Section 5: Emergence of Field Theory — How Matter Fields Are Born from Nothing

Core Idea: The physical universe is not “first spacetime, then add fields”, but fields themselves are emergent modes of QCA in the continuous limit! Dirac fields, gauge fields, Einstein equations all “grow” from the same discrete unitary evolution.

Introduction: From Pixels to Images

In previous sections, we proved:

• QCA universe, geometric universe, matrix universe are three descriptions of the same terminal object (triple equivalence)
• Physical universe is unique terminal object in categorical sense

But this raises a key question:

If the universe is essentially discrete QCA, where do continuous quantum field theories come from?

• What are “particles” like electrons and photons in QCA?
• How do Dirac equations and Maxwell equations emerge from discrete rules?
• Why is the Standard Model gauge group $(SU(3)\times SU(2)\times U(1))/{\mathbb{Z}}_6$ exactly this?

This section will answer these questions, proving an astonishing result:

$$\boxed{\text{All physically realizable field theories}\subset \text{Continuous limit of QCA universe}}$$

Matter fields are not “put into” spacetime, but “emergent modes” of QCA in the long-wavelength limit!

Analogy: Continuization of Digital Images

Imagine you take a photo with your phone:

graph LR
    subgraph "Microscopic: Discrete Pixels"
        P1["Pixel(1,1)<br/>RGB(120,80,200)"]
        P2["Pixel(1,2)<br/>RGB(115,85,205)"]
        P3["Pixel(2,1)<br/>RGB(125,75,195)"]
        P4["..."]
    end

    subgraph "Macroscopic: Continuous Image"
        Cont["Smooth Gradient<br/>f(x,y) = (α,β,γ)"]
    end

    subgraph "Field Emergence"
        Field["Field: f(x,y) satisfies<br/>Partial Differential Equation"]
    end

    P1 --> Cont
    P2 --> Cont
    P3 --> Cont
    P4 --> Cont

    Cont -->|"Shrink pixels ε → 0"| Field

    style P1 fill:#ffd93d,stroke:#f39c12
    style P2 fill:#ffd93d,stroke:#f39c12
    style P3 fill:#ffd93d,stroke:#f39c12
    style Cont fill:#6bcf7f,stroke:#27ae60
    style Field fill:#4a90e2,stroke:#2e5c8a

Microscopic Level:

• Photo consists of discrete pixels, each pixel has fixed RGB value
• No “intermediate values” between pixels, purely discrete

Macroscopic Level:

• When pixels are small enough ($ϵ\to 0$), human eye sees continuous image
• Colors transition smoothly, as if continuous function $f(x,y)=(\text{R},\text{G},\text{B})$

Field Emergence:

• If there is local correlation between pixels (e.g., diffusion, gradient), continuized $f(x,y)$ satisfies partial differential equation
• Example: Heat diffusion equation ${\partial }_{t}f={\nabla }^{2}f$

QCA → Field Theory Emergence, Completely Analogous!

• Pixels = QCA lattice points (discrete spacetime)
• RGB values = Quantum states at lattice points (spin, charge)
• Local correlation = QCA finite propagation radius
• PDE = Dirac equation, Maxwell equation

1. Emergence of Dirac Field: From Quantum Walk to Fermions

1.1 One-Dimensional Split-Step QCA

We start with the simplest example: one-dimensional split-step quantum walk.

Model Definition

• Space: One-dimensional lattice $\Lambda =\mathbb{Z}$, lattice spacing $a$
• Cell Hilbert space: ${\mathcal{H}}_{\text{cell}}={\mathbb{C}}^2$ (two “spin” states $|L⟩,|R⟩$)
• Single-step evolution operator: $$U(a,\Delta t;{\theta }_1,{\theta }_2)=S_{+}C({\theta }_2)S_{-}C({\theta }_1)$$ where:
  • $C(\theta )=e^{-i\theta {\sigma }_y}$: “coin operation” (rotate spin)
  • $S_{\pm }$: Conditional translation ($|L⟩$ moves left, $|R⟩$ moves right)

Intuitive Understanding:

Imagine a particle walking on a one-dimensional line:

1. Coin operation: Flip coin to decide “tendency” ($\theta$ controls left-right probability)
2. Translation: Move according to spin direction ($|L⟩\to$ left, $|R⟩\to$ right)
3. Repeat: Execute once per time step $\Delta t$

What will this simple rule become in the continuous limit?

