Square Root Metric Geometry and Pati-Salam Model in Curved Space-Time

**Square Root Metric Geometry and Pati-Salam Model in Curved Space-Time**

The Riemannian manifold can be described by a metric

$$g(x) = -g_{\mu\nu}(x) dx^\mu \otimes dx^\nu,$$

or given by the inverse metric

$$g^{-1}(x) = -\eta^{ab} \theta_a(x) \theta_b(x),$$

where

$$\eta^{ab} = \text{diag}(1, -1, -1, -1),$$

orthonormal frames describe gravitational field

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

It can be seen that the inverse of the metric is related to $l(x)$ and $\bar{l}(x)$

$$g^{-1}(x) = \frac{1}{4} \text{tr} \left[ \bar{l}(x) l(x) \right].$$

The explicit mathematical formulas of $l(x)$ and $\bar{l}(x)$ are shown in formula

$$l(x) = i \gamma_{ik}^0(x) \gamma_{kj}^a(x) e_j^\dagger \otimes e_i \theta_a(x),$$

$$\bar{l}(x) = i \gamma_{ik}^a(x) \gamma_{kj}^0(x) e_j^\dagger \otimes e_i \theta_a(x),$$

$l(x)$ and $\bar{l}(x)$ form a pair of square root metrics. Since the Dirac matrix has degrees of freedom for phase rotation, the square root metric can be written as

$$l(x) = i \psi_i(x) \gamma^a \psi_j(x) e_j^\dagger \otimes e_i \theta_a(x),$$

$$\bar{l}(x) = i \bar{\psi}_i(x) \gamma^a \bar{\psi}_j(x) e_j^\dagger \otimes e_i \theta_a(x).$$

This pair of square root metrics is anti-Hermitian

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

The connection of this geometric is defined

$$\nabla_\mu \partial_v = \Gamma_{\mu\nu} \partial_\rho,$$

where, eq.(7) is the coefficients of the affine connections on coordinates, eq.(8) coefficients of spin connections on orthonormal frame [9, 10], and eq.(9) is gauge field on the $U(4')$-bundle, eq.(10) is gauge field on the $U(4)$-bundle.

Based on the principle of self-parallel transportation of semi-metric geometry, the equation of motion can be constructed.

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

$$\text{tr} \nabla^2 \left[ \bar{l}(x) l(x) \right] = 0.$$

This pair of equations of motion is generalized covariant, local frame rotation invariant, and $U(4') \times U(4)$ gauge invariant. The relationship between the Lagrangian density and the equation of motion can be given

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

When this geometry satisfies the principle of self-parallel transportation eq.(11), the Lagrangian density is Hermitian.

The Hermitian property of the Lagrangian density ensures the unitary nature of the theory, and the Hermitian property of the Lagrangian density originated from self-parallel transportation principle.

The corresponding effective Lagrangian density for Pati-Salam model in curved space-time is

$$\mathcal{L}_{\text{effective}} = \bar{\psi}_i \gamma^a \left( i \partial_\mu \psi_i + V_\mu \psi_j - \psi_j W_{\mu j} \right) \theta_a^\mu$$

$$+ \bar{\psi}_i \phi \psi_i + V(\phi) - \frac{1}{2} \text{tr} \left( H_{\mu\nu} H_{\mu j} \right) - \frac{\mathcal{L}}{2} F_{\mu j} F_{\mu j},$$

before this Lagrangian density is the Dirac kinetic energy term, followed by the minimum coupling term, then the Yukawa coupling term, and then the Higgs potential and the Yang-Mills term. Where $H_{\mu\nu}$ and $F_{\mu\nij}$ are gauge field strength tensors

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

$$F_{\mu\nij} = \partial_\mu W_{\mu j} - i W_{\mu j} W_{\mu j} + i W_{\mu j} W_{\mu j}.$$

This Lagrangian density eq.(15) describes the Pati-Salam model in curved space-time, and the canonical quantization of this model can be done.

**Sheaf Quantization and Path Integral Quantization**

The Sheaf space is collection of the sections of a bundle. As $l(x), \bar{l}(x)$ are sections of the bundles, respectively. The Sheaf valued section can be written $$\hat{l}(x) = \sum_k \eta_k(x) |k, x\rangle \langle k, x| l_k(x),$$
$$\hat{l} = \sum_k \eta_k(x) |\kappa, x\rangle \langle \kappa, x| \hat{l}_k(x),$$
where $\kappa$ is sheaf space index and evaluated in an abelian group. The density matrix
$$\rho(x) = \sum_k \eta_k(x) |k, x\rangle \langle k, x|,$$
is added before a pair of entity, square root metric $l_k(x), \hat{l}_k(x)$,
to embody superposition principle. The superposition principle at sheaf quantization is, if any two entities $\hat{l}_1(x)$
and $\hat{l}_2(x)$ in $Sh(x)$, there is an entity $\hat{l}(x)$ in $Sh(x)$ equals to the mixing of the two entities
$$\hat{l}(x) = \eta_1(x) \hat{l}_1(x) + \eta_2(x) \hat{l}_2(x),$$
$$\hat{l}_1(x), \hat{l}_2(x) \in Sh(x);$$
$$\Rightarrow \hat{l}(x) \in Sh(x),$$
where the probability of each section
$$\eta_1(x), \eta_2(x) \in [0, 1]$$
and
$$\eta_1(x) + \eta_2(x) = 1.$$
The $Sh(x)$ and $\widetilde{Sh}(x)$ are linear spaces of $l_k(x)$ and $\hat{l}_k(x)$, respectively; and the Sheaf spaces $Sh(x)$ and $\widetilde{Sh}(x)$ are dual to each other. Sheaf quantization switching study objects from single section to all possible sections of the bundle. Sheaf quantization could improve the problem of infinite of quantum field theory. The equations of motion for sheaf valued entities $\hat{l}(x)$ and $\hat{l}(x)$ in Sheaf quantization method are
$$\nu \nabla [\hat{l}(x)] = 0, \nu [\hat{l}(x) \hat{l}(x)] = 0.$$
When we have quantum state of quantum field theory $|\psi(x)\rangle$
, the pure state of density operator is
$$\rho(x) = |\psi(x)\rangle \langle \psi(x)|$$
is shown. As the quantum state spanned by the bases of sheaf space
$$|\psi(x)\rangle = \sum_k \alpha_k(x)|k\rangle$$
the transition amplitude of path integral formulation can be derived [11]

$$\alpha_k(t, \vec{x}) = \int_{t'=t_k(t, \vec{x})}^{t'=\omega t} D\kappa(t', \vec{x}) e^{i\omega t'[\kappa(t', \vec{x})]} \alpha_k(t_0, \vec{x})$$