05. Emergence of Objective Reality: From Quantum Substrate to Classical World

“Objective reality” is not a priori existence, but a macroscopic effective description emerging from the quantum substrate under appropriate limits.

Introduction: The Nature of Reality

Ancient Philosophical Questions

Human inquiry into “Reality” runs throughout the history of philosophy:

• Plato: The world of ideas is real, the sensory world is but shadows
• Aristotle: Substance is an independently existing substrate
• Descartes: I think, therefore I am—the certainty of the subject precedes the object
• Kant: Thing-in-itself (Ding an sich) is unknowable, we can only know phenomena
• Heidegger: Being (Sein) precedes beings (Seiende)

Modern physics makes this question even sharper:

Quantum mechanics tells us:

• Particles are in superposition states before measurement
• Measurement “creates” definite reality
• Observer and observed cannot be separated

Relativity tells us:

• “Simultaneity” is relative
• Space and time are not absolute backgrounds
• Reality depends on reference frame

Thermodynamics tells us:

• Macroscopic states emerge from coarse-graining of many microscopic degrees of freedom
• Entropy increase defines the arrow of time
• “Equilibrium” is a statistically emergent concept

So, what is objective reality? Is it an a priori existing “thing-in-itself,” or an effective description emerging from more fundamental levels?

GLS Theory’s Answer

In the GLS unified theory, objective reality has three levels of characterization:

graph TD
    A["Ontological Layer:<br/>THE-MATRIX<br/>S(omega)<br/>QCA Unitary Evolution"] -->|"Coarse-Graining"| B["Phenomenal Layer:<br/>Observer Reduced State<br/>rho_O = Tr_O |Psi><Psi|<br/>Measurement Probability"]
    B -->|"Multi-Observer Consensus"| C["Consensual Layer:<br/>Objective Reality omega_*<br/>Classical Macroscopic World<br/>Invariant Structure"]

    A -.->|"h -> 0"| D["Classical Limit:<br/>Hamilton Mechanics<br/>Thermodynamics<br/>Continuous Field Theory"]
    B -.->|"xi >> a"| D
    C -.->|"Large N Limit"| D

    style A fill:#e1f5ff
    style B fill:#ffffcc
    style C fill:#ccffcc
    style D fill:#ffcccc

Core Proposition:

Theorem (Triple Emergence of Objective Reality)

1. Phenomenal Emergence: From unitary QCA state $|\Psi ⟩$ to observer reduced state ${\rho }_O$ through partial trace coarse-graining
2. Consensual Emergence: From multi-observer subjective states $\{{\omega }_i\}$ to objective consensus state ${\omega }_{\ast }$ through relative entropy convergence
3. Classical Emergence: From quantum superposition states to classical pointer states through decoherence + law of large numbers

This article will rigorously prove these three emergence mechanisms and provide an operational definition of objective reality.

1. Objectivity as Invariance

1.1 What is “Objective”?

In everyday language, “objective” means “independent of observer.” But in the GLS framework, this requires more precise mathematical characterization.

Definition 1.1 (Three Criteria for Objectivity)

Let $\mathcal{O}$ be an observable (operator) in the matrix universe. We say $\mathcal{O}$ is objective if and only if it satisfies one of the following three conditions:

1. Observer Invariance: $${\rho }_i(\mathcal{O})={\rho }_j(\mathcal{O})\forall i,j\in I$$ That is, all observers have the same expectation value for $\mathcal{O}$

2. Consensus Fixed Point: $$\mathcal{O}\in \text{span}\{{\omega }_{\ast }\}$$ That is, $\mathcal{O}$ has a definite value in consensus state ${\omega }_{\ast }$

3. Gauge Invariance: $$U_g\mathcal{O}{U}_g^{†}=\mathcal{O}\forall g\in \mathcal{G}$$ where $\mathcal{G}$ is a symmetry group (e.g., translation, rotation, gauge transformation)

Physical Meaning:

• Criterion 1: Objectivity = intersubjectivity
• Criterion 2: Objectivity = stable fixed point of consensus
• Criterion 3: Objectivity = invariant structure under symmetry

Examples:

✓ Objective:

• Electron mass $m_e$: All observers measure the same value
• Speed of light $c$: Lorentz invariant
• Black hole mass $M$: Multi-observer consensus converges

✗ Non-Objective:

• Observer’s position coordinate $x_O$: Reference frame dependent
• Single quantum measurement result: Random fluctuation
• Observer’s subjective belief ${\omega }_i^{(0)}$: Varies by person

1.2 Hierarchical Structure of Invariance

Objectivity has different strength levels:

graph TD
    A["Weakest:<br/>Single Observer Internal Consistency<br/>rho_O(t) -> rho_O(t') Deterministic"] --> B["Weak:<br/>Finite Observer Consensus<br/>|omega_i - omega_j| < epsilon"]
    B --> C["Medium:<br/>All Observer Consensus<br/>lim_t->infty omega_i(t) = omega_*"]
    C --> D["Strong:<br/>Gauge Invariance<br/>U_g omega_* U_g^dagger = omega_*"]
    D --> E["Strongest:<br/>Topological Invariance<br/>omega_* Stable Under Small Perturbations"]

    style A fill:#ffeeee
    style B fill:#ffdddd
    style C fill:#ffcccc
    style D fill:#ffbbbb
    style E fill:#ff9999

