Pati-Salam Model in Curved Space-Time from Square Root Lorentz Manifold

Introduction

Four-dimensional pseudo-Riemann geometry with signature $(-,+,+,+)$, Lorentz manifold, is the geometry background of the general relativity, space-time is described by the metric, and the gravitational field is described as the curve of space-time. In general relativity, the geodesic equation describes the trajectories of free particles, and the Einstein equation determines how matter curves space-time. At the last life time of Einstein, he attempted to establish a new geometry unifying electromagnetic interaction and gravity. This idea was developed by Weil into the early idea of gauge invariance and by the Kaluza and Klein into the idea of extra dimensions.

Later, the Yang-Mills theory [1] was confirmed. Yang-Mills theory takes gauge invariance as its basic principle and to be the theoretical framework of electromagnetic, weak and strong interaction in Standard Model (SM) of particle physics [2, 3, 4, 5, 6, 7]. The Yang-Mills theory is the theoretical framework of SM and has a good correspondence with the complex structure group G fiber bundle theory [8]. General relativity can actually be rewritten in the framework of fiber bundle theory also, except that the G structure group of general relativity is real, and the corresponding fiber bundles are the tangent and cotangent bundles. Another way to build unified field theory is introducing extra dimensions to give all fields their geometric positions. And lots of attempts in extra dimension were made. Is it possible to fuse the tangent (cotangent) bundle of general relativity with the complex structure group G-bundle of Yang-Mills theory?

Inspired by the Dirac’s way of finding his equation and

spinors through making square root of the Klein-Gordon

equation, we researched the four-dimensional square root

Lorentz manifold, which similar with the papers in Clifford

algebra or Clifford bundle [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52], spin-gauge theory in

Riemann-Cartan space-time [53, 54], sedenion [55] and

Einstein-Cartan theory [56, 57, 58] etc. Four-dimensional

square root Lorentz manifold has extra

$$ U \left(4 ^ {\prime}\right) \times U (4) $$

principal bundle than Lorentz manifold. Two Lagrangians

based on four-dimension square root Lorentz manifold are

constructed which describe a

( ) ( ) ( ) 4' 4 4 L R U U U × × Pati-

Salam model in curved space-time and a gravity theory,

respectively. In the Pati-Salam model [59], the

( ) '4 SU is

color group with “lepton number as the fourth color”, and the ( ) ( ) 4 4 L R SU SU × is chiral flavor group. The gravity theory is Einstein kind with torsion. We realize an explicit formulation of Sheaf quantization [60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76] scheme which consistent with path integral quantization. The particles spectrum on this model is analyzed.

Geometry and Lagrangian

The notations are introduced here. , , , a b c d represent frame indices and , , , a b c d are equal to 0,1,2,3 . , , , v µ ρ σ represent coordinates indices and , , , v µ ρ σ are equal to 0,1,2,3 . α represent group indices with α equals to

0,1,....,15. , , , , i j k l m are equal to 1,2,3,4 . C is quarks color and equals to ( ) , , 1,2,3 R G B . k is Sheaf space index. Repeated indices are summed by default.

The pseudo-Riemann manifold is described by a metric $$ g (x) = - g _ {\mu \nu} (x) d x ^ {\mu} d x ^ {\nu}, \tag {1} $$ where the metric is symmetric $$ g _ {\mu \nu} (x) = g _ {\nu \mu} (x), \tag {2} $$ $$ \text {a n d} \left\{x \mid x = \left(x ^ {\mu}\right) = \left(t, \vec {x}\right) \right\} i $$

is a four-dimensional topological

space. Here we discuss the four-dimensional pseudo-

Riemann manifold with signature(

) , , ,

−+ + + , Lorentz

manifold. And it can be described by orthonormal frame

(vierbein) formalism as ( ) ( ) ( ) 1 , ab a b g x x x η θ θ − = − (3) where ( ) 1, 1, 1, 1 . ab diag η = −−− orthonormal frames describe gravitational field (Figure 1).

$$ \theta_ {a} (x) = \theta_ {a} ^ {\mu} (x) \frac {\partial}{\partial x ^ {\mu}}. $$ ( ) ( ) . a a x x x

