There have been two standard models in physics, in which one is the standard model of particle physics based on gauge quantum field theory [1], for unifying strong (theoretically approached by QCD), weak and electromagnetic interactions; and another is the standard model of cosmos based on general relativity [2], for explaining expanding cosmos and gravity. The standard models are verified by experiments very well [3]. When the Higgs boson was discovered at 126 GeV by LHC [4,5] recently, the standard model of particle physics is completed.

However, the two kinds of standard models are so different that they can’t reach complete physical unification. Following general relativity, the mass distribution determines the curvature of space-time (equivalence principle). According to QCD, the quarks in hadrons are abided in a small space-time (space radium about 10⁻¹⁸ m) with large mass (around 1 TeV). As we think gravity is too weak to care before, we always ignore gravity in particle physics. But, inside hadrons, since large mass could introduce high curvature of space-time, the effects of general relativity can’t be ignored, i.e. we can’t ignore gravity inside hadrons where large mass would produce high spacetime curvature.

Logically, if the standard model of particle physics and the standard model of cosmos are correct meantime, there will be a combination of gravity and strong interaction in a small space-time, which is just inside hadrons, where quarks possess high mass and act each other in very small space-time. Let us consider the situation.

For measuring physical world, basic physical measures include mass m, energy E and momentum p. According to relativity, under Lorentz invariance, we have their relation

where c is velocity of light. If we choose physical unit and let c = h = 1 where is Planck’s constant, it is

This means mass distribution is proportional to the distribution of energy and momentum, i.e. energy and momentum tensor T_μν.

where g_μν is metric tensor, R_μν Ricci tensor and R scalar curvature.

According to general relativity, T_μν is equivalent to the curvature of space-time. If we introduce the distribution of mass, i.e. the tensor of mass m_μν, we obtain

With introducing the wave function Ψ and considering its normalization, we have

in which ψ_L and ψ_R are left-handed and right-handed parts of a Dirac spinor respectively and <|> are Dirac notations as left-vector and right-vector [5]. Then we obtain

where m is Eigenvalues of m_μν and k is a real constant. As the dimension of curvature is radium⁻¹, is a natural choice for matching physical dimensions, which just combines micro-quantum theory and macro-relativity.

Meanwhile, following the Lagrangian in QCD [1,6], we see

where the first item describes gluons and second item quarks and g is gauge factor, while F means field strength and γ^μ denotes Dirac matrices (c.f. Appendix).

Replacing q with wave function Ψ and introducing Eqs. 6-7, we obtain the Lagrangian of strong interaction with curvature mass in hadrons

This constructs fundamental theoretical analytical framework.

Using multi-vector M_k (k = 0, 1, 2, 3, 4) [7-9] as field measures, where M_k is a multi-vector of grade k. k = 0 is scalar, k = 1 vector, k = 2 bivector, k = 3 pseudovector and k = 4 pseudoscalar, we have

in which Ψ = φ − iθ constructs a complex wave function of matter, while A = V − iU forms a complex vector function of matter particles in space-time and B keeps a bivector as interaction. The Eq.6 means that matter combines wave function and vector function in their interacttion, which is just an image fitting the duality of waveparticle, within combining mass and energy as matter.

The conjuncture of M is

And the reversion of multi-vector M, denoted by, becomes

Introducing differential operators

where Eq.13 means covariant derivative and g denotes gauge factor, we have

So we set up the necessary mathematical foundations. Since we apply the mathematics of geometric algebra and geometric calculus, we can call the following physical unification as geometric field.

Synthesizing above mathematical notations, the mathematical framework for the world is constructed as follows.

The unified Lagrangian contains all elements of geometric field [9,10], characterized by multi-vector and. So, in Eqs. 9-18, all elements of the geometric unified field are included.

When one ignores G and introduce F

and and A are linked by following equation

one can choose Lagrangian

where is current.

As action and, one obtains classical Maxwell’s equations under gauge group U(1)

or equivalent form of geometric algebra

When we notice

the left and right wave functions fit following coupled two-component equations, which are equivalent to Diractype equation as

Let us introduce gauge fields W_μ, stemming from group SU(2), with the field strength tensor

we can choose Lagrangian

Combining Eqs.21 and 26, we have electroweak Lagrangian

If we ignore A, G becomes dominated. Then we have Lagrangian fitting QCD model under SU(3) invariance

where the first item describes gluons and second item quarks, which is just (7), with.

