Observer Worldline Sections—“Now” in Causal Structure

Introduction: What is “Now”?

When you read this sentence, you say “this is now”. But in relativistic 4-dimensional spacetime, there is no absolute “now”—different reference frames have different simultaneity. So what is the “now” observer experiences?

This chapter gives precise answer: Observer’s “now” is a section ${\Sigma }_{\tau }$ of their worldline on unified time scale $\tau$. This section is not arbitrary, but subject to three constraints:

1. Local Causality: Can only see past light cone
2. Dynamical Consistency: Must exist local solution satisfying field equations
3. Record Consistency: Cannot contradict existing memory

Let us start from most basic structure, gradually construct this theory.

Section One: Triplet Structure of Observer

1.1 Observer as Internal Object in Spacetime

In classical physics, observer often viewed as “external perspective”. But in unified framework, observer itself is part of spacetime.

Definition (Observer): Observer is triplet: $$\mathcal{O}=(\gamma ,\Lambda ,{\mathcal{A}}_{\gamma ,\Lambda })$$

where:

• $\gamma :I\to M$: Timelike worldline, parameterized by eigen time $\tau \in I\subset \mathbb{R}$
• $\Lambda >0$: Resolution parameter, characterizes minimum resolvable scale of energy-time, space-momentum
• ${\mathcal{A}}_{\gamma ,\Lambda }\subset {\mathcal{A}}_{\partial }$: Observable subalgebra, corresponding to channels and resolution

Popular Understanding—Three Elements of Telescope:

1. Worldline $\gamma$: Telescope’s position and trajectory
2. Resolution $\Lambda$: Telescope’s lens precision (how clearly can see)
3. Observable algebra ${\mathcal{A}}_{\gamma ,\Lambda }$: Telescope’s observation mode (visible light? Infrared? X-ray?)

Different telescopes (observers) in same spacetime, but see different “worlds”!

graph TB
    A["Observer O"] --> B["Worldline γ(τ)<br/>Spacetime Trajectory"]
    A --> C["Resolution Λ<br/>Minimum Scale"]
    A --> D["Observable Algebra A<br/>Observation Mode"]

    B --> E["Causal Horizon<br/>Past Light Cone"]
    C --> F["Energy-Time<br/>Δω·Δt≥Λ"]
    D --> G["Subalgebra Selection<br/>What to Measure?"]

    E --> H["Observer Section Σ_τ"]
    F --> H
    G --> H

    style A fill:#e1f5ff
    style H fill:#ffe1e1

1.2 Unified Time Scale and Eigen Time

Parameterization of observer worldline uses unified time scale $[\tau ]$. Recall its definition:

Scale Identity: $$\frac{{\phi }^{\prime }(\omega )}{\pi }={\rho }_{\mathrm{rel}}(\omega )=\frac{1}{2\pi }\mathrm{tr}Q(\omega )$$

where:

• $\phi (\omega )$: Total scattering half-phase
• ${\rho }_{\mathrm{rel}}(\omega )$: Relative density of states
• $Q(\omega )=-iS(\omega {)}^{†}{\partial }_{\omega }S(\omega )$: Wigner-Smith group delay matrix

In Boundary Time Geometry (BTG) framework, equivalence of three time scales proven:

1. Scattering Time: From trace of group delay matrix
2. Modular Time: From Tomita-Takesaki modular flow
3. Geometric Time: From Brown-York boundary Hamiltonian

Observer’s eigen time ${\tau }_{\mathrm{prop}}$ defined along worldline $\gamma$, belongs to this equivalence class: $${\tau }_{\mathrm{prop}}\in [\tau ]$$

Popular Analogy—Three Readings of Watch:

• Mechanical watch: Gear rotation (corresponds to scattering group delay)
• Electronic watch: Crystal frequency (corresponds to modular flow)
• Sundial: Sun position (corresponds to geometric time)

They read differently, but conversion relation fixed (affine transformation)—this is “equivalence class”!

1.3 Precise Definition of World Section

Now we can give core definition.