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(a) general relativity (b) Yang-Mills theory and (c) square

root metric. (a) Geometry background of general relativity

is pseudo-Riemann geometry, which is a smooth curved

manifold. (b) Geometry background of Yang-Mills theory

is complex G-bundle with flat base manifold, G is the

gauge group of Yang-Mills theory. (c) Square root metric

geometry in four-dimension has everything of pseudo-

Riemann geometry and extra

$$ U \left(4 ^ {\prime}\right) \times U (4) - b u n d l e. $$ The definition of gamma matrices is $$ \gamma^ {a} \gamma^ {b} + \gamma^ {b} \gamma^ {a} = 2 \eta^ {a b} I _ {4 \times 4}. \tag {4} $$ The Hermiticity conditions for gamma matrices are $$ \gamma^ {a} \gamma^ {b \dagger} + \gamma^ {b \dagger} \gamma^ {a} = 2 I ^ {a b} I _ {4 \times 4}, \tag {5} $$ where ab I is ( ) 1,1,1,1 diag . We define $$ l = i \gamma_ {i k} ^ {0} (x) \gamma_ {k j} ^ {a} (x) e _ {j} ^ {\dagger} \otimes e _ {j} \theta_ {a} (x), \tag {6} $$ $$ \tilde {l} = i \gamma_ {i k} ^ {a} (x) \gamma_ {k j} ^ {0} (x) e _ {j} ^ {\dagger} \otimes e _ {i} \theta_ {a} (x), \tag {7} $$ where ie are the orthogonal bases expanding four-dimension complex space 4 ≤. The orthogonal bases of 4 ≤ satisfy $$ t r \left(e _ {j} ^ {\dagger} \otimes e _ {i}\right) = e _ {i} e _ {j} ^ {\dagger} = \delta_ {i j}. \tag {8} $$ One simple choice of ie is $$ e _ {1} = \left(e ^ {i \theta_ {1}}, 0, 0, 0\right), e _ {2} = \left(0, e ^ {i \theta_ {2}}, 0, 0\right), \tag {9} $$ ( ) ( ) 3 4 3 4 0,0, ,0 , 0,0,0, . i i e e e e θ θ = = (10) After using † 0 0 a a γ γ γ γ = , we find that $$g^{-1}(x) = \frac{1}{4} \text{tr} \left[ \bar{l}(x) l(x) \right]. \tag{11}$$

Then $l(x)$ and $\bar{l}(x)$ are the square root of metrics in some sense. The representation freedom of $\gamma_{ij}^{a}(x)$ can be shown as

$$\gamma_{ik}^{0}(x) \gamma_{kj}^{a}(x) = \psi_{ik}^{\dagger}(x) \gamma_{kl}^{0} \gamma_{lm}^{a} \psi_{mj}(x) = \bar{\psi}_{i}(x) \gamma^{a} \psi_{j}(x),$$

$$\gamma_{ik}^{a}(x) \gamma_{kj}^{0}(x) = \psi_{ik}^{\dagger}(x) \gamma_{kl}^{a} \gamma_{lm}^{0} \psi_{mj}(x) = \bar{\psi}_{i}(x) \gamma^{a \dagger} \psi_{j}(x),$$

where $\psi_{i}$ are the Dirac fermions field with flavor related index $i$ equals to 1,2,3,4. And

$$\bar{\psi}_{i}(x) = \psi_{i}^{\dagger}(x) \gamma^{0}, \psi(x) \in U(4)$$

is $4 \times 4$ matrix. So, the square root metrics are defined as follow

$$l(x) = i \bar{\psi}_{i}(x) \gamma^{a} \psi_{j}(x) e_{j}^{\dagger} \otimes e_{i} \theta_{a}(x), \tag{12}$$

$$\bar{l}(x) = i \bar{\psi}_{i}(x) \gamma^{a \dagger} \psi_{j}(x) e_{j}^{\dagger} \otimes e_{i} \theta_{a}(x). \tag{13}$$

The square root Lorentz manifold is described by square root metric (12), (13). Direct calculations show that the definition (12) and (13) satisfy (11) and

$$l^{\dagger}(x) = -l(x), \bar{l}^{\dagger}(x) = -\bar{l}(x).$$

The coefficients of the affine connections on coordinates, coefficients of spin connections on orthonormal frame [77] and gauge fields on the $U(4) \times U(4)$-bundle are defined as follows

$$\nabla_{\mu} \partial_{v} = \Gamma_{\nu\mu}^{p}(x) \partial_{p}, \tag{14}$$

$$\nabla_{\mu} \theta_{a}(x) = \Gamma_{\nu\mu}^{b}(x) \theta_{h}(x), \tag{15}$$

$$\nabla_{\mu} \left( \gamma^{0} \gamma^{a} \right) = i \left[ V_{\mu}(x) \gamma^{0} \gamma^{a} - \gamma^{0} \gamma^{a} V_{\mu}(x) \right], \tag{16}$$