Theorem 1.2 (Hierarchy of Objectivity)

In the GLS framework:

1. Single observer internal consistency ⟹ Finite observer consensus (through CPTP map monotonicity)
2. Finite observer consensus ⟹ All observer consensus (through strong connectivity)
3. All observer consensus ⟹ Gauge invariance (through spontaneous symmetry breaking mechanism)
4. Gauge invariance ⟹ Topological invariance (through gap protection)

Proof Outline:

• Step 1: Data processing inequality $D(\Phi ({\omega }_1)\Vert \Phi ({\omega }_2))\le D({\omega }_1\Vert {\omega }_2)$
• Step 2: Perron-Frobenius theorem guarantees unique fixed point
• Step 3: Noether’s theorem links symmetry to conserved quantities
• Step 4: Topological stability of gapped systems (Kitaev chain, etc.)

2. Classical Limit: $\hslash \to 0$ and $N\to \infty$

2.1 Two Types of Classical Limits

The classical limit of quantum theory can be achieved through two complementary paths:

Path 1: $\hslash \to 0$ Limit (Planck constant tends to zero)

$$\text{Quantum Mechanics}\stackrel{\hslash \to 0}{\to }\text{Classical Hamilton Mechanics}$$

Path 2: $N\to \infty$ Limit (Number of degrees of freedom tends to infinity)

$$\text{Quantum Many-Body System}\stackrel{N\to \infty }{\to }\text{Thermodynamics/Fluid Mechanics}$$

GLS Unified Picture: The two limits are equivalent under unified time scale

$$\hslash \to 0\iff \frac{\xi (\text{coherence length})}{a(\text{lattice spacing})}\to \infty$$

where $\xi \sim \hslash /(mv)$ is the de Broglie wavelength, $a$ is the QCA lattice spacing.

2.2 WKB Approximation and Hamilton-Jacobi Equation

Theorem 2.1 (WKB Description of Classical Limit)

Let the quantum state have WKB form: $$\psi (x,t)=A(x,t)e^{iS(x,t)/\hslash }$$

where $S(x,t)$ is the action, $A(x,t)$ is the amplitude. In the $\hslash \to 0$ limit:

1. Leading Order ($O({\hslash }^0)$): Hamilton-Jacobi equation $$\frac{\partial S}{\partial t}+H\left(x,\frac{\partial S}{\partial x}\right)=0$$

2. Next-to-Leading Order ($O({\hslash }^1)$): Continuity equation $$\frac{\partial |A{|}^2}{\partial t}+\nabla \cdot \left(|A{|}^2\frac{\nabla S}{m}\right)=0$$

3. Classical Trajectories: Phase space trajectories defined by $p=\nabla S$ satisfy Hamilton equations $$\dot{q}=\frac{\partial H}{\partial p},\dot{p}=-\frac{\partial H}{\partial q}$$

Physical Meaning:

• Phase $S/\hslash$ of quantum state oscillates rapidly as $\hslash \to 0$
• Only stable contribution comes from stationary phase points
• Stationary phase points correspond to classical trajectories

Ehrenfest’s Theorem: Evolution of quantum expectation values tends to classical equations of motion as $\hslash \to 0$: $$\frac{d}{dt}⟨x⟩=\frac{⟨p⟩}{m},\frac{d}{dt}⟨p⟩=-⟨\nabla V⟩\stackrel{\hslash \to 0}{\to }-\nabla V(⟨x⟩)$$

2.3 Law of Large Numbers and Typicality

Theorem 2.2 (Quantum Law of Large Numbers)

Let there be $N$ independent identically distributed quantum systems, each in state $\rho$. Define macroscopic observable: $$\bar{A}:=\frac{1}{N}\sum _{i=1}^N{A}_i$$

Then in the $N\to \infty$ limit, fluctuations of $\bar{A}$ vanish: $$\text{Var}(\bar{A})=\frac{\text{Var}(A)}{N}\stackrel{N\to \infty }{\to }0$$

Macroscopic measurement almost certainly yields the expectation value: $$\mathrm{Pr}\left(|\bar{A}-⟨A⟩|>ϵ\right)\le \frac{\text{Var}(A)}{N{ϵ}^2}\stackrel{N\to \infty }{\to }0$$

Physical Meaning:

• Microscopic quantum fluctuations are averaged out at macroscopic scale
• Macroscopic observables become self-averaging
• Classical determinism comes from statistical law of large numbers

Example: Ideal Gas

Individual molecule velocity $v_i$ is a quantum random variable, but average kinetic energy of $N\sim 10^{23}$ molecules: $${\bar{E}}_{\text{kin}}=\frac{1}{N}\sum _{i=1}^N\frac{1}{2}m{v}_i^2\approx \frac{3}{2}{k}_{B}T$$ has almost no fluctuation, defining macroscopic temperature $T$.