Definition (World Section): On unified time scale $\tau$, world section of observer $\mathcal{O}$ is triplet: $${\Sigma }_{\tau }=(\gamma (\tau ),{\mathcal{A}}_{\gamma ,\Lambda }(\tau ),{\rho }_{\gamma ,\Lambda }(\tau ))$$

where:

1. $\gamma (\tau )\in M$: Observer’s spacetime position at scale $\tau$
2. ${\mathcal{A}}_{\gamma ,\Lambda }(\tau )\subset {\mathcal{A}}_{\gamma ,\Lambda }$: Observable subalgebra readable or activatable at time $\tau$
3. ${\rho }_{\gamma ,\Lambda }(\tau )$: Effective state on this subalgebra, obtained by conditioning and coarse-graining global state $\omega$

Physical Meaning: Section ${\Sigma }_{\tau }$ is a snapshot of world observer can access at “now” $\tau$. But this snapshot not omniscient—limited by:

• Causal Horizon: Can only see past light cone of $\gamma (\tau )$
• Resolution: Details below energy $\Lambda$ cannot see
• Observation Mode: Can only measure operators in ${\mathcal{A}}_{\gamma ,\Lambda }(\tau )$

Popular Analogy—Limitations of Camera Photography:

• Causal horizon: Camera can only photograph objects light has reached (cannot see “now” of Andromeda galaxy)
• Resolution: Lens pixels limited, cannot photograph atomic scale
• Observation mode: Only photograph visible light, cannot photograph infrared or ultraviolet

All sections ${\Sigma }_{\tau }$ constitute section space: $$\Sigma (\tau ;\mathcal{O})=\{\text{All sections satisfying basic measurability}\}$$

But not all sections are “physically allowed”!

Section Two: Causal Consistency—Physically Allowed Sections

2.1 Three Consistency Constraints

Simply giving triplet $(\gamma (\tau ),\mathcal{A},\rho )$ not enough, must satisfy physical constraints.

Definition (Causally Consistent Section): Section ${\Sigma }_{\tau }\in \Sigma (\tau ;\mathcal{O})$ called causally consistent, if satisfies:

Constraint 1: Local Causality For any $A\in {\mathcal{A}}_{\gamma ,\Lambda }(\tau )$, its support located in past causal region of $\gamma (\tau )$: $$\mathrm{supp}(A)\subset J^{-}(\gamma (\tau ))$$

And any operator depending on future region has action on state invisible on ${\mathcal{A}}_{\gamma ,\Lambda }(\tau )$.

Popular Understanding: Cannot “see future”!

Constraint 2: Dynamical Consistency Exists family of local solutions $(g_{ab},\Phi )$ (including geometry and matter fields) defined on $(\tau -ϵ,\tau +ϵ)$, such that:

• For all $t\in (\tau -ϵ,\tau +ϵ)$, this solution satisfies Einstein equations and matter field equations
• Boundary algebra state induced by this solution, when restricted to ${\mathcal{A}}_{\gamma ,\Lambda }(t)$, consistent with some ${\rho }_{\gamma ,\Lambda }(t)$
• ${\rho }_{\gamma ,\Lambda }(\tau )$ is one member

Popular Understanding: Must “evolve self-consistently”—adjacent moments connectable via field equations!

Constraint 3: Record Consistency On subalgebra ${\mathcal{A}}_{\mathrm{mem}}\subset {\mathcal{A}}_{\gamma ,\Lambda }(\tau )$ containing observer internal degrees of freedom and memory, ${\rho }_{\gamma ,\Lambda }(\tau )$ consistent with previous sections at ${\tau }^{\prime }<\tau$ via unitary evolution or CPTP map, no configuration contradicting existing records.

Popular Understanding: Cannot “memory confusion”—things remembered today cannot contradict yesterday!

Set of sections satisfying these three denoted: $${\Gamma }_{\mathrm{causal}}^{\mathrm{dyn}}(\tau ;\mathcal{O})\subset \Sigma (\tau ;\mathcal{O})$$

graph TB
    A["All Possible Sections<br/>Σ(τ;O)"] --> B["Local Causality<br/>supp(A)⊂J⁻(γ)"]
    A --> C["Dynamical Consistency<br/>Exists Field Equation Solution"]
    A --> D["Record Consistency<br/>No Memory Contradiction"]

    B --> E["Causally Consistent Sections<br/>Γ_causal(τ;O)"]
    C --> E
    D --> E

    E --> F["Empirical Sections<br/>Γ_exp(τ;O)"]

    style A fill:#f0f0f0
    style E fill:#e1f5ff
    style F fill:#ffe1e1

2.2 Causal Diamonds and Existence of Local Solutions

Key Question: How to guarantee existence of causally consistent sections?

Answer comes from Jacobson’s entanglement equilibrium hypothesis!

