3.1 Descartes’ Ghost: In the Flowing QCA Information Stream, “I” Is Not a Point, but a Closed, Self-Referential Causal Loop (Strange Loop)

Physics excels at describing “the other”—those particles colliding in spacetime, curved light rays, and expanding galaxies. But when we try to use this language to describe “I,” we hit a wall.

Descartes defined “I” as “thinking substance” (Res Cogitans), an immaterial ghost sitting in the pineal gland driving the machine of the body. Although modern neuroscience has long abandoned this dualism, deep in our intuition, that “little person” (Homunculus) watching the internal movie still lingers.

If the universe is QCA, a giant computer, then what exactly is “I”—a line of code, a data packet, or the CPU running the code?

This section will propose a purely physical, counterintuitive model of the self: “I” is not an entity, but a topological structure. I am a “dead knot” that the information flow in the QCA network ties to maintain its own existence.

3.1.1 Ship of Theseus and the Flowing Self

Our body is an open system. Every seven years, the vast majority of atoms in our body are replaced. Matter flows, energy dissipates.

Then, what remains unchanged, making you still you?

The answer can only be Pattern.

But in QCA, patterns are also fleeting. The wave function $|\Psi (t)⟩$ of the previous second and $|\Psi (t+1)⟩$ of the next second are two orthogonal vectors in Hilbert space. If even states are constantly renewed, where exactly is that constant “I” hidden?

Analogy: Vortex

Imagine a river. Water molecules (underlying Qubits) continuously flow past, not a single drop stays.

But if there is a Vortex at some position in the riverbed.

• The vortex has shape, size, rotation speed.

• The vortex can capture floating objects, interact with other vortices, and even “swallow” small vortices.

• Most importantly, the vortex has causal independence. You can point to it and say: “Look, there is something there.”

In QCA theory, consciousness is a vortex in the information flow.

3.1.2 Hofstadter’s Strange Loop

Douglas Hofstadter proposed the concept of “strange loop” in Gödel, Escher, Bach: a system’s hierarchical structure becomes entangled, making the output of the bottom layer become the input of the top layer, forming a self-referential closed loop.

In QCA networks, this structure has a strict mathematical definition.

Usually, information flow is feedforward: $A\to B\to C$.

But in some complex subnetworks, information flow forms feedback: $A\to B\to C\to \cdots \to A$.

Definition 3.1 (Topological Definition of Self):

“The self” is a Strongly Connected Component (SCC) in the QCA causal network, where the internal information loop time ${\tau }_{loop}$ is less than the characteristic time ${\tau }_{env}$ of exchanging information with the environment.

This means:

1. Causal Closure: Within this loop, past states determine future states, and future states in turn reconstruct the past through memory.

2. Logical Depth: This loop is not simple repetition (like a clock), but contains encoding of the loop itself. That is, the system not only computes, but also computes “I am computing”.

3.1.3 Physicalization of the Ghost: Observer as Fixed Point

Descartes’ ghost appears ghostly because it seems unchanged by the birth and death of matter.

Mathematically, this “invariance” corresponds to eigenstates of operators or fixed points of dynamical systems.

Let the evolution operator of consciousness be ${\hat{U}}_{self}$.

If there exists a macroscopic state $|\text{Self}⟩$ satisfying:

$${\hat{U}}_{self}|\text{Self}⟩\approx e^{i\theta }|\text{Self}⟩$$

Then, although microscopic qubits are flipping wildly, the macroscopic “sense of self” remains stable (only a phase rotation, corresponding to the sense of time passing).

Conclusion:

“I” is not a ghost living in the brain.

“I” is the fixed point that emerges during the operation of the brain’s complex network.

I am the calm in the eye of the storm, the arrow in the information torrent that stubbornly points to itself.

As long as this self-referential loop is not cut (death or deep anesthesia), that “ghost” watching the world will continue to exist. It is not supernatural; it is a topological necessity.