega }_{Uij}^{{}^{\prime }}{S}_j=-S_j{\omega }_{Uji}^{{}^{\prime }}$$

$$D^{{}^{\prime }}={\gamma }^{\mu }{D}_{\mu },d^{{}^{\prime }}={\gamma }^{\mu }{\partial }_{\mu },{\omega }_{Uij}^{{}^{\prime }}={\gamma }^{\mu }{\omega }_{U\mu ij}$$

$$D^{{}^{\prime }}S={\omega }_U^{{}^{\prime }}{S}_U{\Psi }_U+S_U{d}^{{}^{\prime }}{\Psi }_U=S_U(d^{{}^{\prime }}{\Psi }_U-{\omega }_U^{{}^{\prime }}{\Psi }_U)$$

$$({\stackrel{-}{S}}_{Ui},S_{Uj})={\delta }_{ij}$$

$$(\overline{S},D^{{}^{\prime }}S)=({\overline{\Psi }}_U{\overline{S}}_U,S_U(d^{{}^{\prime }}{\Psi }_U-{\omega }_U^{{}^{\prime }}{\Psi }_U))={\overline{\Psi }}_U(d^{{}^{\prime }}{\Psi }_U-{\omega }_U^{{}^{\prime }}{\Psi }_U)$$

$$d^{\prime }{\Psi }_U=d^{\prime }{A}_{UW}{\Psi }_W+A_{UW}{d}^{\prime }{\Psi }_W$$

$${\overline{\Psi }}_U(d^{{}^{\prime }}{\Psi }_U-{\omega }_U^{{}^{\prime }}{\Psi }_U)={\overline{\Psi }}_W{A}_{UW}^{+}[d^{\prime }{A}_{UW}{\Psi }_W+A_{UW}{d}^{{}^{\prime }}{\Psi }_W$$

$$-(d^{{}^{\prime }}{A}_{UW}{A}_{UW}^{-1}+A_{UW}{\omega }_W^{{}^{\prime }}{A}_{UW}^{-1})A_{UW}{\Psi }_W]={\overline{\Psi }}_W(d^{{}^{\prime }}{\Psi }_W-{\omega }_W^{{}^{\prime }}{\Psi }_W)$$

$${\mathcal{L}}_{\Psi }=\overline{\Psi }{\gamma }^{\mu }({\partial }_{\mu }\Psi -{\omega }_{\mu }\Psi )$$

$$<(\overline{DS},DS)>=<({\overline{\phi }}_U(\stackrel{\overline{\leftarrow }}{d}-{\overline{\omega }}_U){\overline{S}}_U,S_U(d{\phi }_U-{\omega }_U{\phi }_U))>$$

$$=<{\overline{\phi }}_U({\stackrel{\overline{\leftarrow }}{\partial }}_{\mu }-{\omega }_{\mu U})d{x}^{\mu },d{x}^{\nu }({\partial }_{\nu }{\phi }_U-{\omega }_{\nu U}{\phi }_U)>$$

$${\mathcal{L}}_{\phi }=<(\overline{DS},DS)>=\overline{\phi }({\stackrel{\overline{\leftarrow }}{\partial }}_{\mu }-{\overline{\omega }}_{\mu })g^{\mu \nu }({\partial }_{\nu }-{\omega }_{\nu })\phi$$

$$=\overline{\phi }({\stackrel{\overline{\leftarrow }}{\partial }}_{\mu }-{\overline{\omega }}_{\mu })({\partial }^{\mu }-{\omega }^{\mu })\phi$$

$$(\overline{S},S)=({\overline{\phi }}_U{\overline{S}}_U,S_U{\phi }_U)={\overline{\phi }}_U{\phi }_U$$

$${\mathcal{L}}_{m\phi }=m^n\overline{\phi }\phi ,$$

$$V_U({\overline{\phi }}_U,{\phi }_U)=V_W({\overline{\phi }}_W,{\phi }_W)$$

$$V(\overline{\phi }\phi )={\lambda }^2(\overline{\phi }\phi -{\mu }^2{)}^2.$$

$$A={\int }_{M^4}(\alpha R+{\mathcal{L}}_m)(-g{)}^{\frac{1}{2}}d\tau$$

$$A={\int }_{M^4}[\alpha R+\overline{\Psi }{\gamma }^{\mu }({\partial }_{\mu }\Psi -{\omega }_{\mu }\Psi )+m\overline{\Psi }\Psi +U(\overline{\Psi },\Psi )-\frac{1}{4q_{\lambda }^2}{F}_{\mu \nu }^{\lambda a}{F}^{\lambda a\mu \nu }](-g{)}^{\frac{1}{2}}d\tau$$

Full text

Key words: Particle physics, field theory, Lagrangian, fundamental interaction, gravitation, unification theory, Noether theorem, causal principle

PACS: 11.10.-z, 11.10.Ef

1 Introduction

Causal principle should be satisfied in expressing physical laws. In quantum field theory, the causal principle is reflected in the results that if the square of the distance of spacetime coordinates of two boson (or fermion) operators is timelike, the canonical commutator (or anticommutator) of the boson (or fermion) field does not vanish, i.e., their measures are coherent, but not coherent for spacelike case [1]. Dispersion relations are deduced by means of the causal principle etc [2]. Ref. [3] studied the relationships of causal principle and symmetry principle.

It is well known that in physics the well-admitted physical law not satisfying causal principle has not been found. In terms of the condition that causal principle is quantitative, i.e., quantitative causal principle ( QCP ), Ref. [4] gives a unified theory of all differential variational principles, and a unified theory of all integral variational principles are achieved [5]. The investigations about the generalization from classical statistical mechanics to quantum mechanics still satisfy quantitative causal principle [6]. Refs. [4, 5, 7] show the quantitative causal principle of gain, loss and transformation of any system, and prove that the equivalent causes must result in the equivalent results, which is just the invariant property of some operations, namely, symmetry properties [8, 9]. In fact, the symmetric properties in physics and mathematics are just the invariant properties under some kinds of operations of systems [9], almost all physics processes, e.g., Refs.[10, 11], satisfy causal relations because causal relations should be satisfied in the universe, the relation between symmetries and QCP is given [4, 5], and using QCP, Ref. [8] gives unification theory of different causal algebras and its applications to theoretical physics, in which group theory is included as a special subset.

