We first define the connection GGE as:

${\omega }_V\left(Y\right)=\mathcal{A}{d}_{g_{UV}{\left(x\right)}^{-1}}{\omega }_U\left(Y\right)+L_{g_{UV}\left(x\right)\ast }^{-1}{g}_{UV\ast }\left(Y\right),\forall x\in U\cap V,Y\in T_{x}M$ (1)

where ( ${\omega }_U$, ${\omega }_V$) represents the Lie algebra g-valued 1-form on the underlying base manifold region ( $U$, $V$), representing a local connection; ( $g_{UV}:U\cap V\to G$) is a transformation function, $G$ is a structure group (e.g., $SO\left(1,3\right)\times U\left(1\right)\times SU\left(2\right)\times SU\left(3\right)$); $\mathcal{A}{d}_{g_{UV}^{-1}}{\omega }_U=g_{UV}^{-1}{\omega }_U{g}_{UV}$ is an adjoint action, a transformation connection; $L_{g_{UV}\left(x\right)\ast }^{-1}{g}_{UV\ast }\left(Y\right)$ is the inverse mapping of the left translation, involving the derivative of ( $g_{UV}$), here $\ast$ means forward mapping [23] . In fact, from reference [23] we can get three clear equivalent definitions of the principal bundle connection, and on this basis we can prove very generally that the connection on the underlying manifold must satisfy relation (1). Here GGE represents the generalized gauge transformation equation, which is a conceptual generalization of the original formula by the author, indicating that the equation of this gauge transformation can be transformed across fundamental interactions. One of the most important manifestations is that the conversion function $g_{UV}\left(x\right)$ can belong to the product of two subgroups, for example: if it is electromagnetic-gravitational conversion, $g_{UV}\in SO\left(1,3\right)\otimes U\left(1\right)$; if it is strong-gravitational conversion, $g_{UV}\in SO\left(1,3\right)\otimes SU\left(3\right)$, and so on. For the derivation and discussion in this regard, see the author’s published references [11] - [22] . On the other hand, from the perspective of quantum field theory (such as Yang-Mills field), the literature [23] also gives a very general proof that if the Lagrangian is to remain unchanged under gauge transformation, the gauge potential must satisfy relationship GGE. In this sense, since the transfer function in the GGE is the gauge transformation, it can be considered that the GGE is the transformation law that the connection (gauge potential) or curvature (gauge field strength) must satisfy. For example, this is the statement in Appendix 3 of reference [24] .

Taking these background conditions into consideration, the author again constructs the cross-fundamental interaction gauge potential or gauge field strength transformation equation GGE in a more general way from the perspective of cross-fundamental interaction gauge theory in Section 8 and Appendix B, that is, GGE is also valid for weak force, strong force, electromagnetic and gravitational gauge fields.

Therefore, this article discusses the gauge field theory from the perspective of the GGE, not from the perspective of the gauge invariance of the Lagrangian, which may be a theoretical optimization approach, because the Lagrangian is not so easy to construct for general physical systems, especially for physical processes across fundamental interactions. In addition, even if the constructed Lagrangian is gauge invariant, it is not necessarily easy to see which elementary particle is transformed into which one. Even if there is certain symmetry, such as the local gauge invariance of the free complex scalar field, the conclusion that the rest mass of the photon is 0 can be “obtained”, but it is still difficult to see which elementary particle is transformed into which one. However, the advantage of GGE across fundamental interactions is obvious, for example: ${\omega }_U\to {\omega }_V$, for $U\cap V\ne \varnothing$. Where ${\omega }_U$ represents the gauge potential of one fundamental physical field (on the bottom manifold) and is converted to ${\omega }_V$ represents the gauge potential of another fundamental physical field (on the bottom manifold). However, the gauge potential $\tilde{\omega }$ of the principal bundle (representing the gauge potential of the cosmic gauge field) is invariant in this gauge transformation, i.e., ${\sigma }_U^{*}\tilde{\omega }={\omega }_U$; ${\sigma }_V^{*}\tilde{\omega }={\omega }_V$, where ${\sigma }_U^{*}$ and ${\sigma }_V^{*}$ represents the pullback mapping of the principal bundle section on the bottom manifold $U$ and $V$, respectively. This is the meaning of gauge symmetry and the meaning of the unification of fundamental interactions, i.e., $\tilde{\omega }$ is the “subject quantity”, ${\omega }_U$ and ${\omega }_V$ are the projections of $\tilde{\omega }$ on the bottom manifold through ${\sigma }_U^{*}$ or ${\sigma }_V^{*}$, respectively, and ${\omega }_U$ and ${\omega }_V$ are converted into each other through the GGE. The meaning of gauge symmetry is very clear. We will discuss this issue again in Section 8.

Therefore, the physical meaning of the above formula: ( ${\omega }_U$, ${\omega }_V$) represents the gauge potential of different regions (such as electromagnetic potential $A_{\mu }^{EM}$, gravitational gauge potential $A_{\mu }^{ab}$).

Formula (1) describes the generalized gauge transformation of the gauge potential in the region ( $U\cap V$), which is applicable to cross-basic interactions (such as electromagnetic to gravitational) [11] - [15] . The above formula (1) can be simplified in matrix expression, that is, when $G$ is a matrix Lie group, we can get

${\omega }_V=g_{UV}^{-1}{\omega }_U{g}_{UV}+g_{UV}^{-1}d{g}_{UV}$ (2)

Here, the first term on the left side of formula (2) is the similarity transformation of the connection; the second term is the exterior differential of the transformation function, reflecting the change of the gauge selection between regions.

