Meta Theory of the Zeckendorf-Hilbert Universe

2.3 Berry Curvature and Geometric Phase: Geometric Origin of Gauge Fields

In Sections 2.1 and 2.2, we established the Riemannian geometric structure of parameter space by analyzing the real part of Quantum Geometric Tensor (QGT)—Quantum Fisher Information Metric $g_{\mu \nu }$. This explains “distance” or distinguishability between physical states. However, QGT is a Hermitian tensor, and its imaginary part also has profound physical meaning.

This section will argue that the antisymmetric imaginary part of quantum geometric tensor—Berry Curvature—constitutes the geometric origin of gauge fields (Gauge Fields) in physics. Just as gravity arises from curvature of Riemannian metric, electromagnetic force and Yang-Mills fields arise from curvature of Hilbert space fiber bundles.

2.3.1 Parallel Transport and Berry Connection

In Riemannian geometry, comparing vectors at two nearby points on a manifold requires introducing “connection” to define parallel transport. In quantum mechanics, when system parameters $\mathbf{\theta }$ change, basis vectors $|\psi (\mathbf{\theta })⟩$ in Hilbert space also rotate accordingly. Due to phase redundancy ($U(1)$ gauge freedom) of quantum states, we need to define what constitutes “no physical change” movement.

Definition 2.3.1 (Quantum Parallel Transport)

Consider a path $\mathcal{C}$ in parameter space. If quantum state $|\psi (t)⟩$ evolving along this path satisfies the following condition, it is said to have undergone parallel transport:

$$⟨\psi (t)|\dot{\psi }(t)⟩=0$$

The geometric meaning of this condition is: the direction of state vector change is perpendicular to the state vector itself, i.e., there is no tangential component along phase, local phase change is minimal.

However, for general parameterized basis vectors $|\psi (\mathbf{\theta })⟩$, the above condition is usually not satisfied. To describe this deviation, we introduce Berry Connection, also called Berry Potential:

$${\mathcal{A}}_{\mu }(\mathbf{\theta })=i⟨\psi (\mathbf{\theta })|{\partial }_{\mu }\psi (\mathbf{\theta })⟩$$

This is a real-valued vector field (for normalized states).

Physical Interpretation:

Berry connection ${\mathcal{A}}_{\mu }$ is mathematically completely equivalent to Gauge Potential (Gauge Potential, i.e., vector potential $\mathbf{A}$) in electromagnetism. It describes the natural phase drift rate induced by local reference frame (basis vectors) when we move in parameter space.

2.3.2 Geometric Phase and Holonomy

When system parameters adiabatically evolve along a closed loop $\mathcal{C}$ and return to the starting point, quantum state $|\psi ⟩$ may not return to the initial state, but may acquire an additional phase factor $e^{i\gamma }$. This phase cannot be completely explained by integration of dynamical Hamiltonian (dynamical phase).

Theorem 2.3.2 (Berry Phase Formula)

The geometric phase (Geometric Phase) accumulated on closed loop $\mathcal{C}$, i.e., Berry phase $\gamma (\mathcal{C})$, is given by line integral of Berry connection:

$$\gamma (\mathcal{C})={\oint }_{\mathcal{C}}{\mathcal{A}}_{\mu }d{\theta }^{\mu }$$

This phase is an element of Holonomy group, reflecting non-trivial topological properties of fiber bundle structure on parameter space.

Proof:

Let evolving state be $|\Psi (t)⟩=e^{-i{\int }_0^{t}E(\tau )d\tau }{e}^{i\gamma (t)}|\psi (\mathbf{\theta }(t))⟩$, where $|\psi (\mathbf{\theta })⟩$ is instantaneous eigenstate. Substituting into Schrödinger equation and using adiabatic approximation, we obtain $\dot{\gamma }=i⟨\psi |\dot{\psi }⟩=i⟨\psi |{\partial }_{\mu }\psi ⟩{\dot{\theta }}^{\mu }$.