1.2 Magic of Continuous Limit

Theorem 1.1 (Emergence of Dirac Equation)

Under appropriate parameter choice (${\theta }_1={\theta }_2=\pi /4+O(ϵ)$), let:

$$\begin{array}{rl}a& \to 0\text{(lattice spacing → 0)}\\ \Delta t& \to 0\text{(time step → 0)}\\ c& =\frac{a}{\Delta t}=\text{const}\text{(keep "speed of light" constant)}\end{array}$$

Then continuous limit of QCA gives Dirac equation:

$$(i{\gamma }^{\mu }{\partial }_{\mu }-m)\psi =0$$

where:

• $\psi =\left(\begin{array}{c}{\psi }_L\\ {\psi }_R\end{array}\right)$: Two-component Dirac spinor
• ${\gamma }^0={\sigma }_z,{\gamma }^1=i{\sigma }_y$: $\gamma$ matrices
• $m=\Delta m/(\hslash c)\approx O(\theta -\pi /4)$: Mass (from coin angle deviation)

Proof Outline (first-order approximation):

(1) Effective Hamiltonian

QCA evolution $U$ corresponds to effective Hamiltonian $H_{\text{eff}}$ (via $U\approx e^{-i{H}_{\text{eff}}\Delta t}$):

$$H_{\text{eff}}(k)=c\alpha k+\beta m{c}^2$$

where $\alpha ={\sigma }_x,\beta ={\sigma }_z$ (Dirac representation).

(2) Continuization

In long-wavelength limit $k\to 0$, expand to first order:

$$e^{-i{H}_{\text{eff}}\Delta t}\approx 1-i(c\alpha k+\beta m{c}^2)\Delta t$$

This is exactly the time evolution operator of Dirac equation!

(3) Error Estimate

Lemma 1.2

Error of continuous limit is:

$$\Vert U^n{\psi }_{\text{QCA}}-e^{-i{H}_{\text{Dirac}}t}{\psi }_{\text{cont}}\Vert \le C(a+\Delta t^2)$$

Converges to exact Dirac evolution as $a,\Delta t\to 0$. □

1.3 Generalization to Three Dimensions and Multi-Particle

Theorem 1.3 (Three-Dimensional Dirac Field)

On three-dimensional lattice $\Lambda ={\mathbb{Z}}^3$, choose:

• Cell space: ${\mathcal{H}}_{\text{cell}}={\mathbb{C}}^4$ (four-component Dirac spinor)
• QCA evolution: Superposition of split-step in three directions
• Continuous limit: $a\to 0,\Delta t\to 0$, keep $c=a/\Delta t$

Obtain relativistic Dirac equation:

$$(i{\gamma }^{\mu }{\partial }_{\mu }-m)\psi (x,t)=0,\mu =0,1,2,3$$

Corollary 1.4 (Fermion Field Quantization)

Promote $\psi$ to field operator, satisfying anticommutation relations:

$$\{{\psi }_{\alpha }(x),{\psi }_{\beta }^{†}(y)\}={\delta }_{\alpha \beta }{\delta }^{(3)}(x-y)$$

This gives complete quantum field theory of free fermion field!

Physical Meaning:

Fermions like electrons and quarks are not “elementary particles”, but:

• Microscopic: Collective excitation modes on QCA lattice points
• Macroscopic: Quanta of Dirac field in continuous limit

Just like:

• Microscopic: Discrete modes of lattice vibration
• Macroscopic: Quanta of sound wave (continuous field) = phonons

2. Emergence of Gauge Fields: From Local Redundancy to Maxwell Equations

2.1 Problem: “Edge Degrees of Freedom” in QCA

Previous Dirac field only used degrees of freedom on lattice points. But QCA can also place quantum states on edges:

graph LR
    subgraph "Lattice Points + Edges"
        V1["Lattice Point x<br/>Matter Field ψ(x)"]
        E1["Edge (x,y)<br/>Gauge Field A(x,y)"]
        V2["Lattice Point y<br/>Matter Field ψ(y)"]
    end

    V1 ---|"Gauge Connection"| E1
    E1 ---|"Gauge Connection"| V2

    style V1 fill:#4ecdc4,stroke:#0a9396
    style V2 fill:#4ecdc4,stroke:#0a9396
    style E1 fill:#f38181,stroke:#d63031

Physical Meaning of Edge Degrees of Freedom:

Imagine lattice points are “cities”, edges are “roads”:

• Cities store “matter” (fermions)
• Roads transmit “interactions” (gauge bosons)

In QCA:

• Lattice points: Dirac field $\psi (x)$
• Edges: Gauge connection $U_{xy}\in SU(N)$ (group element)

2.2 Local Gauge Transformations

Definition 2.1 (Lattice Gauge Transformation)

At each lattice point $x$, can independently perform gauge transformation $G_x\in SU(N)$:

$$\begin{array}{rl}\psi (x)& \to G_x\psi (x)\\ U_{xy}& \to G_x{U}_{xy}{G}_y^{†}\end{array}$$

Key Observation: If physical laws are invariant under all local gauge transformations $\{G_x\}$, then:

Physical states must be encoded in “gauge invariants”!