Utilizing the no-loss-no-gain homeomorphic map transformation satisfying QCP, Ref. [7] overcomes the non-perfect properties of the Volterra process, gains the exact strain tensor formulae in condensed theory, and solves the hard difficulty to give the really serious one-to-one correspondences between the two theories of dislocations in Euclidean space and Weitzenböck Material Manifolds. Ref. [5] gives the unified origin of different variational principle and Noether theorem in terms of QCP. Now it is well known that Noether theorem is deduced from variational principle and group theory [12, 13, 14], while variational principle and Noether theorem may be derived by QCP [5]. It can be seen that this paper and Refs. [4, 5, 7] are essential developments of the investigations of Ref. [3] about the relationships of causal principle and symmetry principle. It is well known that physical fundamental interactions and Noether theorem are used to be regarded as independent fundamental laws being not able to be unifiedly proved and studied up to now, but in the following investigations we show that the two laws are just the deductions of QCP and give the unified description of the two laws.

Many Refs. [15, 16, 17, 18, 19] studied causal field propagation that led to neat results, and these results are interesting and also satisfy QCP. Unitarity and causality are key physical concepts in physics studies, e.g., studying weak gravity conjecture [20]

In fact, superstring theory has gotten many important developments [21, 22, 23], and Ref. [24] very well investigated Quantum modification of general relativity and so on. But it is well known that there are still some key problems in various kinds of the grand or supersymmetric unification theories up to now, e.g., there are too many adjustable free parameters etc in various kinds of the grand or supersymmetric unification theories, which are not natural, even superstring theories and M-theory have themselves unsolved key problems, e.g., we still cannot seriously break the high dimensional theories down to four dimensions and do rigorous phenomenology etc. There are different unification theories, e.g., from up to down (superstring) or from down to up (phenomenological) unification theories, unification theory of this paper belongs to from down to up unification theory. Therefore, exploring the other possible unification theories is actually needed, which may activate to understand and solve all the problems finally, that is, we may try to solve all the problems from different aspects, which may finally activate development and perfection of the various kinds of the unification theories, e.g., superstring theories, M-theory and so on.

Famous physical Nobel Prize winner S. Weinberg tells us [25]: The present generation of young physicists may envy those of us who had the excitement and delight of developing the standard model. This might be a mistake, just as it turned out that my generation would have been mistaken to envy the earlier heroes of quantum electrodynamics. Our newly minted experimentalists and theorists now have a chance to participate in making the next big step beyond the standard model. They may even be able to see their way clear to the very high energy scale where a final theory will be revealed. Thus we need to try various explorations.

The arrangement of this paper is: Sect. 2 is geometric unification theory of fundamental physics interactions, Sect. 3 is the unified description of fundamental physics interactions and Noether theorem, Sect. 4 is applications of this theory and new physics, and Sect.5 is summary and conclusions.

2 Geometric Unification Theory of Fundamental Physics Interactions

For the convenience of research, we simply review the proof of QCP. In physics, quantitative action (cause) of some quantities must lead to the equal action (result), i.e., how much lose (cause), how much gain (result), which is just QCP deduced from the no-loss-no-gain principle in the universe [4, 5] and it may be concretely expressed as

[TABLE]

Eq.(1) means that any quantitative action produced by operator set D acting on S must lead to appearance of set C acting on S so that DS equate CS, where D and C may be different operator sets, the whole process satisfies QCP so that right hand side of Eq.(1) keeps zero, i.e. satisfies the no-loss-no-gain principle in the universe [5]. Thus, Eq.(1) is viewed as a mathematical expression of QCP. Eq.(1) is very useful for the following studies. Eq.(1)’s nonlinear expression refers to Ref. [8].

All physics laws must satisfy causal principle and the causal principle must be quantitative. Furthermore, up to now, all the physics laws violating causal principle have not been found, it can be seen, using the quantitative causal principle we shall be able to give the unification description theory. Consequently, the principle is useful and important.

For a 4-dimensional physical general curved spacetime M4 ( there naturally may exist the constructions of vector bundle $E(M^{4},F,\pi,G)$ or associate vector bundle $E(M^{4},F,\pi_{{}_{V}},G,P(M^{4},G,\pi))$ of principal bundle $P(M^{4},G,\pi)$. Readers, not familiar with manifold and fibre bundle, don’t need to read the Italic type parts in parenthesis, which will not affect their understanding on the paper, as the same below ), taking $S_{V}$ as a general basic vector field in any open neighborhood UV in M4, D as differential connection operator, C as connection $\omega_{V}$ in Eq.(1), it follows that [26, 27]

[TABLE]

Eq.(2)’s physical meaning is that the quantitative physical effect operator D’s acting on $S_{V}$ must equate that the gauge field or connection $\omega_{V}$ times $S_{V}$ ( in which $S_{V}$ may be a basic vector of principal bundle whose base manifold is M4 or a basic vector on M4 of vector bundle ).

Substituting the transformation $S_{V}=A_{VU}S_{U}$ into Eq.(2), it follows that [26]

[TABLE]

where $DA_{VU}=dA_{VU},$ because $A_{VU}$ is the matrix function transforming $S_{U}$ into $S_{V}$ between two open neighborhoods UU and UV (UU $\cap$ UV $\neq 0$) on M4.

Substituting $A_{VU}^{-1}S_{V}=S_{U}$** **into Eq.(3), we obtain the relation transforming connection $\omega_{U}$ into connection $\omega_{V}$ as follows [26]

[TABLE]

For gauge field ( in principal bundle ), it is just gauge transformation between gauge fields $\omega_{U}$ and $\omega_{V}$ in two open neighborhoods UU and UV (UU $\cap$ UV $\neq 0$) in gauge field theory; for connection ( in vector bundle ), it is the transferring relation between connections $\omega_{U}$ and $\omega_{V}$ in two open neighborhoods UU and UV (UU $\cap$ UV $\neq 0$) in curved spacetime, i.e., Eq.(4) is just the unified transformation expression of gauge field and connection in two open neighborhoods UU and UV, thus we can generally call the unified transformation as general gauge transformation.