Integration yields $\gamma =\int {\mathcal{A}}_{\mu }d{\theta }^{\mu }$. $\square$

2.3.3 Berry Curvature: Gauge-Invariant Geometric Tensor

Berry connection ${\mathcal{A}}_{\mu }$ itself is not gauge-invariant. If we perform local gauge transformation $|\psi (\mathbf{\theta })⟩\to e^{i\alpha (\mathbf{\theta })}|\psi (\mathbf{\theta })⟩$, then ${\mathcal{A}}_{\mu }\to {\mathcal{A}}_{\mu }-{\partial }_{\mu }\alpha$. To obtain physically observable quantities, we introduce Berry Curvature.

Definition 2.3.3 (Berry Curvature Tensor)

Berry curvature is the exterior derivative (Exterior Derivative) of Berry connection, defined in component form as antisymmetric tensor ${\Omega }_{\mu \nu }$:

$${\Omega }_{\mu \nu }\equiv {\partial }_{\mu }{\mathcal{A}}_{\nu }-{\partial }_{\nu }{\mathcal{A}}_{\mu }=i\left(⟨{\partial }_{\mu }\psi |{\partial }_{\nu }\psi ⟩-⟨{\partial }_{\nu }\psi |{\partial }_{\mu }\psi ⟩\right)$$

This is precisely the imaginary part of quantum geometric tensor (QGT) (see Section 2.1.3):

$${\chi }_{\mu \nu }=\frac{1}{4}{g}_{\mu \nu }-\frac{i}{2}{\Omega }_{\mu \nu }$$

Properties:

1. Gauge Invariance: ${\Omega }_{\mu \nu }$ does not change with local phase choice of basis vectors; it is a true physical observable.

2. Symplectic Structure: On projective Hilbert space $\mathbb{C}{P}^{N-1}$, ${\Omega }_{\mu \nu }$ defines a symplectic form (Symplectic Form). This shows that quantum state space is not only a Riemannian manifold, but also a symplectic manifold.

2.3.4 Geometric Emergence of Gauge Fields

At this point, the “It from Bit” ontological picture becomes clearer: not only does spacetime metric (gravity) arise from distinguishability of quantum states ($g_{\mu \nu }$), but interactions (gauge fields) also arise from geometric phase structure of quantum states (${\Omega }_{\mu \nu }$).

Proposition 2.3.4 (Geometricization of Electromagnetic Force)

If we interpret parameters $\mathbf{\theta }$ as spacetime coordinates $\mathbf{x}$ of particles, and particles’ internal quantum states (such as spin or band index) adiabatically change with position, then Berry curvature ${\Omega }_{\mu \nu }$ plays exactly the same role in equations of motion as electromagnetic tensor $F_{\mu \nu }$.

Specifically, in semiclassical equations of motion, particles experience an “anomalous velocity” term:

$$\dot{\mathbf{r}}=\frac{\partial E}{\partial \mathbf{k}}-\dot{\mathbf{k}}\times \mathbf{\Omega }(\mathbf{k})$$

This term leads to physical phenomena such as Anomalous Hall Effect.

From this perspective, electromagnetic fields are not some kind of “ether” filling space, but projections of curvature of Hilbert space fiber bundles onto base manifold (spacetime). If we consider degenerate subspaces of non-Abelian groups (such as $SU(N)$), Berry connection will be upgraded to non-Abelian gauge potential (Yang-Mills Field), thereby geometrically explaining weak and strong interactions.

Conclusion 2.3.5

Fundamental forces in physics, at the underlying discrete ontology level, are unified as geometric properties of state space:

• Metric $g_{\mu \nu }$ $\to$ Distance and gravitational effects (rate of information change).

• Curvature ${\Omega }_{\mu \nu }$ $\to$ Phase and gauge field effects (structure and holonomy of information).

Through quantum geometric tensor ${\chi }_{\mu \nu }$, Riemannian geometry and symplectic geometry are unified in an inseparable complex tensor structure, suggesting profound unification of gravity and gauge fields at the level of information geometry. In the next section, we will explore the dynamical flow of this geometric structure on projective space and further reveal its symplectic geometric essence.