Theorem (Local Causally Consistent Extension): Under assumptions:

1. $(M,g)$ stably causal and locally hyperbolic
2. State $\omega$ locally Hadamard state
3. On each small causal diamond satisfies generalized entropy extremum condition

Then for any $p\in M$, exists small causal diamond $D_{p,r}$ containing $p$, and family of local solutions $(g_{ab},\Phi )$ satisfying Einstein equations and matter field equations, such that:

• This solution exists and unique in $D_{p,r}$ (up to local diffeomorphism)
• Induced generalized entropy on boundary satisfies Jacobson-type extremum condition and QNEC/QFC constraints

Proof Strategy:

1. IBVP (initial-boundary value problem) on small causal diamond well-posed in appropriate function space
2. Energy conditions and QNEC/QFC guarantee no pathological focusing
3. Jacobson entanglement equilibrium: $\delta S_{\mathrm{gen}}=0\iff G_{ab}+\Lambda g_{ab}=8\pi G{T}_{ab}$

Popular Understanding—Jigsaw Puzzle Game:

• Each small causal diamond is a “puzzle piece”
• Jacobson condition guarantees each piece “edges match”
• Take covering $\{D_{\gamma ({\tau }_i),r_i}\}$ along worldline $\gamma$, piece together local solutions, obtain complete picture

This guarantees in finite time interval $I=[{\tau }_0,{\tau }_1]$, at least exists one causally consistent section extension family!

2.3 Selection of Observable Subalgebra—Physical Meaning of Observation Mode

Different observation modes correspond to different ${\mathcal{A}}_{\gamma ,\Lambda }(\tau )$. This not “subjective choice”, but has objective physical meaning!

Example 1: Position Measurement vs Momentum Measurement

• Position mode: ${\mathcal{A}}_{\mathrm{pos}}=\{f(\hat{x})\}$
• Momentum mode: ${\mathcal{A}}_{\mathrm{mom}}=\{g(\hat{p})\}$

By Heisenberg uncertainty, cannot simultaneously measure precisely: $$[\hat{x},\hat{p}]=i\hslash \Rightarrow {\mathcal{A}}_{\mathrm{pos}}\not\subset {\mathcal{A}}_{\mathrm{mom}}$$

Example 2: Single Slit vs Double Slit

• No path measurement: ${\mathcal{A}}_{\mathrm{screen}}=\{f({\hat{x}}_{\mathrm{screen}})\}$
• Path measurement: ${\mathcal{A}}_{\mathrm{path}\otimes \mathrm{screen}}=\{P_L,P_R,f({\hat{x}}_{\mathrm{screen}})\}$

Introducing path detector extends observable subalgebra, causes decoherence!

Example 3: Time-Domain Double Slit

• Continuous observation: ${\mathcal{A}}_{\mathrm{cont}}=\{O(t)|t\in \mathbb{R}\}$
• Time window: ${\mathcal{A}}_{\mathrm{window}}=\{O(t)|t\in [t_1,t_1+\delta t]\cup [t_2,t_2+\delta t]\}$

Time window selects two “slits” on unified scale, interference appears in energy spectrum!

Popular Analogy—Camera with Different Filters:

• Black-white filter: Only see brightness (low-dimensional observable algebra)
• RGB filter: See three primary colors (medium-dimensional)
• Full spectrometer: See all wavelengths (high-dimensional)

Choosing different filters, photograph different “worlds”—but all are different projections of same world!

Section Three: Global State and Measure on Section Space

3.1 From Hilbert Space to Section Space

In standard quantum mechanics, global state is density operator $\rho$ on Hilbert space. But observer cannot directly “see” $\rho$—can only see some section ${\Sigma }_{\tau }$!

How to “project” $\rho$ to section space?

Key Idea: Each section ${\Sigma }_{\tau }$ corresponds to an effect operator $E_{{\Sigma }_{\tau }}$.

Definition (Section Effect Operator Family): For each $\tau$, exists mapping: $${\Sigma }_{\tau }\mapsto E_{{\Sigma }_{\tau }}$$

such that:

1. $E_{{\Sigma }_{\tau }}\ge 0$ (positive operator)
2. ${\int }_{{\Gamma }_{\mathrm{causal}}^{\mathrm{dyn}}(\tau ;\mathcal{O})}{E}_{{\Sigma }_{\tau }}\mu (\mathrm{d}{\Sigma }_{\tau })=\mathbb{I}$ (normalization)

Probability Weight: $$p({\Sigma }_{\tau })=\mathrm{Tr}(\rho E_{{\Sigma }_{\tau }})$$

Thus obtain measure $p_{\tau }$ on section space.

Popular Understanding—Voting System:

• Global state $\rho$: “Total opinion” of all voters
• Section ${\Sigma }_{\tau }$: One specific “election result”
• Effect operator $E_{{\Sigma }_{\tau }}$: “Vote counting rule”
• Probability $p({\Sigma }_{\tau })$: “Vote share of this result”

Different vote counting rules (observable subalgebra), get different result distributions!