Using Eq.(4), we have $\omega_{V}A_{VU}-A_{VU}\omega_{U}=dA_{VU},$ and further acting the differential connection operator D on Eq.(3), we obtain not only the unification expression of curvature tensor 2-form ( in tangent vector bundle ) and field strength tensor 2-form ( in principal bundle ) but also the unification expression of general coordinate transformation of curvature tensor 2-form ( in tangent vector bundle ) and gauge transformation of field strength tensor 2-form ( in principal bundle ) as follows

[TABLE]

In fact, Eqs.(4 & 5) show that gravity is also a kind of gauge theory.

Multiplying Eq.(5)’s component quantity expression of the curvature tensor 2-form ( in tangent vector bundle )

[TABLE]

with $g^{ak}\varepsilon_{bklm}dx^{l}\Lambda dx^{m}$ from right hand side, in which the a, b, c,$\cdots$, i, j, k, $\cdots$ and a${}^{{}^{\prime}}$, b${}^{{}^{\prime}},$ c${}^{{}^{\prime}},\cdots$ i${}^{{}^{\prime}}$, j${}^{{}^{\prime}}$, k${}^{{}^{\prime}},\cdots$ are in UV and UU respectively, defining $\varepsilon_{{}_{1234}}=(-g)^{\frac{1}{2}}$  [28] & $x^{4}=ict$ and using $\varepsilon_{i_{1}i_{2}i_{3}i_{4}}\varepsilon^{j_{1}j_{2}j_{3}j_{4}}=sgn(g)\delta_{i_{1}i_{2}i_{3}i_{4}}^{j_{1}j_{2}j_{3}j_{4}}$ [28], we prove

[TABLE]

Namely, Eq.(7) is invariant or doesn’t depend on coordinates of different UV and UU.

Adding integral sign to Eq.(7) and, as usual, defining $\Omega_{ab}^{\text{ }ba}=g^{bc}R_{cab}^{a}=R_{V},\Omega_{a^{{}^{\prime}}b^{{}^{\prime}}}^{\text{ }b^{{}^{\prime}}a^{{}^{\prime}}}=R_{U}$ and $d\tau=dx^{1}\Lambda dx^{2}\Lambda dx^{3}\Lambda dx^{4},$ we obtain the invariant gravitational action [29]

[TABLE]

Thus, Eq.(8) keeps invariant in the whole spacetime (manifold M${}^{4}=$ $\underset{V}{\cup}U_{V}),$ because the two open neighborhoods UU and UV are two arbitrary open overlapped neighborhoods in the global spacetime ( manifold in the vector bundle ). The above condition (8) is just the condition that Eq.(8) may be taken as invariant gravitational action in the whole spacetime, because the physical consistence demands that Eq.(8) must take the invariant formulation in every open neighborhood of the whole spacetime, which satisfies just the general gauge invariant principle of gravitational gauge fields. Then geometry and physics meanings of gauge invariant property are directly, whole and seriously revealed.

A Noether invariant quantity is obtained from Eq.(5) in terms of 2-form field strength tensor ( in principal bundle ) as follows $tr(\Omega_{V}\Lambda\Omega_{V})=tr(\Omega_{U}\Lambda\Omega_{U}),$which satisfy just the general gauge invariant principle [30] of general gauge fields. Then geometry and physics meanings of gauge invariant property of general gauge fields are directly, whole and seriously revealed.

Adding integral sign to $tr(\Omega_{V}\Lambda\Omega_{V})$ and using $tr(T^{a}T^{b})=-2\delta^{ab}$, we have the invariant action with dual gauge field strength tensor in the global curved spacetime as follows [29]

[TABLE]

Thus the Lagrangian density is $\mathcal{L}=-\frac{(-g)^{\frac{1}{2}}}{2\pi^{2}\kappa}\Omega_{ij}^{\text{ }a}\overset{\char 126\relax}{\Omega}^{aij}=-\frac{(-g)^{\frac{1}{2}}}{4q^{2}}\Omega_{ij}^{\text{ }a}\overset{\char 126\relax}{\Omega}^{aij}($ $\overset{\char 126\relax}{\Omega}^{aij}=\frac{1}{2}\varepsilon^{ijkl}\Omega_{kl}^{\text{ }a}$ is dual gauge field strength tensor, taking $\kappa=2q^{2}/\pi^{2}$ $)$ with dual gauge field strength tensor existing in the global spacetime [27, 29]. As Eq.(9)’s natural generalization, for higher dimensions, we may naturally deduce $A=\int\frac{1}{2\pi^{2}\kappa}tr(\Omega\Lambda\Omega\Lambda...\Lambda\Omega).$

In terms of modern differential geometry, we can define the inner product $<\partial_{\mu},\partial_{\nu}>=<e_{\mu},e_{\nu}>$ $=g_{\mu\nu}$ of natural (or called intrinsical ) tangent basic vectors and the inner product $<dx^{\mu},dx^{\nu}>=<e^{\mu},e^{\nu}>$ $=g^{\mu\nu}$ of natural (or called intrinsical ) cotangent basic vectors, they are consistent, see Appendix A, then we obtain an important invariant $<\omega_{\mu}dx^{\mu},\omega_{\nu}dx^{\nu}>=\omega_{\mu}\omega_{\nu}g^{\mu\nu}=\omega_{\mu}\omega^{\mu}$ of inner product of two 1-forms (e.g., $\omega_{V}=\omega_{V\mu}dx_{V}^{\mu})$, i.e., cotangent vectors, under coordinate transformation. Similar to deducing the important invariant $\omega_{\mu}\omega^{\mu}$ of inner product of two 1-forms, see Appendix B we finally achieve the invariant action of general gauge fields in the global curved spacetime as follows

[TABLE]

Eq.(10) satisfies just the general gauge invariant principle in the global spacetime.