We define the curvature by the Cartan second structural equation as:

$\tilde{\Omega }=d\tilde{\omega }+\frac{1}{2}\left[\tilde{\omega },\tilde{\omega }\right]$ (3)

where $\tilde{\Omega }$ and $\tilde{\omega }$ are the principal bundle curvature and principal bundle connection respectively, projected to the ( $U$, $V$) region as ( ${\Omega }_U$, ${\Omega }_V$), then transformed to the bottom manifold as:

${\Omega }_V=\mathcal{A}{d}_{g_{UV}^{-1}}{\Omega }_U=g_{UV}^{-1}{\Omega }_U{g}_{UV}$ (4)

or written as the transformation of the gauge tensor on the bottom manifold as:

${F^{\prime }}_{\mu \nu }=g_{UV}^{-1}{F}_{\mu \nu }{g}_{UV}$ (5)

where $F_{\mu \nu }$ is the gauge field strength in region $U$; ${F^{\prime }}_{\mu \nu }$ is the gauge field strength in region $V$, $U\cap V\ne \varnothing$. The physical meaning of formula (5) is ( ${\Omega }_U$, ${\Omega }_V$) corresponds to the gauge field strength (such as the electromagnetic field $F_{\mu \nu }^{EM}$, the gravitational field strength $F_{\mu \nu }^{ab}$); GGE ensures the consistency of the transformation of curvature under different gauges and supports the unification across interactions [11] [18] [19] [22] .

We construct the structure group as:

$G=SO\left(1,3\right)\times U\left(1\right)\times SU\left(2\right)\times SU\left(3\right)$

Here $SO\left(1,3\right)$ can represent the operating rules of gravity in region $V$; $U\left(1\right)$ expresses the operating rules of electromagnetic force in region $U$; and $SU\left(2\right)$ and $SU\left(3\right)$ express the operating rules of weak and strong interactions in regions $W_2$ and $W_1$ respectively. We can also call these group rules the gauge group representations of the corresponding forces. Obviously, they are related to the scale of the region. In this sense, the so-called quantization is nothing more than a gauge group representation naturally selected by the principal bundle in a sufficiently small regional scale, such as in $W_2$ and $W_1$. However, there are generalized gauge transformations across fundamental interactions. For example, if the generalized gauge transformation $g_{UV}\in SO\left(1,3\right)\times U\left(1\right)$, the gravitational-electromagnetic transformation can be described. Therefore, the four fundamental interactions can be converted into each other through generalized gauge transformations in the intersection region [11] [12] . Of course, choosing $GL(n)$ (general linear group, the group of $n\times n$ invertible matrices over the real or complex fields) as the structure group, including $SO(1,3),U(1),SU(2),SU(3)$ as subgroups, is also mathematical physics possible, because $GL(n)$ is the largest linear transformation group that can cover all possible linear symmetries. However, because in some sense, the basic models of weak and strong interactions with electromagnetical interaction have already been unified [5] [25] [26] , and the key to the unification of the four basic interactions is now the unification of electromagnetism and gravity [4] [8] [27] - [29] , this article will focus on the unified theory of gravity-electromagnetism under generalized gauge transformation, such as $G=SO(1,3)\times U(1)$.

The principal bundle of the gravitational gauge field framework we express is: $P\left(M,SO\left(1,3\right)\right)$, the underlying manifold $M$ is 4-spacetime with Riemann metric $g_{\mu \nu }$; the gauge potential is:

$A_{\mu }=A_{\mu }^{ab}{J}_{ab}$ (7)

Here $A_{\mu }^{ab}$ is defined as the gauge potential component, which is related to the frame $e_{\mu }^a$ or the metric perturbation $h_{\mu \nu }$; $J_{ab}$ is the $SO\left(1,3\right)$ Lie algebra generator, satisfying:

$\left[J_{ab},J_{cd}\right]={\eta }_{bc}{J}_{ad}-{\eta }_{ac}{J}_{bd}-{\eta }_{bd}{J}_{ac}+{\eta }_{ad}{J}_{bc}$ (8)

where ${\eta }_{ab}=\text{diag}\left(-1,1,1,1\right)$, then the corresponding connection expression is:

$\omega =A_{\mu }d{x}^{\mu }=A_{\mu }^{ab}{J}_{ab}d{x}^{\mu }\Rightarrow A_{\mu }=\frac{1}{2}{A}_{\mu }^{ab}{J}_{ab}={\omega }_{\mu }$ (9)

here $\omega$ is a Lie algebra so(1,3)-valued 1-form. Corresponding to the connection GGE, the above $A_{\mu }^{ab}{J}_{ab}$ corresponds to ${\omega }_U$ in formulas (1) and (2), which represents the local gauge potential in region $U$ (gravitational region $V$). Thus, ${\omega }_U\left(Y\right)$ in formula (1) can be expressed on the bottom manifold as follows:

${\omega }_U\left(Y\right)=A_{\mu }^{ab}{J}_{ab}{Y}^{\mu },Y=Y^{\mu }{\partial }_{\mu }$ (10)

The GGE transformation can be expressed as follows:

${\omega }_V\left(Y\right)=A{d}_{g_{UV}^{-1}}{\omega }_U\left(Y\right)+L_{g_{UV}\left(x\right)\ast }^{-1}{g}_{UV\ast }\left(Y\right)$ (11)

The first term of Equation (11) above is:

$A{d}_{g_{UV}^{-1}}{\omega }_U\left(Y\right)=g_{UV}^{-1}\left(A_{\mu }^{ab}{J}_{ab}{Y}^{\mu }\right)g_{UV}$ (12)

If $g_{UV}\in SO\left(1,3\right)$, then $J_{ab}$ is transformed into:

$g_{UV}^{-1}{J}_{ab}{g}_{UV}={\left(g^{-1}\right)}_a^c{\left(g^{-1}\right)}_b^d{J}_{cd}$, $g\in SO\left(1,3\right)$ (13)

Here ${\left(g^{-1}\right)}_b^d$ is a scalar coefficient, $J_{cd}$ is a generator, and the index $d$ is bound to $J_{cd}$ and can be summed. Specifically, ${\left(g^{-1}\right)}_b^d$ is the $b$-th row and $d$-th column element of the $g^{-1}$ matrix; $g^{-1}$ is the matrix representation abbreviation of $g_{UV}^{-1}$.

