23.11 Euler-Lagrange Equations and Computation Worldlines

In previous article, we constructed time-information-complexity action ${\mathcal{A}}_Q$, and derived Euler-Lagrange equations:

• Control: ${\ddot{\theta }}^a+{\Gamma }_{bc}^a{\dot{\theta }}^b{\dot{\theta }}^c=0$ (geodesic);
• Information: ${\ddot{\varphi }}^i+{\Gamma }_{jk}^i{\dot{\varphi }}^j{\dot{\varphi }}^k=-\frac{\gamma }{{\beta }^2}\nabla U_Q$ (geodesic with potential).

These equations describe dynamics of optimal computation worldlines. But like classical mechanics, equations of motion alone are not enough, we also need to understand:

• Conservation Laws: What quantities remain unchanged during evolution?
• Symmetries: What symmetries give rise to conservation laws? (Noether theorem)
• Hamilton Form: How to go from Lagrangian to Hamiltonian? (phase space description)
• Causal Structure: Do computation worldlines have “light cones”?

This article will deeply explore these questions, revealing deep structure of computation worldlines.

Core Questions:

• What conservation laws does time-information-complexity action have?
• How does Noether theorem connect symmetries and conservation laws?
• What do computation worldlines look like in Hamilton form?
• Relationship between causal structure of computation worldlines and physical spacetime?

This article is based on euler-gls-info/05-time-information-complexity-variational-principle.md and classical mechanics Hamilton-Jacobi theory.

1. From Newton’s Laws to Conservation Laws: Everyday Analogy

1.1 Everyday Conservation Law: Money in Wallet

Imagine you keep daily accounts:

• Income: Salary, bonuses (energy input);
• Expenses: Groceries, rent (energy output);
• Balance: Remaining money in wallet (total energy).

Conservation Law: If no income and expenses (isolated system), balance unchanged!

$$\text{Balance}(t)=\text{constant}.$$

This is everyday version of energy conservation.

1.2 Origin of Physical Conservation Laws

In classical mechanics, conservation laws have profound origin: symmetries.

Noether Theorem (1915, Emmy Noether):

Every continuous symmetry corresponds to a conservation law.

Examples:

• Time Translation Symmetry (today’s physics laws = tomorrow’s physics laws) → Energy Conservation;
• Space Translation Symmetry (physics laws here = physics laws there) → Momentum Conservation;
• Space Rotation Symmetry (turn left = turn right) → Angular Momentum Conservation.

graph LR
    A["Symmetry"] --> B["Noether Theorem"]
    B --> C["Conservation Law"]

    D["Time Translation Symmetry<br/>(Physics Laws Don't Change with Time)"] --> E["Energy Conservation<br/>E = Constant"]
    F["Space Translation Symmetry<br/>(Physics Laws Don't Change with Position)"] --> G["Momentum Conservation<br/>p = Constant"]
    H["Space Rotation Symmetry<br/>(Physics Laws Isotropic)"] --> I["Angular Momentum Conservation<br/>L = Constant"]

    style A fill:#e1f5ff
    style B fill:#fff4e1
    style C fill:#ffd4e1
    style D fill:#ffe1e1
    style E fill:#e1ffe1
    style F fill:#e1fff5
    style G fill:#ffe1f5
    style H fill:#f5ffe1
    style I fill:#e1f5ff

1.3 Conservation Laws of Computation Worldlines

In computational universe, we also have action ${\mathcal{A}}_Q$ and Euler-Lagrange equations. Natural questions:

• What symmetries does action ${\mathcal{A}}_Q$ have?
• What conservation laws do these symmetries correspond to?

Core Insight:

• Control-Information Energy: Conserved under specific conditions;
• Information Quality: Conserved under certain evolutions;
• Complexity Momentum: Conserved in spatially uniform systems.

2. Energy Conservation: Time Translation Symmetry

Source Theory: Based on euler-gls-info/05-time-information-complexity-variational-principle.md Section 5 and classical mechanics theory

2.1 Time Translation Invariance of Lagrangian

Recall Lagrangian of joint action:

$$L(\theta ,\dot{\theta };\varphi ,\dot{\varphi })=\frac{1}{2}{\alpha }^2{G}_{ab}(\theta ){\dot{\theta }}^a{\dot{\theta }}^b+\frac{1}{2}{\beta }^2{g}_{ij}(\varphi ){\dot{\varphi }}^i{\dot{\varphi }}^j-\gamma U_Q(\varphi ).$$

Key Observation: $L$ does not explicitly depend on time $t$!

$$\frac{\partial L}{\partial t}=0.$$

This means: Lagrangian invariant under time translation.

Everyday Interpretation:

• Cost of computation today = cost of computation tomorrow;
• Physics laws don’t change with time (assuming no external interference).