$$\nabla_{\mu} e_{i} = i W_{\mu j}(x) e_{j}^{\dagger}. \tag{17}$$

A relation between coefficients of the affine connections on coordinates and coefficients of spin connections on orthonormal frame can be easily found

$$\Gamma_{a\mu}^{b}(x) \theta_{b}^{a}(x) = \partial_{\mu} \theta_{a}^{b}(x) + \theta_{a}^{b}(x) \Gamma_{v\mu}^{p}(x).$$

If the covariant derivative is compatible with metric,

$$\nabla g(x) = 0,$$

then

$$\Gamma_{ab\mu} = -\Gamma_{ba\mu}.$$

The gauge fields on the $U(4) \times U(4)$-bundle are Hermitian

$$V_{ij}^{a}(x) = V_{ij}(x), W_{ij}^{b}(x) = W_{ij}(x).$$

The uniqueness of definition of gauge fields is originated from restriction (4), (5) and (8). The equation as follow can be derived from (16)

$$\nabla_{\mu} \left( \gamma^{a} \gamma^{0} \right) = i \left[ \bar{\nabla}_{\mu}(x) \gamma^{a} \gamma^{0} - \gamma^{a} \gamma^{0} \bar{\nabla}_{\mu}(x) \right], \tag{18}$$

where we write

$$\bar{\nabla}_{\mu}(x) = \gamma^{0} V \gamma^{0}.$$

The gauge field $V_{\mu}(x)$ and $W_{\mu j}(x)$ can be decomposed by the generators of the $U(4)$ group

$$V_{\mu}(x) = V_{\mu}^{a}(x) T^{\alpha}, \tag{19}$$

$$W_{\mu j}(x) = W_{\mu}^{a}(x) T_{ij}^{\alpha}, \tag{20}$$

where $\alpha$ is equals to 0,1,2,...15. There are Hermitian gauge bosons fields

$$V_{\mu}^{\alpha \dagger}(x) = V_{\mu}^{a}(x), W_{\mu}^{\alpha \dagger}(x) = W_{\mu}^{a}(x).$$

The $T^{\alpha}$ are the generators of $U(4)$ and an explicit one can be seen in appendix. An equation is constructed which satisfying the $U(4^{\prime}) \times U(4)$ gauge invariant, locally Lorentz invariant and generally covariant principles

$$\text{tr} \nabla[l(x)] = 0. \tag{21}$$

This equation originated from generalized self-parallel transportation principle. Eliminating index $x$, the explicit formula of equation (21) is

$$\left[ i \partial_{\mu} \bar{\psi}_{i} - \bar{\psi}_{i} \bar{V}_{\mu} + W_{\mu j} \bar{\psi}_{j} \right] \gamma^{a} \psi_{i} + \bar{\psi}_{i} \gamma^{a}$$

$$\left[ i \partial_{\mu} \psi_{i} + V_{\mu} \psi_{i} - \psi_{j} W_{\mu j} \right] \psi_{i} \gamma^{b} \psi_{i} \Gamma_{a\mu}^{\alpha} \theta_{a}^{\alpha}. \tag{22}$$

The last term in Lagrangian (22) is Yukawa coupling term $\psi_{i} \phi \psi_{i}$ and the scalar (Higgs) field is gamma matrix valued and originated from gravitational field.

$$\phi = \frac{i}{2} \gamma^{b} \Gamma_{a\mu}^{\alpha} \theta_{a}^{\alpha}. \tag{23}$$

Then, the Lagrangian (22) describes $U(4') \times U(4)$ Yang-Mills theory in curved space-time (Figure 1). The Lagrangian (22) has relation with (21)

$$\text{tr} \nabla [l(x)] = \mathcal{L} - \mathcal{L}'.$$ (24)

If equation (21) being satisfied, the Lagrangian (22) is Hermitian

$$\mathcal{L} = \mathcal{L}'.$$ (25)