If $g_{UV}$ is a rotation matrix, then

$g_{UV}^{-1}=g^{-1}=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& \cos \theta & \sin \theta & 0\\ 0& -\sin \theta & \cos \theta & 0\\ 0& 0& 0& 1\end{array}\right)$(14)

${\left(g^{-1}\right)}_a^c$ = matrix element, for example ${\left(g^{-1}\right)}_2^2=\cos \theta$; $J_{ab}$ is a $SO\left(1,3\right)$ generator, in matrix form: ${\left(J_{ab}\right)}_f^e={\eta }_{af}{\delta }_b^e-{\eta }_{bf}{\delta }_a^e$, where ${\delta }_a^e=1$, if $e=a$, ${\delta }_a^e=0$, if $e\ne a.$

Next, we calculate the matrix elements:

${\left(g{J}_{ab}{g}^{-1}\right)}_f^e=g_m^e\left({\eta }_{an}{\delta }_b^m-{\eta }_{bn}{\delta }_a^m\right){\left(g^{-1}\right)}_f^n=g_b^e{\eta }_{an}{\left(g^{-1}\right)}_f^n-g_a^e{\eta }_{bn}\text{}{\left(g^{-1}\right)}_f^n$(15)

See if we can achieve the goal:

${\left(g^{-1}\right)}_a^c{\left(g^{-1}\right)}_b^d\left({\eta }_{df}{\delta }_c^e-{\eta }_{cf}{\delta }_d^e\right)$(16)

In fact, if we use $SO\left(1,3\right)$ orthogonality: ${\eta }_{mn}{g}_e^m{g}_f^n={\eta }_{ef}$; and adjust the index, we can get:

$g^{-1}{J}_{ab}g={\left(g^{-1}\right)}_a^c{\left(g^{-1}\right)}_b^d{J}_{cd}$ (17)

The right side of the above formula (17) comes from: ${\left(g^{-1}\right)}_a^c{\left(g^{-1}\right)}_b^d$ is the tensor product of $g^{-1}$, reflecting the transformation of $J_{ab}$ index $a$, $b$ in the adjoint representation, and $J_{cd}$ is the transformed generator basis, and the index $c$, $d$ is reallocated by $g^{-1}$. Therefore, from (12), $A_{\mu }^{ab}$ can be obtained:

$A_{\mu }^{ab}\to {A^{\prime }}_{\mu }^{ab}=g_{UV}^{-1}{A}_{\mu }^{ab}{g}_{UV}={\left(g^{-1}\right)}_a^c{\left(g^{-1}\right)}_b^d{A}_{\mu }^{cd}$ (18)

Note that it will be proved later (see formulas (26) - (28)):

${A^{\prime }}_{\mu }^{ab}=A_{\mu }^{cd}{\left(g^{-1}\right)}_c^a{\left(g^{-1}\right)}_d^b={A^{\prime }}_{\mu }^{ab}=A_{\mu }^{cd}{\left(g^{-1}\right)}_a^c{\left(g^{-1}\right)}_b^d$ (19)

The second term of Equation (11) above by definition is:

$L_{g_{UV}\left(x\right)\ast }^{-1}{g}_{UV\ast }\left(Y\right)=g_{UV}^{-1}\left({\partial }_{\mu }{g}_{UV}\right)Y^{\mu }$ (20)

If $g_{UV}$ is a matrix, then there is

${\omega }_V=g_{UV}^{-1}{A}_{\mu }{g}_{UV}d{x}^{\mu }+g_{UV}^{-1}{\partial }_{\mu }{g}_{UV}d{x}^{\mu }$ (21)

which is consistent with the connection GGE (2).

The following is the core consistency verification of the gauge potential transformation in the generalized gauge theory (GGE). We need to confirm whether the derived gauge potential transformation formula (19) ${A^{\prime }}_{\mu }^{ab}=A_{\mu }^{cd}{\left(g^{-1}\right)}_c^a{\left(g^{-1}\right)}_d^b$, which can directly derive ${\omega }_V=g_{UV}^{-1}{A}_{\mu }{g}_{UV}d{x}^{\mu }+g_{UV}^{-1}{\partial }_{\mu }{g}_{UV}d{x}^{\mu }$ and verify its consistency with the GGE transformation ${\omega }_V=g_{UV}^{-1}{\omega }_U{g}_{UV}+g_{UV}^{-1}d{g}_{UV}$. This step is crucial because it connects the transformation form of the gravitational gauge field based on $SO\left(1,3\right)$ and the quantum gauge field, such as electromagnetic gauge field [11] [15] [19] [24] [30] [31] , reflecting the consistency of the gauge fields theory.

1) Understand the matrix form of $A_{\mu }$

In order to verify the target formula, we first need to clarify the meaning of $A_{\mu }$ in ${\omega }_V=g_{UV}^{-1}{A}_{\mu }{g}_{UV}d{x}^{\mu }+g_{UV}^{-1}{\partial }_{\mu }{g}_{UV}d{x}^{\mu }$, and then give the definition of gauge potential, that is, in region $U$, ${\omega }_U=A_{\mu }^{ab}{J}_{ab}d{x}^{\mu }$, $A_{\mu }^{ab}$ is the component (tensor) of gauge potential, $J_{ab}$ is the generator (matrix) of $SO\left(1,3\right)$ Lie algebra, so $A_{\mu }^{ab}{J}_{ab}$ can be regarded as the Lie algebra value matrix:

$A_{\mu }=A_{\mu }^{ab}{J}_{ab}$ (22)

where $A_{\mu }$ is a matrix, representing the connection component on the Lie algebra; $A_{\mu }^{ab}$ is the component of gauge potential, with indices $a$ and $b$ corresponding to the basis $J_{ab}$ of $SO\left(1,3\right)$ Lie algebra. Mathematically, $A_{\mu }^{ab}$ is a tensor (double index, antisymmetric) whose values are scalars (or numerical coefficients) and does not carry a matrix structure. For fixed $\mu$, the $a$, $b$ are numbers (e.g., real or complex) and do not participate in matrix multiplication. $J_{ab}$ is a Lie algebra generator of the $SO\left(1,3\right)$, represented as a matrix (in some representations, such as the adjoint representation or the fundamental representation). $J_{ab}$ satisfies the Lie algebra commutation relation: $\left[J_{ab},J_{cd}\right]={\eta }_{bc}{J}_{ad}-{\eta }_{ac}{J}_{bd}-{\eta }_{bd}{J}_{ac}+{\eta }_{ad}{J}_{bc}$; $J_{ab}$ is a matrix and is therefore subject to matrix multiplication by $g_{UV}^{-1}$. And $A_{\mu }=A_{\mu }^{ab}{J}_{ab}$ is a linear combination of Lie algebra elements: $A_{\mu }={\displaystyle {\sum }_{a,b}{A}_{\mu }^{ab}{J}_{ab}}$; (Einstein’s summation convention, $a,b=0,1,2,3$, but $J_{ab}=-J_{ba}$, only sums over independent combinations). Here, $A_{\mu }^{ab}$ is the coefficient, $J_{ab}$ is the basis, and $A_{\mu }$ is a matrix (Lie algebra value). Because the target formula $g_{UV}^{-1}{A}_{\mu }{g}_{UV}$ implies that $A_{\mu }$ is in matrix form, so it is: $A_{\mu }=A_{\mu }^{ab}{J}_{ab}$, where $g_{UV}=g\in SO\left(1,3\right)$, is a group element (here in matrix form), acting on the Lie algebra space.

2) Verify the target formula

We start from the contact GGE transformation formula to derive ${\omega }_V$ and check whether we get the target form. From GGE transformation (2), we get the first term:

${\omega }_U=A_{\mu }^{ab}{J}_{ab}d{x}^{\mu }$ (23)

Substituting, we get:

$g_{UV}^{-1}{\omega }_U{g}_{UV}=g_{UV}^{-1}\left(A_{\mu }^{ab}{J}_{ab}d{x}^{\mu }\right)g_{UV}$ (24)

Since $d{x}^{\mu }$ is a scalar, we have:

$g_{UV}^{-1}{\omega }_U{g}_{UV}=g_{UV}^{-1}\left(A_{\mu }^{ab}{J}_{ab}{g}_{UV}\right)d{x}^{\mu }$ (25)

Using the known formula (17):

$g_{UV}^{-1}{J}_{ab}{g}_{UV}={\left(g^{-1}\right)}_a^c{\left(g^{-1}\right)}_b^d{J}_{cd}$

Then we can get:

$g_{UV}^{-1}{A}_{\mu }^{ab}{J}_{ab}{g}_{UV}=A_{\mu }^{ab}\left(g_{UV}^{-1}{J}_{ab}{g}_{UV}\right)=A_{\mu }^{ab}{\left(R^{-1}\right)}_a^c{\left(R^{-1}\right)}_b^d{J}_{cd}$ (26)

Rename indicator $c,d\to a,b$:

$A_{\mu }^{ab}{\left(R^{-1}\right)}_a^c{\left(R^{-1}\right)}_b^d{J}_{cd}=A_{\mu }^{cd}{\left(R^{-1}\right)}_c^a{\left(R^{-1}\right)}_d^b{J}_{ab}$ (27)

defining:

${A^{\prime }}_{\mu }^{ab}=A_{\mu }^{cd}{\left(R^{-1}\right)}_c^a{\left(R^{-1}\right)}_d^b$ (28)

hence we obtain:

$g_{UV}^{-1}{A}_{\mu }^{ab}{J}_{ab}{g}_{UV}={A^{\prime }}_{\mu }^{ab}{J}_{ab}$ (29)

So the first term becomes:

$g_{UV}^{-1}{\omega }_U{g}_{UV}=g_{UV}^{-1}\left(A_{\mu }^{ab}{J}_{ab}d{x}^{\mu }\right)g_{UV}=\left({A^{\prime }}_{\mu }^{ab}{J}_{ab}\right)d{x}^{\mu }$ (30)

where note that the matrix form $A_{\mu }=A_{\mu }^{ab}{J}_{ab}$ then having ${\omega }_U=A_{\mu }d{x}^{\mu }$; so the first term is:

$g_{UV}^{-1}{\omega }_U{g}_{UV}=g_{UV}^{-1}\left(A_{\mu }d{x}^{\mu }\right)g_{UV}=\left(g_{UV}^{-1}{A}_{\mu }{g}_{UV}\right)d{x}^{\mu }$

here we have verified: $g_{UV}^{-1}{A}_{\mu }{g}_{UV}=g_{UV}^{-1}\left(A_{\mu }^{ab}{J}_{ab}\right)g_{UV}=A_{\mu }^{ab}\left(g_{UV}^{-1}{J}_{ab}{g}_{UV}\right)$.

Why can $A_{\mu }^{ab}$ be extracted? Because $A_{\mu }^{ab}$ is a scalar coefficient and does not participate in matrix operations, and again consider $a,b\to c,d$, the above formula is

$g_{UV}^{-1}{A}_{\mu }{g}_{UV}=A_{\mu }^{ab}{\left(R^{-1}\right)}_a^c{\left(R^{-1}\right)}_b^d{J}_{cd}=A_{\mu }^{cd}{\left(R^{-1}\right)}_c^a{\left(R^{-1}\right)}_d^b{J}_{ab}={A^{\prime }}_{\mu }^{ab}{J}_{ab}$ (31)

Therefore we arrive at

$g_{UV}^{-1}{A}_{\mu }{g}_{UV}={A^{\prime }}_{\mu }^{ab}{J}_{ab}$ (32)

The first term matches:

$g_{UV}^{-1}{A}_{\mu }{g}_{UV}d{x}^{\mu }={A^{\prime }}_{\mu }^{ab}{J}_{ab}d{x}^{\mu }$ (33)

The second item is:

$g_{UV}^{-1}d{g}_{UV}=g_{UV}^{-1}\left({\partial }_{\mu }{g}_{UV}\right)d{x}^{\mu }$ (34)

This term is in Lie algebra value 1-form: $g_{UV}^{-1}{\partial }_{\mu }{g}_{UV}\in Lie\left(SO\left(1,3\right)\right)$, which can be expanded as:

$g_{UV}^{-1}{\partial }_{\mu }{g}_{UV}=B_{\mu }^{ab}{J}_{ab}$ (35)

where $B_{\mu }^{ab}$ is the coefficient that depend on the specific form of $g_{UV}$.