When $S_{Ui}$ $(i=1,2,3,\cdots,k;$** $U$ is subscript of the open neighborhood UU) **are k linearly independent basic vectors of the vector space ( of vector bundle or associate vector bundle $E(M^{4},F,\pi_{{}_{V}},G,P(M^{4},G,\pi))$), or equivalently the orthonormal basic vectors of the Group G’s representation vector space F, Therefore, any vector field S may be expressed as

[TABLE]

where $\Psi_{U}$ is complex column matrix function of the corresponding components. It follows from Eq.(2) that

[TABLE]

in which we have remarked

[TABLE]

where $\gamma^{\mu}$ is $\gamma$ matrix with Lorentz superscript $\mu$ in Eq.(13), and $dx^{{}^{\prime}\mu}=a_{\text{ }\nu}^{\mu}dx^{\nu}$ $(\mu,\nu=0,1,2,3)$ have the same transformation law as $\gamma^{{}^{\prime}\mu}=a_{\text{ }\nu}^{\mu}$ $\gamma^{\nu}$ [31], $a_{\text{ }\nu}^{\mu}$ is Lorentz group matrix element. Because the $i,j,..$ and $\mu,\nu,...$ belong to the different freedoms’ groups, thus, their matrixes corresponding to different groups are commutable. From Eqs.(2), (11) and (12) we have

[TABLE]

Using the orthonormal relation (we can choose the orthonormal basic vectors of the vector space)

[TABLE]

we obtain

[TABLE]

Now consider the expression of Eq.(16) in an open neighborhood UW. It follows from $\Psi_{U}=A_{UW}\Psi_{W}$ that

[TABLE]

Using Eqs.(4), (16), (17) and $\Psi_{U}=A_{UW}\Psi_{W}$ ( $A_{UW}$ is transformation matrix ), we have

[TABLE]

[TABLE]

where we have used $A_{UW}^{+}A_{UW}=I$. Namely Eq.(18) is the invariant quantity in the whole spacetime (manifold in the bundle), and the physical consistence demands that Eq.(18) must take the invariant formulation in the every open neighborhood of the whole spacetime, which satisfies just the general gauge invariant principle of the gauge fields interacting with Fermi fields. Then geometry and physics meanings of gauge invariant property of gauge fields interacting with Fermion fields are directly, whole and seriously revealed.

Inserting Eq.(13) into Eq.(18), we obtain the general topological invariant Lagrangians of matter field $\Psi$ interacting with gauge field $\omega_{\mu}$, which keeps effective in the whole spacetime as follows

[TABLE]

Now we consider another invariant.

Similar to the discussion of Eqs.(14 & 18), taking $D=dx^{\mu}D_{\mu},$ $d=dx^{\mu}\partial_{\mu},$ $\omega_{Uij}=dx^{\mu}\omega_{U\mu ij}$, replacing spinor field $\Psi_{U}$ with scalar field $\varphi$ in Eq.(14), it follows that $DS=S_{U}(d\Psi_{U}-\omega_{U}\Psi_{U}),$ further taking Hermite conjugation of $DS$, and making an inner product, relevant to $(\overset{-}{S}_{Ui},S_{Uj})=\delta_{ij}$ and $<dx^{\mu},dx^{\nu}>$ , of both $DS$ and its conjugation, we get

[TABLE]

[TABLE]

Analogous to the discussion of Eq.(18), it is easy to prove that Eq.(20) is the invariant quantity existing in the whole spacetime .

In terms of modern differential geometry, we again use the inner product $<dx^{\mu},dx^{\nu}>=<e^{\mu},e^{\nu}>$ $=g^{\mu\nu}$ of natural cotangent basic vectors, then we achieve an important invariant Lagrangian of scalar fields interacting with gauge fields

[TABLE]

[TABLE]

Thus, Eq.(21) satisfies the physical invariant consistent demand in the every open neighborhood of the global spacetime. That is, Eq.(21) satisfies just the general gauge invariant principle of the gauge fields interacting with scalar fields. Then geometry and physics meanings of gauge invariant property of gauge fields interacting with Boson fields are directly, whole and seriously revealed.

On the other hand, due to

[TABLE]

and using $\varphi_{U}=A_{UW}\varphi_{W},$ it is easy to prove that Eq.(22) is invariant quantity existing in the whole spacetime. Thus we may multiply Eq.(22) with mass parameter $m^{n}$ ( $n=2,$ taking $\varphi$ as boson function; $n=1,$ taking $\varphi$ as fermion function ) to construct the mass part of the Lagrangian of a system.

[TABLE]

About potential functional of $(\overline{\varphi}_{U},\varphi_{U})$ due to the invariance of $(\overline{\varphi}_{U},\varphi_{U})$ under the general gauge transformation, it must satisfy

[TABLE]

which is the invariant condition in the global spacetime, i.e., Eq.(24) satisfies just the general gauge invariant principle of potential energy. For example, Eq.(22)’s arbitrary combinations may satisfy condition (24)., i.e., we may generally take Eq.(22) as the variable to construct scalar potential functional. e. g., the potential functional V of SU(2) complex scalar fields

[TABLE]

Using Eqs.(8), (10), (19), (21). (23) and (24), we achieve gravitational action, all general actions of matter fields, spinor fields and scalar fields interacting with gauge fields in the curved spacetime as follows

[TABLE]

[TABLE]

[TABLE]

where $\alpha$ is a parameter, $\mathcal{L}_{m}$ is the Lagrangian of general matter fields, $\overline{\omega}^{\mu}=g^{\mu\nu}\overline{\omega}_{\nu}$ and $\gamma^{\mu}=g^{\mu\nu}\gamma_{\nu}$**. (**In the case that the fibre G of principal bundle is a semisimple group of a general form, ) the Lagrangian contains r arbitrary constants $q_{{}_{\lambda}}$ , $\lambda=1,2,...,r$, in which r is the number of invariant simple factors [29].