So, the unitary principle of quantum field theory (25) consistent with generalized self-parallel transportation principle (21). The equations of motion for the Lagrangian (22) are

$$\gamma^a \left( i \partial_\mu \psi_i + V_\mu \psi_j - \psi_j W_{\mu j} \right) \theta_a^a + \frac{i}{2} \gamma^b \psi_i \Gamma_a^b \theta_a^a = 0,$$ (26)

and this equation’s conjugate transpose. We point out that (21) is density matrix version of (26). The effective equation of motion of (26) has signature (1, -1, -1, -1). For example, the massless Dirac equation in curved space-time of this model is

$$i \gamma^a \theta_a^a \partial_\mu \psi_i = 0.$$ (27)

The square of equation (27) is massless Klein-Gordon equation in curved space-time

$$\eta^{ab} \theta_a^a \theta_b^a \partial_\mu \partial_v \psi_i = 0.$$ (28)

The signature of equation (28) is $(1, -1, -1, -1)$ and consistent with special relativity. Then, a Lagrangian (22) which describes the $U(4') \times U(4)$ Yang-Mills theory in curved space-time is all other fields (except Higgs field) dynamical background which satisfies the characteristic of the gravitational field in our real world.

Lagrangian (22) is $U(4') \times U(4)$ gauge invariant, locally Lorentz invariant and generally covariant. So, Lagrangian (22) is demanded invariant under the transformations

$$\psi_i = \tilde{U} \psi_j U_{ji},$$

$$\gamma^a = \tilde{U} \gamma^b \tilde{U}^\dagger \Lambda_b^a,$$ (29)

$$\theta_a^a = \Lambda_b^a \theta_b^a,$$

Then, the transformation rules have to be derived as follows

$$V_\mu = \tilde{U} V_\mu \tilde{U}^\dagger - \left( \partial_\mu \tilde{U} \right) \tilde{U}^\dagger,$$ (30)

$$V_\mu = U_{ki} W_\mu U_{ij} + U_{ki} \left( \partial_\mu W_{ij} \right),$$ (31)

$$\Gamma_{b\mu}^b = \Lambda_a^a \Gamma_{c\mu}^d \Lambda_d^b - \Lambda_a^a \partial_\mu \Lambda_c^b,$$ (32)

where the transformation matrices satisfy

$$\tilde{U} \tilde{U}^\dagger = I, U_{ji} U_{ji}^* = \delta_{ik}, \Lambda_a^b \Lambda_b^c = \delta_a^c.$$

Then the transformation elements

$$\tilde{U} \in U(4'),$$

$$\left( U_{ij} \right) \in U(4),$$

$$\Lambda_b^a \in O(1,3),$$

and $U(4)$ is color group, $U(4)$ is flavor group, $O(1,3)$ is locally Lorentz group. Then we complete the proof of the Lagrangian (22) is $U(4') \times U(4)$ gauge invariant, locally Lorentz invariant and generally covariant.

The gauge field strength tensors and curvature tensor are defined as follows

$$H_{\mu v} = \partial_\mu V_\mu - \partial_v V_\mu - i V_\mu V_v,$$

$$F_{\mu vj} = \partial_\mu W_{vj} - \partial_v W_{vj} - i W_{\mu k} W_{vkj},$$

$$R_a^b_{b\mu} = \partial_\mu \Gamma_a^b_{b\mu} - \partial_v \Gamma_a^b_{b\mu} + \Gamma_b^c \Gamma_c^a - \Gamma_c^b \Gamma_c^a.$$

If the covariant derivative is compatible with metric, the curvature tensor satisfies

$$R_{ab\mu\nu} = -R_{bau\nu}.$$

The gauge fields strengths on the $U(4') \times U(4)$-bundle match the Hermiticity conditions

$$H^\dagger_{\mu v} = H_{\mu v}, F_{\mu vj}^* = F_{\mu vj}.$$

The gauge field strength can be decomposed by the $U(4)$ generators

$$H_{\mu v} = H_{\mu v}^a T_a^a, F_{\mu vj} = F_{\mu v}^a T_{ij}^a.$$

After the torsion being defined

$$T_{vp}^a = 2 \Gamma_a^a_{[vp]},$$ (33)

we have the Ricci identity and Bianchi identity [78] on this geometry structure as follows

$$\partial_{\mu} H_{vp} = H_{[vp} V_\rho] - V_{\mu} H_{vp},$$ (34)

$$\partial_{\mu} F_{vp} = F_{[vp] k} W_\rho^k - W_{\mu k} F_{vp} k_j,$$ (35)

$$T_a^a_{\rho} T_{vp} = R_a^a_{\rho} + \nabla_{\rho} T_{a\nu}.$$ (36)

$$ \nabla_ {[ \rho} R _ {| b | \mu \nu ]} ^ {a} = R _ {b \sigma [ \rho} ^ {a} T _ {\mu \nu ]}. \tag {37} $$ There is Yang-Mills Lagrangian for gauge bosons in this model $$ \mathcal {L} _ {Y} = \frac {- 1}{2} \boldsymbol {t r} \left(H ^ {\mu \nu} H _ {\mu \nu}\right) - \frac {- \zeta}{2} F _ {i j} ^ {\mu \nu} F _ {\mu \nu j i}, $$ (38) where R ζ ∈ is constant.