3.2 Consistent Histories Framework and Decoherence

Measure structure on section space closely related to consistent histories framework.

Consistent Histories Review: Take finite time sequence ${\tau }_0<{\tau }_1<\cdots <{\tau }_n\subset I$, at each ${\tau }_k$ select POVM decomposition $\{E_{{\alpha }_k}^{(k)}\}$, define history operator: $$C_{\mathbf{\alpha }}=E_{{\alpha }_n}^{(n)}\cdots E_{{\alpha }_1}^{(1)}{E}_{{\alpha }_0}^{(0)}$$

Decoherence Condition: If for $\mathbf{\alpha }\ne \mathbf{\beta }$ have: $$\omega (C_{\mathbf{\alpha }}{C}_{\mathbf{\beta }}^{†})\approx 0$$

Then interference terms negligible, history probability well-defined: $$p(\mathbf{\alpha })=\omega (C_{\mathbf{\alpha }}{C}_{\mathbf{\alpha }}^{†})$$

Connection with Section Theory:

• History $\mathbf{\alpha }$ corresponds to section family $\{{\Sigma }_{{\tau }_k}{\}}_{k=0}^n$
• Decoherence condition guarantees different histories “orthogonal” on observable subalgebra
• Conditionalization $p({\Sigma }_t|{\Sigma }_{\tau })$ defines prediction of future sections

Popular Analogy—Multiple Parallel Universes vs Single Branch Movie:

• Global state $\rho$: “Superposition” of all parallel universes
• Decoherence: Different universes “lose connection”
• Observer experience: In one of universe lines (single branch)

Section Four: Empirical Section Family—World Observer Actually “Sees”

4.1 Definition of Empirical Section

Observer cannot simultaneously “see” all sections—can only see one single branch path among them.

Definition (Empirical Section): Given observer $\mathcal{O}$ and scale $\tau$, if ${\Sigma }_{\tau }\in {\Gamma }_{\mathrm{causal}}^{\mathrm{dyn}}(\tau ;\mathcal{O})$ satisfies:

1. $p({\Sigma }_{\tau })>0$ (non-zero probability)
2. On memory subalgebra ${\mathcal{A}}_{\mathrm{mem}}$, ${\rho }_{\gamma ,\Lambda }(\tau )$ consistent with actual observer’s memory
3. Exists at least one causally consistent section extension family defined on $t\ge \tau$

Then ${\Sigma }_{\tau }$ called observer’s empirical section at scale $\tau$.

All such sections constitute: $${\Gamma }^{\exp }(\tau ;\mathcal{O})\subset {\Gamma }_{\mathrm{causal}}^{\mathrm{dyn}}(\tau ;\mathcal{O})$$

Conditional State: $${\omega }_{{\Sigma }_{\tau }}(A)=\frac{\omega (E_{{\Sigma }_{\tau }}A{E}_{{\Sigma }_{\tau }})}{\omega (E_{{\Sigma }_{\tau }})},A\in {\mathcal{A}}_{\gamma ,\Lambda }(\tau )$$

This is observer’s empirical world at this section!

Definition (Empirical Section Family): If exists mapping $\tau \mapsto {\Sigma }_{\tau }\in {\Gamma }^{\exp }(\tau ;\mathcal{O})$, such that holds for almost all $\tau$ (relative to some natural measure), then $\{{\Sigma }_{\tau }{\}}_{\tau \in I}$ called observer’s empirical section family.

Popular Analogy—“Choose Your Own Adventure” Game Book:

• Each page: One section ${\Sigma }_{\tau }$
• All possible paths: Section space ${\Gamma }_{\mathrm{causal}}$
• Path you actually flip: Empirical section family $\{{\Sigma }_{\tau }\}$
• Probability $p({\Sigma }_{\tau })$: “Plot reasonableness” of reaching this page

You can only play along one path, but all paths “exist” in book!

graph LR
    A["Global State ρ"] --> B["Section Space<br/>Γ_causal(τ)"]
    B --> C["Measure p(Σ_τ)"]

    C --> D1["Empirical Section Family 1<br/>{Σ_τ⁽¹⁾}"]
    C --> D2["Empirical Section Family 2<br/>{Σ_τ⁽²⁾}"]
    C --> D3["..."]
    C --> Dn["Empirical Section Family n<br/>{Σ_τ⁽ⁿ⁾}"]

    D1 --> E["Single Branch World<br/>Observer O Actually Sees"]

    F["Future Prediction"] --> G["p(Σ_t|Σ_τ)<br/>t>τ"]
    E --> F

    style A fill:#e1f5ff
    style B fill:#f0f0f0
    style E fill:#ffe1e1
    style G fill:#fff4e1

4.2 Core Theorem: Existence of Empirical Section Family

This is most important theorem of this chapter!

