03. Quantum Field Theory, Scattering, Modular Flow: Trio of Dynamics

Introduction: From Static Framework to Dynamic Evolution

Previous article established “static foundation” of universe:

• $U_{\text{evt}}$: Causal framework of events
• $U_{\text{geo}}$: Geometric stage of spacetime
• $U_{\text{meas}}$: Measure lighting of probability

But universe is not static photograph, but dynamic movie. Next three components describe “mechanism of evolution”:

1. Quantum Field Theory Layer $U_{\text{QFT}}$: Defines “how physical fields excite”
2. Scattering and Spectrum Layer $U_{\text{scat}}$: Defines “how particles collide, how long delayed”
3. Modular Flow and Thermal Time Layer $U_{\text{mod}}$: Defines “how thermodynamic time flows”

Relationship among these three is similar to:

• Orchestra Performance (Quantum Field Theory): Each instrument (field mode) vibrates independently
• Concert Recording (Scattering Matrix): Records all information of “input sound → output sound”
• Metronome (Modular Flow): Unifies “beat” of all instruments (thermodynamic time)

They are tightly locked through unified time scale formula: $$\boxed{\kappa (\omega )=\frac{{\phi }^{\prime }(\omega )}{\pi }={\rho }_{\text{rel}}(\omega )=\frac{1}{2\pi }\text{tr}Q(\omega )}$$

Part I: Quantum Field Theory Layer $U_{\text{QFT}}$

1.1 Intuitive Picture: Infinite-Dimensional String Orchestra

Imagine an infinitely large string orchestra:

• Each string = a field mode (e.g., some frequency of electromagnetic field)
• Vibration of string = excitation of particle (photon, electron, etc.)
• Fundamental tone of string = vacuum zero-point energy
• Overtones of string = multi-particle states

But this “string orchestra” must perform on curved stage (spacetime $M$), curvature affects:

• Natural frequency of strings
• Coupling between strings
• Tuning standard of strings (vacuum state definition)

graph TD
    A["Spacetime (M, g)"] --> B["Field Operator φ̂(x)"]
    B --> C["Fock Space F"]
    C --> D["Vacuum State |0⟩"]
    D --> E["Excited State a†|0⟩"]
    E --> F["Particle Interpretation"]

    style A fill:#ffe6e6
    style C fill:#e6f3ff
    style F fill:#e6ffe6

1.2 Strict Mathematical Definition

Definition 1.1 (Quantum Field Theory Layer): $$U_{\text{QFT}}=(\mathcal{A}(M),{\omega }_0,\{{\mathcal{H}}_{\Sigma }{\}}_{\Sigma },\mathcal{U})$$

where:

(1) Local Algebra $\mathcal{A}(M)$:

For each open region $\mathcal{O}\subseteq M$ in spacetime, assign a $C^{\ast }$-algebra $\mathcal{A}(\mathcal{O})$, satisfying:

Isotony: $${\mathcal{O}}_1\subseteq {\mathcal{O}}_2\Rightarrow \mathcal{A}({\mathcal{O}}_1)\subseteq \mathcal{A}({\mathcal{O}}_2)$$

Microcausality (Key Constraint): $${\mathcal{O}}_1\perp {\mathcal{O}}_2(\text{spacelike separated})\Rightarrow [A_1,A_2]=0,\forall A_i\in \mathcal{A}({\mathcal{O}}_i)$$

Physical Meaning: Spacelike separated observation operators commute—no superluminal signals.

(2) Vacuum State ${\omega }_0$:

State (normalized linear functional) on $\mathcal{A}(M)$: $${\omega }_0:\mathcal{A}(M)\to \mathbb{C}$$ satisfying:

• Positivity: ${\omega }_0(A^{\ast }A)\ge 0$
• Normalization: ${\omega }_0(1)=1$
• Hadamard Condition (regularity): $${\omega }_0(\varphi (x)\varphi (y))\sim \frac{1}{4{\pi }^2\sigma (x,y)}+\text{smooth terms}$$ where $\sigma (x,y)$ is geodesic interval.

Physical Meaning: ${\omega }_0$ defines “what is vacuum”—not unique in curved spacetime!

(3) Family of Hilbert Spaces on Cauchy Surfaces $\{{\mathcal{H}}_{\Sigma }{\}}_{\Sigma }$:

For each Cauchy hypersurface $\Sigma$, define Fock space: $${\mathcal{H}}_{\Sigma }=\underset{n=0}{\overset{\infty }{⨁}}{\mathcal{H}}_{\Sigma }^{(n)}$$ where ${\mathcal{H}}_{\Sigma }^{(n)}$ is $n$-particle state space.