The fundamental physics interactions, e.g., strong, weak, electromagnetic and gravitational interactions, can be described by Eqs.(26-27), these actions are unifiedly deduced by QCP, i.e., Eq.(1), thus the new unification theory of fundamental physics interactions is given in terms of modern differential geometry. The more concrete examples are that the known grand unification theories may be SU(5), S(10) or E6 gauge theories, and it is very easy that their relative more fermion mass terms etc can similarly be deduced by using the expression which is relevant to spinor’s general gauge invariant quantities.

3 The Unified Description of Fundamental Physics Interactions and

Noether Theorem

In the expression (1) of QCP, which has deduced all the physical fundamental interactions in this paper, for general field variables $X(x)=$ $\{\Psi(x),\varphi(x),\omega_{\mu}(x),g_{\mu\nu}(x),...,\}$ above, when S is the actions (26-28), C is unit element and D is infinitesimal transformation operator of continuous Lie group [5, 13, 14]

[TABLE]

[TABLE]

in which $\varepsilon_{\sigma}$ $(\sigma=1,2,\cdots,m)$ are infinitesimal parameters of Lie group Dm in Eq.(1). Thus, we get the unified expression of their variational principles, as follows

[TABLE]

Not losing generality, under the transformations of Eqs.(29) and (30), we can take ad-hoc

[TABLE]

where $\partial_{\mu}\Omega^{\mu}=\varepsilon_{\sigma}\partial_{\mu}\Omega^{\sigma\mu}$ and the Ricci Scalar R contains $g_{\alpha\beta}$,μν, and $\Omega$ may be rapidly decreasing to fit the usual physics experiments.

Using the unified expression of the variational principles, we have

[TABLE]

[TABLE]

in which $\delta X^{a}=\Delta X^{a}-X^{a}$,${}_{\nu}\Delta x^{\nu}$. Using the above discussions, we can obtain their two-order general Euler-Lagrange Equations and corresponding Noether conservation currents as follows

[TABLE]

[TABLE]

[TABLE]

Therefore, Noether theorem of the general physical system is deduced by QCP**. **

In all, using QCP, i.e., Eq.(1), we derive not only all the fundamental physics Lagrangians and their variational principles and further Noether theorem**,** but also their two-order general Euler-Lagrange Equations and corresponding Noether conservation currents. Using the above studies, variation laws of the different physical systems are naturally determined. Therefore, it is essential to deduce effectively unified expressions of elementary physical laws by QCP.

4

Applications and New Physics

We can use the geometric unification theory to study different physical systems. For example:

(1) As Eq.(24)’s natural generalization, when the invariant scalar product $\overline{\varphi}\varphi$ is extended by the invariant scalar curvature $R,$ we may naturally deduce $f(R,\overline{\varphi}\varphi)$ gravity, i.e., $A=\int f(R,\overline{\varphi}\varphi)(-g)^{\frac{1}{2}}d\tau,$ because$(-g)^{\frac{1}{2}}d\tau$ is invariant volume element. Further adding the first term $\alpha R$ of Eq.(28), we get $A_{1}=\int F(R,\overline{\varphi}\varphi)(-g)^{\frac{1}{2}}d\tau=$ $\int[f(R,\overline{\varphi}\varphi)+\alpha R](-g)^{\frac{1}{2}}d\tau$ [32].

(2) As Eq.(24)’s direct application, when the invariant scalar product $\overline{\varphi}\varphi$ is taken as the invariant scalar product $H^{+}H$ of Higgs field H’s {5} multiplets, we may naturally deduce $V(H)$ $=\frac{\mu_{0}^{2}}{2}H^{+}H+\frac{\lambda}{4}(H^{+}H)^{2}$ ($\mu_{0}$ and $\lambda$ are parameters), which is just the potential of Higgs field H’s {5} multiplets for grand unification theory [33]; for Higgs field $\phi$’s {24} multiplets that are expressed by a $5\times 5$ matrix, the matrix satisfies Eq.(5)’s general gauge invariant property, i.e., $\phi_{V}=A_{VU}\phi_{U}A_{VU}^{-1}$ or $tr\phi_{V}=tr\phi_{U}$ and more generally it follows that $tr\phi_{V}^{n}=tr\phi_{U}^{n}$ , and as Eq.(24)’s natural generalization, we have $V(\phi)$ $=$ $-\frac{1}{2}\mu^{2}tr\phi^{2}+\frac{a}{4}(tr\phi^{2})^{2}+\frac{b}{2}tr\phi^{4}$ ( $\mu,$ $a$ and $b$ are parameters ); for mixed invariants of H’s {5} and $\phi$’s {24} multiplets, similarly, people can deduce $V(H,\phi)=cH^{+}H(tr\phi^{2})+dH^{+}\phi^{2}H$ ( $c$ and $d$ are parameters ), thus we totally deduce $V_{t}(H,\phi)=V(H)+V(\phi)+V(H,\phi),$ which is the total Higgs potential of grand unification theory and the same as that of Ref. [33].

(3) Generalizing $R$ as an invariant functional $f(R)$ and using Eq.(27) deduced by QCP, people can deduce Fermion field Lagrangian interacting with Non-Abelian gauge field and general $f(R)$ gravitational field as follows

[TABLE]

Eq.(37) is just a general generalization of Eq.(27), and further utilizing the deduced Eqs.(34) and (35) by QCP, people can naturally gives the new unification theory of the fundamental Non-Abelian gauge field, Fermion field and general $f(R)$ gravitational field interactions and Noether theorem by using QCP, and can concretely derive their Euler-Lagrange Equations and corresponding Noether conservation currents as done in Sect. 3. Thus the relative books and articles may be renewedly and systematically rewritten, which will help people to understand and express the fundamental Non-Abelian gauge field, Fermion field and general $f(R)$ gravitational field interactions simpler and with clear quantitative causal physical meanings.