Combining (33) and (35) we get:

$\begin{array}{c}{\omega }_V=\left(g_{UV}^{-1}{A}_{\mu }{g}_{UV}\right)d{x}^{\mu }+g_{UV}^{-1}\left({\partial }_{\mu }{g}_{UV}\right)d{x}^{\mu }\\ =\left({A^{\prime }}_{\mu }^{ab}{J}_{ab}+B_{\mu }^{ab}{J}_{ab}\right)d{x}^{\mu }\\ =\left({A^{\prime }}_{\mu }^{ab}+B_{\mu }^{ab}\right)J_{ab}d{x}^{\mu }\end{array}$ (36)

and

$A_{\mu }^{\text{'}ab}=A_{\mu }^{ab}{\left(R^{-1}\right)}_a^c{\left(R^{-1}\right)}_b^d+B_{\mu }^{ab}$ (37)

also can directly obtain:

${\omega }_V={A^{\prime }}_{\mu }^{ab}{J}_{ab}d{x}^{\mu }+g_{UV}^{-1}\left({\partial }_{\mu }{g}_{UV}\right)d{x}^{\mu }$ (38)

where ${A^{\prime }}_{\mu }^{ab}=A_{\mu }^{ab}{\left(R^{-1}\right)}_a^c{\left(R^{-1}\right)}_b^d$, and $g_{UV}^{-1}{\partial }_{\mu }{g}_{UV}=B_{\mu }^{ab}{J}_{ab}$, namely:

${\omega }_V=\left({A^{\prime }}_{\mu }^{ab}{J}_{ab}+B_{\mu }^{ab}{J}_{ab}\right)d{x}^{\mu }=\left({A^{\prime }}_{\mu }^{ab}+B_{\mu }^{ab}\right)J_{ab}d{x}^{\mu }$ (39)

This is consistent with the target formula (2):

${\omega }_V=g_{UV}^{-1}{A}_{\mu }{g}_{UV}d{x}^{\mu }+g_{UV}^{-1}\left({\partial }_{\mu }{g}_{UV}\right)d{x}^{\mu }$ (40)

Conclusion: Equation (40) accurately reflects the GGE transformation. ${A^{\prime }}_{\mu }^{ab}=A_{\mu }^{cd}{\left(R^{-1}\right)}_c^a{\left(R^{-1}\right)}_d^b$ is the coefficient transformation of the first term $g_{UV}^{-1}{A}_{\mu }{g}_{UV}$.

In Yang-Mills gauge theory or gravitational gauge theory, the transformation of the gauge potential always consists of two parts:

For $A_{\mu }=A_{\mu }^{ab}{J}_{ab}$, we have $g^{-1}{A}_{\mu }g={A^{\prime }}_{\mu }^{ab}$, ${A^{\prime }}_{\mu }^{ab}=A_{\mu }^{cd}{\left(R^{-1}\right)}_c^a{\left(R^{-1}\right)}_d^b$, and inhomogeneous term $g_{UV}^{-1}{\partial }_{\mu }{g}_{UV}=B_{\mu }^{ab}{J}_{ab}$, so the complete transformation is:

$A_{\mu }\to {A^{\prime }}_{\mu }=g_{UV}^{-1}{A}_{\mu }{g}_{UV}+g_{UV}^{-1}{\partial }_{\mu }{g}_{UV}$

which corresponding to

${A^{\prime }}_{\mu }^{ab}{J}_{ab}=\left(A_{\mu }^{cd}{\left(g^{-1}\right)}_c^a{\left(g^{-1}\right)}_d^b\right)J_{ab}+\left(B_{\mu }^{ab}{J}_{ab}\right)$

where ${A^{\prime }}_{\mu }^{ab}$ is the transformation result of the first term, while the second term retains its Lie algebraic form. So the connection ${\omega }_U=A_{\mu }d{x}^{\mu }=A_{\mu }^{ab}{J}_{ab}d{x}^{\mu }$ of the gravitational gauge field (based on $SO\left(1,3\right)$) follows the same gauge transformation rules as the quantum gauge field, i.e. the connection GGE, and the same transformation form as the quantum gauge field (based on general Lie groups, such as $SU\left(N\right)$, except that the Lie algebra is different:

$g_{UV}^{-1}{\omega }_U{g}_{UV}=A_{\mu }^{cd}{\left(g^{-1}\right)}_c^a{\left(g^{-1}\right)}_d^b$

The adjoint representation transformation corresponding to the quantum gauge field is still:

$A_{\mu }^a\to {\left(g^{-1}\right)}_b^a{A}_{\mu }^b$

The second term $g_{UV}^{-1}{\partial }_{\mu }{g}_{UV}$ in the connection GGE (40) above ensures the non-homogeneous nature of the gauge transformation, which is consistent with the quantum gauge field. From this we can conclude that: $A_{\mu }d{x}^{\mu }=A_{\mu }^{ab}{J}_{ab}d{x}^{\mu }={\omega }_U$, which is completely consistent with the connection GGE transformation as a gauge potential.

In fact, the transformation form of the gravitational gauge field is consistent with the quantum gauge field, only the Lie algebra and group ( $SO\left(1,3\right)$ vs. $SU\left(N\right)$) are different. So it can maintain

${\omega }_V=g_{UV}^{-1}{A}_{\mu }{g}_{UV}d{x}^{\mu }+g_{UV}^{-1}{\partial }_{\mu }{g}_{UV}d{x}^{\mu }$

The above formula can be directly derived from the GGE ${\omega }_V=g_{UV}^{-1}{\omega }_U{g}_{UV}+g_{UV}^{-1}d{g}_{UV}$, based on the idea that connection is gauge potential and gauge potential is connection [16] [23] , where $g_{UV}^{-1}{A}_{\mu }{g}_{UV}=A_{\mu }^{cd}{\left(g^{-1}\right)}_c^a{\left(g^{-1}\right)}_d^b$.