Field Operator Decomposition: $$\hat{\varphi }(\mathbf{x},t){|}_{\Sigma }=\sum _k\left({\hat{a}}_k{f}_k(\mathbf{x})+{\hat{a}}_k^{†}{f}_k^{\ast }(\mathbf{x})\right)$$ where:

• ${\hat{a}}_k,{\hat{a}}_k^{†}$: Annihilation/creation operators
• $f_k(\mathbf{x})$: Positive frequency mode functions

Canonical Commutation Relations: $$[{\hat{a}}_k,{\hat{a}}_{k^{\prime }}^{†}]={\delta }_{k{k}^{\prime }},[{\hat{a}}_k,{\hat{a}}_{k^{\prime }}]=0$$

(4) Unitary Evolution Family $\mathcal{U}$: $$\mathcal{U}({\Sigma }_1\to {\Sigma }_2):{\mathcal{H}}_{{\Sigma }_1}\to {\mathcal{H}}_{{\Sigma }_2}$$ satisfying:

• Unitarity: ${\mathcal{U}}^{†}\mathcal{U}=1$
• Compositionality: $\mathcal{U}({\Sigma }_2\to {\Sigma }_3)\mathcal{U}({\Sigma }_1\to {\Sigma }_2)=\mathcal{U}({\Sigma }_1\to {\Sigma }_3)$
• Schrödinger Equation: $$i\hslash \frac{\partial }{\partial t}\mathcal{U}(t)=\hat{H}(t)\mathcal{U}(t)$$

1.3 Core Properties: Haag Theorem and Unruh Effect

Property 1.1 (Haag Theorem):

In curved spacetime or with interactions, does not exist Fock representation unitarily equivalent to free field.

Mathematical Statement: $$\nexists U:{\mathcal{H}}_{\text{free}}\to {\mathcal{H}}_{\text{interacting}},U^{†}{\hat{\varphi }}_{\text{int}}U={\hat{\varphi }}_{\text{free}}$$

Physical Meaning: Vacuum state, particle concept are observer-dependent—no absolute “vacuum”.

Property 1.2 (Unruh Effect):

Minkowski vacuum as seen by accelerated observer (acceleration $a$) is thermal state: $${\omega }_{\text{Rindler}}(A)=\text{tr}\left(\frac{e^{-\beta {\hat{H}}_{\text{Rindler}}}}{Z}A\right),\beta =\frac{2\pi }{a}$$

Temperature Formula: $$T_{\text{Unruh}}=\frac{\hslash a}{2\pi c{k}_B}\approx 4\times 10^{-23}\text{K}\cdot \left(\frac{a}{1{\text{m/s}}^2}\right)$$

Physical Meaning: Acceleration = sensing vacuum radiation—particle concept depends on motion state.

Property 1.3 (Hawking Radiation):

Temperature of black hole horizon (surface gravity ${\kappa }_H$): $$T_H=\frac{\hslash {\kappa }_H}{2\pi c{k}_B}=\frac{\hslash c^3}{8\pi GM{k}_B}\approx 6\times 10^{-8}\text{K}\cdot \left(\frac{M_{\odot }}{M}\right)$$

Physical Meaning: Black holes not completely black, slowly evaporate—combination of quantum field theory + general relativity.

1.4 Example: Fock Space Construction of Scalar Field

Setting: Massless scalar field $\varphi$ in Minkowski spacetime.

(1) Klein-Gordon Equation: $$\square \varphi =\left(-\frac{{\partial }^2}{\partial t^2}+{\nabla }^2\right)\varphi =0$$

(2) Mode Expansion: $$\varphi (t,\mathbf{x})=\int \frac{d^{3}k}{(2\pi {)}^3\sqrt{2{\omega }_k}}\left(a_{\mathbf{k}}{e}^{i(\mathbf{k}\cdot \mathbf{x}-{\omega }_{k}t)}+a_{\mathbf{k}}^{†}{e}^{-i(\mathbf{k}\cdot \mathbf{x}-{\omega }_{k}t)}\right)$$ where ${\omega }_k=|\mathbf{k}|$.

(3) Canonical Quantization: $$[\hat{\varphi }(t,\mathbf{x}),\hat{\pi }(t,\mathbf{y})]=i\hslash {\delta }^3(\mathbf{x}-\mathbf{y})$$ where $\hat{\pi }={\partial }_t\hat{\varphi }$ is conjugate momentum.

(4) Fock Space: $$\mathcal{H}=\mathbb{C}|0⟩\oplus \underset{n=1}{\overset{\infty }{⨁}}{\text{Sym}}^n(L^2({\mathbb{R}}^3,d^{3}k))$$

Vacuum state: $${\hat{a}}_{\mathbf{k}}|0⟩=0,\forall \mathbf{k}$$

Single-particle state: $$|\mathbf{k}⟩={\hat{a}}_{\mathbf{k}}^{†}|0⟩$$

1.5 Challenge of Curved Spacetime: Bogoliubov Transformation

In curved spacetime, mode functions not unique. For example in Schwarzschild black hole:

(1) Boulware Vacuum: Vacuum far from black hole (2) Hartle-Hawking Vacuum: Thermal equilibrium at horizon (3) Unruh Vacuum: Natural state of collapsing black hole

Different vacua connected through Bogoliubov transformation: $${\hat{b}}_k=\sum _j({\alpha }_{kj}{\hat{a}}_j+{\beta }_{kj}{\hat{a}}_j^{†})$$

Key coefficient: $$|{\beta }_{kj}{|}^2=\text{particle production rate}$$

Physical Meaning: Spacetime curvature “produces” particles—vacuum instability.

