Modular Time: Intrinsic Evolution of Quantum States

“Modular time can be viewed as the quantum state’s own clock.”

🎯 Core Proposition

Definition (Tomita-Takesaki Modular Flow):

For a von Neumann algebra $\mathcal{M}$ and a faithful state $\omega$, there exists a unique one-parameter automorphism group:

$$\boxed{{\sigma }_t^{\omega }:\mathcal{M}\to \mathcal{M},t\in \mathbb{R}}$$

called the modular flow, generated by the modular operator ${\Delta }_{\omega }$:

$${\sigma }_t^{\omega }(A)={\Delta }_{\omega }^{it}A{\Delta }_{\omega }^{-it}$$

Thermal Time Hypothesis (Connes-Rovelli, 1994):

This hypothesis proposes that the parameter $t$ of the modular flow can be physically interpreted as time.

KMS Condition:

The modular flow corresponds to a thermal equilibrium state at temperature ${\beta }^{-1}$:

$$\omega (AB)=\omega (B{\sigma }_{i\beta }^{\omega }(A))$$

Physical Meaning:

• ${\sigma }_t^{\omega }$: “intrinsic evolution” of state $\omega$
• $t$: “intrinsic time” parameter independent of external clocks
• $\beta$: establishes connection to geometric time (e.g., Unruh temperature)

graph TB
    OMEGA["Quantum State<br/>ω"] --> TT["Tomita-Takesaki<br/>Construction"]
    TT --> DELTA["Modular Operator<br/>Δ_ω"]
    DELTA --> FLOW["Modular Flow<br/>σ_t = Δ^(it) · Δ^(-it)"]
    FLOW --> TIME["Modular Time<br/>t_mod"]

    TIME --> KMS["KMS Condition<br/>Thermal Equilibrium"]
    TIME --> PHY["Physical Time<br/>t_phys ~ t_mod"]

    style OMEGA fill:#e1f5ff
    style FLOW fill:#fff4e1,stroke:#ff6b6b,stroke-width:3px
    style TIME fill:#e1ffe1,stroke:#ff6b6b,stroke-width:3px

💡 Intuitive Image: Self-Rotation of Quantum Systems

Analogy: Earth’s Rotation

Earth has two types of time:

• External Time: Solar system time (orbital period)
• Internal Time: Earth’s rotation (24 hours)

Analogy:

• Earth → quantum state $\omega$
• Rotation → modular flow ${\sigma }_t^{\omega }$
• Rotation period → KMS temperature $\beta$

Key Point: Even without the Sun (external reference), Earth’s rotation still defines a “day”.

Modular Time Perspective: Quantum states possess “intrinsic rotation,” thereby defining their own time parameter.

“Memory” of Quantum States

Imagine a quantum system:

• Pure State $|\psi ⟩$: no memory, trivial modular flow (${\sigma }_t=\text{id}$)
• Mixed State $\rho$: has “entanglement memory,” non-trivial modular flow

Example: Half-Space Entangled State

• Global: pure state $|\Psi ⟩$
• Half-space A: reduced state ${\rho }_A={\text{tr}}_B|\Psi ⟩⟨\Psi |$
• Modular flow of ${\rho }_A$ → “intrinsic time” of half-space A

Physical Interpretation: Entanglement structure may encode time information.

graph LR
    PURE["Pure State<br/>|ψ⟩"] --> TRIV["Trivial Modular Flow<br/>σ_t = id"]
    MIX["Mixed State<br/>ρ = tr_B|Ψ⟩⟨Ψ|"] --> ENT["Entanglement Structure"]
    ENT --> MOD["Non-Trivial Modular Flow<br/>σ_t(A) ≠ A"]
    MOD --> TIME["Intrinsic Time<br/>t_mod"]

    style PURE fill:#e1f5ff
    style MIX fill:#ffe1e1
    style TIME fill:#fff4e1,stroke:#ff6b6b,stroke-width:3px

📐 Tomita-Takesaki Theory

Mathematical Construction

Setup:

• $\mathcal{M}$: von Neumann algebra (observable algebra)
• $\omega$: faithful normal state (quantum state)
• $\Omega$: cyclic separating vector in GNS representation

Tomita Operator:

Define an anti-linear operator $S_0$:

$$S_0:A\Omega \mapsto A^{\ast }\Omega ,A\in \mathcal{M}$$

Polar Decomposition:

$$S_0=J{\Delta }^{1/2}$$

where:

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

Modular Flow:

$${\sigma }_t^{\omega }(A)={\Delta }^{it}A{\Delta }^{-it}$$

Tomita-Takesaki Theorem:

$${\sigma }_t^{\omega }(\mathcal{M})=\mathcal{M}$$

That is, the modular flow preserves the algebra structure.

graph TB
    STATE["State<br/>ω"] --> GNS["GNS Construction<br/>(M, H, Ω)"]
    GNS --> S["Tomita Operator<br/>S₀: AΩ → A*Ω"]
    S --> POLAR["Polar Decomposition<br/>S₀ = J Δ^(1/2)"]
    POLAR --> J["Modular Conjugation<br/>J"]
    POLAR --> DELTA["Modular Operator<br/>Δ"]
    DELTA --> FLOW["Modular Flow<br/>σ_t = Δ^(it) · Δ^(-it)"]

    style STATE fill:#e1f5ff
    style DELTA fill:#fff4e1,stroke:#ff6b6b,stroke-width:3px
    style FLOW fill:#e1ffe1,stroke:#ff6b6b,stroke-width:3px

KMS Condition

Definition (KMS State):

A state $\omega$ is a KMS state at temperature ${\beta }^{-1}$ if for all $A,B\in \mathcal{M}$:

$$\omega (AB)=\omega (B{\sigma }_{i\beta }^{\omega }(A))$$

Physical Meaning:

• Mathematical form of thermal equilibrium condition
• $\beta =1/(k_{B}T)$: inverse temperature
• ${\sigma }_{i\beta }$: imaginary time evolution (analytic continuation)

Example: Canonical Ensemble

$$\omega (A)=\frac{1}{Z}\text{tr}(e^{-\beta H}A)$$

Its modular operator:

$${\Delta }_{\omega }=e^{-\beta H}\otimes e^{\beta H}$$

Modular flow:

$${\sigma }_t^{\omega }(A)=e^{itH}A{e}^{-itH}$$

Observation: In this example, the modular flow is formally identical to the normal time evolution $U(t)=e^{-iHt}$.

Modular Hamiltonian

Definition:

$$K_{\omega }=-\ln {\Delta }_{\omega }$$

called the modular Hamiltonian.

Modular Flow Rewritten:

$${\sigma }_t^{\omega }(A)=e^{it{K}_{\omega }}A{e}^{-it{K}_{\omega }}$$

Physical Analogy:

• $K_{\omega }$: “energy” generating “intrinsic time evolution”
• $t$: modular time
• Form identical to $e^{-iHt}A{e}^{iHt}$

Difference: $K_{\omega }$ is not necessarily a local Hamiltonian.

graph LR
    OMEGA["State<br/>ω"] --> DELTA["Modular Operator<br/>Δ_ω"]
    DELTA --> K["Modular Hamiltonian<br/>K = -ln Δ"]
    K --> FLOW["Modular Flow<br/>σ_t = e^(itK) · e^(-itK)"]
    FLOW --> ANALOGY["Analogy<br/>U(t) = e^(-iHt)"]

    style OMEGA fill:#e1f5ff
    style K fill:#fff4e1,stroke:#ff6b6b,stroke-width:3px
    style FLOW fill:#e1ffe1

🌀 Thermal Time Hypothesis

Connes-Rovelli Proposal (1994)

Core Idea:

In the context of quantum gravity without external clocks, the modular flow parameter $t$ can be identified as physical time.

Argument Logic:

1. In generally covariant theories, there is no external time parameter
2. Given state $\omega$, modular flow ${\sigma }_t^{\omega }$ is intrinsically defined
3. For thermal equilibrium states, $t$ is proportional to the “time” measured by temperature
4. Inference: A correspondence exists between physical time and modular time

Mathematical Form:

Physical time flow ${\alpha }_t$ is equivalent to modular flow:

$${\alpha }_t\sim {\sigma }_t^{\omega }$$

in the sense of the outer automorphism group $\text{Out}(\mathcal{M})$.