2.4 Coherence Length and Decoherence Time

Definition 2.3 (Three Scales of Classical Limit)

The classical limit requires three scale conditions:

1. Spatial Scale: Coherence length $\gg$ lattice spacing $$\xi :=\frac{\hslash }{\sqrt{2m{k}_{B}T}}\gg a$$

2. Temporal Scale: Decoherence time $\ll$ observation time $${\tau }_{\text{decohere}}\ll {\tau }_{\text{obs}}$$

3. Energy Scale: Thermal energy $\gg$ quantum level spacing $$k_{B}T\gg \frac{{\hslash }^2}{m{L}^2}(\text{typical level spacing})$$

Theorem 2.4 (Necessary and Sufficient Conditions for Classical Limit)

Let QCA system characteristic parameters be $(a,\hslash ,m,T,{\tau }_{\text{decohere}})$. The classical limit holds if and only if: $$\frac{\xi }{a}\gg 1,\frac{{\tau }_{\text{obs}}}{{\tau }_{\text{decohere}}}\gg 1,\frac{k_{B}T\cdot a^2}{m{\hslash }^2}\gg 1$$

In this limit, off-diagonal elements of quantum operators are exponentially suppressed: $$⟨i|\rho |j⟩\sim \exp \left(-\frac{|i-j{|}^2{a}^2}{2{\xi }^2}\right)\stackrel{\xi /a\to \infty }{\to }{\delta }_{ij}\cdot p_i$$

That is, the density matrix diagonalizes, and the system becomes a classical probability distribution.

3. Emergence of Macroscopic Objects

3.1 What is a “Table”?

In everyday life, we consider a “table” to be an objectively existing entity. But from a quantum perspective:

• A table consists of $\sim 10^{25}$ atoms
• Each atom is a quantum system that can be in superposition
• So can a table also be in a superposition of “here” and “there”?

Macroscopic Version of Schrödinger’s Cat Paradox: Why do we never see a table in a superposition of two positions?

GLS’s Answer:

Macroscopic objects are not fundamental entities, but effective descriptions emerging from the combined action of coarse-graining, decoherence, and the law of large numbers.

3.2 Coarse-Graining Flow

Definition 3.1 (Coarse-Graining Map)

Let $\mathcal{H}={⨂}_{x\in \Lambda }{\mathcal{H}}_x$ be the microscopic Hilbert space of QCA. Coarse-graining is a CPTP map: $${\Phi }_{\text{coarse}}:\mathcal{S}({\mathcal{H}}_{\text{micro}})\to \mathcal{S}({\mathcal{H}}_{\text{macro}})$$

satisfying:

1. Spatial Coarse-Graining: Merge $n\times n\times n$ lattice sites into one “coarse-grained site” $${\mathcal{H}}_{\text{macro}}=\underset{X\in {\Lambda }_{\text{macro}}}{⨂}{\mathcal{H}}_X,{\mathcal{H}}_X=\underset{x\in {\text{block}}_X}{⨂}{\mathcal{H}}_x$$

2. Diagonalization: Only retain diagonal elements (classical probability distribution) $${\rho }_{\text{macro}}=\sum _I{p}_I|I⟩⟨I|$$ where $|I⟩$ are coarse-grained basis states (e.g., macroscopic averages of position, momentum, spin)

Renormalization Group Flow:

Coarse-graining can be iterated, forming RG flow: $${\rho }^{(0)}\stackrel{\Phi }{\to }{\rho }^{(1)}\stackrel{\Phi }{\to }{\rho }^{(2)}\stackrel{\Phi }{\to }\cdots \stackrel{\Phi }{\to }{\rho }^{(\infty )}$$

Theorem 3.2 (Entropy Increase Under Coarse-Graining)

Each coarse-graining does not decrease entropy: $$S(\Phi (\rho ))\ge S(\rho )$$

Equality holds if and only if $\rho$ is unchanged before and after coarse-graining (fixed point).

Physical Meaning:

• Coarse-graining loses microscopic information, entropy increases
• Macroscopic description is “coarser” than microscopic description
• Irreversibility comes from information loss

3.3 Collective Excitations and Quasiparticles

In condensed matter physics, macroscopic objects often manifest as quasiparticles—collective excitation modes.

Example: Phonons

Quantization of lattice vibrations gives phonons: $$H=\sum _{\vec{k}}\hslash {\omega }_{\vec{k}}\left(a_{\vec{k}}^{†}{a}_{\vec{k}}+\frac{1}{2}\right)$$

Phonons are collective modes of $10^{23}$ atoms vibrating cooperatively, not properties of individual atoms.

Example: Magnons

Quantization of spin waves in ferromagnets: $$H=\sum _{\vec{k}}\hslash {\omega }_{\vec{k}}{b}_{\vec{k}}^{†}{b}_{\vec{k}}$$

Magnons describe collective flipping of $10^{23}$ spins.