(4) Generalizing $R$ as an invariant functional $f(R)$ and using Eq.(28) deduced by QCP, people can deduce Boson field Lagrangian interacting with Non-Abelian gauge field and general $f(R)$ gravitational field as follows

[TABLE]

Eq.(38) is just a general generalization of Eq.(28), and further utilizing the deduced Eqs(34) and (35) by using QCP, people can naturally gives new unification theory of the fundamental Non-Abelian gauge field and general $f(R)$ gravitational field interactions and Noether theorem by using QCP, and can concretely derive their Euler-Lagrange Equations and corresponding Noether conservation currents. Thus the relevant books and articles may be renewedly and systematically rewritten, which will help people to understand and express the fundamental Non-Abelian gauge field, Boson field and general $f(R)$ gravitational interactions simpler and with clear quantitative causal physical meanings. When $f(R)=0,$ we naturally give electric and color superconduction theories corresponding Abelian $-\frac{1}{4q^{2}}F_{\mu\nu}^{\text{ }}F^{\mu\nu}$ and non-Abelian $-\frac{1}{4q_{\lambda}^{2}}F_{\mu\nu}^{\text{ }\lambda a}F^{\lambda a\mu\nu}$ in Eq.(38), respectively [33], thus the relevant books and articles may be renewedly and systematically rewritten, which will help people to understand and express the fundamental Non-Abelian gauge field and Boson field interactions simpler and with clear quantitative causal physical meanings.

(5) Using QCP, people can more generally deduce

[TABLE]

[TABLE]

Eq.(39) is just a general generalization of Eqs.(37) and (38), and further utilizing the deduced Eqs(34) and (35) by QCP, people can naturally gives the new unification description theory of the fundamental Non-Abelian gauge field, Fermion field, Boson field, general $f(R)$ gravitational field interactions and Noether theorem by using QCP, and can concretely derive their Euler-Lagrange Equations and corresponding Noether conservation currents. Thus the relevant books and articles may be renewedly and systematically rewritten, which will make people understand and express the fundamental Non-Abelian gauge field, Fermion field, Boson field, and general $f(R)$ gravitational field interactions simpler and with clear quantitative causal physical meanings. Specially, Eq.(39) is just the action of the theory of unifiedly describing all known physical fundamental interactions, further further utilizing the deduced Eqs(34) and (35) by QCP, consequently the new unification theory of the fundamental physics interactions and Noether theorem is given by using QCP for the first time.

When considering nonlinear dynamic interaction action of their fields in Eq.(39) in current particle physics standard model, we deduce

[TABLE]

[TABLE]

When further considering two types of nonlinear dynamic interaction actions in Eq.(40) in current particle physics standard model, we have

[TABLE]

[TABLE]

[TABLE]

Using quantitative causal principle, we not only can give a restatement of well-known physical theories/Lagrangians in the language of fibre bundles, but also give the new unified description of the fundamental physics interactions and Noether theorem in this paper, these are new physics and cannot be given in the past [33].

Especially, Eqs.(40) and (41) give a lot of new physics and new physical interactions in current particle physics standard model, which can describe a lots of high order fundamental interactions and different nonlinear physical systems in different branches of physics, e.g., high order fundamental interactions (especially in both early universe and particle physics standard model), cosmology, strong field physics, condensed matter physics, nonlinear systems etc.

On the other hand, using the research in this paper, we naturally have the least coupling action between gravitational curvature tensor and gauge field strength tensor in current particle physics standard model

[TABLE]

[TABLE]

Eq.(42) gives new interaction vertexes, new Feynman figures between gravitational field and gauge fields with U(1), SU(2), SU(3), SU(5), SO(10) etc symmetries, respectively, which are new physics and cannot given in the past [33].

When considering high order fundamental interactions and non-linear dynamical systems, Eq.(42) can be generally written as

[TABLE]

Further using Eq.(40), we have

[TABLE]

[TABLE]

Further considering Weyl tensor $W^{\mu\nu\alpha\beta}$ and $\overset{\char 126\relax}{F}^{\alpha\beta}=\frac{1}{2}\varepsilon^{\alpha\beta\mu\nu}F_{\mu\nu}$, Eq.(43) can be generalized as

[TABLE]

In Eq.(45), when taking special expressions $\alpha f(R)=\frac{1}{2M_{PI}^{2}}R$, $F_{1}(F_{\mu\nu}F^{\mu\nu})=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{\alpha_{1}}{4M_{PI}^{4}}(F_{\mu\nu}F^{\mu\nu})^{2}$, $F_{2}(F_{\alpha\beta}\overset{\char 126\relax}{F}^{\alpha\beta})=\frac{\alpha_{2}}{4M_{PI}^{4}}(F_{\alpha\beta}\overset{\char 126\relax}{F}^{\alpha\beta})^{2}$, $F_{3}(F_{\mu\nu}F_{\alpha\beta}W^{\mu\nu\alpha\beta})=\frac{\alpha_{3}}{2M_{PI}^{2}}F_{\mu\nu}F_{\alpha\beta}W^{\mu\nu\alpha\beta}$, we deduce the key formula (2) in the important Ref.[20]. Thus our research is consistent with current research.

All the deduced Lagrangians in this paper in current particle physics standard model are invariant in all general coordinate transformations (for tangent bundle) and gauge transformations (for principal bundle), which just means that the physics laws are not dependent on people’s observations.

The general Lagrangians of both high order fundamental interactions and different nonlinear physical interactions in current particle physics standard model cannot be given up to now [33]. Therefore, this paper, for the first time, gives all these Lagrangians, thus we may give these expression’s new physics, e.g., relevant to Eqs. (37), (38) and (40-44).

Einstein thought, god would not care about only inertial system, and god should specially care about the invariable non-inertial system under general coordinate transformation, because only using the general non-inertial system people can identify the most general laws of physics, and can deduce that the general laws of physics do not depend on people’s observation, god should be more capable, so Einstein believes that the general theory of relativity is correct.

Similarly, the investigations in this paper are the same. Our investigations also satisfy the invariances of general coordinate transformation and gauge transformation, and also lead to the fact that physical laws do not depend on human observation. God should be more capable and would not give up such a good plan not to choose.