2.2 Energy Conservation Law

Theorem 2.1 (Energy Conservation)

If Lagrangian $L$ does not explicitly depend on time $t$, then there exists conserved quantity (energy):

$$E=\sum _a\frac{\partial L}{\partial {\dot{\theta }}^a}{\dot{\theta }}^a+\sum _i\frac{\partial L}{\partial {\dot{\varphi }}^i}{\dot{\varphi }}^i-L,$$

i.e.,

$$\frac{\mathrm{d}E}{\mathrm{d}t}=0.$$

Calculation:

For our Lagrangian:

$$\frac{\partial L}{\partial {\dot{\theta }}^a}={\alpha }^2{G}_{ab}{\dot{\theta }}^b,\frac{\partial L}{\partial {\dot{\varphi }}^i}={\beta }^2{g}_{ij}{\dot{\varphi }}^j.$$

Therefore energy is

$$E={\alpha }^2{G}_{ab}{\dot{\theta }}^a{\dot{\theta }}^b+{\beta }^2{g}_{ij}{\dot{\varphi }}^i{\dot{\varphi }}^j-L.$$

Substitute expression of $L$:

$$E={\alpha }^2{G}_{ab}{\dot{\theta }}^a{\dot{\theta }}^b+{\beta }^2{g}_{ij}{\dot{\varphi }}^i{\dot{\varphi }}^j-(\frac{1}{2}{\alpha }^{2}G{\dot{\theta }}^2+\frac{1}{2}{\beta }^2{g}_Q{\dot{\varphi }}^2-\gamma U_Q),$$

$$E=\frac{1}{2}{\alpha }^2{G}_{ab}{\dot{\theta }}^a{\dot{\theta }}^b+\frac{1}{2}{\beta }^2{g}_{ij}{\dot{\varphi }}^i{\dot{\varphi }}^j+\gamma U_Q(\varphi ).$$

Physical Meaning:

$$E=\underset{\text{Control Kinetic Energy}}{\underbrace{\frac{1}{2}{\alpha }^{2}G{\dot{\theta }}^2}}+\underset{\text{Information Kinetic Energy}}{\underbrace{\frac{1}{2}{\beta }^2{g}_Q{\dot{\varphi }}^2}}+\underset{\text{Information Potential Energy}}{\underbrace{\gamma U_Q(\varphi )}}.$$

Everyday Interpretation:

• Energy = Kinetic Energy + Potential Energy;
• Control kinetic energy: Cost of control changes;
• Information kinetic energy: Cost of information changes;
• Information potential energy: Current information quality.

Conservation Condition: Along solutions of Euler-Lagrange equations, $E$ remains unchanged!

graph TD
    A["Time Translation Symmetry<br/>dL/dt = 0"] --> B["Noether Theorem"]
    B --> C["Energy Conservation<br/>dE/dt = 0"]

    C --> D["Energy Components"]
    D --> E["Control Kinetic Energy<br/>½alpha²G·theta_dot²"]
    D --> F["Information Kinetic Energy<br/>½beta²g_Q·phi_dot²"]
    D --> G["Information Potential Energy<br/>gamma U_Q(phi)"]

    E --> H["Total Energy E = Constant"]
    F --> H
    G --> H

    H --> I["Physical Meaning:<br/>Total Computational Resources Conserved"]

    style A fill:#e1f5ff
    style B fill:#fff4e1
    style C fill:#ffd4e1
    style D fill:#ffe1e1
    style E fill:#e1ffe1
    style F fill:#e1fff5
    style G fill:#ffe1f5
    style H fill:#f5ffe1
    style I fill:#e1f5ff

2.3 Application of Energy Conservation: Velocity Constraints

Energy conservation gives constraints on velocity. If initial energy is $E_0$, then at any time:

$$\frac{1}{2}{\alpha }^{2}G{\dot{\theta }}^2+\frac{1}{2}{\beta }^2{g}_Q{\dot{\varphi }}^2=E_0-\gamma U_Q(\varphi ).$$

Physical Interpretation:

• If information potential $U_Q(\varphi )$ increases (enter high information regions), kinetic energy must decrease (velocity decreases);
• If $U_Q(\varphi )$ decreases (information quality drops), kinetic energy increases (velocity increases);
• This is similar to “mountain climbing”: Higher climb, slower speed.

Example: If $\varphi$ fixed (don’t optimize information), then

$$\frac{1}{2}{\alpha }^{2}G{\dot{\theta }}^2=\text{constant},$$

i.e., control velocity constant (uniform motion).