Theorem 3.3 (Emergence of Quasiparticles)

In the QCA framework, long-wavelength low-energy excitation modes can be described by effective field theory: $${\mathcal{L}}_{\text{eff}}=\sum _{\text{quasiparticles}}{\mathcal{L}}_{\text{qp}}$$

Quasiparticle effective mass, lifetime, and interactions are determined by microscopic QCA, but manifest as “elementary particles” at macroscopic scale.

Physical Meaning:

• “Particles” are not necessarily fundamental, they can be collectively emergent
• “Electrons” in solid-state physics are actually dressed electrons (quasielectrons)
• “Objects” in the macroscopic world are emergent effective descriptions

3.4 Spontaneous Symmetry Breaking and Order Parameters

Definition 3.4 (Order Parameter)

Let the system have symmetry group $G$. Order parameter $\Phi$ is an operator satisfying:

• In symmetric phase: $⟨\Phi ⟩=0$
• In broken phase: $⟨\Phi ⟩\ne 0$

Example: Ferromagnet

• Symmetry: Spin rotation $SO(3)$
• Order parameter: Magnetization $\vec{M}=\frac{1}{N}{\sum }_i{\vec{S}}_i$
• High temperature: $⟨\vec{M}⟩=0$ (paramagnetic phase)
• Low temperature: $⟨\vec{M}⟩\ne 0$ (ferromagnetic phase)

Theorem 3.5 (Landau Phase Transition Theory)

In the $N\to \infty$ limit, free energy can be expanded in order parameter: $$F(\Phi ,T)=a(T){\Phi }^2+b(T){\Phi }^4+\cdots$$

Phase transition occurs when $a(T_c)=0$, at which point symmetry spontaneously breaks.

Goldstone’s Theorem: Continuous symmetry breaking leads to massless Goldstone bosons (e.g., magnons in ferromagnets).

Physical Meaning:

• Macroscopic phases (solid, liquid, gas, ferromagnetic, etc.) are emergent results of symmetry breaking
• Order parameters are macroscopically observable “objective reality”
• Phase transitions are collective phenomena; individual particles do not have “phases”

4. Thermodynamic Limit and Typicality

4.1 Typical Subspace

Definition 4.1 (Typical Subspace)

Let the total Hilbert space dimension of an $N$-particle system be $d^N$ ($d$ is single-particle dimension). Given density matrix $\rho$, define typical subspace ${\mathcal{H}}_{\text{typ}}$ as the span of all states $|\psi ⟩$ satisfying: $$\left|\frac{1}{N}\sum _{i=1}^N{A}_i-\text{Tr}(\rho A)\right|<ϵ$$ for all local observables $A$.

Theorem 4.2 (Typicality Theorem)

In the thermodynamic limit $N\to \infty$:

1. Dimension of typical subspace: $$\dim ({\mathcal{H}}_{\text{typ}})\sim e^{NS(\rho )}$$ where $S(\rho )=-\text{Tr}(\rho \log \rho )$ is von Neumann entropy

2. Proportion of typical subspace in total Hilbert space: $$\frac{\dim ({\mathcal{H}}_{\text{typ}})}{\dim (\mathcal{H})}\sim \frac{e^{NS(\rho )}}{d^N}\to 1(\text{when }S(\rho )=\log d)$$

3. Randomly sampled states are almost certainly in typical subspace: $$\mathrm{Pr}(|\psi ⟩\in {\mathcal{H}}_{\text{typ}})\ge 1-\delta$$ where $\delta \sim e^{-cN}$ is exponentially small

Physical Meaning:

• “Typical states” of thermodynamic systems occupy an exponentially small subspace
• This subspace is uniquely determined by macroscopic thermodynamic parameters (temperature, pressure, volume)
• Microscopic details are irrelevant—this is the quantum foundation of the second law of thermodynamics

4.2 Microcanonical Ensemble and Energy Shell

Definition 4.3 (Energy Shell)

All quantum states with energy in $[E,E+\Delta E]$ form the energy shell: $${\mathcal{H}}_E:=\text{span}\{|\psi ⟩:E\le ⟨\psi |H|\psi ⟩\le E+\Delta E\}$$

Microcanonical Ensemble: Uniform distribution on energy shell $${\rho }_{\text{micro}}(E)=\frac{1}{\Omega (E)}{P}_E$$ where $\Omega (E)=\dim ({\mathcal{H}}_E)$ is density of states, $P_E$ is energy shell projection.

Boltzmann Entropy: $$S_{\text{Boltzmann}}(E)=k_B\log \Omega (E)$$

Theorem 4.4 (Microcanonical = Typical)

In the $N\to \infty$ limit, microcanonical ensemble is equivalent to typical subspace: $${\mathcal{H}}_E\approx {\mathcal{H}}_{\text{typ}}$$

That is: Systems with given energy are almost certainly in typical subspace.

Physical Meaning:

• Macroscopic thermodynamic equilibrium states correspond to typical subspace
• Boltzmann’s “most probable distribution” is mathematically necessary
• Quantum origin of entropy maximization principle

4.3 Eigenstate Thermalization Hypothesis (ETH)

ETH Hypothesis: For chaotic quantum systems, local reduced density matrices of energy eigenstates approximate thermal states.