For the gravity, the Einstein-Hilbert action in Lorentz manifold be showed as follow $$ S = \int \omega R, \tag {39} $$ where is the Ricci scalar curvature in Lorentz manifold, ω is volume form $$ \omega = \sqrt {- g _ {v}} d x ^ {0} \wedge d x ^ {1} \wedge d x ^ {2} \wedge d x ^ {3} $$ and $$ g _ {v} = \det \left(g _ {\mu v} (x)\right). $$ In this geometry framework, the equations can be derived as follows * [ ] 1( 2 $$ \nabla_ {[ \mu} \nabla_ {\nu ]} l = \frac {- 1}{2} \left(\psi_ {i} \gamma^ {a} \psi_ {k} F _ {\mu \nu k j} - F _ {\mu \nu k i} ^ {*} \bar {\psi} _ {k} \gamma^ {a} \psi_ {j} + \bar {\psi} _ {i} \tilde {H} _ {\mu \nu} \gamma^ {a} \psi_ {j}\right) $$ $$ - \bar {\psi} _ {i} \gamma^ {a} H _ {\mu \nu} \psi_ {j} + \frac {i}{2} \bar {\psi} _ {i} \gamma^ {b} \psi_ {j} R _ {b \mu \nu} ^ {a}) e _ {j} ^ {\dagger} \otimes e _ {i} \theta^ {a}, \tag {40} $$ † ) , 2 † * † † [ ] 1( 2 $$ \nabla_ {[ \mu} \nabla_ {\nu ]} \tilde {l} = \frac {- 1}{2} \left(\psi_ {i} \gamma^ {a ^ {\dagger}} \psi_ {k} F _ {\mu \nu k j} - F _ {\mu \nu k i} ^ {*} \bar {\psi} _ {k} \gamma^ {a ^ {\dagger}} \psi_ {j} + \bar {\psi} _ {i} \tilde {H} _ {\mu \nu} \gamma^ {a ^ {\dagger}} \psi_ {j}\right) $$ $$ - \bar {\psi} _ {i} \gamma^ {a \dagger} H _ {\mu \nu} \psi_ {j} + \frac {i}{2} \bar {\psi} _ {i} \gamma^ {b \dagger} \psi_ {j} R _ {b \mu \nu} ^ {a}) e _ {j} ^ {\dagger} \otimes e _ {i} \theta^ {a}, \tag {41} $$

where we write † † † ) , 2 $$ \tilde {H} _ {\mu \nu} = \gamma^ {0} H _ {\mu \nu} \gamma^ {0}. $$ We define $$ \nabla^ {2} = \nabla_ {[ \mu} \nabla_ {\nu ]} d x ^ {\mu} \wedge d x ^ {\nu}, $$

the equation of a gravity theory is constructed $$ \boldsymbol {t r} \nabla^ {2} \left[ \tilde {l} (x) l (x) \right] = 0. \tag {42} $$ This equation (42) is obviously (4 ) ') (4 U U × gauge invariant, locally Lorentz invariant and generally covariant. The explicit formula of equation (42) is ( ) † † † † ( abij a b b a i i j i R i F ψ ψ ψ γ γ γ γ ψ = − $$ - \psi_ {i} ^ {\dagger} H _ {a b} \left(\gamma^ {a} \gamma^ {b} - \gamma^ {b \dagger} \gamma^ {a \dagger}\right) \psi_ {i}, \tag {43} $$ where we use the notations as follow, $$ \partial_ {\mu} d x ^ {\nu} \otimes d x ^ {\rho} \partial_ {\sigma} = \delta_ {\mu} ^ {\nu} \delta_ {\sigma} ^ {\rho}, $$ $$ d x ^ {\mu} \otimes d x ^ {\nu} \partial_ {\rho} \partial_ {\sigma} = \delta_ {\rho} ^ {\nu} \delta_ {\sigma} ^ {\mu}, $$ $$ F _ {a b i j} = F _ {a b i j} \theta_ {a} ^ {u} \theta_ {b} ^ {v}, $$ $$ H _ {a b} = H _ {\mu \nu} \theta_ {a} ^ {u} \theta_ {b} ^ {v}. $$ So, we define a