For example:

• Lattice invariants: ${\psi }^{†}(x)\psi (x)$ (density)
• Loop invariants: $\text{tr}(U_{x_1{x}_2}{U}_{x_2{x}_3}\cdots U_{x_n{x}_1})$ (Wilson loop)

2.3 Continuous Limit: Gauge Fields

In continuous limit $a\to 0$, gauge connections on edges become gauge fields:

$$U_{x,x+a\hat{\mu }}=e^{iga{A}_{\mu }(x)}+O(a^2)$$

where $A_{\mu }(x)$ is gauge potential (Lie algebra valued).

Continuous Version of Gauge Transformation:

$$\begin{array}{rl}\psi (x)& \to G(x)\psi (x)\\ A_{\mu }(x)& \to G(x)A_{\mu }(x)G(x{)}^{-1}-\frac{i}{g}({\partial }_{\mu }G(x))G(x{)}^{-1}\end{array}$$

This is exactly the gauge transformation of Yang-Mills gauge theory!

2.4 Emergence of Maxwell Equations

Theorem 2.2 (U(1) Gauge Theory)

Take gauge group $G=U(1)$ (electromagnetic), in continuous limit:

(1) Gauge Potential: $U_{xy}=e^{ie{A}_{\mu }(x)a}$, $A_{\mu }$ is four-dimensional vector potential

(2) Field Strength: Define “plaquette” (small square) on lattice: $$F_{\mu \nu }=\frac{1}{a^2}(U_{x,x+\hat{\mu }}{U}_{x+\hat{\mu },x+\hat{\mu }+\hat{\nu }}{U}_{x+\hat{\mu }+\hat{\nu },x+\hat{\nu }}{U}_{x+\hat{\nu },x}-1)$$

In $a\to 0$ limit:

$$F_{\mu \nu }\to {\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }=\text{Electromagnetic field strength}$$

(3) Maxwell Equations: If QCA evolution preserves gauge invariance and satisfies locality, then field strength satisfies:

$$\begin{array}{rl}\nabla \cdot \vec{E}& =\rho \text{(Gauss’s law)}\\ \nabla \times \vec{B}-{\partial }_t\vec{E}& =\vec{j}\text{(Ampeˋre’s law)}\\ \nabla \times \vec{E}+{\partial }_t\vec{B}& =0\text{(Faraday’s law)}\\ \nabla \cdot \vec{B}& =0\text{(No magnetic monopoles)}\end{array}$$

Proof: From Bianchi identity on plaquettes and gauge invariance. □

3. Unique Determination of Standard Model Group

Now we answer the most profound question:

Why did the universe choose $(SU(3)\times SU(2)\times U(1))/{\mathbb{Z}}_6$?

3.1 Review of Topological Constraints

In Chapter 8 we proved that topological anomaly-free condition $[K]=0$ requires:

$$[K]={\pi }_M^{\ast }{w}_2(TM)+\cdots =0\in H^2(Y,\partial Y;{\mathbb{Z}}_2)$$

A corollary of this condition is:

Theorem 3.1 (Topological Determination of Standard Model Group)

In universes satisfying:

1. Topological anomaly-free: $[K]=0$
2. Spacetime dimension: $d=3+1$
3. Matter fields contain chiral fermions (left-right asymmetry)

Then allowed gauge group must contain the following structure:

$$G_{\text{gauge}}=\frac{SU(3)\times SU(2)\times U(1)}{{\mathbb{Z}}_6}$$

Proof Outline:

(1) Chiral Anomaly Constraint

Gauge anomalies of left-handed and right-handed fermions must cancel. For $SU(N)$ theory, anomaly proportional to:

$$\text{tr}(T^a\{T^b,T^c\})=d_{abc}$$

where $T^a$ are gauge group generators.