The gauge tensor (field strength) can be defined as:

$F_{\mu \nu }=F_{\mu \nu }^{ab}{J}_{ab}$ (41)

$F_{\mu \nu }^{ab}={\partial }_{\mu }{A}_{\nu }^{ab}-{\partial }_{\nu }{A}_{\mu }^{ab}+{\left[A_{\mu },A_{\nu }\right]}^{ab}$ (42)

and

${\left[A_{\mu },A_{\nu }\right]}^{ab}=A_{\mu }^{ac}{A}_{\nu c}^b-A_{\nu }^{ac}{A}_{\mu c}^b$ (43)

then we have the curvature Cartan second structural equation:

$F=dA+A\wedge A$ (44)

In the coordinate base:

$F=\frac{1}{2}{F}_{\mu \nu }^{ab}{J}_{ab}d{x}^{\mu }\wedge d{x}^{\nu }$ (45)

In this way, $F_{\mu \nu }^{ab}$ is usually the associated frame curvature or gravitational field strength, but it is not directly equal to the Ricci curvature $R_{\mu \nu }$ or the Weyl tensor $C_{\mu \nu \rho \sigma }$, but needs to be mapped to the metric through the frame $g_{\mu \nu }=e_{\mu }^a{e}_{\nu }^b{\eta }_{ab}$.

Therefore, the Cartan transform (46) corresponding to the curvature GGE [23] [24] is:

$\Omega =d\omega +\frac{1}{2}\left[\omega ,\omega \right]$ (46)

While the curvature component on the bottom manifold is:

${\Omega }_U=\frac{1}{2}{F}_{\mu \nu }^r{e}_{r}d{x}^{\mu }\wedge d{x}^{\nu }$ (47)

$F_{\mu \nu }^r={\partial }_{\mu }{A}_{\nu }^r-{\partial }_{\nu }{A}_{\mu }^r+k{C}_{st}^r{A}_{\mu }^s{A}_{\nu }^t$ (48)

where $C_{st}^r$ is the Lie algebra structure constant.

The curvature GGE (4) or (5) is:

${\Omega }_V=g_{UV}^{-1}{\Omega }_U{g}_{UV}$

where, the gravitational gauge field is introduced

${F^{\prime }}_{\mu \nu }^r={\left(g^{-1}\right)}_s^r{F}_{\mu \nu }^s,g\in G$ (49)

here the Lie algebra basis $J_{ab}$ corresponds to $e_r$, the structure constant: $\left[J_{ab},J_{cd}\right]=C_{ab,cd}^{ef}{J}_{ef}$ is equivalent to $C_{st}^r$; the transformation corresponding to $F_{\mu \nu }^{ab}$ is:

${F^{\prime }}_{\mu \nu }^{ab}={\left(g^{-1}\right)}_a^c{\left(g^{-1}\right)}_b^d{F}_{\mu \nu }^{cd}$

which is consistent with ${\Omega }_V=g_{UV}^{-1}{\Omega }_U{g}_{UV}$, and the detailed derivation can be found in Appendix A and B. Therefore, the gauge tensor $F_{\mu \nu }^{ab}$ has the same form as the curvature ${\Omega }_U$ on the manifold of the principal bundle, both of which are

Lie algebraic valued 2-forms, and $F_{\mu \nu }^{ab}{J}_{ab}$ and ${\Omega }_U=\frac{1}{2}{F}_{\mu \nu }^{ab}{J}_{ab}d{x}^{\mu }\wedge d{x}^{\nu }$ are also

Lie algebraic valued 2-forms, and their forms are consistent (the factor 1/2 is the exterior differential convention). Here, the definition of $F_{\mu \nu }^{ab}$ includes ${\partial }_{\mu }{A}_{\nu }^{ab}-{\partial }_{\nu }{A}_{\mu }^{ab}+{\left[A_{\mu },A_{\nu }\right]}^{ab}$, see formula (42) above, which is also consistent with GGE’s

$F_{\mu \nu }^r={\partial }_{\mu }{A}_{\nu }^r-{\partial }_{\nu }{A}_{\mu }^r+k{C}_{st}^r{A}_{\mu }^s{A}_{\nu }^t$

and its structural constant $C_{st}^r$ corresponds to $\left[J_{ab},J_{cd}\right]$.

Although the above $A_{\mu }^{ab}{J}_{ab}$ and $F_{\mu \nu }^{ab}{J}_{ab}$ directly correspond to ${\omega }_U$ and ${\Omega }_U$, and the transformation formula is compatible with the connection GGE and the curvature GGE, the above framework still needs to be extended for the following reasons:

We expand the structure group:

$G=SO\left(1,3\right)\times U\left(1\right)\times SU\left(2\right)\times SU\left(3\right)$ (50)

and define the gauge potential that unifies the four fundamental interactions:

$A_{\mu }=A_{\mu }^{EM}{T}_{EM}+A_{\mu }^W{T}_W+A_{\mu }^s{T}_s+A_{\mu }^{ab}{J}_{ab}$ (51)

Here $T_{EM}$, $T_W$, $T_s$ are the generators of U(1), SU(2), SU(3) respectively; $A_{\mu }^{ab}{J}_{ab}$ is the gravitational gauge potential, then the general connection (gauge potential) can be defined as:

${\omega }_U=A_{\mu }d{x}^{\mu }=\left(A_{\mu }^{EM}{T}_{EM}+A_{\mu }^W{T}_W+A_{\mu }^s{T}_s+A_{\mu }^{ab}{J}_{ab}\right)d{x}^{\mu }$ (52)

According to the principal bundle theory of generalized gauge transformations, the general connection (gauge potential) GGE holds:

${\omega }_V=g_{UV}^{-1}{\omega }_U{g}_{UV}+g_{UV}^{-1}d{g}_{UV}$ (53)

where ${\omega }_U\in$ any fundamental gauge potential (e.g. ${\omega }_U=A_{\mu }^W{T}_W$) in the region $U\cap V\ne \varnothing$, ${\omega }_V\in$ another fundamental gauge potential (e.g. ${\omega }_V=A_{\mu }^{EM}{T}_{EM}$); $g_{UV}\in$ the product of the corresponding two or four subpopulations (e.g. $g_{UV}=g_{U\left(1\right)}\times g_{SU\left(2\right)}$), or

$g_{UV}=g_{SO\left(1,3\right)}\times g_{U\left(1\right)}\times g_{SU\left(2\right)}\times g_{SU\left(3\right)}$ (54)

Although according to the generalized gauge transformation theory, we believe that the GGE of connection and curvature is generally valid for the mutual conversion between the four basic interactions at the intersection of the two, and the certain framework will be shown in section 8, but considering the unification of weak, strong and electromagnetic interactions, there is already a model that has laid the foundation. The problem of the unification of the four basic interactions in modern times is mainly the unification of electromagnetism and gravity, so this article still focuses on how the electromagnetic force is converted into gravity through the generalized gauge transformation. In fact, for the gravitational-electromagnetic transformation, $g_{UV}\in SO\left(1,3\right)\times U\left(1\right)$, there is

${\omega }_V=g_{UV}^{-1}\left(A_{\mu }^{EM}{T}_{EM}+A_{\mu }^{ab}{J}_{ab}\right)g_{UV}+g_{UV}^{-1}{\partial }_{\mu }{g}_{UV}d{x}^{\mu }$ (55)

We can verify that the connection GGE (2) holds, that is, according to the previous definition, if we set:

${\omega }_U=A_{\mu }^{ab}{J}_{ab}d{x}^{\mu },{\omega }_V={A^{\prime }}_{\mu }^{ab}{J}_{ab}d{x}^{\mu }$ (56)

then the connection GGE is transformed into (gravity is transformed into gravity):

${A^{\prime }}_{\mu }^{ab}{J}_{ab}=g_{UV}^{-1}\left(A_{\mu }^{cd}{J}_{cd}\right)g_{UV}+g_{UV}^{-1}{\partial }_{\mu }{g}_{UV}$ (57)

If $g_{UV}\in SO\left(1,3\right)$, then we can obtain:

${A^{\prime }}_{\mu }^{ab}={\left(g^{-1}\right)}_a^c{\left(g^{-1}\right)}_b^d{A}_{\mu }^{cd},{J^{\prime }}_{ab}=g_{UV}^{-1}{J}_{ab}{g}_{UV}$ (58)

For interactions across electromagnetic and gravitational ties:

${A^{\prime }}_{\mu }^{ab}{J}_{ab}=g_{UV}^{-1}\left(A_{\mu }^{EM}{T}_{EM}\right)g_{UV}+g_{UV}^{-1}{\partial }_{\mu }{g}_{UV}$ (59)

From the above analysis, we can find that it is necessary to define a mapping from $T_{EM}$ to $J_{ab}$, that is, if $J_{ab}=g_{UV}^{-1}{T}_{EM}{g}_{UV}$, $A_{\mu }^{ab}$ can indeed be embedded in the unified gauge potential, connection GGE (2).

Further adjustment formally unifies the curvature of the four basic interactions into:

$F_{\mu \nu }=F_{\mu \nu }^{EM}{T}_{EM}+F_{\mu \nu }^W{T}_W+F_{\mu \nu }^s{T}_s+F_{\mu \nu }^{ab}{J}_{ab}$ (60)

$F_{\mu \nu }^{ab}={\partial }_{\mu }{A}_{\nu }^{ab}-{\partial }_{\nu }{A}_{\mu }^{ab}+A_{\mu }^{ac}{A}_{\nu c}^b-A_{\nu }^{ac}{A}_{\mu c}^b$ (61)

Then the curvature GGE is transformed into:

${F^{\prime }}_{\mu \nu }^{ab}={\left(g^{-1}\right)}_a^c{\left(g^{-1}\right)}_b^d{F}_{\mu \nu }^{cd}$ (62)

or written as:

${F^{\prime }}_{\mu \nu }=g_{UV}^{-1}{F}_{\mu \nu }{g}_{UV}$(63)

For gravity-electromagnetism the formula (63) becomes:

${F^{\prime }}_{\mu \nu }^{ab}{J}_{ab}=g_{UV}^{-1}\left(F_{\mu \nu }^{EM}{T}_{EM}\right)g_{UV}$ (64)

In the physical mapping, it is also necessary to make $F_{\mu \nu }^{ab}$ associated with the metric, namely

$F_{\mu \nu }^{ab}~e_{\rho }^a{e}_{\sigma }^b{R}_{\sigma \mu \nu }^{\rho }$ (65)

where $R_{\sigma \mu \nu }^{\rho }$ is the Riemann curvature, i.e. the adjustment:

$F_{\mu \nu }^{ab}=e_{\rho }^a{e}_{\sigma }^b\left({\partial }_{\mu }{\Gamma }_{\sigma \nu }^{\rho }-{\partial }_{\nu }{\Gamma }_{\sigma \mu }^{\rho }+{\Gamma }_{\lambda \mu }^{\rho }{\Gamma }_{\sigma \nu }^{\lambda }-{\Gamma }_{\lambda \nu }^{\rho }{\Gamma }_{\sigma \mu }^{\lambda }\right)$ (66)

here ${\Gamma }_{\sigma \mu }^{\rho }$ is the Christoffel symbol, and the associated Vielbein:

$A_{\mu }^{ab}~e_{\rho }^a{\partial }_{\mu }{e}^{b\rho }$ (67)

In short, for curvature GGE, the curvature transformation is ${\Omega }_V=g_{UV}^{-1}{\Omega }_U{g}_{UV}$; for gravity only, Equation (62) and Equation (63) hold; for the curvature interaction across electromagnetic gravity, Equation (64) holds, and the expression of $T_{EM}\to J_{ab}$ defined must satisfy