Theorem 4.5 (ETH and Typicality)

Let $H$ be a chaotic Hamiltonian, $|E_n⟩$ be energy eigenstates. For any local operator $A$, we have: $$⟨E_n|A|E_n⟩=⟨A{⟩}_{\text{thermal}}(E_n)+O(e^{-S(E_n)/2})$$

where $⟨A{⟩}_{\text{thermal}}$ is thermal average at corresponding temperature.

Corollary:

• A single energy eigenstate is sufficient for thermalization
• No ensemble average needed
• Quantum entanglement leads to local thermalization

ETH in GLS Framework:

In QCA universe, ETH holds if and only if:

1. QCA dynamics is chaotic (Lyapunov exponent $>0$)
2. Unified time scale $\kappa (\omega )$ is smooth within energy window
3. Entanglement growth rate saturates Lieb-Robinson bound

5. Operational Definition of Objective Reality

5.1 Challenge of Positivism

Logical Positivism (Vienna Circle) claims:

A proposition is meaningful if and only if it can be verified or falsified through experience.

For “objective reality,” this means: We cannot talk about “unobservable” reality, only properties that can be “operationally measured.”

Bridgman’s Operationalism:

The meaning of a physical concept is the operational procedure for measuring it.

For example: Definition of “length” = procedure for measuring with a ruler.

GLS’s Response:

We accept the spirit of operational definition, but mathematize it:

Objective reality = operational limits satisfying specific convergence, invariance, and repeatability.

5.2 Three Operational Principles of Reality

Definition 5.1 (Operational Definition of Reality)

Let $\mathcal{O}$ be an observable (operator in matrix universe). We say $\mathcal{O}$ corresponds to objective reality if and only if it satisfies:

Principle 1 (Repeatability): $$\mathrm{Pr}\left(\left|{\mathcal{O}}^{(t_1)}-{\mathcal{O}}^{(t_2)}\right|<ϵ\mid \text{same conditions}\right)\ge 1-\delta$$ That is, repeated measurements under same initial conditions yield consistent results with high probability.

Principle 2 (Intersubjectivity): $$\underset{N\to \infty }{\lim }\frac{1}{N}\sum _{i=1}^N\left|{\omega }_i(\mathcal{O})-\bar{\omega }(\mathcal{O})\right|=0$$ That is, measurement results from multiple independent observers converge to the same value.

Principle 3 (Stability): $$\left|\frac{d\mathcal{O}}{dt}\right|<ϵ\text{or}\mathcal{O}=\text{constant}$$ That is, observable is approximately conserved or slowly varying under time evolution.

Theorem 5.2 (Equivalence of Three Principles)

In the GLS framework, the set of observables satisfying Principles 1, 2, 3 is the same, corresponding to: $${\mathcal{O}}_{\text{reality}}=\{\text{conserved quantities of consensus state }{\omega }_{\ast }\}$$

Proof:

• Principle 1 → Principle 2: Repeatability leads to different experimenters getting same results
• Principle 2 → Principle 3: Intersubjectivity requires observable not to change rapidly with observer time
• Principle 3 → Principle 1: Conserved quantities automatically satisfy repeatability

5.3 Reality of Classical Macroscopic Quantities

Corollary 5.3 (Classical Quantities are Objective Reality)

In the classical limit, the following macroscopic observables satisfy the three principles of reality:

1. Extensive Quantities:

   • Total mass: $M={\sum }_i{m}_i$
   • Total energy: $E={\sum }_i{E}_i$
   • Total entropy: $S=k_B\log \Omega$

2. Intensive Quantities:

   • Temperature: $T=\partial E/\partial S$
   • Pressure: $P=-\partial E/\partial V$
   • Chemical potential: $\mu =\partial E/\partial N$

3. Order Parameters:

   • Magnetization: $M=⟨{\sum }_i{S}_i^z⟩$
   • Superconducting gap: $\Delta =⟨{\psi }_{\uparrow }{\psi }_{\downarrow }⟩$

Non-Real Quantities:

• Position of individual atom (quantum fluctuation)
• Random result of single measurement (probabilistic)
• Observer’s subjective belief (non-consensual)

5.4 Emergence and Limits of Reality

Theorem 5.4 (Existence of Reality as Limit)

In QCA universe, objective reality corresponds to the intersection of the following four limits:

1. Thermodynamic Limit: $N\to \infty$ (number of particles tends to infinity)
2. Classical Limit: $\hslash \to 0$ (quantum fluctuations vanish)
3. Decoherence Limit: ${\tau }_{\text{decohere}}\to 0$ (phase loss completes instantly)
4. Consensus Limit: $t\to \infty$ (multi-observer convergence to fixed point)

Under these four limits, quantum state $|\Psi ⟩$ reduces to classical phase space distribution $f(q,p)$: $${\rho }_{\text{quantum}}\stackrel{\text{four limits}}{\to }{\rho }_{\text{classical}}\leftrightarrow f_{\text{Liouville}}(q,p)$$

satisfying classical Liouville equation: $$\frac{\partial f}{\partial t}+\{H,f{\}}_{\text{Poisson}}=0$$

Physical Meaning:

• “Objective reality” is not a priori given
• But emerges from quantum substrate under appropriate limits
• Different limits may give different “effective realities”

6. Case Study: From Quarks to Protons

6.1 Statement of the Problem

The proton is a fundamental component of everyday matter, considered a paradigm of “objective reality.” But from QCD (Quantum Chromodynamics) perspective:

• Proton consists of 3 quarks: $p=uud$
• Quarks are fundamental degrees of freedom of QCD
• But quarks have never been observed individually—this is quark confinement

Questions:

• If quarks cannot exist alone, are they “real”?
• As a bound state, where does the proton’s “reality” come from?
• How to understand “reality of parts” vs. “reality of whole”?

