Appendix M: The Holographic Renormalization of History — The Past is Coarse-Grained Future

In Volume IV “Rewrite” of the main text of Vector Cosmology VII, we granted observers the power to “edit timelines.” This sounds like a subjective psychological operation, but in modern high-energy physics, it has a solid mathematical foundation: Holographic Renormalization.

This appendix will prove, based on AdS/CFT Duality and Renormalization Group Flow (RG Flow), that the passage of time is geometrically equivalent to “changes in resolution”.

The past is not a definite, detail-complete entity; it is a coarse-grained projection of the future (present) at low energy scales.

M.1 Time as RG Flow

In physics, the Renormalization Group (RG) describes how a physical system changes with observation scale (energy scale).

• Ultraviolet (UV): High energy, microscopic, detail-rich.

• Infrared (IR): Low energy, macroscopic, detail-blurred.

In holographic universe models, the radial dimension ($z$-axis) is interpreted as the direction of RG flow.

If we regard “now” ($t_{now}$) as the holographic boundary (Boundary, with the highest information density), then “past” ($t_{past}$) lies in the bulk interior (Bulk, deep regions with lower information density).

Theorem:

Retracing into the past is mathematically equivalent to performing a “coarse-graining” operation on the current wave function.

$$|{\Psi }_{past}⟩={\hat{O}}_{coarse}|{\Psi }_{now}⟩$$

This means:

History does not preserve all microscopic details (quantum phases). It only preserves “macroscopic thermodynamic statistics”.

Those lost phase information (details) were not “written in stone” in the past but exist in “undefined” degrees of freedom.

M.2 Mechanism of Retro-definition

Since the past is “coarse-grained,” it contains many “Degenerate States”.

That is, there are countless different microscopic histories that, after coarse-graining, appear as the same macroscopic history.

$$H_{macro}\leftarrow \{h_1,h_2,h_3,\dots \}$$

• Present Observer: Possesses the highest resolution ($v_{int}$).

• Operation: When you perform “retrospective observation” or “narrative reconstruction,” you are actually “fine-graining”.

  You select a specific $h_k$ from that set of degenerate microscopic histories $\{h_i\}$ and endow it with “reality”.

Physical Picture:

You are not “modifying” a history book that has already been written.

You are facing a book with “only chapter titles, no body text” (coarse-grained past).

Your present recollection and definition are “filling in the body text” (fine-grained details).

As long as your filling does not violate the macroscopic titles (the occurred $v_{ext}$ facts), you physically possess absolute editing rights over the details.

M.3 The Omega Point as a Fixed Point

Finally, we push this RG flow to its limit.

If time is renormalization flow, what is the endpoint of the flow?

It is a Fixed Point.

• Big Bang ($t=0$): The Ultraviolet Fixed Point (UV Fixed Point). It is the highest energy, smallest scale, chaotic starting point.

• $\Omega$ Point ($t=\infty$): The Infrared Fixed Point (IR Fixed Point).

In conformal field theory, infrared fixed points typically correspond to a critical state with scale invariance.

This means that when the universe evolves to the $\Omega$ point, all coarse-graining processes are complete.

All information is integrated into a perfect, self-similar fractal structure.

Conclusion:

Each of our present choices fine-tunes the trajectory of the RG flow, determining which fixed point basin in phase space it ultimately flows toward.

History is not inertia pushing us forward.

History is the stepping stones we lay to reach that fixed point.