$J_{ab}=g_{UV}^{-1}{T}_{EM}{g}_{UV}$ (68)

The goal of this adjustment is to support the gravitational-electromagnetic transformation by defining the hybrid representation as:

$g_{UV}=\left(\begin{array}{cc}{R}_{SO\left(1,3\right)}& 0\\ 0& e^{i\theta \left(x\right)}\end{array}\right)$ (69)

where $R_{SO\left(1,3\right)}\in SO\left(1,3\right)$, $e^{i\theta \left(x\right)}\in U\left(1\right)$; then we have:

${\omega }_V=\left(\begin{array}{cc}{A^{\prime }}_{\mu }^{ab}{J}_{ab}& 0\\ 0& {A^{\prime }}_{\mu }^{EM}{T}_{EM}\end{array}\right)$ (70)

where ${A^{\prime }}_{\mu }^{ab}={\left(g^{-1}\right)}_a^c{\left(g^{-1}\right)}_b^d{A}_{\mu }^{cd}$, ${A^{\prime }}_{\mu }^{EM}=A_{\mu }^{EM}-{\partial }_{\mu }\theta \left(x\right)$.

${F^{\prime }}_{\mu \nu }^{ab}={\left(g^{-1}\right)}_a^c{\left(g^{-1}\right)}_b^d{F}_{\mu \nu }^{cd},{F^{\prime }}_{\mu \nu }^{EM}=F_{\mu \nu }^{EM}$ (71)

The adjusted (69), (70) and (71) support the above Equation (53) and Equation (64), and it can be determined that the adjusted gravitational electromagnetic gauge field framework supports $SO\left(1,3\right)\times U\left(1\right)$, satisfying the connection and curvature GGE [32] - [35] , the relevant details can be seen in Appendix A and B.

First, we define the polarization matrix of the gauge potential of the gravitational soliton is expressed as:

${\omega }_V=h_{ij}={\mathrm{sech}}^2\left(ku\right)\left(\begin{array}{cc}A& B\\ B& -A\end{array}\right)$ (72)

where ( $u,v,x,y$) are light cone coordinates. It can be proved that $h_{ij}$ satisfies the vacuum Einstein Equation [19] Equation [22] , that is, $R_{\mu \nu }=0$, indicating a local solution of gravitational waves, and the polarization tensor is:

${ϵ}_{ij}=\left(\begin{array}{cc}A& B\\ B& -A\end{array}\right)$ (73)

Moreover, we are very happy to find that this gravitational soliton can be connected with two polarization states of optical solitons through rotation transformation. Further, in the weak field approximation, these two optical solitons can be transformed into two photons, and the corresponding gravitational soliton can be transformed into graviton. They can still be transformed through the original rotational gauge transformation through the GGE equation. For detailed derivation and application on curvature engine spacecraft, please refer to references [19] [22] . Therefore, this should be regarded as an example of the existence of the generalized gauge transformation across fundamental interactions that we proposed. Further analysis is given below.

We define the electromagnetic gauge potential as the polarization state of two optical solitons:

${\omega }_U~A_{\mu }^{EM}{T}_{EM}d{x}^{\mu }$ (74)

here the light cone coordinates are used here, and $A_{\mu }^{EM}~e_{\mu }{\mathrm{sech}}^2\left(ku\right)$ is taken to represent the polarization representation of the electromagnetic wave corresponding to the two optical solitons:

${\omega }_U={\mathrm{sech}}^2\left(ku\right)\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)=A_{\mu }^{EM}{T}_{EM}$ (75)

where the polarization vector $e_{\mu },\mu =x,y$, then the connection GGE (2) is:

${\omega }_V=g_{UV}^{-1}{\omega }_U{g}_{UV}+g_{UV}^{-1}d{g}_{UV}$

the gauge transformation matrix is selected as the rotation transformation:

$g_{UV}=\left(\begin{array}{cc}\cos \theta \left(u\right)& -\sin \theta \left(u\right)\\ \sin \theta \left(u\right)& \cos \theta \left(u\right)\end{array}\right)$ (76)

then following calculation can be divided into two parts:

1) Prove that it satisfies the connection GGE, that is, starting from the connection GGE

${\omega }_V=g_{UV}^{-1}\left(A_{\mu }^{EM}{T}_{EM}\right)g_{UV}+g_{UV}^{-1}{\partial }_{\mu }{g}_{UV}d{x}^{\mu }$

Assuming the generator $T_{EM}\to J_{ab}$ through the spinor mapping: $A_{\mu }^{EM}{e}_{\mu }\to h_{\mu \nu }{ϵ}_{\sigma }^{\nu }$, the proof result can be obtained

${\omega }_V~h_{\mu \nu }{J}_{\sigma }^{\nu }d{x}^{\mu }~{\mathrm{sech}}^2\left(ku\right)\left(\begin{array}{cc}A& B\\ B& -A\end{array}\right)J_{ij}d{x}^i$ (77)

2) Curvature verification, that is, through the gravitational tensor $F_{\mu \nu }^{ab}$ of the soliton and the electromagnetic tensor $F_{\mu \nu }^{EM}$ of the optical soliton constructed above, it is finally proved that the curvature GGE holds.

The relevant detailed derivations are given as follows:

1) Proof of contact with GGE

(1) Electromagnetic gauge potential and input

Assume the electromagnetic gauge potential is

${\omega }_U=A_{\mu }^{EM}{T}_{EM}d{x}^{\mu }$

$A_{\mu }^{EM}~e_{\mu }{\mathrm{sech}}^2\left(ku\right)$ (78)

where $T_{EM}$ is the generator of the U(1) Lie algebra, usually in scalar or matrix form; $e_{\mu }$ is the polarization vector of the electromagnetic wave, satisfying the transverse condition (for example, $k^{\mu }{e}_{\mu }=0$); ${\mathrm{sech}}^2\left(ku\right)$ is the soliton waveform, $k^{\mu }$ is the wave vector, and $u=k_{\mu }{x}^{\mu }$.