1.6 Analogy Summary: Piano with Adjustable Pitch

Imagine $U_{\text{QFT}}$ as a special piano:

• Keys = field modes
• Pressing key = producing particle
• Tuning of piano = vacuum state selection
• Stage tilt = spacetime curvature

On flat stage (Minkowski), tuning standard unique; on curved stage (black hole), different observers hear different pitches (Unruh/Hawking radiation).

Part II: Scattering and Spectrum Layer $U_{\text{scat}}$

2.1 Intuitive Picture: “Slow-Motion Replay” of Particle Collision

Imagine a high-speed camera filming particle collision:

• Incoming particles = initial state on left side of camera
• Outgoing particles = final state on right side of camera
• Scattering matrix $S$ = transformation rule “initial → final”
• Delay time $Q$ = “how long particles stay in interaction region”

Key insight: Phase $\phi (\omega )$ of $S$ matrix completely encodes scattering information, and its derivative ${\phi }^{\prime }(\omega )/\pi$ is exactly density of states—this is core of unified time scale!

graph LR
    A["In-State |in⟩"] --> B["S Matrix<br/>S(ω)"]
    B --> C["Out-State |out⟩"]
    B --> D["Phase φ(ω)"]
    D --> E["Wigner Delay<br/>Q(ω) = dS†/dω · S"]
    E --> F["Density of States<br/>ρ_rel(ω) = φ'(ω)/π"]

    style B fill:#ffe6cc
    style E fill:#cce6ff
    style F fill:#ccffcc

2.2 Strict Mathematical Definition

Definition 2.1 (Scattering and Spectrum Layer): $$U_{\text{scat}}=(S(\omega ),Q(\omega ),\phi (\omega ),\{{\lambda }_i{\}}_i,{\rho }_{\text{rel}})$$

where:

(1) Scattering Matrix $S(\omega )$: $$S:{\mathbb{R}}_{+}\to U(N)(\text{unitary matrix-valued function})$$ satisfying:

• Unitarity: $S^{†}(\omega )S(\omega )=1$
• Symmetry (time reversal): $S(\omega )=S^T(\omega )$ (symmetric scattering)
• Smoothness: $S(\omega )\in C^{\infty }({\mathbb{R}}_{+}\setminus \{\text{thresholds}\})$

Physical Meaning: $S_{ij}(\omega )$ is “probability amplitude of incoming wave at frequency $\omega$, scattering from channel $i$ to channel $j$”.

(2) Wigner-Smith Delay Matrix $Q(\omega )$: $$Q(\omega ):=-i\hslash S^{†}(\omega )\frac{dS(\omega )}{d\omega }$$

Key Properties:

• Hermiticity: $Q^{†}=Q$
• Positive semidefiniteness: $Q\ge 0$ (causality guarantee)
• Eigenvalues ${\tau }_i(\omega )$: Delay time of channel $i$

Physical Meaning: $Q$ measures “average residence time of particles in interaction region”.

(3) Scattering Phase $\phi (\omega )$:

Define total phase: $$\phi (\omega ):=-\text{arg}\det S(\omega )=-\sum _i{\theta }_i(\omega )$$ where ${\theta }_i$ are eigenphases of $S$.

Levinson Theorem (bound state counting): $$\frac{\phi (0)-\phi (\infty )}{\pi }=N_{\text{bound}}$$ where $N_{\text{bound}}$ is number of bound states.

(4) Relative Density of States ${\rho }_{\text{rel}}(\omega )$: $$\boxed{{\rho }_{\text{rel}}(\omega ):=\frac{1}{\pi }\frac{d\phi (\omega )}{d\omega }}$$

Unified Time Scale Formula (Core Identity): $$\boxed{\kappa (\omega )=\frac{{\phi }^{\prime }(\omega )}{\pi }={\rho }_{\text{rel}}(\omega )=\frac{1}{2\pi }\text{tr}Q(\omega )}$$

Physical Meaning:

• ${\phi }^{\prime }(\omega )/\pi$: Rate of phase change (geometric)
• ${\rho }_{\text{rel}}(\omega )$: Density of states (statistical)
• $\text{tr}Q(\omega )/2\pi$: Average delay time (dynamical)

Three are completely equivalent!

2.3 Core Properties: Optical Theorem and Friedel Sum Rule

Property 2.1 (Optical Theorem):

Relation between total scattering cross-section ${\sigma }_{\text{tot}}$ and forward scattering amplitude $f(0)$: $${\sigma }_{\text{tot}}=\frac{4\pi }{k}\text{Im}f(0)$$

Scattering Matrix Statement: $$\text{Im}\text{tr}S(\omega )=-\frac{{\sigma }_{\text{tot}}(\omega )\omega }{2\pi }$$

Physical Meaning: Total scattering probability determined by imaginary part of forward amplitude—manifestation of probability conservation.

Property 2.2 (Friedel Sum Rule):

Change in total particle number: $$\Delta N=\frac{1}{\pi }{\int }_0^{E_F}\sum _i\frac{d{\theta }_i(\omega )}{d\omega }d\omega =\frac{1}{\pi }\sum _i{\theta }_i(E_F)$$

Physical Meaning: Scattering phase shift directly measures “number of particles bound by scatterer” (e.g., electron cloud around impurity).