Theorem (Existence of Empirical Section Family): Under assumptions:

1. Geometry-entropy consistency (stably causal, Hadamard state, generalized entropy extremum)
2. Section effect operator family well-defined

For any observer $\mathcal{O}$ and bounded interval $I=[{\tau }_0,{\tau }_1]$, exists non-empty empirical section family $\{{\Sigma }_{\tau }{\}}_{\tau \in I}$, satisfying:

1. For almost all $\tau \in I$, ${\Sigma }_{\tau }\in {\Gamma }^{\exp }(\tau ;\mathcal{O})$
2. For any finite time sequence ${\tau }_0<\cdots <{\tau }_n\subset I$, conditional states induced by $\{{\Sigma }_{{\tau }_k}\}$ consistent with consistent history probabilities
3. Any two empirical section families if consistent on ${\mathcal{A}}_{\mathrm{mem}}$ at some moment ${\tau }_{\ast }$, then on $[{\tau }_{\ast },{\tau }_1]$ almost everywhere give same observable probability predictions

Proof Outline:

1. Geometry Part: Use small causal diamond covering and Jacobson entanglement equilibrium, obtain local solution family
2. Probability Part: Use consistent history decoherence condition, construct conditional measure on section space
3. Memory Part: Record consistency guarantees when different empirical families consistent on memory, predictions also consistent

Physical Meaning: This theorem precisely expresses core claim: Observer’s empirical world can be characterized as single branch conditionalization path of causally consistent section family, while global superposition only manifests in probability distribution of future sections!

4.3 Re-understanding Superposition and “Collapse”

Theorem (Superposition Only Manifests in Probability of Future Sections): For any $\tau \in I$ and empirical section ${\Sigma }_{\tau }\in {\Gamma }^{\exp }(\tau ;\mathcal{O})$, exists conditional measure $p(\cdot |{\Sigma }_{\tau })$ defined on future section space ${\Gamma }_{\mathrm{causal}}^{\mathrm{dyn}}(>\tau ;\mathcal{O})$, such that:

1. Observer’s experience at moment $\tau$ determined by single state ${\omega }_{{\Sigma }_{\tau }}$, independent of other sections
2. So-called “superposition” only appears in conditional probabilities $p({\Sigma }_t|{\Sigma }_{\tau })$ for $t>\tau$ section predictions
3. If for some future event $F$ have $p(F|{\Sigma }_{\tau })=1$, then for all empirical section families consistent with ${\Sigma }_{\tau }$ records, $F$ necessarily occurs in experience

Popular Understanding—Weather Forecast vs Actual Weather:

• Global superposition: Weather model gives “tomorrow 60% sunny, 40% rainy”
• Observer’s experience today: Single state (e.g., “today cloudy”)
• “Collapse”: When tomorrow arrives, from probability distribution becomes single result
• But this not “true collapse”—just from “prediction distribution” to “actual experience”

In section theory:

• “Now”: Single section ${\Sigma }_{\tau }$
• “Future”: Probability distribution $p({\Sigma }_t|{\Sigma }_{\tau }),t>\tau$
• “Collapse”: When ${\Sigma }_t$ becomes “new now”, other possibilities no longer “future”, but “unrealized possibilities”

Section Five: Section Reformulation of Double-Slit Experiment

Now let us re-understand classic experiments of quantum mechanics using section language.

5.1 Spatial Double Slit: No Path Measurement Case

Experimental Setup:

• Single particle incident state: $|\psi ⟩=\frac{1}{\sqrt{2}}(|L⟩+|R⟩)$
• Observer observable subalgebra: ${\mathcal{A}}_{\gamma ,\Lambda }^{\mathrm{pos}}=\{f({\hat{x}}_{\mathrm{screen}})\}$ (only screen position)

Section Analysis:

1. At each moment ${\tau }_k$, observer’s section ${\Sigma }_{{\tau }_k}$ contains:

   • Position $\gamma ({\tau }_k)$
   • Observable algebra ${\mathcal{A}}^{\mathrm{pos}}$ (does not contain path information)
   • Effective state: ${\rho }_{\gamma ,\Lambda }({\tau }_k)$ on ${\mathcal{A}}^{\mathrm{pos}}$ maintains path coherence

2. Single click event corresponds to effect operator: $E_x\approx |x⟩⟨x|$

3. After many repetitions, click point cloud density: $$P(x)=|{\psi }_L(x)+{\psi }_R(x){|}^2=|{\psi }_L(x){|}^2+|{\psi }_R(x){|}^2+2\mathrm{Re}[{\psi }_L^{\ast }(x){\psi }_R(x)]$$ Has interference term!