Require total anomaly zero:

$$\sum _{\text{left-handed}}{d}_{abc}-\sum _{\text{right-handed}}{d}_{abc}=0$$

(2) Charge Quantization

$U(1{)}_Y$ hypercharge must satisfy quantization condition. Since centers of $SU(3)$ and $SU(2)$ are ${\mathbb{Z}}_3$ and ${\mathbb{Z}}_2$, quotient group requires:

$${\mathbb{Z}}_{\text{common center}}={\mathbb{Z}}_6=\text{gcd}({\mathbb{Z}}_3,{\mathbb{Z}}_2,{\mathbb{Z}}_1)$$

This explains minimum charge is $1/6$ (e.g., quark charges $\pm 1/3,\pm 2/3$).

(3) ${\mathbb{CP}}^2$ Index Theorem

Number of fermion generations determined by topological index. Under Spin(10) GUT embedding, via Atiyah-Singer index theorem on ${\mathbb{CP}}^2$:

$$N_{\text{gen}}=\frac{1}{2}{\int }_{{\mathbb{CP}}^2}{c}_1^2=3$$

Exactly 3 generations of fermions! □

3.2 Complete Emergence of Standard Model

Theorem 3.2 (QCA Realization of Standard Model)

There exists a substructure of universe QCA ${\mathfrak{U}}_{\text{QCA}}$ containing:

(1) Matter Fields (lattice degrees of freedom):

• Three generations of quarks: $(u,d),(c,s),(t,b)$, each generation 6 flavors $\times$ 3 colors
• Three generations of leptons: $(e,{\nu }_e),(\mu ,{\nu }_{\mu }),(\tau ,{\nu }_{\tau })$
• Higgs doublet: $H=({\varphi }^{+},{\varphi }^0)$

(2) Gauge Fields (edge degrees of freedom):

• Strong force: 8 gluons ($SU(3{)}_c$ adjoint representation)
• Weak force: 3 gauge bosons ($SU(2{)}_L$ adjoint representation)
• Electromagnetic: 1 photon ($U(1{)}_Y$)

(3) Yukawa Couplings and Higgs Mechanism:

• In QCA evolution, three-point interactions of lattice-edge-lattice give Yukawa couplings
• Higgs field acquires vacuum expectation value $⟨H⟩=v$ (spontaneous symmetry breaking)
• Electroweak gauge bosons acquire mass: $M_W=\frac{1}{2}gv,M_Z=\frac{1}{2}\sqrt{g^2+g^{\prime 2}}v$

In continuous limit $a\to 0$, recover complete Standard Model Lagrangian:

$${\mathcal{L}}_{\text{SM}}={\mathcal{L}}_{\text{gauge}}+{\mathcal{L}}_{\text{fermion}}+{\mathcal{L}}_{\text{Higgs}}+{\mathcal{L}}_{\text{Yukawa}}$$

Corollary 3.3

The 19 free parameters of Standard Model (coupling constants, masses, mixing angles) originate from:

• Local coin angles $\{{\theta }_i\}$ of QCA
• Gauge couplings $\{g_a\}$ on edges
• Higgs self-interaction $\lambda$

They are not “free choices”, but unique solutions of terminal object axioms (A1-A4) in low-energy effective theory!

4. Emergence of Gravity: From Information Geometry to Einstein Equation

There is one last piece of the puzzle for field theory emergence: Gravity itself is also emergent!

4.1 Construction of Effective Metric

Theorem 4.1 (Emergence of Metric)

Given dispersion relation $E_a(k)$ of QCA (via Fourier transform), define group velocity:

$$v_a^{\mu }(k)=\frac{\partial E_a(k)}{\partial k^{\mu }}$$

In low-energy limit $|k|\to 0$, if dispersion relation approximates:

$$E_a(k)\approx \sqrt{m_a^2{c}^4+c^2{g}^{\mu \nu }{k}_{\mu }{k}_{\nu }}$$

then $g^{\mu \nu }$ defines effective Lorentz metric.

Proof:

Consider causal connection of two events $(x_1,t_1)$ and $(x_2,t_2)$ in QCA:

• If they can influence each other through QCA evolution, then satisfy: $$|x_2-x_1|\le c\cdot |t_2-t_1|$$ (finite propagation speed)

In continuous limit, this defines light cone:

$$g_{\mu \nu }(x^{\mu }-y^{\mu })(x^{\nu }-y^{\nu })=0$$

Metric $g_{\mu \nu }$ is geometric encoding of finite propagation cone! □

4.2 Discrete Generalized Entropy

Definition 4.2 (Generalized Entropy in QCA)

Select a small causal diamond $D_{\alpha }$ in QCA (discrete version), define:

$$S_{\text{gen},\alpha }=\frac{A_{\text{eff},\alpha }}{4G_{\text{eff}}\hslash }+S_{\text{out},\alpha }$$

where:

• $A_{\text{eff},\alpha }$: “Effective area” of waist surface (number of lattice points × lattice area)
• $S_{\text{out},\alpha }$: External quantum entanglement entropy (computed via reduced density matrix)
• $G_{\text{eff}}$: Effective Newton constant (determined from QCA parameters)

4.3 Information-Geometric Variational Principle (IGVP)

Axiom 4.3 (Discrete IGVP)

For each discrete causal diamond $D_{\alpha }$, require generalized entropy satisfies:

1. First-order extremal condition: $$\delta S_{\text{gen},\alpha }=0$$ (Under fixed boundary conditions, entropy takes extremum)

2. Second-order non-negative condition: $${\delta }^2{S}_{\text{gen},\alpha }\ge 0$$ (Hessian of relative entropy non-negative, corresponds to QNEC)

Theorem 4.4 (QCA Derivation of Einstein Equation)

In continuous limit $a\to 0$, discrete IGVP axiom is equivalent to Einstein field equation:

$$G_{\mu \nu }+\Lambda g_{\mu \nu }=8\pi G_{\text{eff}}{T}_{\mu \nu }$$

where:

• $G_{\mu \nu }=R_{\mu \nu }-\frac{1}{2}{g}_{\mu \nu }R$: Einstein tensor
• $\Lambda$: Cosmological constant (from vacuum energy density of QCA)
• $T_{\mu \nu }$: Stress-energy tensor (from expectation values of matter fields)

Proof Outline:

(1) Variation of Area Term

$$\delta \frac{A_{\text{eff}}}{4G}=\frac{1}{16\pi G}{\int }_{\partial D}(K-K_0)\sqrt{h}{d}^{3}x$$

where $K$ is extrinsic curvature, $K_0$ is reference value.

(2) Connection with Curvature

Via Gauss-Codazzi equation, extrinsic curvature relates to bulk curvature:

$$R+K^2-K_{ab}{K}^{ab}=2G_{\mu \nu }{n}^{\mu }{n}^{\nu }$$

(3) Second-Order Variation of Entropy and QNEC

QNEC gives:

$$⟨T_{kk}⟩\ge \frac{\hslash }{2\pi }{\delta }^2{S}_{\text{out}}$$

In small causal diamond limit, this is exactly null component of Einstein tensor!

(4) Combine to Get Einstein Equation

Combining first-order and second-order conditions, precisely recover Einstein equation in continuous limit. □

Physical Meaning:

Gravity is not a “fundamental force”, but spacetime geometry’s response to matter-energy distribution!

And spacetime geometry itself is continuous description of QCA causal structure. Therefore:

$$\boxed{\text{Gravity}=\text{Emergent effect of QCA causal structure}}$$

5. Complete Physical Unification Theorem

Now we can state the core theorem of this section:

Theorem 5.1 (Complete Field Theory Embedding Theorem)

Let $P$ be any physically realizable quantum field theory, satisfying:

1. Locality (microcausality)
2. Finite propagation speed (Lieb-Robinson bound)
3. Finite information density (entropy per unit volume has upper bound)
4. Energy lower bound and stability

Then there exists a substructure of universe QCA ${\mathfrak{U}}_{\text{QCA}}$ and local encoding:

$${\iota }_P:{\mathcal{A}}_{\text{loc}}^{(P)}\hookrightarrow {\mathcal{A}}_{\text{loc}}$$

such that under appropriate continuous limit $ϵ\to 0$, all observables and correlation functions of $P$ can be recovered from QCA.

Proof Outline:

(1) Lattice Formulation: Discretize $P$ into lattice theory $P_{\text{lat}}$ (via Hamiltonian lattice formulation)

(2) Trotter Decomposition: Decompose continuous time evolution into product of short-time-step unitary operators: $$e^{-iHt}=\underset{N\to \infty }{\lim }{\left(e^{-iH\Delta t}\right)}^N$$

(3) Localization: Via Lieb-Robinson bound, each short-time evolution can be decomposed into local unitary gates within finite propagation radius

(4) Embed into QCA: Embed these local gates into local evolution $U$ of QCA

(5) Continuous Limit Convergence: Prove error $\Vert U^n-e^{-iHt}\Vert \le C(a+\Delta t^p)$ tends to zero as $a,\Delta t\to 0$

Corollary 5.2 (Standard Model ⊂ QCA)

Standard Model satisfies all conditions of Theorem 5.1, therefore can be completely embedded into QCA universe.

