slides 2.pdf

I am very glad to introduce the geometry structure of "square root metric" at Summer  Institute 2015. My name is De-Sheng Li, I come from IHEP.

Einstein used to say, "To grasp the entity by mind, which is physics." This lead me believe our world is described by a geometry structure.

Review the geometry background of general relativity is Lorentz manifold and geometry background of Standard Model is similar with Minkowski space-time with G-structure. The most naive idea is Curved manifold with G-structure might describe real-world space-time naturelly. That is the motivation to study "square root metric" manifold. Because "square root metric" manifold not only can be curved but also equiped with U(4)-bundle structure. Another motivation is square root something usual lead to unusual. For example, square root of minus one equals i, and square root of Klein-Corden equation lead us find Dirac equation and his spinor. So, why not study square root metric? What is the geometry structure of suqare root metric?

At first, the notations are introduced. a, b, c, d represent frame indices; $\mu$, $\nu$, $\rho$, $\sigma$ represent coordinates indices; $\alpha$, $\beta$, $\gamma$, $\tau$ represent group indices; $\alpha$ equals zero to fifteen; $\beta$ equals ont to eight; $\gamma$ equals nine, eleven and thirteen and $\tau$ equals ten, twelveand forteen. Repeated indices are summed by default.

Then, we begin at metric. Here is the definition of metric. And the inverse metric can be defined as follows. Inverse metric can be described by orthonormal frame formalism as equation 2.3. Beause $\eta^{ab}$ is constant matrix, $\theta_{a}=\theta_{a}^{\ \mu}\partial_{\mu}$ describe gravitational field.

Then, we analyse freedom of gamma matrices. Equation 2.4 is the definition of gamma matrices, Equation 2.5 is the hermiticity conditions for gamma matrices. Under those two constrains, a U(4) unitary phase transformation can be made for $\gamma^{a}$. After the transformation, $\gamma^{a}$ prime still are gamma matrices also.

Then, a new entiry l can be defined as follows,

                            $l=u^{\dagger}\gamma^{a}u\theta_{a}.$

And we find that l is the suqare root of inverse metric in somesense.

Then, the connection on this manifold can be defined as follows. Because I believe space-time equiped with pure G-sructure, the connections on U(4) principal bundle is defined in equation 2.9c. And some identities are found.

Then, a U(4) gauge invariant, locally Lorentz invariant and generally covariant Lagrangian is constructed as

                            $\mathcal{L}= \nabla l.$

This Lagrangian describe U(4) Yang-Mills theory in curved space-time. The explicit formalism of this Lagrangian is equation 2.11. In this Lagrangian, u might describe fermions field, $A_{\mu}$ describe gauge field, $\theta_{a}^{\ \mu}$ describe gravitational field. $u^{\dagger}\phi u$ is Yukawa coupling term, and $\phi=\gamma^{a}\Gamma^{b}_{\ a\mu}\theta_{a}^{\ \mu}$. After preliminary decomposition, the chiral properties and something more of this Lagrangian appear.

The Lagrangian's symmetry of U(4) gauge invariant, locally Lorentz invariant and generally covariant can be proved exactly. And the transformation rules for connections and scalar have to be derived as follows.

The curvature tensor and gauge field strength tensor can be defined in Equation 2.16b. Then, some identies can be found, espacially Ricci identity and Bianchi identity.

For gravity, Einstein-Hilbert action be showed in equation 2.18. The variation of this action give us Einstein tensor. For Einstein euquation, Einstein used to say:" The reason for the formalism of left hand is to let its divergence identically zero in the meaning of covariant derivative. The right hand of equation are the sum up of all the things still problems in the meaning of field theory.

In this geometry framework of "suqare root metric", a Lagrangian $\mathcal{L}_{g}$ might describe gravity is constructed

                            $\mathcal{L}_{g}=\nabla^{2} l^{2}.$

The explicit formalism of Lagrangian $\mathcal{L}_{g}$ is equation 2.21. This Lagrangian is obviously U(4) gauge invariant, locally Lorentz invariant and generally covariant. The last term correspond to Einstein-Hilbert action naturally, the leftover terms might describe Lagrangian of matter in gravity theory.

$A_{\mu}$ are U(4) gauge fields and can be expanded by generators of U(4). The $\mathcal{T}_{0}$ generator correpond to a new particle, Fiona. And the left over SU(4) generators correspond to 8 gluons, 1 gamma and 6 mystery particles, X and $\bar{X}$. Electrodynamics like, Chromodynamics theory appear in U(1) and SU(3) part of this geometry framework in flat space-time naturally. From 6 mystery particles, 3 color-singlet, composite particles can be constructed. Those 3 particle have nonzero mass correspond to $W^{\pm}$ and Z. The explicit formalism of $W^{\pm}$ and Z are exhibited in equation 3.3a and 3.3b. The 3 boson vertexes in flat space-time are determined by SU(4) structure constants and we show it on Fig.5.

Fermionic like fields u transfer as U(4) fundamental representation according to equation 2.14. So fermions are filled into SU(4) fundamental representation $4\otimes  6$ naturally as Table 1. Antifermions are filled into $\bar{4}\otimes 6$ similarly. Here are weight diagram of SU(4) fundamental representation 4 and 6. Those weight diagram are Polato polyhedron and Achimedean solids.

$\phi$ is scalar field based on equation 2.12 and 2.15c. Corresponding particle of scalar field is Higgs. Equation 2.12 tell us that scalar field origin from gravitational field. This is amazing but inescapable result in this geometry framework. In Strandard Model, Yukawa coupling and gravity interaction without repulsion forces either but gauge interaction not at least on tree level, this might hint that there are deep connections between scalar field and gravitational field.

So, the "suqare root" of 4 dimensional Lorentzian manifold have extra U(4)-bundle structure than 4 dimensional Lorentzian maniford. Fermionic like fields origin from sections of U(4) bundle , gauge fields are conncetions of U(4)-bundle, gravitational field is described by orthonormal frame, scalar field origin from gravitational field. Fermionic like fields transfer as fundamental representation and give quarks and leptons particles spectrum. Gauge fields transfer as adjoint representation and give 8 gluons, color singlet, composite particles $W^{\pm}$ and Z, 1 most familiar particle $\gamma$. Scalar field correspond to Higgs.

This theory unify 64 "entities" into a single "entity" l, and $l=u^{\dagger}\gamma^{a}u \theta_{a}$. The interactions between those fields might be determined by those two Lagrangians

              $\mathcal{L}= \nabla l,  \mathcal{L}_{g}=\nabla^{2} l^{2}.$

and some identities. Freedom condense bring our universe structure appear.

New physics prediction on this unified field theory(UFT) are listed as follows. Firstly, $W^{\pm}$ and Z are composite particles. Secondly, scalar field origin from gravitational field. Thirdly, right handed neutrinos should be existed. At last, Fiona is exist but can not be detected by  directly by collider experiments, but have important gravity effect for our universe.