$$ U \left(4 ^ {\prime}\right) \times U (4) $$

gauge invariant, locally Lorentz

invariant, generally covariant Lagrangian ( ) † † † † abij a b b a g i i j i † † † ( R i F ψ ψ ψ γ γ γ γ ψ $$ = R \psi_ {i} ^ {\dagger} \psi_ {i} - i \left(F ^ {a b i j} \psi_ {j} ^ {\dagger} \left(\gamma^ {a} \gamma^ {b} - \gamma\right\right) $$ Λ (44) ( ) a b b a i ab i .

i H

ψ γ γ γ γ ψ + −

The Lagrangian (44) is Hermitian

$$ \mathcal {L} _ {g} = \mathcal {L} _ {g} ^ {\dagger}. \tag {45} $$

The † i i Rψ ψ in Lagrangian (44) gives us the Einstein-Hilbert action. The equation (43) and the Einstein tensor can be derived from the Lagrangian (44). The gravity theory in this model is Einstein-Cartan kind with torsion.

Sheaf Quantization and Path Integral Quantization

The entities ( ) l x and ( ) l x  are two sections of the two bundles, respectively, where these two bundles are dual to each other. Further, the Sheaf valued entities ( ) ˆl x and ( ) l x  are superposition of sections ( ) l x and ( ) l x  (Figure 2).

$$ \hat {l} (x) = \sum_ {\kappa} \eta_ {\kappa} (x) \left| \kappa , x \right\rangle \langle \kappa , x \mid l _ {\kappa} (x), \tag {46} $$ $$ \hat {\tilde {l}} (x) = \sum_ {\kappa} \eta_ {\kappa} (x) | \kappa , x \rangle \langle \kappa , x | \tilde {l} _ {\kappa} (x), \tag {47} $$

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where ( ) [ ] 0,1 x κ η ∈ are probability of corresponding section ( ) l x κ and ( ), l x κ κ is Sheaf space index and evaluated in an abelian group. The density matrix corresponds to ( ) ˆl x and ( ) l x  is $$ \rho (x) = \sum_ {\kappa} \eta_ {\kappa} (x) \left| \kappa , x \right\rangle \langle \kappa , x |. \tag {48} $$ We have orthogonal bases in Sheaf space. The orthogonal bases in Sheaf space satisfy probability complete formulas $$ \left\langle \kappa , x \mid \kappa^ {\prime}, x ^ {\prime} \right\rangle = \delta \left(x - x ^ {\prime}\right) \delta \left(\kappa - \kappa^ {\prime}\right). \tag {49} $$ $$ \operatorname {t r} \rho (x) = \sum_ {\hat {\mathrm {e}}} \eta_ {\kappa} (x) = 1. \tag {50} $$ ( ) ( ) In mathematics, a Sheaf is a collection of sections, the index κ of each section correspond to an abelian group element. In physics, the Sheaf spaces ( ) Sh x and ±( ) Sh x are expanded by all possible sections ( ) l x and ( ) l x  of the two bundles, respectively. The ( ) Sh x and ±( ) Sh x are linear spaces, which means, for example, any two entities ( ) 1ˆl x and ( ) 2ˆl x in ( ) Sh x , there is an entity ( ) ˆl x in ( ) Sh x equals to the mixing of the two entities ˆ ˆ ˆ ( ) ( ) ( ) ( ) ( ) , l x x l x x l x $$ ) = \eta_ {1} (x) \hat {l} _ {1} (x) + \eta_ {2} () $$

1 1 2 2 ˆ ˆ

( ) ( ) ( )

, ;

l x l x Sh x ∈

(51)