3. General Form of Noether Theorem

Source Theory: Based on classical variational principle, combined with euler-gls-info/05-time-information-complexity-variational-principle.md

3.1 Definition of Continuous Symmetry

Definition 3.1 (Continuous Symmetry)

Let action be $\mathcal{A}={\int }_0^{T}L(\theta ,\dot{\theta };\varphi ,\dot{\varphi };t)\mathrm{d}t$. If there exists one-parameter transformation

$${\theta }^a\to {\theta }^a+ϵ\delta {\theta }^a,{\varphi }^i\to {\varphi }^i+ϵ\delta {\varphi }^i,$$

such that when $ϵ\to 0$, change in action is

$$\delta \mathcal{A}=O({ϵ}^2),$$

then this transformation is called symmetry transformation.

Examples:

• Time Translation: $t\to t+ϵ$, $L$ doesn’t explicitly depend on $t$;
• Parameter Translation: ${\theta }^a\to {\theta }^a+ϵc^a$ (constant), $L$ periodic in ${\theta }^a$;
• Rotation: Rotate parameters in some subspace, $L$ rotation invariant.

3.2 Noether Theorem

Theorem 3.2 (Noether Theorem, 1915)

For every continuous symmetry transformation $(\delta {\theta }^a,\delta {\varphi }^i)$, there exists conserved current:

$$J=\sum _a\frac{\partial L}{\partial {\dot{\theta }}^a}\delta {\theta }^a+\sum _i\frac{\partial L}{\partial {\dot{\varphi }}^i}\delta {\varphi }^i-L\delta t,$$

satisfying

$$\frac{\mathrm{d}J}{\mathrm{d}t}=0.$$

Applications:

1. Time Translation $\delta t=ϵ$, $\delta \theta =\delta \varphi =0$: $$J=-L\Rightarrow E=-L+\sum _a{p}_a{\dot{\theta }}^a+\sum _i{\pi }_i{\dot{\varphi }}^i.$$ (Energy conservation)

2. Parameter Translation $\delta {\theta }^a=ϵc^a$: $$J=\sum _a\frac{\partial L}{\partial {\dot{\theta }}^a}{c}^a.$$ (Momentum conservation)

3. Information Manifold Rotation $\delta {\varphi }^i=ϵR_j^i{\varphi }^j$: $$J=\sum _i\frac{\partial L}{\partial {\dot{\varphi }}^i}{R}_j^i{\varphi }^j.$$ (Information angular momentum conservation)

graph TD
    A["Symmetry Transformation<br/>(theta,phi) → (theta+delta theta,phi+delta phi)"] --> B["Check:<br/>delta A = O(epsilon²)?"]
    B --> C["Yes → Symmetry"]
    B --> D["No → Not Symmetry"]

    C --> E["Noether Theorem"]
    E --> F["Conserved Current J"]
    F --> G["dJ/dt = 0"]

    G --> H["Specific Forms"]
    H --> I["Time Translation<br/>→ Energy E"]
    H --> J["Space Translation<br/>→ Momentum p"]
    H --> K["Rotation<br/>→ Angular Momentum L"]

    I --> L["Conservation Law:<br/>Physical Quantity Doesn't Change with Time"]
    J --> L
    K --> L

    style A fill:#e1f5ff
    style C fill:#fff4e1
    style E fill:#ffd4e1
    style F fill:#ffe1e1
    style G fill:#e1ffe1
    style L fill:#ffe1f5

4. Hamilton Form: From Lagrangian to Hamiltonian

Source Theory: Based on classical mechanics Hamilton theory, combined with euler-gls-info/05-time-information-complexity-variational-principle.md

4.1 Why Do We Need Hamilton Form?

Lagrangian form uses $(\theta ,\dot{\theta },\varphi ,\dot{\varphi })$ as variables (position + velocity).

Hamilton form uses $(\theta ,p_{\theta },\varphi ,p_{\varphi })$ as variables (position + momentum).

Advantages:

• Phase Space Description: Lift $2n$-dimensional configuration space to $4n$-dimensional phase space;
• Symplectic Geometric Structure: Hamilton equations have beautiful symplectic symmetry;
• Quantization: Hamilton form is starting point of quantum mechanics (operators replace classical variables).

4.2 Legendre Transform: From Velocity to Momentum

Definition 4.1 (Conjugate Momentum)

Define control conjugate momentum:

$$p_a=\frac{\partial L}{\partial {\dot{\theta }}^a}={\alpha }^2{G}_{ab}(\theta ){\dot{\theta }}^b.$$

Define information conjugate momentum:

$${\pi }_i=\frac{\partial L}{\partial {\dot{\varphi }}^i}={\beta }^2{g}_{ij}(\varphi ){\dot{\varphi }}^j.$$

Physical Meaning:

• $p_a$ is “control momentum”, measures control velocity times “mass” (metric);
• ${\pi }_i$ is “information momentum”, measures information velocity times “mass” (Fisher metric).