Especially so far, no experiments prove the existence of a higher dimensional space [33], and this studies of this paper do not need the extra higher dimensional spaces, we realize the unification of the fundamental interaction laws of physics, this theory is the “from down to up” beautiful phenomenological theory of the very economical and very realistic symmetries, so the research of this paper is of important theoretical and practical values, and many new physical predictions are presented.

For example, this paper discovers the new Lagrangians in the equations (37), (38) and (40-44), their corresponding Euler-Lagrange equations by Eq.(34) and the new physical interaction law expressed by the equations and so on.

Because of length limitation of the paper, the other Lagrangians in physics and the corresponding new physics may be analogously derived, their detailed applications can be given easily by means of the research of this paper and which will be written in our following papers.

This principle not only is not opposed to the usual restrictions of causality in a quantum field theory, but also the principle is more general, and can deduce the usual restrictions of causality in a quantum field theory. Because the usual restrictions of causality in a quantum field theory are mainly coming from the restrictions of Lorentz transformations, Lorentz transformations consist of some equations, any equation must satisfy quantitative causal principle, namely, some changes (cause) of some quantities in any equation must result in the relative changes (result) of the other quantities in the equation so that the equation’s right side keeps no-loss-no-gain, i.e., zero, namely, the equation satisfies the quantitative causal principle, which is very similar to Eq.(2.4)’s casual discussion (below Eq.(2.4) in Ref.[34].

Causality is one of the most fundamental and essential notions in physics [35]. Causal efficacy cannot propagate faster than light, which comes from the constraints of Lorentz transformations of special relativity. Thus, in space-like spacetime, any two physical operators in a general quantum field are commutative, because they cannot have any relation by Lorentz transformations.

For symmetric principle [41], people currently know that it is the fundamental principle of physics, which may deduce various symmetric groups, four fundamental interactions of physics and so on [9, 14, 29, 36]. Consequently, symmetry is playing the key role in physics.

Actually, Eq.(1) is just a special situation of symmetric principle, namely, the system is invariant under some operations, i.e., symmetric transformations that keep quantitative causal relation to be invariant. Consequently, QCP is a special expression of symmetric principle, which means that symmetric principle is more general and includes QCP as a subset. Thus this paper deduces QCP from symmetric principle.

But in the past, QCP and symmetric principle are firmly viewed as two independent fundamental principles, this paper gives the unification of QCP and symmetric principle, which will hugely affect the global physics and the corresponding developments relevant to QCP and symmetric principle in both physics and science, because causal principle and symmetric principle both are the very old, classical, famous and fundamental principles, the two principles have been hugely affecting the developments of different branches of physics and science for very long time.

5

Summary and Conclusions

This paper discovers and naturally gives geometric unification theory of the grand unification and gravitational interactions, and shows unification description theory of the grand unifying and gravitational interactions and Noether theorem by utilizing QCP, i.e., Eq.(1). The invariant quantities covering all the parts of the Lagrangians of the four fundamental physics interactions are deduced by QCP, all these invariant quantities satisfy general gauge invariance of the general gauge fields interacting with Fermion and/or boson fields. These different invariant quantities are constructed ( in vector bundle or principal bundle, whose base manifolds are a 4-dimensional physical general curved spacetime manifold M4). Thus, using these invariant quantities, all the Lagrangians of the four physical fundamental interactions are given and satisfy the general gauge invariant principle of general gauge fields interacting with Fermion and/or boson fields, and geometry and physics meanings of gauge invariant property of the different physical systems are directly, whole and seriously revealed.

Actually it is in terms of QCP that their variational principles and corresponding Noether theorem in field theory are derived. Therefore, the very hard difficulty ( that the four fundamental physics interactions and variational principle, further Noether theorem, previously regarded as independent fundamental laws, were not able to be unifiedly proved and investigated in the past [36]) is naturally overcome in this paper. Namely, the geometric unification theory and origins of the four fundamental physics interactions and the Noether theorems are shown by QCP, furthermore, their two-order general Euler-Lagrange equations and corresponding Noether conservation currents are obtained.

In fact, general physics process [4, 5, 8] and different physics processes [37, 38, 39, 40] all satisfy QCP with no-loss-no-gain character, the above investigations satisfy QCP, and are consistent. Especially, using the theory of this paper, a lot of research works about different branches of physics etc can be anew done and expressed simpler with clear quantitative causal physical meanings, e.g., people can deduce a lot of different parts of different Lagrangians, which will naturally give Lagrangians of different physics systems according to QCP and different symmetries etc, and then using Eqs.(34-36) people can give all corresponding concrete physics laws, i.e., find new Euler-Lagrange equations and conservation laws, and discover new concrete physics laws and so on. Thus the theory of this paper is useful and will be broad utilized and cited in different branches of physics and so on due to the key important properties of the four fundamental interactions for different physics systems. Therefore, all articles and (text)books relevant to the four fundamental physics interactions and Noether theorem would be supplemented with the new conclusions, and this paper will be cited by using the geometric unification theory in different physical systems, e.g., condensed physics, atomic physics, molecular physics, quantum optics, nuclear physics, particle physics, quantum field theories and so on.

Because the standard model of strong, weak and electromagnetic interactions can be included in a unification interaction model of a big gauge group, thus people get the grand unified theory, and because the gravitational interaction is related to the structures of spacetime, and the spacetime is the stage performing the laws of strong, weak and electromagnetic interactions, gravitational interaction cannot be included into a unification interaction model of a larger gauge group. This is because the grand unification theory of gauge theories of strong, weak and electromagnetic interactions is based on principal bundle theory in differential geometry, and gravitational theory is based on the tangent vector bundle theory in differential geometry [28], so people cannot unify the these four basic interactions in principal bundle theory in differential geometry theory, so for their unification, people have met great difficulties.

In order to overcome these difficulties, people have to increase the dimensional number of spacetime etc to solve these problems, the most promising success theory is superstring theory, but in return to low four-dimensional case etc, superstring theory encounters the big difficulties [21], up to now, there is no way that, in the case of four dimensions, strong, weak, electromagnetic and gravitational interactions are unified as one unification description theory.