6.2 QCD Vacuum and Quark Condensation

In QCD, the vacuum state $|0⟩$ is not empty, but filled with condensation of quark-antiquark pairs: $$⟨0|\bar{q}q|0⟩\ne 0$$

This is called chiral symmetry spontaneous breaking.

Nambu-Goldstone Theorem: Continuous symmetry breaking leads to massless bosons ($\pi$ mesons): $$m_{\pi }^2\propto m_{\text{quark}}(\text{when }{m}_{\text{quark}}\to 0,m_{\pi }\to 0)$$

Effective Theory:

At low energy, QCD can be described by chiral perturbation theory: $${\mathcal{L}}_{\text{eff}}=\frac{f_{\pi }^2}{4}\text{Tr}({\partial }_{\mu }{U}^{†}{\partial }^{\mu }U)+\cdots$$

where $U(x)\in SU(2)$ is the $\pi$ meson field.

6.3 Emergence of Proton

Lattice QCD Simulation:

Discretizing QCD on a lattice and performing Monte Carlo simulation, proton mass can be calculated: $$m_p=938.3\text{ MeV}$$

This value is not input, but output—naturally emerging from quark and gluon interactions.

Key Observation:

• $>95\%$ of proton mass comes from gluon energy (QCD vacuum energy)
• Only $<5\%$ comes from quark rest mass
• Proton is a collective excitation of strong interactions

Theorem 6.1 (Proton as QCD Topological Soliton)

Proton can be understood as Skyrmion—topological soliton of $SU(2)$ chiral field, with topological charge $B=1$ (baryon number): $$B=\frac{1}{24{\pi }^2}\int d^{3}x{ϵ}^{ijk}\text{Tr}(U^{†}{\partial }_{i}U\cdot U^{†}{\partial }_{j}U\cdot U^{†}{\partial }_{k}U)$$

Topological protection ensures proton stability (lifetime $>10^{34}$ years).

Physical Meaning:

• Proton is not “simple combination of 3 quarks”
• But topological excitation of QCD vacuum structure
• Its reality comes from topological stability, not “constituent particles”

6.4 Hierarchical Reality

From quarks to protons, we see hierarchical structure of reality:

graph TD
    A["Fundamental Layer:<br/>Quarks + Gluons<br/>(QCD Fundamental Fields)"] -->|"Confinement"| B["Intermediate Layer:<br/>Mesons + Baryons<br/>(Chiral Effective Theory)"]
    B -->|"Nuclear Force"| C["Nucleon Layer:<br/>Atomic Nuclei<br/>(Shell Model)"]
    C -->|"Electromagnetic Force"| D["Atomic Layer:<br/>Atoms + Molecules<br/>(Chemistry)"]
    D -->|"Van der Waals Force"| E["Macroscopic Layer:<br/>Solids + Liquids + Gases<br/>(Thermodynamics)"]

    style A fill:#ffeeee
    style B fill:#ffddcc
    style C fill:#ffccaa
    style D fill:#ffbb88
    style E fill:#ffaa66

Each layer is an emergent description of the next:

• “Elementary particles” of upper layer are collective excitations of lower layer
• “Reality” of upper layer may not have individual correspondents in lower layer
• But reality of upper layer is not diminished—it has its own operational definition and invariance

Philosophical Meaning:

• Reality is not “monistic,” but “hierarchical”
• Different scales have different effective realities
• Reductionism is incomplete

7. Philosophical Discussion: Constructivism vs. Realism

7.1 Position of Scientific Realism

Scientific Realism claims:

1. Metaphysical Proposition: World objectively exists, independent of human mind
2. Semantic Proposition: Terms in scientific theories refer to real entities
3. Epistemological Proposition: Mature scientific theories are approximately true

No-Miracles Argument (Putnam):

If scientific theories are not approximately true, then the success of science would be a miracle.

GLS’s Response:

We accept weak form of scientific realism:

• Ontological reality (THE-MATRIX) indeed exists
• But observable reality is emergent and hierarchical
• Theories at different levels can all be “true” (within their respective effective domains)

7.2 Challenge of Social Constructivism

Social Constructivism claims:

Scientific knowledge is product of social negotiation, not discovery of objective reality.