1 2 ˆ

( ) ( ) , l x Sh x ⇒ ∈

where the probability of each section ( ) ( ) [ ] 1 2 , 0,1 x x η η ∈ and $$ \eta_ {1} (x) + \eta_ {2} (x) = 1. $$ The Sheaf spaces ( ) Sh x and ±( ) Sh x are dual to each other. We call it Sheaf quantization which switching study objects from single section to all possible sections of the bundle. The equation (50) derives to equation of motion for density matrix $$ d \left(\boldsymbol {t r} \rho (x)\right) = \boldsymbol {t r} \left(d \rho (x)\right) = 0, \tag {52} $$ where d is exterior differential derivative. The equations of motion for entities ( ) ˆl x and ( ) l x  in Sheaf quantization method are $$ \boldsymbol {t r} \nabla \left[ \hat {l} (x) \right] = 0, \boldsymbol {t r} \left[ \hat {\hat {l}} (x) \hat {l} (x) \right] = 0. \tag {53} $$ The corresponding total Lagrangian density is , , , ˆ g Y g g κ κ κ κ = + + ∑  Λ Λ Λ Λ (54) where g and g are Lagrangian multipliers with , . g g ∈   For pure state $$ \rho (x) = \left| \Psi (x) \Psi (x) \right|, \tag {55} $$ $$ \left| \Psi (x) \right\rangle = \sum_ {\kappa} \alpha_ {\kappa} (x) | \kappa \rangle , \tag {56} $$ where the ( ) x Ψ is the quantum state of quantum field theory. The transition amplitude can be defined through path integral formula , 0 t t ,t ρ ρ ρ ρ Λ (57) ˆ , , , , i t x t x D t x e t x ω κ κ κ α κ α  ( ) ( ) ( ) ( ) ( ) $$ ) = \int_ {t ^ {\prime} \in \left(t _ {0}, t\right)} D \kappa \left(t ^ {\prime}, \vec {x}\right) e ^ {i \omega \hat {\mathcal {L}} \left[ \kappa \left(t, \vec {x}\right) \right]} d $$ ∈ ′

0 where ω is volume form.

Particles Spectrum

µ V α and µ

W α (α is equals to 0,1 ,

bosons fields. The interactions related with the gauge bosons

fields µ

W α always preserves the possibility of chiral symmetry

breaking in the Lagrangian (54) such that the gauge group

can be decomposed to

$$ U \left(4 ^ {\prime}\right) \times U \left(4\right) _ {L} \times U \left(4\right) _ {R}, \text {w h e r e} U \left(4 ^ {\prime}\right) $$

is color group and

( ) ( ) 4 4 L R U U ×

is chiral flavor group. The

0 µ

V is dark photon and

0 µ

W is Fiona particle.

0 µ V and 0 µ W are related with ( ) 1 U ′ and ( ) 1 U

gauge group in

( ) 4 U ′ and ( ) 4 U

, respectively. The left-over gauge group is a Pati-Salam gauge

group ( ) ( ) ( ) 4 4 4 L R SU SU SU × × ′

(Figure 3) [59, 79]. The

( ) 4 SU

′ gauge group can be decomposed as follow $$ U \left(4 ^ {\prime}\right) = S U \left(3 ^ {\prime}\right) \oplus U \left(1 ^ {\prime}\right) + U _ {X ^ {+}} + U _ {X ^ {-}}. \tag {58} $$

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adjoint representation is

$$ 1 5 = 8 \otimes 1 + 3 + 3 ^ {*}. \tag {a} $$

The wight diagram of V α

µ which contained gauge bosons

are gluons, photon and

$$ \left[ X _ {v ^ {0}} ^ {\mu} ^ {\pm C}. \right] $$

(b) The wight diagram of W α µ

related gauge bosons are

$$ Y ^ {0}, Y ^ {1}, Y ^ {2}, Y _ {*} ^ {1}, Y _ {*} ^ {2} \mathrm {a n d} W ^ {\pm}, Z. $$ The ( ) 3 SU ′ is the gauge group of quantum chramodynamics (QCD) and the corresponding gauge bosons µ V α (α is equals to 1, 2

$$ X ^ {+ C} \mathrm {a n d} X ^ {- C} \mathrm {a r e} 1 / 3 \mathrm {a n d} $$

−1/3, respectively. The chiral gauge group

( ) , 4 L R SU can be

decomposed as (Figure 4).

$$ S U \left(4\right) _ {L, R} = S U \left(3\right) _ {Y} \oplus U \left(1\right) _ {Z} + U _ {W ^ {+}} + U _ {W ^ {-}}, \tag {60} $$ and related gauge bosons µ V α (with α equals to1, 2,

9 10 11 12 W W iW W iW (61)

µ µ µ µ µ

13 14, W iW

= ± µ µ $$ Z _ {\mu} = W _ {\mu} ^ {1 5}. \tag {62} $$

Click to enlarge

The left over gauge bosons are 0 1 2 1 2 * * , , , , Y Y Y Y Y with 0 electric charge. The gauge bosons 1 2 1 2 * * , , , Y Y Y Y transport non-SM flavor changing neutral currents (FCNCs) and (63) $$ \text {The} X ^ {\pm C} \text {and} Y ^ {1}, Y ^ {2}, Y _ {*} ^ {1}, Y _ {*} ^ {2} $$

Y Y must be super heavy from the

restrictions of experimental data.