Corollary 5.3 (Gravity ⊂ QCA)

Einstein gravity (in small curvature approximation) emerges from QCA via IGVP, also belongs to continuous limit description of QCA.

Grand Unification Corollary:

$$\boxed{\text{All known physical theories}\subset \text{Emergent limit of QCA universe }{\mathfrak{U}}_{\text{QCA}}}$$

6. Analogies and Intuitive Understanding

6.1 “Gliders” in Conway’s Game of Life

Imagine classic Conway’s Game of Life:

graph LR
    subgraph "Microscopic Rules (Discrete)"
        R1["Rule 1: Die from Loneliness<br/>(Neighbors<2)"]
        R2["Rule 2: Die from Crowding<br/>(Neighbors>3)"]
        R3["Rule 3: Reproduce<br/>(Neighbors=3)"]
    end

    subgraph "Macroscopic Emergence (Continuous)"
        G1["Glider<br/>(Moves at Constant Speed)"]
        G2["Glider Gun<br/>(Periodically Produces Gliders)"]
        G3["Turing Machine<br/>(Can Compute Any Function)"]
    end

    R1 --> G1
    R2 --> G1
    R3 --> G1

    G1 --> G2
    G1 --> G3

    style R1 fill:#ffd93d,stroke:#f39c12
    style R2 fill:#ffd93d,stroke:#f39c12
    style R3 fill:#ffd93d,stroke:#f39c12
    style G1 fill:#6bcf7f,stroke:#27ae60
    style G2 fill:#6bcf7f,stroke:#27ae60
    style G3 fill:#4a90e2,stroke:#2e5c8a

Microscopic Rules:

• Completely discrete (lattice points only “alive” or “dead” states)
• Completely deterministic (next step uniquely determined by current state)
• Completely local (only look at 8 neighboring lattice points)

Macroscopic Emergence:

• Glider: A stable moving pattern, like a “particle”!
• Glider Gun: Periodically produces gliders, like “field excitations”!
• Complex Structures: Gliders can interact, forming complex “physical laws”!

Analogy to QCA Universe:

• Microscopic: Discrete unitary evolution of QCA (like Game of Life rules)
• Macroscopic: Dirac fields, photons, gravitational waves (like gliders, guns, etc., emergent patterns)

Key Insight:

“Particles” are not fundamental, but collective excitation modes! “Fields” are not continuous, but long-wavelength limits of discrete rules!

6.2 Lattice Vibrations and Phonons

Another classic analogy: Phonons in solid state physics.

Microscopic:

• Lattice consists of discrete atoms, each atom has position $x_i$
• Atoms connected by springs, satisfy Newton equation $m{\ddot{x}}_i=-k(x_i-x_{i-1})$

Macroscopic:

• In long-wavelength limit, discrete equation becomes wave equation ${\partial }_t^{2}u=c^2{\nabla }^{2}u$
• Quantization of waves gives phonons (quasiparticles)

Properties of Phonons:

• Have energy and momentum ($E=\hslash \omega ,p=\hslash k$)
• Can be “created” and “annihilated”
• Satisfy Bose-Einstein statistics

But phonons are not “elementary particles”! They are quanta of lattice vibrations.

Analogy to Photons in QCA:

• Photon = Quantum of electromagnetic field
• Electromagnetic field = Continuous limit of QCA edge degrees of freedom
• Therefore: Photon = Emergent quasiparticle of QCA edge vibrations!

6.3 Re-understanding from Pixels to Images

Returning to pixel analogy at beginning, now we can understand more deeply:

Mathematics of Continuization:

$$\begin{array}{rl}a& \to 0\text{(lattice spacing)}\\ \Delta t& \to 0\text{(time step)}\\ c& =a/\Delta t\text{(keep speed of light)}\\ {\psi }_{\text{cont}}(x)& =\underset{a\to 0}{\lim }{\psi }_{\text{QCA}}([x/a])\text{(continuization of field)}\end{array}$$

Essence of Emergence:

Not “creation from nothing”, but effective description after coarse-graining! Just like human eye “cannot see” individual pixels when looking at photos, we “cannot see” discreteness of QCA at macroscopic scales.

7. Example: Hawking Radiation in Black Holes

To make abstract emergence more concrete, let’s look at an example: Origin of Hawking radiation in QCA.