Invert Velocity:

$${\dot{\theta }}^a=\frac{1}{{\alpha }^2}{G}^{ab}{p}_b,{\dot{\varphi }}^i=\frac{1}{{\beta }^2}{g}^{ij}{\pi }_j.$$

4.3 Definition of Hamiltonian

Definition 4.2 (Hamiltonian)

Define Hamiltonian through Legendre transform:

$$H(\theta ,p_{\theta };\varphi ,p_{\varphi })=\sum _a{p}_a{\dot{\theta }}^a+\sum _i{\pi }_i{\dot{\varphi }}^i-L.$$

Substitute:

$$H=p_a{\dot{\theta }}^a+{\pi }_i{\dot{\varphi }}^i-(\frac{1}{2}{\alpha }^2{G}_{ab}{\dot{\theta }}^a{\dot{\theta }}^b+\frac{1}{2}{\beta }^2{g}_{ij}{\dot{\varphi }}^i{\dot{\varphi }}^j-\gamma U_Q).$$

Using $p_a={\alpha }^2{G}_{ab}{\dot{\theta }}^b$:

$$p_a{\dot{\theta }}^a={\alpha }^2{G}_{ab}{\dot{\theta }}^b{\dot{\theta }}^a={\alpha }^{2}G{\dot{\theta }}^2,$$

Similarly ${\pi }_i{\dot{\varphi }}^i={\beta }^2{g}_Q{\dot{\varphi }}^2$. Therefore:

$$H={\alpha }^{2}G{\dot{\theta }}^2+{\beta }^2{g}_Q{\dot{\varphi }}^2-\frac{1}{2}{\alpha }^{2}G{\dot{\theta }}^2-\frac{1}{2}{\beta }^2{g}_Q{\dot{\varphi }}^2+\gamma U_Q,$$

$$H=\frac{1}{2}{\alpha }^{2}G{\dot{\theta }}^2+\frac{1}{2}{\beta }^2{g}_Q{\dot{\varphi }}^2+\gamma U_Q.$$

Expressed in momentum:

$$H=\frac{1}{2{\alpha }^2}{G}^{ab}{p}_a{p}_b+\frac{1}{2{\beta }^2}{g}^{ij}{\pi }_i{\pi }_j+\gamma U_Q(\varphi ).$$

Physical Meaning: $H$ is exactly energy $E$!

$$H=\underset{\text{Control Kinetic Energy}}{\underbrace{\frac{1}{2{\alpha }^2}{G}^{ab}{p}_a{p}_b}}+\underset{\text{Information Kinetic Energy}}{\underbrace{\frac{1}{2{\beta }^2}{g}^{ij}{\pi }_i{\pi }_j}}+\underset{\text{Information Potential Energy}}{\underbrace{\gamma U_Q(\varphi )}}.$$

4.4 Hamilton Canonical Equations

Theorem 4.3 (Hamilton Canonical Equations)

Euler-Lagrange equations equivalent to Hamilton canonical system:

$$\{\begin{array}{l}{\dot{\theta }}^a=\frac{\partial H}{\partial p_a}=\frac{1}{{\alpha }^2}{G}^{ab}{p}_b\\ {\dot{p}}_a=-\frac{\partial H}{\partial {\theta }^a}=-\frac{1}{2{\alpha }^2}({\partial }_a{G}^{bc})p_b{p}_c\\ {\dot{\varphi }}^i=\frac{\partial H}{\partial {\pi }_i}=\frac{1}{{\beta }^2}{g}^{ij}{\pi }_j\\ {\dot{\pi }}_i=-\frac{\partial H}{\partial {\varphi }^i}=-\frac{1}{2{\beta }^2}({\partial }_i{g}^{jk}){\pi }_j{\pi }_k-\gamma {\partial }_i{U}_Q\end{array}$$

Everyday Interpretation:

• Equations 1, 3: Momentum determines velocity (velocity is function of momentum);
• Equations 2, 4: Gradients of potential and metric drive momentum changes.

graph TD
    A["Lagrangian Form<br/>L(theta,theta_dot,phi,phi_dot)"] --> B["Legendre Transform"]
    B --> C["Hamiltonian Form<br/>H(theta,p,phi,pi)"]

    A --> D["Variables:<br/>(theta,theta_dot,phi,phi_dot)<br/>Position + Velocity"]
    C --> E["Variables:<br/>(theta,p,phi,pi)<br/>Position + Momentum"]

    D --> F["Euler-Lagrange Equations<br/>Second-Order Differential Equations"]
    E --> G["Hamilton Canonical Equations<br/>First-Order Differential System"]

    F --> H["Advantages:<br/>Intuitive, Variational Principle"]
    G --> I["Advantages:<br/>Symplectic Geometry, Quantization"]

    C --> J["Energy:<br/>H = T + V"]
    J --> K["Control Kinetic + Information Kinetic + Information Potential"]

    style A fill:#e1f5ff
    style B fill:#fff4e1
    style C fill:#ffd4e1
    style E fill:#ffe1e1
    style G fill:#e1ffe1
    style I fill:#e1fff5
    style J fill:#ffe1f5
    style K fill:#f5ffe1

5. Symplectic Geometry: Geometric Structure of Phase Space

Source Theory: Based on classical symplectic geometry theory

5.1 Phase Space and Symplectic Form

Definition 5.1 (Phase Space)

Phase space is space of all possible $(\theta ,p,\varphi ,\pi )$:

$$\mathcal{P}=T^{\ast }\mathcal{M}\times T^{\ast }{\mathcal{S}}_Q,$$

where $T^{\ast }\mathcal{M}$ is cotangent bundle of control manifold, $T^{\ast }{\mathcal{S}}_Q$ is cotangent bundle of information manifold.

