Appendix C: Quantum Mechanics of Synchronicity — Semantic Entanglement and Resonance

In Volume II “Synchronicity” of the main text of Vector Cosmology VII, we proposed a view that transcends classical causality: Meaning is a physical force. Similar souls, similar thoughts, no matter how far apart in spacetime, attract each other due to resonance of internal structures.

This view is often misunderstood as psychological “survivor bias.” This appendix will provide a rigorous physical explanation for “synchronicity” based on Quantum Perturbation Theory and Hilbert Space Geometry. We will prove that in the quantum world, “Similarity” is indeed the fundamental cause of dramatic increases in “Interaction Cross-section”.

C.1 Semantic Distance: Neighbors in Hilbert Space

In physical space ($v_{ext}$), distance is defined by the Euclidean metric $d=\sqrt{\Delta x^2+\Delta y^2+\Delta z^2}$.

But in quantum state space ($v_{int}$), the definition of distance is completely different.

For two complex quantum systems A and B (such as two people’s states of consciousness), we define their Semantic Distance as the Fubini-Study angle in projective Hilbert space:

$$d_{sem}(A,B)=\arccos \left(\frac{|⟨{\psi }_A|{\psi }_B⟩|}{\sqrt{⟨{\psi }_A|{\psi }_A⟩⟨{\psi }_B|{\psi }_B⟩}}\right)$$

• Completely orthogonal ($⟨A|B⟩=0$)：$d_{sem}=\pi /2$. Even if two people sit face to face, if their thinking patterns are completely orthogonal (no common language), they are infinitely far apart in semantic space.

• Complete resonance ($⟨A|B⟩\approx 1$)：$d_{sem}\approx 0$. If two people’s thinking structures are highly similar (same frequency), they are coincident in semantic space.

Physical Corollary:

In quantum topology, interactions only occur between “neighbors”.

Since “resonant individuals” are neighbors in semantic space, the probability of quantum entanglement and information exchange between them is far higher than between physical neighbors.

C.2 Fermi’s Golden Rule and Resonant Enhancement

Why does similarity produce “attraction”? This can be explained using Fermi’s Golden Rule in quantum mechanics.

This rule describes the rate $\Gamma$ at which a system transitions from state $|i⟩$ to state $|f⟩$ (or two systems interact):

$${\Gamma }_{i\to f}=\frac{2\pi }{\hslash }|⟨f|H^{\prime }|i⟩{|}^2\rho (E_f)$$

where:

• $H^{\prime }$ is the interaction Hamiltonian.

• $\rho (E_f)$ is the final state density.

• $⟨f|H^{\prime }|i⟩$ is the Matrix Element.

For two people A and B trying to establish a connection (synchronicity event):

• If their frequencies ($v_{int}$ eigenvalues) don’t match (${\omega }_A\ne {\omega }_B$), the wave function phase oscillates rapidly, causing the integral $\int e^{i({\omega }_A-{\omega }_B)t}dt$ to approach zero. Interaction is suppressed.

• If their frequencies match (${\omega }_A\approx {\omega }_B$), the phase factor remains constant, the integral grows linearly with time, and the probability amplitude $|\dots {|}^2$ grows quadratically with time. Interaction is resonantly amplified.

Conclusion:

“Speak of the devil and he appears” is resonant tunneling of the wave function.

When you strongly miss someone, you adjust your $v_{int}$ to the same frequency band as theirs. This dramatically increases the interaction matrix element between you.

An encounter that originally had extremely low probability in physical space (chance event) becomes a statistical necessity due to the strong resonant channel in semantic space.

C.3 Geometry of Attractors in Phase Space

Finally, we use dynamical systems theory to describe “fate.”

If we regard cosmic evolution as a flow ${\Phi }_t$ in phase space, then an Attractor is the endpoint set $\mathcal{A}$ of the flow.

$$\underset{t\to \infty }{\lim }\text{dist}({\Phi }_t(x),\mathcal{A})=0$$

In Vector Cosmology, the Hamiltonian $H$ determines the terrain of phase space.

Since $H$ is locked by future boundary conditions (retro-causality), there exist preset “deep valleys” in phase space.

• Potential Landscape:

  Certain specific geometric configurations (such as “love,” “awakening,” “completion”) correspond to minima of the potential surface.

• Convergence:

  Regardless of initial conditions (wherever you were born, whatever you experienced), as long as your kinetic energy is insufficient to escape this gravitational basin, your trajectory will eventually spiral into that unique geometric shape.

This is the mathematical proof of “All roads lead to Rome”.

Fate is not a line; fate is a funnel.

As long as you are at the edge of this funnel, no matter which direction you run, gravity (yearning) will eventually bring you to the center of the funnel—that singularity called $\Omega$.