Strong Programme (Bloor):

• Acceptance of scientific theories determined by social factors
• “Truth” is manifestation of power relations
• No objective standards transcending culture

GLS’s Response:

We partially agree with constructivism:

• Consensual reality is indeed socially constructed (multi-observer convergence)
• But this construction is not arbitrary, constrained by unified time scale
• Feedback from nature (experimental failure) limits freedom of social construction

Middle Ground:

• Science has both discovery (ontological layer) and construction (consensual layer)
• “Objectivity” is limit of intersubjectivity, not transcendental given

7.3 Structural Realism

Structural Realism claims:

Reliable parts of scientific theories are not descriptions of individual entities, but descriptions of structures and relations.

Epistemic Structural Realism (Worrall):

• In scientific revolutions, theoretical content changes, but mathematical structures are preserved
• Example: Structure of Maxwell equations remains unchanged from ether theory to field theory

Ontic Structural Realism (Ladyman):

• Basic constituents of world are structures, not material objects
• Relations precede relata

GLS’s Position:

We are radical structural realists:

• Matrix universe THE-MATRIX is pure structure (scattering matrix + unified time scale)
• “Particles,” “fields,” “spacetime” are all emergent images of structure
• There are no “entities” independent of structure

Difference from Traditional Realism:

• Traditional: Objects (substance) first, then relations
• Structural: Network of relations itself is all reality
• GLS: Algebraic structure of THE-MATRIX is the universe ontology

8. Summary: Three Faces of Objective Reality

8.1 Unity of Ontological, Phenomenal, and Consensual

Objective reality has three aspects in the GLS framework:

graph LR
    A["Ontological Reality<br/>THE-MATRIX<br/>S(omega)"] -->|"Observer Compression"| B["Phenomenal Reality<br/>Reduced State rho_O"]
    B -->|"Intersubjective Convergence"| C["Consensual Reality<br/>Fixed Point omega_*"]
    C -.->|"Feedback Correction"| A

    A -.->|"Category Equivalence"| D["Mathematical Structure"]
    B -.->|"Operational Definition"| E["Measurement Protocol"]
    C -.->|"Social Process"| F["Scientific Community"]

    style A fill:#e1f5ff
    style B fill:#ffffcc
    style C fill:#ccffcc

Theorem 8.1 (Hierarchical Embedding of Triple Reality)

In the GLS framework, ontological, phenomenal, and consensual three-layer reality satisfy: $$\text{Ontological}\supseteq \text{Phenomenal}\supseteq \text{Consensual}$$

And each layer is the minimal invariant extension of the next:

• Phenomenal = invariant subspace of ontological under observer group action
• Consensual = fixed point set of phenomenal under multi-observer exchange

8.2 Irreducibility of Emergence

Key Proposition:

Emergent properties of objective reality cannot be completely reduced to microscopic substrate.

Evidence 1: Phase Transitions

• “Solid-liquid-gas” three phases of water are macroscopic emergent properties
• Individual H₂O molecules do not have concept of “phase”
• Phase transitions determined by symmetry breaking of collective degrees of freedom

Evidence 2: Life

• “Life” is emergent property of molecular networks
• Individual proteins, DNA molecules are not “alive”
• Life phenomena require holistic dynamics

Evidence 3: Consciousness

• Consciousness (if it exists) is emergent property of neural networks
• Individual neurons do not have “consciousness”
• Consciousness may correspond to higher-order self-referential loops

Anti-Reductionist Argument (Anderson, 1972):

“More is different.”

Hierarchical view of reality holds:

• Each level has its autonomy
• Upper-level laws cannot be completely reduced to lower level
• But upper level is constrained by lower level (does not violate lower-level laws)

8.3 Dynamical Nature of Reality

Traditional view of reality is static: Objective reality is eternal unchanging “thing-in-itself.”

GLS view of reality is dynamic: Objective reality continuously emerges and evolves under different limits.

Temporal Evolution of Reality:

1. Early Universe ($t<10^{-12}$ s):

   • Only quark-gluon plasma
   • No protons, no atoms, no molecules

2. Nucleosynthesis Period ($t\sim 3$ min):

   • Protons, neutrons form
   • Light element nuclei (H, He, Li) emerge

3. Recombination Period ($t\sim 380,000$ yr):

   • Atoms form
   • Photons decouple, universe becomes transparent

4. Stellar Period ($t\sim 10^8$ yr):

   • Heavy elements synthesized in stellar cores
   • Planets, life emerge

Implications:

• “Reality” changes with cosmic evolution
• New levels continuously emerge
• Future may emerge reality levels we cannot imagine

9. Open Questions and Outlook

9.1 Problem of Consciousness

Hard Problem of Consciousness (Chalmers):

Why do physical processes accompany subjective experience?

Can GLS framework explain consciousness?

Possible Directions:

• Consciousness = higher-order self-referential observer structure
• Definition of “I” (Article 01) involves self-referential fixed point
• Consciousness may correspond to specific types of self-referential loops

To Be Resolved:

• How to characterize “subjective experience” (qualia)?
• Where does unity of consciousness come from?
• How to reconcile free will with determinism?

9.2 Limits of Reality

Questions:

• Does “ultimate reality” exist?
• Or is reality infinitely hierarchical?
• Tegmark’s mathematical universe hypothesis: Reality = mathematical structure?