Symplectic Form: Define symplectic 2-form on phase space:

$$\omega =\sum _a\mathrm{d}{p}_a\wedge \mathrm{d}{\theta }^a+\sum _i\mathrm{d}{\pi }_i\wedge \mathrm{d}{\varphi }^i.$$

Properties:

• $\omega$ is closed form: $\mathrm{d}\omega =0$;
• $\omega$ is non-degenerate: For any nonzero vector $v$, there exists $w$ such that $\omega (v,w)\ne 0$.

Physical Meaning: Symplectic form encodes “conjugate relationship” between canonical variables $(\theta ,p)$ and $(\varphi ,\pi )$.

5.2 Hamilton Flow and Symplectic Preservation

Definition 5.2 (Hamilton Vector Field)

Hamiltonian $H$ induces a vector field $X_H$, satisfying:

$$\omega (X_H,\cdot )=\mathrm{d}H.$$

In coordinates:

$$X_H=\frac{\partial H}{\partial p_a}\frac{\partial }{\partial {\theta }^a}-\frac{\partial H}{\partial {\theta }^a}\frac{\partial }{\partial p_a}+\frac{\partial H}{\partial {\pi }_i}\frac{\partial }{\partial {\varphi }^i}-\frac{\partial H}{\partial {\varphi }^i}\frac{\partial }{\partial {\pi }_i}.$$

Theorem 5.3 (Hamilton Flow Preserves Symplectic Form)

Hamilton flow (integral curves along $X_H$) preserves symplectic form:

$${\mathcal{L}}_{X_H}\omega =0,$$

where $\mathcal{L}$ is Lie derivative.

Physical Meaning: Hamilton evolution preserves “volume” of phase space (generalization of Liouville theorem).

5.3 Poisson Bracket

Definition 5.4 (Poisson Bracket)

For any two phase space functions $f,g$, define Poisson bracket:

$$\{f,g\}=\sum _a(\frac{\partial f}{\partial {\theta }^a}\frac{\partial g}{\partial p_a}-\frac{\partial f}{\partial p_a}\frac{\partial g}{\partial {\theta }^a})+\sum _i(\frac{\partial f}{\partial {\varphi }^i}\frac{\partial g}{\partial {\pi }_i}-\frac{\partial f}{\partial {\pi }_i}\frac{\partial g}{\partial {\varphi }^i}).$$

Properties:

• Antisymmetric: $\{f,g\}=-\{g,f\}$;
• Jacobi identity: $\{\{f,g\},h\}+\{\{g,h\},f\}+\{\{h,f\},g\}=0$;
• Leibniz rule: $\{f,gh\}=g\{f,h\}+\{f,g\}h$.

Evolution Equation: Time evolution of any physical quantity $f(\theta ,p,\varphi ,\pi )$ is:

$$\frac{\mathrm{d}f}{\mathrm{d}t}=\{f,H\}.$$

In particular, Hamilton equations can be written as:

$${\dot{\theta }}^a=\{{\theta }^a,H\},{\dot{p}}_a=\{p_a,H\},{\dot{\varphi }}^i=\{{\varphi }^i,H\},{\dot{\pi }}_i=\{{\pi }_i,H\}.$$

graph TD
    A["Phase Space P<br/>(theta,p,phi,pi)"] --> B["Symplectic Form omega"]
    B --> C["omega = dp∧dtheta + dpi∧dphi"]

    C --> D["Properties"]
    D --> E["Closed Form:<br/>domega = 0"]
    D --> F["Non-Degenerate:<br/>omega(v,w)≠0"]

    A --> G["Hamiltonian H"]
    G --> H["Hamilton Vector Field X_H"]
    H --> I["Hamilton Flow<br/>(Time Evolution)"]

    I --> J["Symplectic Preservation:<br/>L_X_H omega = 0"]
    J --> K["Liouville Theorem:<br/>Phase Space Volume Conserved"]

    A --> L["Poisson Bracket<br/>{f,g}"]
    L --> M["Evolution Equation<br/>df/dt = {f,H}"]