Property 2.3 (Spectral Decomposition of Delay Matrix):

$$Q(\omega )=\sum _i{\tau }_i(\omega )|i⟩⟨i|$$ where ${\tau }_i(\omega )\ge 0$ are characteristic delay times.

Trace Formula: $$\text{tr}Q(\omega )=\sum _i{\tau }_i(\omega )=-i\hslash \text{tr}\left(S^{†}\frac{dS}{d\omega }\right)$$

2.4 Example: Potential Barrier Scattering

Problem: One-dimensional potential barrier $V(x)=V_0\theta (a-|x|)$, particle incident from left.

(1) Scattering Matrix (single channel): $$S(\omega )=e^{2i\delta (\omega )}$$ where phase shift: $$\tan \delta (\omega )=\frac{k\sin (2Ka)}{k\cos (2Ka)-K\sin (2Ka)}$$ here:

• $k=\sqrt{2mE}/\hslash$ (external wavevector)
• $K=\sqrt{2m(E-V_0)}/\hslash$ (internal wavevector)

(2) Wigner Delay: $$\tau (\omega )=-\hslash \frac{d\delta }{d\omega }=\frac{2\hslash K}{k}\cdot \frac{a{\cos }^2(Ka)}{[k\cos (2Ka)-K\sin (2Ka){]}^2+k^2{\sin }^2(2Ka)}$$

Physical Interpretation:

• $E\ll V_0$: $\tau \to 0$ (total reflection, no residence)
• $E\approx V_0$: $\tau$ extremely large (resonance, long residence)
• $E\gg V_0$: $\tau \to 2a/v$ (classical traversal time)

(3) Density of States: $${\rho }_{\text{rel}}(\omega )=\frac{1}{\pi }\frac{d\delta }{d\omega }=\frac{\tau (\omega )}{2\pi \hslash }$$

2.5 Multi-Particle Scattering and Unitarity of S Matrix

In multi-particle scattering, $S$ matrix becomes operator: $$\hat{S}:{\mathcal{H}}_{\text{in}}\to {\mathcal{H}}_{\text{out}}$$

LSZ Reduction Formula: $$⟨p_1^{\prime }\cdots p_n^{\prime }|S|p_1\cdots p_m⟩=\int \prod _{i=1}^n{d}^4{x}_i^{\prime }{e}^{i{p}_i^{\prime }\cdot x_i^{\prime }}\prod _{j=1}^m{d}^4{y}_j{e}^{-i{p}_j\cdot y_j}⟨0|T\{\varphi (x_1^{\prime })\cdots \varphi (y_m)\}|0⟩$$

Unitarity Constraint: $${\hat{S}}^{†}\hat{S}=1\Rightarrow \sum _{\text{all final states}}|⟨f|S|i⟩{|}^2=1$$

Physical Meaning: Total probability conserved—particles must scatter to some out-state.

2.6 Analogy Summary: Acoustic Measurement of Echo Wall

Imagine scattering process as echo wall experiment:

• Clap = incoming particle
• Echo = scattered wave
• Echo time = Wigner delay $\tau$
• Echo pitch change = phase shift $\delta (\omega )$

By analyzing “echo delays at different frequencies”, can reconstruct “internal structure of wall” (potential energy distribution)—this is essence of scattering theory.

Part III: Modular Flow and Thermal Time Layer $U_{\text{mod}}$

3.1 Intuitive Picture: Clock of Thermodynamics

Imagine an hourglass timer:

• Sand flow = entropy increase
• Hourglass flip = time reversal
• Sand flow rate = temperature (fast = high temperature, slow = low temperature)

But in quantum systems, “hourglass” is abstract clock defined by modular Hamiltonian: $$\rho (t)=e^{-itK}\rho (0)e^{itK}$$ where $K$ is modular operator (not necessarily energy!).

Key insight: “Flow rate” of modular flow is exactly another representation of unified time scale $\kappa (\omega )$.

graph TD
    A["Quantum State ρ"] --> B["Modular Hamiltonian K"]
    B --> C["Modular Flow σ_t(ρ)"]
    C --> D["KMS Condition<br/>(Thermal Equilibrium)"]
    D --> E["Modular Time t_mod"]
    E --> F["Time Scale<br/>κ(ω)"]

    style B fill:#fff0e6
    style D fill:#e6f0ff
    style F fill:#e6ffe6

3.2 Strict Mathematical Definition

Definition 3.1 (Modular Flow and Thermal Time Layer): $$U_{\text{mod}}=(\rho ,K,\{{\sigma }_t^{\omega }{\}}_t,\beta (\omega ),t_{\text{mod}})$$

where:

(1) Density Matrix $\rho$:

State on $\mathcal{H}$, already defined in $U_{\text{meas}}$. Here focus on thermal state: $${\rho }_{\beta }=\frac{e^{-\beta \hat{H}}}{Z(\beta )},Z(\beta )=\text{tr}{e}^{-\beta \hat{H}}$$