The fermionic fields

i

ψ transfer as the

$$ U \left(4 ^ {\prime}\right) \times U (4) $$

fundamental representation according to (29). So, fermions

are filled into the

( ) 4 SU fundamental representation ⊗ 4 6

naturally. The fundamental representation 4 corresponds to

3 colors and 1 lepton and leads us reobtain “Lepton number as the fourth color” [59]. The fundamental representation 6 corresponds to 6 flavors of quarks and leptons. The weight diagram coordinates in Chevalley basis of representation 6 (see in Table 1) have good correspondence with the quark quantum number [80]. The antifermions be filled into the representation ⊗ 6 4 similarly. The weight diagram of fermions is shown in Figure 4. Both left-handed and right- handed fermions for all quarks and leptons are existed. Especially, the existence of right-handed neutrinos is predicted. This is compatible with experimental results [81] and the well know Seesaw mechanism [82, 83, 84, 85, 86]. The 2 zI being used in Table I are not no reasons because we have the Gell-Mann-Nishijima formula [80].

$$ Q = I _ {z} + \frac {\mathfrak {P} + S + C + B + T}{2}. \tag {64} $$

An explicit fermions representation [79] in this model might be $$ = \left( \begin{array}{c c c c} \sqrt {2} u _ {R} & \sqrt {2} c _ {R} & \sqrt {2} t _ {R} & d _ {R} ^ {\prime} \\ \sqrt {2} u _ {G} & \sqrt {2} c _ {G} & \sqrt {2} t _ {G} & d _ {G} ^ {\prime} \\ \sqrt {2} u _ {B} & \sqrt {2} c _ {B} & \sqrt {2} t _ {B} & d _ {B} ^ {\prime} \\ e & \mu & \tau & \nu^ {\prime} \end{array} \right), $$ '

2 2 2 u c t d R R R R

'

2 2 2 , 2 2 2

u c t d (65) ψ G G G G i       

' u c t d e B B B B

µ τ ν where , , u c t and d′ are quarks fields, , , e µ τ and ' ν are electron, mu, tau and neutrinos fields. The corresponding fermions electric charges of (65) are

2 2 2 1 3 3 3 3 2 2 2 1 . 3 3 3 3 2 2 2 1 3 3 3 3 1 1 1 0

$$ Q = \left( \begin{array}{c c c c} \frac {2}{3} & \frac {2}{3} & \frac {2}{3} & - \frac {1}{3} \\ \frac {2}{3} & \frac {2}{3} & \frac {2}{3} & - \frac {1}{3} \\ \frac {2}{3} & \frac {2}{3} & \frac {2}{3} & - \frac {1}{3} \\ 1 & 1 & 1 & 0 \end{array} \right). $$ Q (66) The quarks states like , , d s b and neutrinos states , , e u τ ν ν ν are eigen states of the Lagrangian [79].

Conclusion and Discussion

A Pati-Salam model and a gravity theory from square root Lorentz manifold are derived. A Sheaf quantization scheme which consistent with path integral quantization is shown. The particles spectrum in this model is discussed.

Some possible new physics on this model are listed as follows. The gravity theory has torsion. There are exotic gauge bosons such as dark photon, Fiona, $X^{zC}$ and $Y^{0}, Y^{1}, Y^{2}, Y^{3}, Y^{4}$. The $X^{zC}$ transports semi-leptonic processes, the $Y^{1}, Y^{2}, Y^{3}, Y^{4}$ transport non-SM FCNCs. The right-handed neutrinos are existed. The Higgs field is gamma matrix valued.

Acknowledgments

We thank Professor Chao-Guang Huang for long time discussions and powerful help. Without Professor Huang’s help, this work is impossible. We thank Jun-Bao Wu, Zhao Li, Cai-Dian Lv, Ming Zhong, Yong-Chang Huang, Qi-Shu Yan, Yu Tian, De-Shan Yang, Yang-Hao Ou for valuable discussions. We thank Jean Thierry-Mieg for critical comments. We thank Yu Lu for computer drawing.