    M --> N["Quantization:<br/>[f,g] = iℏ{f,g}"]

    style A fill:#e1f5ff
    style B fill:#fff4e1
    style C fill:#ffd4e1
    style G fill:#ffe1e1
    style I fill:#e1ffe1
    style J fill:#e1fff5
    style L fill:#ffe1f5
    style M fill:#f5ffe1
    style N fill:#e1f5ff

6. Causal Structure of Computation Worldlines

Source Theory: Based on euler-gls-info/05-time-information-complexity-variational-principle.md and GLS causal theory

6.1 Timelike, Lightlike, Spacelike Worldlines

In relativity, worldlines are classified by norm of four-velocity:

• Timelike: $\Vert \dot{x}{\Vert }^2<0$ (massive particles);
• Lightlike: $\Vert \dot{x}{\Vert }^2=0$ (photons);
• Spacelike: $\Vert \dot{x}{\Vert }^2>0$ (unreachable, faster than light).

In computational universe, we have joint metric $\mathbb{G}={\alpha }^{2}G\oplus {\beta }^2{g}_Q$. Define “velocity norm squared” of worldline:

$$\Vert \dot{z}{\Vert }_{\mathbb{G}}^2={\alpha }^2{G}_{ab}{\dot{\theta }}^a{\dot{\theta }}^b+{\beta }^2{g}_{ij}{\dot{\varphi }}^i{\dot{\varphi }}^j.$$

Analogously, we can define:

Definition 6.1 (Causal Type of Computation Worldline)

• Timelike Computation: $\Vert \dot{z}{\Vert }_{\mathbb{G}}^2=2E$ (finite energy);
• Lightlike Computation: $\Vert \dot{z}{\Vert }_{\mathbb{G}}^2=0$ (zero energy, limiting case);
• Spacelike Computation: $\Vert \dot{z}{\Vert }_{\mathbb{G}}^2<0$ (negative energy, usually excluded).

Physical Meaning:

• Timelike computation: Normal computation process, consumes finite energy;
• Lightlike computation: Adiabatic limit (infinitely slow), energy tends to zero;
• Spacelike computation: Non-physical (violates energy conservation).

6.2 “Light Cone” of Computation

In Minkowski spacetime, light cone defines causally reachable region: Only timelike or lightlike paths can connect two events.

In computational universe, similar causal structure defined by complexity metric $G$ and information metric $g_Q$.

Definition 6.2 (Computation Light Cone)

Given initial point $z_0=({\theta }_0,{\varphi }_0)$, future computation light cone from $z_0$ is set of all points $z=(\theta ,\varphi )$ satisfying:

$$\exists \text{ worldline }z(t),z(0)=z_0,z(T)=z,E[z]\le E_{\max },$$

where $E_{\max }$ is available total energy budget.

Everyday Interpretation:

• Computation light cone is “all states reachable from initial state under given energy”;
• Similar to “reachable set of goods you can buy with 100 dollars”.

6.3 Causal Structure and Computational Complexity

In GLS theory, causal structure of computational universe closely related to causal structure of physical spacetime (through QCA realization).

Key Insight:

• Complexity Distance $d_G$ similar to “proper time” in spacetime;
• Information Distance $d_{g_Q}$ similar to “distance in information space”;
• Energy Constraint $E\le E_{\max }$ limits reachable region, similar to speed of light limiting causally reachable region.

graph TD
    A["Computation Worldline z(t)"] --> B["Velocity Norm Squared<br/>||z_dot||²_G"]

    B --> C["Timelike Computation<br/>||z_dot||² = 2E > 0<br/>(Normal Computation)"]
    B --> D["Lightlike Computation<br/>||z_dot||² = 0<br/>(Adiabatic Limit)"]
    B --> E["Spacelike Computation<br/>||z_dot||² < 0<br/>(Non-Physical)"]

    C --> F["Finite Energy<br/>Consumes Resources"]
    D --> G["Zero Energy<br/>Infinitely Slow"]
    E --> H["Negative Energy<br/>Violates Physics"]

    A --> I["Causal Structure"]
    I --> J["Computation Light Cone:<br/>Reachable Region Under<br/>Energy Constraint"]

    J --> K["Similar to Physics:<br/>Speed of Light Limits<br/>Causally Reachable Region"]

    style A fill:#e1f5ff
    style B fill:#fff4e1
    style C fill:#ffd4e1
    style F fill:#ffe1e1
    style I fill:#e1ffe1
    style J fill:#e1fff5
    style K fill:#ffe1f5

7. Connection with Physical Universe: QCA Worldlines

Source Theory: Based on GLS unified theory framework (preview from euler-gls-info/06-categorical-equivalence-computational-physical-universes.md)

7.1 QCA (Quantum Cellular Automaton) Universe

In GLS theory, physical universe is modeled as Quantum Cellular Automaton (QCA):

• Spacetime is discrete grid;
• Each lattice point has quantum state;
• Evolution controlled by unitary operators (preserve probability sum).