(2) Modular Hamiltonian $K$:

Defined through Tomita-Takesaki theory. For algebra $\mathcal{A}$ and state $\omega$, define:

GNS Construction: $$({\mathcal{H}}_{\omega },{\pi }_{\omega },|\Omega ⟩)\text{such that}\omega (A)=⟨\Omega |{\pi }_{\omega }(A)|\Omega ⟩$$

Antilinear Operator $S$: $$S\pi (A)|\Omega ⟩=\pi (A^{\ast })|\Omega ⟩$$

Polar Decomposition: $$S=J{\Delta }^{1/2}$$ where:

• $J$: Modular conjugation (anti-unitary)
• $\Delta$: Modular operator (positive definite)

Modular Hamiltonian: $$K:=-\log \Delta$$

(3) Modular Flow ${\sigma }_t^{\omega }$: $${\sigma }_t^{\omega }(A):={\Delta }^{it}A{\Delta }^{-it},\forall A\in \mathcal{A}$$

Tomita Theorem: ${\sigma }_t$ is automorphism of $\mathcal{A}$.

(4) KMS Condition (thermal equilibrium criterion):

State $\omega$ at temperature ${\beta }^{-1}$ in thermal equilibrium $\iff$ for any $A,B\in \mathcal{A}$, exists analytic function $F_{AB}(z)$ satisfying: $$F_{AB}(t)=\omega (A{\sigma }_t(B)),F_{AB}(t+i\beta )=\omega ({\sigma }_t(B)A)$$

Physical Meaning: Time translation $t\to t+i\beta$ equivalent to operator exchange order—this is quantum formulation of thermodynamics.

(5) Modular Time $t_{\text{mod}}$:

Define thermodynamic time flow: $$\frac{d{t}_{\text{mod}}}{d{t}_{\text{geo}}}:=\kappa (\omega )=\frac{1}{2\pi }\text{tr}Q(\omega )$$

Physical Meaning: Thermodynamic time flows in units of “delay time”—rate of entropy increase.

3.3 Core Properties: Multi-Faceted Nature of Time

Property 3.1 (Geometric Time vs Modular Time):

In flat spacetime and no external field: $$t_{\text{mod}}=t_{\text{geo}}(\text{trivial case})$$

But in accelerated frame or near black hole: $$\frac{d{t}_{\text{mod}}}{d{t}_{\text{geo}}}=\sqrt{-g_{00}}\ne 1$$

Example: Schwarzschild spacetime, radial free fall: $$\frac{d{t}_{\text{mod}}}{d\tau }={\left(1-\frac{2GM}{r}\right)}^{-1/2}$$ At horizon $r\to 2GM$, $t_{\text{mod}}\to \infty$ (modular time “freezes”).

Property 3.2 (Geometric Meaning of Inverse Temperature):

For Rindler horizon (acceleration $a$) or black hole horizon (surface gravity ${\kappa }_H$): $$\beta =\frac{2\pi }{{\kappa }_{\text{surface}}}$$

Unified Formula: $${\beta }_{\text{Rindler}}=\frac{2\pi c}{a},{\beta }_{\text{Hawking}}=\frac{8\pi GM}{c^3}$$

Physical Meaning: “Temperature” of horizon determined by geometry (surface gravity)—thermodynamics of gravity.

Property 3.3 (Modular Flow and Time Reversal):

$${\sigma }_{-i\beta /2}(A)=JA{J}^{-1}$$

Physical Meaning: Imaginary time translation $-i\beta /2$ equivalent to time reversal—this is algebraic version of CPT theorem.

3.4 Example: Modular Flow of Quantum Harmonic Oscillator

Setting: Single-mode harmonic oscillator, Hamiltonian: $$\hat{H}=\hslash \omega ({\hat{a}}^{†}\hat{a}+\frac{1}{2})$$

(1) Thermal State: $${\rho }_{\beta }=\frac{e^{-\beta \hat{H}}}{Z},Z=\sum _{n=0}^{\infty }{e}^{-\beta \hslash \omega (n+1/2)}=\frac{e^{-\beta \hslash \omega /2}}{1-e^{-\beta \hslash \omega }}$$

(2) Modular Operator: $$K=\beta \hat{H}$$

(3) Modular Flow: $${\sigma }_t(\hat{a})=e^{it\beta \hslash \omega }\hat{a},{\sigma }_t({\hat{a}}^{†})=e^{-it\beta \hslash \omega }{\hat{a}}^{†}$$

(4) Verify KMS: $$\omega ({\hat{a}}^{†}{\sigma }_t(\hat{a}))=⟨{\hat{a}}^{†}{e}^{it\beta \hslash \omega }\hat{a}⟩=e^{it\beta \hslash \omega }⟨{\hat{a}}^{†}\hat{a}⟩$$ $$\omega ({\sigma }_t(\hat{a}){\hat{a}}^{†})=e^{it\beta \hslash \omega }⟨\hat{a}{\hat{a}}^{†}⟩=e^{it\beta \hslash \omega }(1+⟨{\hat{a}}^{†}\hat{a}⟩)$$

Analytic continuation to $t\to t+i\beta$: $$e^{i(t+i\beta )\beta \hslash \omega }⟨{\hat{a}}^{†}\hat{a}⟩=e^{it\beta \hslash \omega }{e}^{-{\beta }^2\hslash \omega }⟨n⟩$$

Requires: $$⟨n⟩=\frac{1}{e^{\beta \hslash \omega }-1}$$ Exactly Bose-Einstein distribution!

