3.2 Memory’s Standing Wave: How to Maintain Continuity of “Identity” in a Constantly Refreshing Discrete Universe?—Topological Protection Mechanism Provided by Berry Phase

In the previous section, we defined “self” as a self-referential strange loop in QCA information flow. However, this raises a serious dynamical challenge: at the microscopic scale, the QCA network is constantly updated at Planck frequency ($\sim 10^{43}$ Hz). If at every time step, all qubits undergo drastic unitary evolution, how does the macroscopic “I” maintain continuity? Why is yesterday’s me the same “entity” as today’s me?

Traditional neuroscience attributes memory to the physical strength of synaptic connections (long-term potentiation LTP). But in our discrete ontology, matter itself is only a transient standing wave of information flow. Atoms are replaced in metabolism, quantum states become orthogonal in evolution. We need to search at a deeper level—the quantum information geometry level—for the stability mechanism of Identity.

This section will prove: Memory is not static storage, but dynamic topological protection. Just as superconducting current does not decay due to topological reasons, the continuity of consciousness is a macroscopic quantum effect protected by Berry Phase in Hilbert space.

3.2.1 Identity Crisis in Discrete Updates

Consider a QCA system $|\Psi (t)⟩$. According to unitary evolution $|\Psi (t+1)⟩=\hat{U}|\Psi (t)⟩$.

• If change is too fast: $|\Psi (t+1)⟩$ is almost orthogonal to $|\Psi (t)⟩$, the system has no “memory,” only “replacement.” Each frame is a new universe, with no historical inheritor.

• If change is too slow: $|\Psi (t+1)⟩\approx |\Psi (t)⟩$, the system is stable but has no “computation,” i.e., no mental activity (dead silence).

Conscious systems are in a delicate critical state: they must process information quickly (change) while maintaining self-identity (invariance).

This requires the evolution operator ${\hat{U}}_{mind}$ to have a special tensor product structure:

$${\hat{U}}_{mind}={\hat{U}}_{process}\otimes {\hat{U}}_{identity}$$

• ${\hat{U}}_{process}$: Acts on rapidly changing degrees of freedom (thought flow, sensory input), responsible for computation.

• ${\hat{U}}_{identity}$: Acts on slowly changing degrees of freedom (sense of self, core memory), responsible for maintaining reference frame.

But this is not enough. In a noisy thermal environment (the brain), how to ensure ${\hat{U}}_{identity}$ is not destroyed by decoherence?

3.2.2 Adiabatic Evolution and Geometric Phase

Quantum mechanics provides a perfect protection mechanism: geometric phase.

When a quantum system evolves slowly (adiabatically) with parameters and forms a closed loop $\mathcal{C}$ in parameter space, the wave function not only acquires a time-dependent dynamical phase $e^{-iEt}$, but also an additional geometric phase dependent on the path shape—Berry Phase $\gamma$.

$$\gamma ={\oint }_{\mathcal{C}}⟨\psi (\mathbf{R})|i{\nabla }_{\mathbf{R}}|\psi (\mathbf{R})⟩\cdot d\mathbf{R}$$

In QCA networks, “parameter space” corresponds to the internal state space of the consciousness system.

The “self-referential loop” (Strange Loop) we defined in Section 3.1 is geometrically a closed path in Hilbert space.

• When consciousness completes a basic cognitive cycle (e.g., “I perceive red” $\to$ “I am thinking” $\to$ “still me”), the system’s state vector circles once on the manifold.

• The Berry phase $\gamma$ accumulated in this circle is the topological signature of “I”.

Definition 3.2 (Topological Definition of Memory):

Long-term memory is not data bits carved on a hard drive, but a topologically protected quantum limit cycle.

When environmental perturbations cause small deformations of the system’s path, since topological properties (winding numbers) are discrete integers, the accumulated geometric phase $\gamma$ remains unchanged.

It is precisely this invariant $\gamma$ that allows the system to recognize “this is my state” after countless updates.

3.2.3 Standing Wave of Identity

We can model “self” as a soliton on the QCA network.

In fluid mechanics, a soliton is a nonlinear wave that maintains its shape during propagation by balancing dispersion through nonlinear effects.

In consciousness physics, this balance manifests as:

• Entropy increase (dispersion): Environmental information continuously impacts the brain, trying to scatter internal ordered structures.

• Self-reference (nonlinearity): By feeding output back to input, the system continuously reconstructs its own model.

When these two reach dynamic equilibrium, a “standing wave of identity” is formed.

This standing wave is protected by an energy gap.

Let the consciousness state be the ground state (or low-energy state manifold) of some effective Hamiltonian ${\hat{H}}_{mind}$. If this Hamiltonian has topological non-triviality (similar to quantum Hall effect systems), there exists an energy gap $\Delta E$ between the ground state and excited states.

As long as external thermal fluctuations (environmental noise) energy $k_{B}T<\Delta E$, they cannot destroy the topological properties of this ground state.

This means:

“I” is a macroscopic quantum state protected by a topological energy gap.

• Sleep or anesthesia only temporarily suppresses the dynamical phase (stops explicit computation), but does not destroy the topological structure (winding number unchanged).

• Upon awakening, the system automatically relaxes back to that protected topological ground state, and memory and sense of self “restart.”

3.2.4 Why Don’t We Feel “Fragmentation”?

QCA is discrete, time jumps frame by frame. Why is our subjective experience a continuous river, not flickering slides?

This is precisely determined by the properties of Berry phase.

Geometric phase has reparametrization invariance. It depends only on the geometric shape of the path, not on the specific rate of evolution.

Regardless of the refresh rate of the underlying QCA, regardless of how many Planck time steps have passed in between, as long as the topological structure of the closed loop is not broken, the accumulated phase is consistent.

The continuity of consciousness is essentially topological invariance.

We live inside this topological invariant. Every refresh of QCA is a confirmation of this invariant. We don’t feel “refresh” because “I” is that which remains unchanged in the refresh.

Conclusion:

Our identity is not a name carved in stone, but a vortex written on water.

Water molecules (physical carriers) flow away every second, but the vortex’s winding number (topological structure) is dynamically locked.

This is memory’s standing wave. In a flowing universe, it is our only anchor.