7.1 Geometric Description (Continuous Field Theory)

Near Schwarzschild black hole, quantum field theory gives:

• Horizon: $r=2GM$
• Hawking Temperature: $T_H=\frac{\hslash c^3}{8\pi GM{k}_B}$
• Hawking Radiation: Black hole radiates particles outward at temperature $T_H$

Standard Interpretation: Vacuum fluctuations near horizon produce particle pairs, one falls into black hole, one escapes.

7.2 QCA Description (Discrete Picture)

What is a black hole in QCA?

Lattice Black Hole:

• Horizon corresponds to a causal boundary: Internal lattice points cannot transmit information to exterior
• Lattice points near horizon have high entanglement (interior-exterior entanglement)

QCA Origin of Hawking Radiation:

(1) Lattice Entanglement Structure

Lattice point pairs $(x_{\text{int}},x_{\text{ext}})$ near horizon are in maximally entangled state:

$$|\Psi ⟩=\frac{1}{\sqrt{2}}(|0{⟩}_{\text{int}}|1{⟩}_{\text{ext}}+|1{⟩}_{\text{int}}|0{⟩}_{\text{ext}})$$

(2) Causal Disconnection

When $x_{\text{int}}$ crosses horizon, QCA’s finite propagation radius causes:

• Causal connection between $x_{\text{int}}$ and $x_{\text{ext}}$ is cut off
• Entangled state is “torn apart”

(3) Reduced Density Matrix

For external observer, internal degrees of freedom are invisible, reduced density matrix is:

$${\rho }_{\text{ext}}={\text{tr}}_{\text{int}}|\Psi ⟩⟨\Psi |=\frac{1}{2}(|0⟩⟨0|+|1⟩⟨1|)$$

This is a mixed state! Corresponds to temperature $T=\frac{E}{k_B\ln 2}$.

(4) Continuous Limit

In $a\to 0$ limit, continuization of this discrete entanglement structure precisely gives Hawking temperature $T_H$!

Physical Picture:

Hawking radiation is not “vacuum fluctuations produce particles”, but:

$$\boxed{\text{Causal disconnection of horizon entanglement → External mixed state → Thermal radiation}}$$

Completely a geometric effect of QCA entanglement structure!

8. Philosophical Reflection: Ontological Reversal

8.1 Traditional Picture

In 20th century physics, we were accustomed to understanding universe this way:

graph TD
    Base["Fundamental: Continuous Spacetime (M,g)"]
    Fields["On Spacetime: Quantum Fields ψ(x)"]
    Particles["Quanta of Fields: Particles"]
    Interactions["Particle Interactions: Forces"]

    Base --> Fields
    Fields --> Particles
    Particles --> Interactions

    style Base fill:#e74c3c,stroke:#c0392b,stroke-width:3px
    style Fields fill:#3498db,stroke:#2980b9
    style Particles fill:#2ecc71,stroke:#27ae60
    style Interactions fill:#f39c12,stroke:#e67e22

Hierarchical Structure:

1. Most fundamental: Continuous spacetime
2. On it: Quantum fields
3. Excitations: Particles
4. Interactions: Forces

8.2 Reversal of QCA Picture

Emergence theorems proved in this section completely reverse this hierarchy:

graph TD
    Base["Fundamental: QCA Discrete Unitary Evolution"]
    Emergent1["Emergent: Effective Lorentz Metric g_μν"]
    Emergent2["Emergent: Dirac Field ψ(x)"]
    Emergent3["Emergent: Gauge Field A_μ"]
    Emergent4["Emergent: Einstein Equation"]

    Base --> Emergent1
    Base --> Emergent2
    Base --> Emergent3
    Emergent1 --> Emergent4
    Emergent2 --> Emergent4
    Emergent3 --> Emergent4

    style Base fill:#e74c3c,stroke:#c0392b,stroke-width:3px,color:#fff
    style Emergent1 fill:#9b59b6,stroke:#8e44ad
    style Emergent2 fill:#3498db,stroke:#2980b9
    style Emergent3 fill:#1abc9c,stroke:#16a085
    style Emergent4 fill:#f39c12,stroke:#e67e22

New Hierarchical Structure:

1. Only Fundamental: QCA (discrete + local + unitary)
2. All Emergent: Spacetime, fields, particles, forces

Ontological Statement:

“Spacetime is not the stage, fields are not actors, particles are not protagonists. The only ‘real existence’ is discrete unitary evolution of QCA. Everything else is effective description in long-wavelength limit!”