Understanding from Section Perspective:

• Each click: One point in empirical section family
• Interference pattern: Result of statistical accumulation of empirical sections
• Key: Because ${\mathcal{A}}^{\mathrm{pos}}$ does not contain path operators, section family inherits global coherence!

Popular Analogy—Projector Focus Mode:

• No path measurement: Projector focuses on “screen” (position)
• Pattern seen: Interference fringes (coherence information preserved)

5.2 Spatial Double Slit: Path Detection Case

Experimental Setup:

• System-environment joint state: $|\Psi ⟩=\frac{1}{\sqrt{2}}(|L⟩\otimes |E_L⟩+|R⟩\otimes |E_R⟩)$
• Environment pointer states $|E_L⟩,|E_R⟩$ approximately orthogonal
• Observable subalgebra extended: ${\mathcal{A}}_{\gamma ,\Lambda }^{\mathrm{path}\otimes \mathrm{pos}}=\{f(\hat{x}),P_L\otimes \mathbb{I},P_R\otimes \mathbb{I}\}$

Section Analysis:

1. Section ${\Sigma }_{{\tau }_k}$ now contains path record

2. After partial trace over environment: $${\rho }_{\mathrm{screen}}\approx \frac{1}{2}(|L⟩⟨L|+|R⟩⟨R|)$$ No interference term! (decoherence)

3. Click point cloud density: $$P_{\mathrm{decoh}}(x)\propto |{\psi }_L(x){|}^2+|{\psi }_R(x){|}^2$$

Understanding from Section Perspective:

• Introducing path detector → Extends ${\mathcal{A}}_{\gamma ,\Lambda }$
• Extended section family contains path records
• Different path histories “orthogonal” on ${\mathcal{A}}_{\mathrm{mem}}$ → Decoherence
• Interference terms suppressed to negligible

Popular Analogy—Labeled Projector:

• Path measurement: Projector simultaneously displays “path label” and “position”
• Label information “contaminates” coherence
• Pattern seen: Two independent Gaussian peaks (no interference)

graph TB
    A["Double-Slit Experiment"] --> B["No Path Measurement"]
    A --> C["Path Measurement"]

    B --> D["Observable Algebra<br/>A_pos={f(x)}"]
    C --> E["Observable Algebra<br/>A_path⊗pos"]

    D --> F["Section Family Maintains<br/>Path Coherence"]
    E --> G["Section Family Contains<br/>Path Records"]

    F --> H["Interference Pattern<br/>P(x) Has Interference Term"]
    G --> I["No Interference Pattern<br/>P(x) No Interference Term"]

    style B fill:#e1ffe1
    style C fill:#ffe1e1

5.3 Wheeler Delayed Choice Experiment

Experimental Setup: After particle passes first beam splitter (BS1), decide whether to insert second beam splitter (BS2) in spatially separated region.

Two Configurations:

1. Insert BS2: Observe interference, observable algebra ${\mathcal{A}}_{\mathrm{interference}}$
2. No BS2: Observe path, observable algebra ${\mathcal{A}}_{\mathrm{path}}$

“Paradox”: Decision made after particle “already passed BS1”, seems to “backwardly influence past”?

Section Theory Answer:

1. Decision event $D$ occurs at moment ${\tau }_D$
2. Detection event $E$ occurs at moment ${\tau }_E>{\tau }_D$
3. Local Causality requires: $D$ must be in past light cone of $E$

Key Insight:

• “Delayed choice” changes structure of future section space ${\Gamma }_{\mathrm{causal}}(>{\tau }_D)$
• Different configurations correspond to different ${\mathcal{A}}_{\gamma ,\Lambda }({\tau }_E)$
• But this does not violate causality: Decision $D$ in past of $E$, completely conforms to $\mathrm{supp}(A)\subset J^{-}(E)$!

