5.2 Matter as Topological Knots: Self-Referential Loops and Winding Numbers

In the previous section, we established that photons are pure translation modes in QCA networks—they are unidirectional information flows with no internal echo, hence no mass. Now, we face a more challenging question: How to construct a particle that can “stop”?

In a medium with constant fundamental propagation speed $c$, rest seems impossible. The only solution is looping. If a wave packet no longer propagates in a straight line but is trapped by some mechanism rotating in a closed path, then its average position can remain stationary or move at speeds below $c$.

This section will prove: Matter particles are essentially topological knots in spacetime networks. This “knot” is not a physical rope knot, but phase winding of quantum states in parameter space (momentum space). It is precisely this topological non-triviality that prevents wave packet dispersion and endows particles with “solid” identity—namely rest mass.

5.2.1 Schrödinger’s Trembling and “Photon in a Box” Model

To intuitively understand the origin of mass, let’s revisit a classic physical picture: photon in a box.

Imagine a box made of perfect mirrors containing a photon. The photon bounces back and forth at speed $c$.

• For observers outside the box, the photon as a whole has average velocity 0.

• The photon’s energy $E=h\nu$ is confined in the box. According to mass-energy equation, the box exhibits increased inertial mass $m=E/c^2$.

In the microscopic world of QCA, there are no “mirrors” as macroscopic objects. So what acts as mirrors, bouncing information flow back?

The answer is spatial topological structure or chirality coupling.

In a one-dimensional Dirac-QCA model (see Chapter 3, Section 3.2), evolution operator contains a mixing term $m{c}^2{\sigma }_x$. This term flips particle chirality: turning “left-moving wave” into “right-moving wave” and vice versa.

$$L\stackrel{\text{mass term}}{\to }R\stackrel{\text{mass term}}{\to }L$$

This constant flipping causes particles to perform Zitterbewegung (trembling) at microscopic scales. Like a drunkard, though each step is taken at light speed, macroscopic displacement is very slow due to constant direction changes.

Physical Picture 5.2:

Mass is not the quantity of some “matter”; it is the frequency at which information flow undergoes “self-referential scattering.” A particle is a photon that traps itself.

5.2.2 Topology of Momentum Space: Winding Number

Why is this “trapped” state stable? Why don’t particles suddenly disintegrate into photons flying away? This requires introducing the concept of topological protection.

In QCA, due to translation invariance, we can diagonalize evolution operator $\hat{U}$ in momentum space (Brillouin zone, topologically homeomorphic to circle $S^1$).

For a two-component system (such as fermions), effective Hamiltonian $\hat{H}(k)$ can be written as linear combination of Pauli matrices:

$$\hat{H}(k)=\mathbf{d}(k)\cdot \mathbf{\sigma }=d_x(k){\sigma }_x+d_y(k){\sigma }_y+d_z(k){\sigma }_z$$

where vector $\mathbf{d}(k)$ is a periodic function of momentum $k$.

As $k$ traverses the entire Brillouin zone (from $-\pi /a$ to $\pi /a$), vector $\mathbf{d}(k)$ traces a closed curve in three-dimensional space.

More importantly, if we focus on normalized vector $\hat{\mathbf{n}}(k)=\mathbf{d}(k)/|\mathbf{d}(k)|$, it defines a mapping from circle $S^1$ to unit sphere $S^2$ (or its equator circle $S^1$).

Definition 5.2 (Winding Number):

For one-dimensional systems, if Hamiltonian is constrained by chiral symmetry (i.e., $\mathbf{d}$ restricted to $xz$ plane), mapping degenerates to $S^1\to S^1$. We can define integer topological invariant $\mathcal{W}$:

$$\mathcal{W}=\frac{1}{2\pi }{\oint }_{BZ}\left({\hat{n}}_z\frac{d{\hat{n}}_x}{dk}-{\hat{n}}_x\frac{d{\hat{n}}_z}{dk}\right)dk$$

Intuitively, this is the number of times vector $\mathbf{d}(k)$ rotates around the origin.

5.2.3 Mass Generation Theorem

Now we state the core theorem of this chapter, establishing the necessary connection between topology and mass.

Theorem 5.2 (Topological Mass Generation Theorem):

In a local unitary QCA system, if an excited state has winding number $\mathcal{W}\ne 0$, then this excited state necessarily has non-zero rest mass (energy gap).

That is: Non-trivial topology $⟹$ massive particle.

Proof:

1. Proof by Contradiction: Assume particle is massless.

2. According to Definition 5.1, massless means evolution is pure translation, or can be continuously deformed to pure translation.

3. For pure translation operator ${\hat{U}}_{trans}=e^{ik{\hat{\sigma }}_z}$, corresponding $\mathbf{d}(k)$ vector always points in $z$-axis direction (north pole), or monotonically changes along $z$-axis with $k$ without rotating around origin.

4. In this case, winding number $\mathcal{W}=0$.

5. Since winding number is a discrete integer, it cannot change continuously with parameters. As long as Hamiltonian’s energy gap doesn’t close (i.e., $|\mathbf{d}(k)|\ne 0$), $\mathcal{W}$ is a topological invariant.

6. Therefore, if we want to construct a system with $\mathcal{W}\ne 0$ (e.g., $\mathbf{d}(k)$ winds once in $xz$ plane), it’s impossible to deform it into massless photons without going through phase transition (gap closing).

7. Non-zero energy gap $\min |\mathbf{d}(k)|>0$ is precisely rest mass $m_0$.

Physical Interpretation:

Winding number $\mathcal{W}$ is like a “dead knot” tied on the wave function.

• Photon ($\mathcal{W}=0$): A straight rope. It can smoothly flow through space.

• Electron ($\mathcal{W}=1$): A knotted rope. When you try to pull it, the knot must move as a whole. To untie this knot microscopically, you need extremely high energy (reaching Planck energy scale, collapsing lattice structure). Therefore, electrons are stable.

5.2.4 Geometric Meaning of Self-Reference

We call this structure “self-referential loop” because mathematically, it requires different components of wave function to mutually “see” each other.

In feedback loops described by Riccati equations (see Section 5.4), system output is fed back to input.

$$x_{next}=f(x_{now},x_{neighbor})$$

For massless particles, $x_{next}$ only depends on $x_{neighbor}$ (only neighbors tell me what happened).

For massive particles, $x_{next}$ strongly depends on $x_{now}$ (I must remember my state from previous moment).

This memory or consistency geometrically manifests as non-trivial covering of Brillouin zone. Particle “knows” it’s a whole because it “touches” its boundary in momentum space and finds it has wound around once.

5.2.5 Summary

We arrive at a startling conclusion: Matter is dead loops of information.

• The universe’s background is massless information flow (light-speed propagation).

• When local information flow undergoes topological entanglement, forming non-trivial winding numbers, information is forced to rotate in place.

• The frequency of this in-place rotation macroscopically manifests as mass.

• The topological robustness of this rotation macroscopically manifests as particle stability (charge conservation, baryon number conservation are essentially topological charge conservation).

We are not made of “atoms”; we are made of “knots tied by light rays.” In the next section, we will further explore how this “knot” responds to external forces, thereby explaining the essence of inertia.