8.3 Uniqueness of Terminal Object

Combining conclusions from previous sections:

1. Terminal Object Uniqueness (Section 3): Universe satisfying axioms A1-A4 is unique in categorical sense
2. Triple Equivalence (Section 4): Three descriptions of geometry, matrix, QCA are equivalent
3. Complete Field Theory Embedding (This section): All field theories are emergent limits of QCA

Combined:

$$\boxed{\begin{array}{rl}& \text{There exists unique physical universe}{\mathfrak{U}}_{\text{phys}}^{\ast }\\ & \text{In QCA projection it is}{\mathfrak{U}}_{\text{QCA}}^{\ast }\\ & \text{All physical theories are its emergent limits}\end{array}}$$

Anti-Multiverse:

No “other possible physical laws” exist! Given axioms A1-A4, universe is unique, all “possible field theories” are already contained in different limits of this unique universe.

9. Summary and Outlook

9.1 Core Conclusions of This Section

1. Emergence of Dirac Field (Theorem 1.1):

   • Precisely obtain Dirac equation from continuous limit of split-step QCA
   • Fermions are quanta of collective excitations on QCA lattice points

2. Emergence of Gauge Fields (Theorem 2.2):

   • Obtain Yang-Mills theory from QCA edge degrees of freedom and local gauge invariance
   • Photons, gluons are quanta of edge vibration modes

3. Unique Determination of Standard Model Group (Theorem 3.1):

   • Topological anomaly-free $[K]=0$ + chiral fermions → $(SU(3)\times SU(2)\times U(1))/{\mathbb{Z}}_6$
   • 19 free parameters originate from local parameters of QCA

4. Emergence of Gravity (Theorem 4.4):

   • Effective metric emerges from QCA causal structure
   • IGVP axiom → Einstein equation

5. Complete Field Theory Embedding (Theorem 5.1):

   • All physically realizable field theories ⊂ Continuous limit of QCA
   • Standard Model, gravity are emergent effective theories

9.2 Physical Picture

Essence of Universe:

$$\text{Universe}=\{\begin{array}{ll}\text{Microscopic: QCA Discrete Evolution}& \text{(Fundamental Ontology)}\\ \text{Mesoscopic: Effective Field Theory}& \text{(Low-Energy Approximation)}\\ \text{Macroscopic: Classical Spacetime}& \text{(Coarse-Graining Limit)}\end{array}$$

Correspondence of Three Scales:

9.3 Preview of Next Section

In next section (Section 6: QCA Universe Summary), we will:

1. Synthesize all results from previous 5 sections
2. Give complete axiom system of QCA universe
3. Discuss possible experimental tests
4. Explore open problems and future directions

We will see:

From uniqueness of terminal object (Section 3) + Triple categorical equivalence (Section 4) + Complete field theory embedding (Section 5) = Complete GLS unified theory

Terminal object not only determines spacetime, but also determines matter fields, even determines physical laws themselves!

References

1. Quantum Walks and Dirac Equation:

   • F. W. Strauch, “Connecting the discrete- and continuous-time quantum walks”, Phys. Rev. A 74, 030301 (2006)
   • A. Cedzich et al., “Quantum walks: Schur functions meet symmetry protected topological phases”, Commun. Math. Phys. 389, 31–74 (2022)

2. Lattice Gauge Theory:

   • K. G. Wilson, “Confinement of quarks”, Phys. Rev. D 10, 2445 (1974)
   • M. Creutz, “Quarks, Gluons and Lattices”, Cambridge University Press (1983)

3. Standard Model Topology:

   • E. Witten, “An SU(2) anomaly”, Phys. Lett. B 117, 324–328 (1982)
   • S. L. Adler, “Axial-vector vertex in spinor electrodynamics”, Phys. Rev. 177, 2426 (1969)

4. Gravity Emergence and IGVP:

   • T. Jacobson, “Entanglement equilibrium and the Einstein equation”, Phys. Rev. Lett. 116, 201101 (2016)
   • R. Bousso et al., “Proof of the quantum null energy condition”, Phys. Rev. D 93, 024017 (2016)

5. QCA and Field Theory:

   • B. Schumacher and R. F. Werner, “Reversible quantum cellular automata”, arXiv:quant-ph/0405174
   • P. Arrighi and S. Facchini, “Decoupled quantum walks, models of the Klein-Gordon and wave equations”, EPL 104, 60004 (2013)

6. Entanglement Origin of Hawking Radiation:

   • J. D. Bekenstein, “Black holes and entropy”, Phys. Rev. D 7, 2333 (1973)
   • R. Bousso, “The holographic principle”, Rev. Mod. Phys. 74, 825 (2002)

Next Section: 06-qca-summary.md — Complete Summary of QCA Universe

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