Popular Analogy—Plot Branch in Game:

• At level 5 (BS1) choose “left path” or “right path”
• But at level 6 (decision point) can choose “watch plot” or “skip plot”
• This does not change “what you chose at level 5”, only changes “what you see at level 7 (detection)”

Mathematical Formulation:

• Insert BS2: Section family $\{{\Sigma }_{\tau }^{\mathrm{int}}\}$, where ${\mathcal{A}}_{\gamma ,\Lambda }^{\mathrm{int}}({\tau }_E)$ contains phase information
• No BS2: Section family $\{{\Sigma }_{\tau }^{\mathrm{path}}\}$, where ${\mathcal{A}}_{\gamma ,\Lambda }^{\mathrm{path}}({\tau }_E)$ contains path information

Two agree on $\tau <{\tau }_D$, differ on $\tau \ge {\tau }_E$—this is selection of future sections, not change of past!

5.4 Time-Domain Double Slit Experiment

Experimental Setup: Open two “windows” in time $[{\tau }_1,{\tau }_1+\delta \tau ]$, $[{\tau }_2,{\tau }_2+\delta \tau ]$ via pump pulses, detect optical field energy spectrum.

Section Analysis:

1. Observable subalgebra selects time windows: $${\mathcal{A}}_{\gamma ,\Lambda }^{\mathrm{window}}=\{O(\omega )|\omega \in \text{frequency domain}\}$$

2. Interval between two windows on unified time scale $\tau$: $\Delta \tau ={\tau }_2-{\tau }_1$

3. Energy spectrum interference fringe spacing: $$\Delta \omega \cdot \Delta \tau \sim 2\pi$$

Understanding from Section Perspective:

• Time windows: “Sampling” of unified scale $[\tau ]$
• Two windows: Two section families $\{{\Sigma }_{\tau }^{(1)}\}$, $\{{\Sigma }_{\tau }^{(2)}\}$
• Energy spectrum interference: Coherent superposition of two section families on frequency reading subalgebra

Popular Analogy—Shutter Time Control:

• Spatial double slit: Two spatial positions’ “slits”
• Time double slit: Two time positions’ “slits”
• Spatial interference: Fringes in position
• Time interference: Fringes in frequency (Fourier dual)

Key Role of Unified Time Scale: Whether measure $\Delta \tau$ by eigen time, group delay, or modular time, its relationship with $\Delta \omega$ consistent within equivalence class! This makes time double slit direct physical realization of boundary time geometry.

Section Six: Engineering Implementation Schemes

How to test section theory in laboratory?

6.1 Section Engineering in Microwave Scattering Network

Goal: Explicitly construct different observable subalgebras in controllable scattering network, observe effects of section selection.

Experimental Steps:

1. Design Dual-Path Network:

   • Left path ${\mathcal{N}}_L$, right path ${\mathcal{N}}_R$
   • Tunable scattering phases ${\varphi }_L(\omega ),{\varphi }_R(\omega )$

2. Configuration 1: No Path Measurement

   • Only measure output port power $P_{\mathrm{out}}(\omega )$
   • Corresponds to ${\mathcal{A}}_{\gamma ,\Lambda }^{\mathrm{pos}}=\{P_{\mathrm{out}}\}$

3. Configuration 2: Path Detection

   • Insert absorber/amplifier in each path, introduce path markers
   • Corresponds to ${\mathcal{A}}_{\gamma ,\Lambda }^{\mathrm{path}\otimes \mathrm{pos}}=\{P_L,P_R,P_{\mathrm{out}}\}$

4. Measurement:

   • Compare $P_{\mathrm{out}}(\omega )$ under two configurations
   • Verify configuration 1 has interference, configuration 2 no interference

5. Group Delay Consistency:

   • Measure Wigner-Smith group delay $Q(\omega )$
   • Verify $\frac{1}{2\pi }\mathrm{tr}Q(\omega )=\frac{{\phi }^{\prime }(\omega )}{\pi }$
   • Confirm connection between section selection and unified time scale

Prediction: Changes in group delay scale, interference visibility can be unified explained under scale identity, no super-causality needed!

6.2 ITO Optical Time Double Slit

Goal: Realize double slit in time domain, test role of unified time scale in time double slit.

Experimental Steps:

1. Use ITO thin film, induce two refractive index jumps via pump pulses
2. Time interval $\Delta \tau$ tunable
3. Measure probe light energy spectrum $S(\omega )$
4. Observe energy spectrum interference fringes, verify $\Delta \omega \propto 1/\Delta \tau$

Section Theory Prediction:

• Energy spectrum fringe spacing and time window interval satisfy Fourier relation
• Whether measure $\Delta \tau$ by eigen time, group delay, or modular time, relation consistent within equivalence class

6.3 Time-Frequency Interference in Atomic Quantum Memory

Goal: Realize time-frequency double slit in quantum memory, directly test memory sections.