Worldlines of QCA are trajectories on lattice, satisfying:

• Locality: Each step only affects neighboring lattice points;
• Unitarity: Total probability conserved;
• Causality: Future lattice points only depend on past light cone lattice points.

7.2 Computation Worldline ↔ QCA Worldline

Core Proposition 7.1 (Computation-Physics Correspondence, from GLS Framework)

There exists functor $F:{\mathbf{CompUniv}}^{\mathrm{phys}}\to {\mathbf{PhysUniv}}^{\mathrm{QCA}}$, such that:

1. Object Level: Computational universe $U_{\mathrm{comp}}$ corresponds to a QCA universe $U_{\mathrm{QCA}}$;
2. Trajectory Level: Computation worldline $z(t)=(\theta (t),\varphi (t))$ corresponds to worldline $x_{\mathrm{QCA}}(t)$ on QCA lattice;
3. Causal Level: Computation light cone ↔ QCA light cone;
4. Energy Level: Computation energy $E$ ↔ Expectation value of QCA Hamiltonian.

Everyday Interpretation:

• Computation worldline is “abstract algorithm trajectory”;
• QCA worldline is “concrete physical particle trajectory”;
• Both completely equivalent (categorically equivalent) under GLS theory framework!

7.3 Quantum Entanglement and Information Manifold

In QCA, quantum entanglement is core feature. In computational universe framework, entanglement corresponds to non-trivial structure on information manifold.

Examples:

• Unentangled State: $\rho ={\rho }_A\otimes {\rho }_B$, information manifold is product manifold;
• Maximally Entangled State: Bell state, information manifold has non-trivial topology;
• Entanglement Entropy: $S({\rho }_A)=-\mathrm{Tr}({\rho }_A\log {\rho }_A)$, corresponds to volume on information manifold.

Key Insight:

• Information potential $U_Q(\varphi )$ can encode entanglement information;
• Optimal computation worldlines tend to “utilize entanglement” (enter high $U_Q$ regions).

graph TD
    A["Computational Universe<br/>CompUniv^phys"] --> B["Functor F"]
    B --> C["Physical Universe<br/>PhysUniv^QCA"]

    A --> D["Computation Worldline<br/>z(t) = (theta(t),phi(t))"]
    C --> E["QCA Worldline<br/>x_QCA(t)"]

    D -->|"F"| E

    A --> F["Computation Light Cone<br/>(Energy Constraint)"]
    C --> G["QCA Light Cone<br/>(Speed of Light Constraint)"]

    F -->|"F"| G

    A --> H["Computation Energy<br/>E = T + V"]
    C --> I["QCA Hamiltonian<br/>⟨H⟩"]

    H -->|"F"| I

    E --> J["Physical Realization:<br/>Particle Trajectories, Wave Functions"]
    I --> K["Quantum Entanglement:<br/>Encoded in Information Manifold"]

    style A fill:#e1f5ff
    style B fill:#fff4e1
    style C fill:#ffd4e1
    style D fill:#ffe1e1
    style E fill:#e1ffe1
    style F fill:#e1fff5
    style G fill:#ffe1f5
    style H fill:#f5ffe1
    style J fill:#e1f5ff
    style K fill:#fff4e1

8. Complete Picture: From Action to Worldlines

8.1 Theoretical Structure Summary

graph TD
    A["Time-Information-Complexity<br/>Action A_Q"] --> B["Variational Principle<br/>delta A_Q = 0"]

    B --> C["Euler-Lagrange Equations"]
    C --> D["Control: Geodesic<br/>theta_ddot + Gamma·theta_dot² = 0"]
    C --> E["Information: Geodesic with Potential<br/>phi_ddot + Gamma·phi_dot² = -nabla U_Q"]

    A --> F["Symmetry Analysis"]
    F --> G["Time Translation Symmetry<br/>dL/dt = 0"]
    F --> H["Parameter Translation Symmetry"]
    F --> I["Rotation Symmetry"]

    G --> J["Noether Theorem"]
    H --> J
    I --> J

    J --> K["Conservation Laws"]
    K --> L["Energy Conservation<br/>E = T + V"]
    K --> M["Momentum Conservation<br/>p = Constant"]
    K --> N["Angular Momentum Conservation<br/>L = Constant"]

    A --> O["Legendre Transform"]
    O --> P["Hamiltonian H"]
    P --> Q["Hamilton Canonical Equations<br/>theta_dot = dH/dp<br/>p_dot = -dH/dtheta"]

    P --> R["Symplectic Geometry"]
    R --> S["Symplectic Form omega"]
    R --> T["Poisson Bracket {f,g}"]
    R --> U["Symplectic Preserving Flow"]

    Q --> V["Computation Worldline<br/>z(t) = (theta(t),phi(t))"]
    D --> V
    E --> V