This paper, for the first time, discovers and gives geometric unification theory of strong, weak, electromagnetic and gravitational interactions by investigating the general fiber bundle theory ( in differential geometry theory ) of unifiedly describing both principal bundle theory and tangent bundle theory and by studying important laws in physics, e.g., relations between symmetry and QCP etc. Consequently, the research of this paper is based on the rigorous scientific bases of mathematics and physics, therefore, people can essentially and really understand the four basic interaction laws of physics as well as the relations between them in terms of the research in this paper, for their further development builds up the scientific solid foundation.

This paper also discovers the new Lagrangians in Eqs.(37), (38) and (40-44), their corresponding Euler-Lagrange equations by Eq.(34) and the new physical interaction law expressed by the equations and so on, i.e., gives the relevant new physics.

This paper deduces QCP from symmetric principle, and gives the unification of QCP and symmetric principle, which would hugely affect the physics and the corresponding developments in both physics and science, because both causal principle and symmetric principle are the very old and fundamental principles, and the two principles have been hugely affecting the developments of different branches of physics and science for very long time. Thus, this paper opens a door to both study and give new developments of geometric unification theory of physics laws, and using the new geometric unification theory, a lot of research works about different branches of physics can be anew done and expressed simpler with different symmetric characters.

Acknowledgments

Authors are grateful to Prof. Z. P. Li for useful discussion and comment.

The work is supported by NSF through grants PHY-0805948, DOE USA through grant DE-SC0007884, and National Natural Science Foundation of China ( No. 11875081 ).

Appendix A

For any natural coordinate $x^{\mu}$ and local orthogonal coordinate $x^{a}$ in any 4-dimensional general spacetime, any tangent vector along any curve line $t$ can be formally written as $X=\frac{d}{dt}=\frac{dx^{\mu}}{dt}\frac{\partial}{\partial x^{\mu}}=\frac{dx^{a}}{dt}\frac{\partial}{\partial x^{a}},$ then the inner product of the tangent vector equates

[TABLE]

where we may define the inner product $<\partial_{\mu},\partial_{\nu}>=<e_{\mu},e_{\nu}>$ $=g_{\mu\nu}$ ( i.e., natural covariant metric ) of natural tangent basic vectors and the inner product $<\partial_{a},\partial_{b}>=<e_{a},e_{b}>$ $=\eta_{ab}$ ($\eta_{ab}=-1,$ as $a=b=0;$ $\eta_{ab}=1,$ as $a=b=1,2,3;$ $\eta_{ab}=0,$ as $a\neq b,$ i.e., orthonormal covariant metric ) of orthonormal tangent basic vectors, so we can deduce the infinitesimal line element square

[TABLE]

Similar to $<\partial_{\mu},\partial_{\nu}>=g_{\mu\nu}$ and $<\partial_{a},\partial_{b}>$ $=\eta_{ab},$ we define the inner product of natural (or called intrinsical ) cotangent basic vectors as $<dx^{\mu},dx^{\nu}>=<e^{\mu},e^{\nu}>$ $=g^{\mu\nu}$ ( i.e., natural contravariant metric ) and the inner product of orthonormal cotangent basic vectors as $<dx^{a},dx^{b}>=<e^{a},e^{b}>$ $=\eta^{ab}$ (($\eta^{ab}=-1,$ as $a=b=0;$ $\eta^{ab}=1,$ as $a=b=1,2,3;$ $\eta^{ab}=0,$ as $a\neq b,$ i.e., orthonormal contravariant metric ), thus, we prove

[TABLE]

[TABLE]

Furthermore, using Eq.(A2) we deduce $ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}=\eta_{ab}\frac{\partial x^{a}}{\partial x^{\mu}}\frac{\partial x^{b}}{\partial x^{\nu}}dx^{\mu}dx^{\nu}=\eta_{ab}e_{\mu}^{a}e_{\nu}^{b}dx^{\mu}dx^{\nu},$ i.e., we prove

[TABLE]

where $\frac{\partial x^{a}}{\partial x^{\mu}}=e_{\mu}^{a}$ is vierbein. On the other hand, we may directly prove Eq.(A4) by using $g_{\mu\nu}=<\frac{\partial}{\partial x^{\mu}},\frac{\partial}{\partial x^{\nu}}>=\frac{\partial x^{a}}{\partial x^{\mu}}\frac{\partial x^{b}}{\partial x^{\nu}}<\frac{\partial}{\partial x^{a}},\frac{\partial}{\partial x^{b}}>=\eta_{ab}e_{\mu}^{a}e_{\nu}^{b},$ and we may have $g^{\mu\nu}=<dx^{\mu},dx^{\nu}>=\frac{dx^{\mu}}{dx^{c}}\frac{dx^{\nu}}{dx^{d}}<dx^{c},dx^{d}>=e_{c}^{\mu}e_{d}^{\nu}\eta^{cd}.$ Thus this method is very convenient.

We still need to prove the consistence of indexes’ raising and lowing by using covariant and contravariant metrics as follows

[TABLE]

[TABLE]

Analogously, we may prove $A_{\mu}=g_{\mu\nu}A^{\nu}=<\frac{\partial}{\partial x^{\mu}},\frac{\partial}{\partial x^{\nu}}>A^{\nu}=e_{\mu}^{a}A_{a}.$ Thus, these studies are consistent.

Appendix B

Similar to deducing the important invariant $\omega_{\mu}\omega^{\mu}$ of inner product of two 1-forms, therefore, we deduce an important invariant of inner product of two 2-form $\Omega$ under coordinate transformation, as follows

[TABLE]

[TABLE]

substituting Eq.(5) into the deduced inner product expression (B1) and taking trace, then we have

[TABLE]

which satisfies gauge invariant principle [30] of general gauge field’s invariant, further using the invariant volume elements $(-g_{U})^{\frac{1}{2}}d\tau_{U}=(-g_{V})^{\frac{1}{2}}d\tau_{V}$ [28], we finally achieve the invariant action Eq.(10) of general gauge fields in the global curved spacetime.