3.5 Proof of Unified Time Scale

Theorem 3.1 (Time Scale Unification):

For state ${\omega }_{\beta }$ satisfying KMS condition, following three are equal: $$\kappa (\omega )=\frac{{\phi }^{\prime }(\omega )}{\pi }={\rho }_{\text{rel}}(\omega )=\frac{1}{2\pi }\text{tr}Q(\omega )=\frac{1}{\beta }\frac{\partial \log Z}{\partial \omega }$$

Proof Outline:

(1) Scattering Phase Shift and Modular Flow:

From KMS condition, phase shift satisfies: $$\frac{d\phi }{d\omega }=\beta ⟨n(\omega )⟩$$ where $⟨n⟩$ is average occupation number.

(2) Delay Matrix and Density of States:

From causality ($Q\ge 0$) and optical theorem: $$\text{tr}Q(\omega )=-i\hslash \text{tr}(S^{†}dS/d\omega )=2\pi \hslash {\rho }_{\text{rel}}(\omega )$$

(3) Partition Function Relation: $$Z(\beta )=\int d\omega e^{-\beta \omega }\rho (\omega )\Rightarrow \frac{\partial \log Z}{\partial \omega }{|}_{\beta }=-\beta +\beta \frac{{\rho }^{\prime }(\omega )}{\rho (\omega )}$$

Combining above three equations, obtain unified formula. ∎

3.6 Analogy Summary: Global Clock with Multiple Time Zones

Imagine modular flow as global time zone system:

• Greenwich Time = geometric time $t_{\text{geo}}$
• Local Time = modular time $t_{\text{mod}}$
• Time Difference = time scale $\kappa (\omega )$

Different “regions” (energy scale $\omega$) have different “time differences”, but through unified “conversion formula” $\kappa (\omega )$ can align all clocks—this is “standard time” of universe.

Part IV: Deep Unification of the Three

4.1 Unified Time Scale: Physical Meaning of Core Identity

$$\boxed{\kappa (\omega )=\frac{{\phi }^{\prime }(\omega )}{\pi }={\rho }_{\text{rel}}(\omega )=\frac{1}{2\pi }\text{tr}Q(\omega )}$$

This formula reveals three seemingly unrelated physical quantities are actually different formulations of same reality:

Philosophical Meaning: Time is not single concept, but common scale of multi-level phenomena.

graph TD
    A["Scattering Phase Shift φ'(ω)"] --> E["Unified Scale κ(ω)"]
    B["Density of States ρ_rel(ω)"] --> E
    C["Delay Matrix tr Q(ω)"] --> E
    D["Modular Flow Rate"] --> E
    E --> F["Essence of Time"]

    style E fill:#ffcccc
    style F fill:#ccffcc

4.2 Compatibility Conditions: Dynamical Closure

Condition 4.1 (Quantum Field Theory → Scattering Matrix):

From unitary evolution $\mathcal{U}$ of $U_{\text{QFT}}$, define: $$S(\omega )=\underset{t\to \pm \infty }{\lim }{e}^{i\omega t}\mathcal{U}(+\infty \to -\infty )e^{-i\omega t}$$

Physical Meaning: $S$ matrix is evolution between asymptotic free states—“encodes” all dynamics as a matrix function.

Condition 4.2 (Scattering Matrix → Modular Flow):

KMS condition gives: $$\beta =\frac{2\pi }{\kappa (\omega )}\cdot \frac{1}{\text{tr}Q(\omega )}$$

Physical Meaning: Inverse temperature determined by delay time—scattering dynamics equivalent to thermodynamics.

Condition 4.3 (Modular Flow → Quantum Field Theory):

Modular operator $K$ generates time translation of $\mathcal{A}(M)$: $${\sigma }_t(A)=e^{itK}A{e}^{-itK}=\mathcal{U}(t)A{\mathcal{U}}^{†}(t)$$

Physical Meaning: Thermodynamic time equivalent to quantum evolution time (in KMS state).

These three conditions form dynamical closed loop:

graph LR
    A["U_QFT"] -- "LSZ Reduction" --> B["U_scat"]
    B -- "KMS Relation" --> C["U_mod"]
    C -- "Tomita Flow" --> A

    style A fill:#e6f3ff
    style B fill:#ffe6cc
    style C fill:#fff0e6

4.3 Core Theorem: Uniqueness of Dynamical Triplet

Theorem 4.1 (Determinacy of Dynamical Triplet):

Given:

1. Spacetime geometry $(M,g)$
2. Matter content (types of fields and coupling constants)
3. Boundary conditions (asymptotic states or initial states)

Then dynamical triplet: $$(U_{\text{QFT}},U_{\text{scat}},U_{\text{mod}})$$ uniquely determined by unified time scale condition.