Experimental Steps:

1. Couple two time-separated tunable write pulses to cold atom ensemble
2. Generate two coherent collective spin-wave modes
3. During readout, measure time distribution and energy spectrum
4. Incorporate storage-readout process into ${\mathcal{A}}_{\mathrm{mem}}$, analyze “memory sections”

Section Theory Prediction:

• Long-lived memory degrees of freedom correspond to extension of ${\mathcal{A}}_{\mathrm{mem}}$
• Consistency of empirical section family on memory subalgebra can be tested via readout statistics

Chapter Summary

This chapter established complete theory of observer worldline sections:

Core Concepts:

1. Observer: Triplet $(\gamma ,\Lambda ,{\mathcal{A}}_{\gamma ,\Lambda })$
2. World Section: ${\Sigma }_{\tau }=(\gamma (\tau ),{\mathcal{A}}_{\gamma ,\Lambda }(\tau ),{\rho }_{\gamma ,\Lambda }(\tau ))$
3. Causal Consistency: Local causality, dynamical consistency, record consistency
4. Empirical Section Family: Single branch path observer actually sees

Core Theorems:

1. Existence: Under Jacobson entanglement equilibrium hypothesis, causally consistent section family necessarily exists
2. Uniqueness (mod memory): Two empirical families if memory consistent, then predictions consistent
3. Superposition Reformulation: Superposition only manifests in probability of future sections $p({\Sigma }_t|{\Sigma }_{\tau })$

Experimental Applications:

• Double-slit experiment: Different ${\mathcal{A}}_{\gamma ,\Lambda }$ correspond to different section families
• Delayed choice: Changes future section structure, does not violate causality
• Time double slit: Direct verification of unified time scale

Philosophical Implications:

• “Now” not absolute, but section on observer worldline
• “Collapse” not physical process, but cognitive transition from probability distribution to single experience
• “Many worlds” and “single branch” not contradictory: Measure exists on section space, experience exists on single branch

graph TB
    A["Observer O<br/>=(γ,Λ,A)"] --> B["Worldline γ(τ)<br/>Unified Time Scale"]

    B --> C["Section Σ_τ<br/>=(γ(τ),A(τ),ρ(τ))"]

    C --> D["Local Causality"]
    C --> E["Dynamical Consistency"]
    C --> F["Record Consistency"]

    D --> G["Causally Consistent Sections<br/>Γ_causal(τ)"]
    E --> G
    F --> G

    G --> H["Global State ρ"]
    H --> I["Measure p(Σ_τ)"]

    I --> J["Empirical Section Family<br/>{Σ_τ}"]

    J --> K["Single Branch Empirical World"]

    style A fill:#e1f5ff
    style G fill:#f0f0f0
    style J fill:#ffe1e1
    style K fill:#fff4e1

Poetic Ending:

Observer not bystander outside spacetime, But a worldline inside spacetime. Each “now” not absolute time slice, But section under causal-dynamical-memory constraints. Global superposition not disappeared, Just retreated to probability cloud of future. We proceed along empirical section family, On unified time scale, Physics and experience finally reconcile.

Quick Reference of Core Formulas:

Observer: $$\mathcal{O}=(\gamma :I\to M,\Lambda ,{\mathcal{A}}_{\gamma ,\Lambda })$$

World Section: $${\Sigma }_{\tau }=(\gamma (\tau ),{\mathcal{A}}_{\gamma ,\Lambda }(\tau ),{\rho }_{\gamma ,\Lambda }(\tau ))$$

Causally Consistent Conditions:

1. $\mathrm{supp}(A)\subset J^{-}(\gamma (\tau ))$
2. Exists local solution $(g_{ab},\Phi )$ satisfying Einstein equations
3. ${\rho }_{\gamma ,\Lambda }(\tau )$ consistent with previous states on ${\mathcal{A}}_{\mathrm{mem}}$

Conditional State: $${\omega }_{{\Sigma }_{\tau }}(A)=\frac{\omega (E_{{\Sigma }_{\tau }}A{E}_{{\Sigma }_{\tau }})}{\omega (E_{{\Sigma }_{\tau }})}$$

Future Prediction: $$p({\Sigma }_t|{\Sigma }_{\tau })=\frac{p({\Sigma }_t\cap {\Sigma }_{\tau })}{p({\Sigma }_{\tau })},t>\tau$$

Theoretical Sources:

• observer-world-section-structure-causality-conditionalization.md
• Jacobson entanglement equilibrium: arxiv.org/abs/1505.04753
• QNEC/QFC: arxiv.org/abs/1509.02542

Next chapter we will deeply explore mathematical definition of consciousness, giving five structural criteria of consciousness using quantum Fisher information, mutual information, and causal controllability!