When Eqs.21, 27 and 28 act together in a particle system, SU_C(3) × SU_L(2) × U_Y(1) symmetry had been embedded. Meanwhile, as the m is substituted by curvature of space-time according to, the unified physics covers gravity, strong and electroweak interaction in it, with holding unified Lagrangian as

Now, we have the geometric unified field, where mass can be replaced by curvature of space-time, which provides a mathematical structure for unified physics.

While the standard model of particles is characterized by gauge symmetry with SU_C(3) × SU_L(2) × U_Y(1), the standard model of cosmos and general relativity are characterized by big-bang cosmology and the mass being equivalent to the curvature of space-time, based on the general equivalence principle. They constructed a theoretical frame-work for understanding physical reality [11, 12].

Therefore, all fields and their phenomena will be interpreted by the unified mechanism, and the unified physical mechanism can be induced into ACC (Action, Connection and Construction) principles [13].

The action principle acts on action S, revealing the dynamic mechanism of a physical system

The action principle is the first physical principle in geometric unified field, which originates from the Lagrange-Hamilton principle and the Nöther theorem and determines the dynamic evolution of a physical system.

The first key physical idea retains the relativity principle of local gauge invariance, which includes that a phase change and a rotation or a displacement never change physical laws, in following gauge trans-formation

where Λ denotes an arbitrary function of space and time and g is gauge factor.

The connection principle links physical potential with mathematical connection, physical field strength and mathematical curvature, as shown with the following mathematical structure

in which P is multi-vector potential, ω multi-vector connection, and k₁ a real constant; F multi-vector field strength, Ω multi-vector curvature, and k₂ another real constant.

The connection principle is the second physical principle in geometric unified field, which originates in Newton’s 2nd law and Einstein’s equivalence principle and encompass the kinematical characteristics and the dynamic structure of a physical system.

The second key physical idea focuses on matter and energy being equivalent to the curvature of space-time locally. This is the general equivalence principle in both micro-world and macro-cosmos. As we knew that the world was a duality combining wave with particle, as well as space-time curvature, when we apply M as the measure of matter and energy, Ω as curvature of spacetime, the core idea becomes Distribution of mass = Curvature of space-time The construction principle links inner geometric structure with outer topological index as

where Ω is curvature and ω connection, while χ denotes characteristic index and M as manifold on multivector.

Originated from the Gauss-Bonnet formula and Stokes theorem, the construction principle is the third physical principle in geometric unified field, which links inner structure with surface parameters, so that it reveals the constructure of the world for unified physics.

One can apply the above three principles to any physical system as a general approach to physical unification, which constructs the overall dynamics of the world.

The above physical unification retains all ideas from both quantum field theory and general relativity in it, while the duality of wave and particle as well as mass and curvature are combined in a unified physics.

Applying Lagrange equation to the Lagrangian (29), we find

And remember energy-momentum tensor to be

As and

, supposing and, we obtain

And varying the in Eq.29, we have

The variation causes

as

and, Eq.39 could be divided into two parts, particle phase and wave phase, and result in following combined equations

Eqs.37 and 40 unify strong interaction and gravity while Eq.41 links electroweak interaction where Higgs mechanism remains.

If we introduce linkage between and as

(42)

electroweak model is maintained by linking to strong interaction.

Based on Eq.37, we can also provide a theoretical interpretation of dark matter and dark energy [14], where p (Ψ) item indicates dark matter and E(Ψ) item denotes dark energy. Thus, unknown dark matter and dark energy originate from theoretical flaws, and they are phenomenological effects in the physical unification of geometric field.

When we combine with, we obtain

So, we know that the essences of dark matter and dark energy are effects of scalar and vector functions mathematically. Meanwhile, complement to Higgs mechanism, the mass also originated from curvature inside hadrons.

Combining quantum field theory and general relativity, geometric field did unify the two standard models together, which provided another way to consider physical unification, where the distribution of matter-energy is proportional to the curvature of space-time and waveparticle duality is embedded into multi-vector functions. The unified physical framework looks good in both mathematics and physics, with retaining and matching all results of present physics, which is another U-theory [15].

While Higgs boson was found in LHC, we would provide this way to introduce mass inside hadrons. Following this nonlinear physical unification, dark matter and energy might also be naturally interpreted, which could stimulate further studies.

The author is grateful to the anonymous reviewer for helpful comments.

Dirac matrices are defined as

Mostly their Weyl or chiral representation of Dirac matrices are applied as

The standard complex Pauli matrices with conjuncture keep following forms

For numerical estimation of curvature mass, as the curvature is proportional to the radium⁻¹, we can estimate the gravity strength with mass in a space with radium at 10⁻¹⁵ m referring to Eq.6

.