Corollary 4.1 (No Parameter Freedom):

Cannot independently adjust:

• Scattering phase shift $\phi (\omega )$
• Density of states ${\rho }_{\text{rel}}(\omega )$
• Temperature $\beta (\omega )$

Three locked into one.

4.4 Practical Application: Black Hole Thermodynamics

Problem: Calculate entropy of Schwarzschild black hole.

Solution:

(1) Horizon Surface Gravity: $${\kappa }_H=\frac{c^4}{4GM}$$

(2) Hawking Temperature (from modular flow): $$T_H=\frac{\hslash {\kappa }_H}{2\pi c{k}_B}=\frac{\hslash c^3}{8\pi GM{k}_B}$$

(3) Inverse Temperature: $${\beta }_H=\frac{1}{k_B{T}_H}=\frac{8\pi GM}{c^3}$$

(4) Unified Time Scale:

From scattering near horizon (matter falling into black hole), delay time: $${\tau }_{\text{horizon}}\sim \frac{GM}{c^3}$$

Verification: $$\kappa (\omega )=\frac{1}{2\pi \hslash \tau }=\frac{c^3}{2\pi \hslash GM}=\frac{k_B{T}_H}{\hslash }$$ Consistent with modular flow rate!

(5) Bekenstein-Hawking Entropy:

From statistical mechanics (Component 7 $U_{\text{ent}}$ will derive in detail): $$S_{\text{BH}}=\frac{A}{4{\ell }_P^2}=\frac{k_B{c}^{3}A}{4G\hslash }$$ where $A=16\pi G^2{M}^2/c^4$ is horizon area.

Conclusion: Thermodynamics of black hole completely determined by geometry (${\kappa }_H$) and scattering delay ($\tau$)—victory of unified time scale!

Part V: Physical Picture and Philosophical Meaning

5.1 Triple Identity of Time

In traditional physics, time has multiple “avatars”:

• Newtonian Absolute Time: Background flowing independently
• Special Relativistic Proper Time: Geometric length along worldline
• Thermodynamic Time: Direction of entropy increase
• Quantum Evolution Parameter: $t$ in Schrödinger equation

GLS theory reveals: these “avatars” unified at deep level:

$$t_{\text{Newton}}\sim {\tau }_{\text{SR}}\sim t_{\text{thermal}}\sim t_{\text{quantum}}(\text{affinely equivalent})$$

Core Mechanism: Unified time scale $\kappa (\omega )$ as “conversion coefficient”.

5.2 Triangular Relationship of Causality, Evolution, Entropy Increase

graph TD
    A["Causal Structure<br/>(U_evt)"] --> D["Unified Time [τ]"]
    B["Quantum Evolution<br/>(U_QFT, U_scat)"] --> D
    C["Entropy Increase Direction<br/>(U_mod)"] --> D
    D --> E["Uniqueness of Time Arrow"]

    style D fill:#ffcccc
    style E fill:#ccffcc

Three Mutually Constrain:

• Causal structure excludes “time reversal” (no closed causal chains)
• Quantum evolution guarantees unitarity (probability conservation)
• Entropy increase defines “future” direction (second law)

Philosophical Question: Is time arrow emergent or fundamental?

GLS answer: Both emergent and fundamental—bridged between microscopic (reversible scattering) and macroscopic (irreversible entropy increase) through $\kappa (\omega )$.

5.3 Relativity of Observer and Vacuum

Amazing facts revealed by three components:

1. Vacuum state not unique (Haag theorem): Vacuum in accelerated observer’s eyes = thermal bath in inertial observer’s eyes
2. Particle concept relative (Unruh effect): $T_{\text{Unruh}}=\hslash a/(2\pi k_B)$
3. Time flow rate relative (modular flow): $d{t}_{\text{mod}}/d{t}_{\text{geo}}=\kappa (\omega )$

Implication: No “God’s eye view” absolute physical quantities, everything is relational—this paves way for $U_{\text{obs}}$.

5.4 Hints for Quantum Gravity

Unified time scale formula: $$\kappa (\omega )=\frac{1}{2\pi \hslash }\text{tr}Q(\omega )$$

At quantum gravity scale ($\omega \sim {\ell }_P^{-1}$), possibly: $$\text{tr}Q(\omega )\sim {\ell }_P\Rightarrow \kappa \sim \frac{{\ell }_P}{\hslash }=\frac{1}{m_{P}c}$$

Conjecture: “Quantization” unit of time scale is Planck time $t_P={\ell }_P/c\sim 10^{-44}\text{s}$.

Physical Meaning: At scales $t<t_P$, “time” itself loses meaning—needs entirely new framework of quantum gravity (see $U_{\text{cat}},U_{\text{comp}}$).

Part VI: Advanced Topics and Open Problems

6.1 Modular Flow of Non-Equilibrium States

Current theory mainly handles equilibrium states (KMS states). But real universe often in non-equilibrium:

• Black hole evaporation (net flux)
• Cosmic expansion (time-dependent background)
• Quantum quench (suddenly changing parameters)

Challenge: How to generalize modular flow to non-equilibrium?