    V --> W["Causal Structure"]
    W --> X["Timelike, Lightlike, Spacelike"]
    W --> Y["Computation Light Cone"]

    V --> Z["Physical Realization"]
    Z --> AA["QCA Worldline"]
    Z --> AB["Quantum Entanglement"]

    style A fill:#e1f5ff
    style B fill:#fff4e1
    style F fill:#ffd4e1
    style J fill:#ffe1e1
    style K fill:#e1ffe1
    style P fill:#e1fff5
    style R fill:#ffe1f5
    style V fill:#f5ffe1
    style W fill:#e1f5ff
    style Z fill:#fff4e1

8.2 Core Formula Quick Reference

9. Summary

This article deeply explored dynamics and geometric structure of computation worldlines:

9.1 Core Concepts

1. Energy Conservation: Time translation symmetry → Energy $E=T+V$ conserved

   • Control kinetic energy + Information kinetic energy + Information potential energy = constant

2. Noether Theorem: Symmetry ↔ Conservation Law

   • Time translation → Energy conservation
   • Space translation → Momentum conservation
   • Rotation → Angular momentum conservation

3. Hamilton Form: Lagrangian → Hamiltonian

   • Variables: (position, velocity) → (position, momentum)
   • Equations: Second-order differential equations → First-order canonical system

4. Symplectic Geometry: Geometric structure of phase space

   • Symplectic form $\omega =\mathrm{d}p\wedge \mathrm{d}\theta +\mathrm{d}\pi \wedge \mathrm{d}\varphi$
   • Poisson bracket $\{f,g\}$
   • Symplectic preserving flow (Liouville theorem)

5. Causal Structure: Computation light cone

   • Timelike computation: Normal energy
   • Lightlike computation: Adiabatic limit
   • Energy constraint defines reachable region

6. Connection with Physical Universe: Computation worldline ↔ QCA worldline

   • Functor $F:\mathbf{CompUniv}\to \mathbf{PhysUniv}$
   • Entanglement ↔ Information manifold structure

9.2 Core Insights

• Conservation Laws Arise from Symmetries: Noether theorem is bridge connecting geometry and dynamics;
• Hamilton Form Unifies Dynamics: First-order canonical equations, symplectic geometric structure, paves way for quantization;
• Causal Structure Geometrized: Energy constraint defines computation light cone, similar to speed of light limiting physical causality;
• Computation Is Physics: Computation worldlines and QCA worldlines equivalent under GLS framework;
• Information Geometry Encodes Entanglement: Non-trivial structure of information manifold corresponds to quantum entanglement.

9.3 Everyday Analogy Review

• Wallet Balance: Everyday version of conservation law (balance unchanged when income = expenses);
• Mountain Climbing: Slow speed when potential high (energy conservation);
• Phase Space: “Complete description” of position + momentum;
• Light Cone: Reachable region limited by energy/speed of light;
• Abstract Algorithm ↔ Concrete Physics: Computation worldline ↔ particle trajectory.

9.4 Connections with Previous and Subsequent Chapters

Connection with Articles 23.1-10:

• Articles 23.3-5: Complexity geometry → Control manifold $(\mathcal{M},G)$
• Articles 23.6-7: Information geometry → Information manifold $({\mathcal{S}}_Q,g_Q)$
• Article 23.10: Joint action ${\mathcal{A}}_Q$ and Euler-Lagrange equations
• This Article: Conservation laws, Hamilton form, symplectic geometry, causal structure

Preview of Article 23.12: Next article will construct functor structure of Physical Universe ↔ Computational Universe:

• Functor $F:{\mathbf{CompUniv}}^{\mathrm{phys}}\to {\mathbf{PhysUniv}}^{\mathrm{QCA}}$
• Preservation of morphisms (simulation maps ↔ QCA maps)
• Correspondence of complexity distance ↔ spacetime distance
• Paving way for categorical equivalence theorem

Preview of Next Article: 23.12 Physical Universe ↔ Computational Universe: Functor Structure

In next article, we will:

1. Construct Functor $F:\mathbf{CompUniv}\to \mathbf{PhysUniv}$: Mapping from computation to physics;
2. Object Level: Computational universe $U_{\mathrm{comp}}$ ↔ QCA universe $U_{\mathrm{QCA}}$;
3. Morphism Level: Simulation maps ↔ QCA morphisms (preserving locality, unitarity);
4. Distance Preservation: Complexity distance $d_{\mathrm{comp}}$ ↔ Spacetime distance $d_{\mathrm{spacetime}}$;
5. Inverse Functor $G:\mathbf{PhysUniv}\to \mathbf{CompUniv}$: From physics to computation;
6. Natural Isomorphism: $F\circ G\simeq \mathrm{id}$, $G\circ F\simeq \mathrm{id}$ (paving way for categorical equivalence).