Part IX: Geometric Unification of Interactions

Chapter 16: Total Space Geometry

In Volumes II and III, we established a gravitational theory purely based on quantum information: spacetime is a network of quantum entanglement, and gravity is an entropic force maintaining entanglement entropy balance. However, the physical world contains more than gravity. The Standard Model describes electromagnetic, weak, and strong forces, governed by Gauge Fields.

If “geometry is information” is a universal truth, then gauge fields must also have geometric origins. Historically, Kaluza-Klein (KK) theory attempted to unify gravity and electromagnetism by introducing “extra dimensions.” In this chapter, we reconstruct this idea in the framework of modern fiber bundles and QCA discrete ontology. We will prove that gauge fields are not “matter” filling spacetime, but intrinsic geometric properties of Total Space—i.e., “spacetime + internal information space.” Gravity and gauge forces achieve perfect unification in the higher-dimensional total space geometry.

16.1 Modern Fiber Bundle Formulation of Kaluza-Klein Ideas

In 1921, Theodor Kaluza discovered that if general relativity is extended to five-dimensional spacetime, and the fifth dimension is assumed to curl into a tiny circle (compactification), then five-dimensional Einstein equations naturally decompose into four-dimensional Einstein equations and Maxwell equations. This miraculous discovery suggested: electromagnetic force is gravity in the fifth dimension.

In modern mathematical physics, this idea is elevated to fiber bundle geometry. This section will rigorously define this geometric structure and argue how it provides geometric explanations for gauge interactions in QCA universes.

16.1.1 From Extra Dimensions to Internal Symmetry

In classical KK theory, the fifth dimension is a physically existing spatial dimension. But in quantum mechanics, particles have not only spatial position $x^{\mu }$ but also internal degrees of freedom (such as phase, spin, flavor).

Definition 16.1.1 (Internal Space)

In QCA discrete ontology, each spacetime point (lattice point) $x$ carries a local Hilbert space ${\mathcal{H}}_x$. If ${\mathcal{H}}_x$ has some symmetry group $G$ (such as $U(1)$ phase symmetry or $SU(N)$ flavor symmetry), we can regard the group manifold $G$ as the “internal space” or fiber attached to point $x$.

Unlike classical KK theory, the “extra dimensions” here need not be macroscopic or spatial—they are algebraic. The geometrization of gauge field theory essentially studies how these internal algebraic spaces are “twisted” and connected in spacetime.

16.1.2 Geometric Construction of Principal Fiber Bundles

To uniformly describe external spacetime (base manifold $M$) and internal space (structure group $G$), we introduce Principal Fiber Bundles.

Definition 16.1.2 (Principal Bundle $P(M,G)$)

Total space $P$ is a differential manifold composed of base manifold $M$ and structure group $G$, satisfying:

1. Projection: There exists a smooth map $\pi :P\to M$ such that for any $x\in M$, the preimage ${\pi }^{-1}(x)$ (fiber) is diffeomorphic to group $G$.

2. Group Action: Group $G$ has a right action $R_g:P\to P$ on $P$, and this action preserves fibers (moves points within fibers).

3. Local Triviality: $P$ locally looks like $M\times G$, but globally may have non-trivial topological structure (like Möbius strip versus cylinder).

Physical Picture:

Total space $P$ is the complete stage of physical reality. Spacetime $M$ is just a projection or slice of $P$. A physical event is determined not only by its position $x^{\mu }$ but also by its internal state (point $p$ on the fiber).

16.1.3 Connection and Geometrization of Gauge Potentials

On total space $P$, the core of geometric structure is how to define “horizontal” directions.

For a point $u$ in $P$, its tangent space $T_{u}P$ can be decomposed into two subspaces:

1. Vertical Subspace $V_u$: Tangent to fiber direction. Naturally defined by group generators. This corresponds to pure internal gauge transformations.

2. Horizontal Subspace $H_u$: Tangent to base manifold direction. But this has no natural definition and requires specifying a rule.

Definition 16.1.3 (Ehresmann Connection)

A connection $\Gamma$ is a smooth direct sum decomposition of tangent space $T_{u}P=V_u\oplus H_u$, satisfying invariance under group action.

This decomposition defines a Lie algebra $\mathfrak{g}$-valued 1-form $\omega$ (connection form), which vanishes on horizontal vectors.

In local coordinates of base manifold $M$, the pullback of connection form $\omega$ is precisely the gauge potential $A_{\mu }$ in physics:

$${\sigma }^{\ast }\omega =A_{\mu }d{x}^{\mu }$$

where $\sigma :U\to P$ is a local section (gauge fixing).

Theorem 16.1.4 (Gauge Fields as Geometric Connections)

Yang-Mills gauge fields $A_{\mu }^a$ in physics are geometrically equivalent to horizontal distributions in total space $P$.

• Parallel Transport: Given a path $\gamma$ on $M$, the connection defines how to “horizontally” move points on fibers to neighboring fibers. This corresponds to covariant derivative $D_{\mu }$ in gauge theory.

• Curvature: If horizontal transport along a closed loop returns to the original fiber but at a different position (producing group element difference $g$), then total space has curvature. This corresponds to field strength tensor $F_{\mu \nu }$ (or Berry curvature) in physics.

16.1.4 QCA Perspective: Internal Registers and Entanglement Connections

In QCA discrete networks, this abstract geometry has concrete constructive interpretations.

Each lattice point $x$ has Hilbert space decomposed as ${\mathcal{H}}_x={\mathcal{H}}_{spin}\otimes {\mathcal{H}}_{internal}$.

• Base Manifold $M$: Defined by lattice network and its nearest-neighbor relations.

• Fiber $G$: Defined by basis transformation group of internal registers ${\mathcal{H}}_{internal}$.

• Connection $A_{\mu }$: In Section 4.3, we introduced connection variables (Link Variables) ${\mathcal{U}}_{xy}$. In total space geometry, ${\mathcal{U}}_{xy}$ precisely defines discrete parallel transport from fiber ${\pi }^{-1}(x)$ to ${\pi }^{-1}(y)$.

Corollary 16.1.5 (Unification of Forces)

In Riemannian geometry of total space $P$, geodesic equations describe particle motion.

$$\frac{d^2{X}^A}{d{\tau }^2}+{\Gamma }_{BC}^A\frac{d{X}^B}{d\tau }\frac{d{X}^C}{d\tau }=0$$

where $X^A=(x^{\mu },y^i)$ are total space coordinates.

The four-dimensional projection of this equation is the Lorentz force equation (including gravitational and gauge force terms):

$$\frac{d^2{x}^{\mu }}{d{\tau }^2}+\{\dots {\}}_{\text{grav}}=\frac{q}{m}{F}^{\mu }{}_{\nu }\frac{d{x}^{\nu }}{d\tau }$$

This shows that electromagnetic force is not a separate force, but the geometric effect of motion in total space.