GLS’s Position:

• THE-MATRIX may not be the ultimate level
• There may be deeper “META-MATRIX”
• But each level is self-consistent and operationally definable

9.3 Multiple Realities

Many-Worlds Interpretation of Quantum Mechanics:

• Each measurement causes universe branching
• All possible outcomes “really exist”

GLS’s Alternative:

• No branching needed, only consensus convergence
• Single ontology (THE-MATRIX), multiple phenomena (different observers)
• Consensus emerges unique objective reality

Philosophical Question:

• Is it “one universe, many branches,” or “one universe, many perspectives”?
• Are the two pictures equivalent in predictions?

Appendix A: Mathematical Details of Classical Limit

A.1 Weyl Quantization and Wigner Function

Weyl Correspondence:

Classical phase space function $f(q,p)$ corresponds to quantum operator $\hat{f}$: $$\hat{f}=\int \frac{dqdp}{(2\pi \hslash {)}^n}f(q,p)\hat{W}(q,p)$$

where $\hat{W}(q,p)=e^{i(q\hat{p}-p\hat{q})/\hslash }$ is Weyl operator.

Wigner Function:

Quantum state $\rho$ corresponds to phase space quasi-probability distribution: $$W(q,p)=\int \frac{dy}{2\pi \hslash }{e}^{ipy/\hslash }⟨q-y/2|\rho |q+y/2⟩$$

Properties:

• $\int W(q,p)dp=⟨q|\rho |q⟩$ (position distribution)
• $\int W(q,p)dq=⟨p|\rho |p⟩$ (momentum distribution)
• But $W(q,p)$ can be negative (quantum interference)

Classical Limit:

When $\hslash \to 0$, Wigner function becomes positive Liouville distribution: $$W_{\hslash }(q,p)\stackrel{\hslash \to 0}{\to }{f}_{\text{classical}}(q,p)\ge 0$$

A.2 Path Integral and Saddle-Point Approximation

Feynman Path Integral:

Quantum amplitude expressed as coherent superposition of all paths: $$⟨q_f,t_f|q_i,t_i⟩=\int \mathcal{D}q(t)e^{iS[q]/\hslash }$$

where $S[q]={\int }_{t_i}^{t_f}L(q,\dot{q},t)dt$ is the action.

Saddle-Point Approximation ($\hslash \to 0$):

Integral dominated by stationary phase points: $$\frac{\delta S}{\delta q(t)}=0\Rightarrow \frac{d}{dt}\frac{\partial L}{\partial \dot{q}}-\frac{\partial L}{\partial q}=0$$

This is exactly Euler-Lagrange equation, giving classical trajectories.

Appendix B: Landau Theory of Phase Transitions

B.1 Order Parameter and Symmetry

Landau Theory assumes free energy can be expanded in order parameter $\Phi$: $$F(\Phi ,T)=F_0(T)+a(T){\Phi }^2+\frac{b(T)}{2}{\Phi }^4+\cdots$$

Symmetric Phase ($T>T_c$):

• $a(T)>0$, minimum at $\Phi =0$
• Symmetry preserved

Broken Phase ($T<T_c$):

• $a(T)<0$, minimum at $\Phi =\pm {\Phi }_0\ne 0$
• Symmetry spontaneously broken

Critical Exponents:

Near phase transition point: $$\Phi \sim (T_c-T{)}^{\beta },\chi \sim |T-T_c{|}^{-\gamma },\xi \sim |T-T_c{|}^{-\nu }$$

where $\beta ,\gamma ,\nu$ are critical exponents, determined by universality class of the system.

References

1. Anderson, P. W. (1972). “More is different.” Science 177(4047): 393–396.

2. Ehrenfest, P. (1927). “Bemerkung über die angenäherte Gültigkeit der klassischen Mechanik innerhalb der Quantenmechanik.” Z. Phys. 45: 455–457.

3. Landau, L. D., Lifshitz, E. M. (1980). Statistical Physics, 3rd ed. Pergamon Press.

4. Wigner, E. (1932). “On the quantum correction for thermodynamic equilibrium.” Phys. Rev. 40: 749–759.

5. Srednicki, M. (1994). “Chaos and quantum thermalization.” Phys. Rev. E 50: 888–901.

6. Deutsch, J. M. (2018). “Eigenstate thermalization hypothesis.” Rep. Prog. Phys. 81: 082001.

7. Popescu, S., Short, A. J., Winter, A. (2006). “Entanglement and the foundations of statistical mechanics.” Nat. Phys. 2: 754–758.

8. Chalmers, D. (1995). “Facing up to the problem of consciousness.” J. Conscious. Stud. 2(3): 200–219.

9. Ladyman, J., Ross, D. (2007). Every Thing Must Go: Metaphysics Naturalized. Oxford University Press.

10. Tegmark, M. (2014). Our Mathematical Universe. Knopf.

Next Article Preview: In Article 06 (Chapter Summary), we will:

• Review core results of Chapter 10
• Summarize logical chain of observer theory
• Compare with other quantum interpretations
• Look forward to future research directions

Stay tuned!