Possible Scheme: $${\sigma }_t^{\text{neq}}=\underset{ϵ\to 0^{+}}{\lim }{e}^{-ϵ\hat{H}}{e}^{it{\hat{K}}_{\text{eff}}}(\text{finite temperature correction})$$

6.2 Holographic Duality of Scattering Theory

In AdS/CFT, boundary scattering matrix $S_{\text{CFT}}$ dual to bulk geometry: $$S_{\text{CFT}}(\omega )\leftrightarrow \text{Witten diagram}(\text{bulk})$$

Problem: What does unified time scale $\kappa (\omega )$ correspond to in bulk?

Conjecture: Corresponds to localization length (size of “cloud” of matter in bulk): $$\kappa (\omega )\sim \frac{1}{{\ell }_{\text{bulk}}(\omega )}$$

6.3 Modular Flow in Topological Field Theory

In topological quantum field theory (TQFT), Hamiltonian $\hat{H}=0$, traditional modular flow definition fails.

Generalization: Use topological entanglement entropy to define “topological temperature”: $$S_{\text{topo}}=\gamma ,{\beta }_{\text{topo}}:=\frac{\partial S_{\text{topo}}}{\partial E_{\text{topo}}}$$

Physical Meaning: “Phase transition temperature” of topological order—may relate to categorical structure of $U_{\text{cat}}$.

Part VII: Learning Path and Practical Suggestions

7.1 Steps for Deep Understanding of Three Components

Stage 1: Master algebraic QFT foundations (3-4 weeks)

• Haag-Kastler axioms
• GNS construction
• Wightman functions

Stage 2: Learn scattering theory (2-3 weeks)

• Unitarity of S matrix
• Optical theorem
• Levinson theorem

Stage 3: Tomita-Takesaki theory (4-5 weeks, difficult)

• Definition of modular operator
• KMS condition
• Modular conjugation

Stage 4: Unified time scale (2-3 weeks)

• Derive multiple formulations of $\kappa (\omega )$
• Verify Unruh/Hawking temperature
• Calculate concrete examples

7.2 Recommended References

Classical Textbooks:

1. Haag, Local Quantum Physics (algebraic QFT)
2. Taylor, Scattering Theory (scattering matrix)
3. Bratteli & Robinson, Operator Algebras and Quantum Statistical Mechanics (modular theory)

Modern Reviews:

1. Jacobson, Thermodynamics of Spacetime (thermodynamics of gravity)
2. Rovelli, Thermal Time Hypothesis (thermodynamic origin of time)
3. Witten, APS Medal Lecture (scattering amplitudes and geometry)

GLS Specific:

1. Chapter 5 of this tutorial (scattering matrix and time)
2. Chapter 6 of this tutorial (modular flow and thermal time)
3. Source theory: docs/euler-gls-union/scattering-spectral-density.md

7.3 Common Misconceptions

Misconception 1: “Modular flow only meaningful in thermal equilibrium”

• Correction: Non-equilibrium states also have modular flow (though more complex), key is Tomita operator always exists.

Misconception 2: “Wigner delay is just technical quantity”

• Correction: $Q(\omega )$ is direct manifestation of essence of time, connecting quantum and thermodynamics.

Misconception 3: “Unified time scale is just coincidence”

• Correction: This is manifestation of deep symmetry, possibly originating from ultimate theory of quantum gravity.

Summary and Outlook

Core Points Review

1. Quantum Field Theory Layer $U_{\text{QFT}}$: Defines field operators and vacuum states (observer-dependent)
2. Scattering and Spectrum Layer $U_{\text{scat}}$: Encodes all dynamics as $S(\omega )$ and $Q(\omega )$
3. Modular Flow and Thermal Time Layer $U_{\text{mod}}$: Defines thermodynamic time flow $t_{\text{mod}}$

Three locked through unified time scale formula: $$\kappa (\omega )=\frac{{\phi }^{\prime }(\omega )}{\pi }={\rho }_{\text{rel}}(\omega )=\frac{1}{2\pi }\text{tr}Q(\omega )$$

Connections with Subsequent Components

• $U_{\text{ent}}$: Calculate generalized entropy $S_{\text{gen}}$ from ${\rho }_{\text{mod}}$, reverse derive geometry through IGVP
• $U_{\text{obs}}$: Assign different vacuum states ${\omega }_{\alpha }$ to different observers, resolve Wigner friendship paradox
• $U_{\text{cat}}$: Categorify entire structure, handle topological phase transitions
• $U_{\text{comp}}$: Treat evolution as “computation”, introduce realizability constraints

Philosophical Implication

Time is not “external container”, but common emergence of dynamics, causality, thermodynamics:

• Quantum evolution defines “change”
• Scattering delay defines “duration”
• Entropy increase defines “direction”

Three resonate in $\kappa (\omega )$, this may be answer to “why there is time”.

Next Article Preview:

• 04. Entropy, Observer, Category: Three Pillars of Information Geometry
  • $U_{\text{ent}}$: How does generalized entropy unify black holes and quantum information?
  • $U_{\text{obs}}$: How do multiple observers reach consensus?
  • $U_{\text{cat}}$: How does category theory unify all structures?