Corollary:

Modular flows of different states $\omega ,{\omega }^{\prime }$ are related by rescaling:

$${\sigma }_t^{{\omega }^{\prime }}=\text{Ad}(u_t)\circ {\sigma }_{f(t)}^{\omega }$$

where $u_t$ is an inner automorphism, $f(t)=\alpha t+\beta$.

Time Scale Equivalence Class:

$$[t_{\text{mod}}]=\{\alpha t+\beta \mid \alpha >0\}$$

Connection to Geometric Time

Unruh Effect:

An accelerating observer experiences temperature in vacuum:

$$T_{\text{Unruh}}=\frac{\hslash a}{2\pi c{k}_B}$$

where $a$ is the proper acceleration.

Rindler Wedge:

• Rindler coordinates: $d{s}^2=-(ax{)}^{2}d{t}^2+d{x}^2+d{y}^2+d{z}^2$
• Minkowski vacuum $|0⟩$ reduced to Rindler wedge
• Reduced state ${\rho }_R$ is a thermal state at temperature $T=a/(2\pi )$

Modular Hamiltonian:

$$K=2\pi {\int }_{\text{Rindler}}{T}_{00}\xi \cdot k$$

where $\xi =ax{\partial }_t$ is the Killing vector.

Modular Time and Killing Time:

$$t_{\text{mod}}=2\pi t_{\text{Killing}}$$

Conclusion: A clear proportionality exists between the two.

graph TB
    MINK["Minkowski Vacuum<br/>|0⟩"] --> RIND["Reduce to Rindler Wedge<br/>ρ_R = tr_L|0⟩⟨0|"]
    RIND --> TEMP["Thermal State<br/>T = a/2π"]
    TEMP --> KMS["KMS Condition<br/>β = 2π/a"]
    KMS --> MOD["Modular Flow<br/>σ_t"]
    MOD --> K["Modular Hamiltonian<br/>K ~ ∫T_00"]
    K --> TIME["Modular Time<br/>t_mod = 2π t_K"]

    style MINK fill:#e1f5ff
    style TEMP fill:#ffe1e1
    style MOD fill:#fff4e1,stroke:#ff6b6b,stroke-width:3px
    style TIME fill:#e1ffe1,stroke:#ff6b6b,stroke-width:3px

Half-Space Entanglement

Setup:

• Vacuum state $|0⟩$ in Minkowski space
• Partition: $A=\{x>0\}$, $B=\{x<0\}$
• Reduced state: ${\rho }_A={\text{tr}}_B|0⟩⟨0|$

Modular Hamiltonian (Bisognano-Wichmann, 1976):

$$K_A=2\pi {\int }_{x>0}{T}_{00}xdxdydz$$

Physical Meaning:

• $K_A$ is the Rindler boost generator
• Modular flow corresponds to Lorentz boost
• Modular time corresponds to boost parameter (rapidity)

Relation to Proper Time:

Along Rindler orbit $x=x_0$:

$$d\tau =x_{0}d\eta$$

where $\eta$ is the boost parameter (rapidity).

Modular Time:

$$t_{\text{mod}}=2\pi \eta$$

Conclusion: Modular time is proportional to boost rapidity.

🔑 Relative Entropy and Time Arrow

Relative Entropy Monotonicity

Definition (Relative Entropy):

$$S({\rho }_1\Vert {\rho }_2)=\text{tr}({\rho }_1\ln {\rho }_1-{\rho }_1\ln {\rho }_2)$$

Monotonicity Theorem:

For inclusion relation ${\mathcal{A}}_1\subset {\mathcal{A}}_2$:

$$S({\rho }_1{|}_{{\mathcal{A}}_1}\Vert {\rho }_2{|}_{{\mathcal{A}}_1})\le S({\rho }_1{|}_{{\mathcal{A}}_2}\Vert {\rho }_2{|}_{{\mathcal{A}}_2})$$

Time Arrow:

Under modular flow evolution, relative entropy exhibits monotonicity (non-increasing or non-decreasing, depending on direction).

ANEC/QNEC Connection:

Relative entropy monotonicity $\iff$ Quantum Null Energy Condition (QNEC)

$$S_{\text{out}}^{\prime \prime }\ge \frac{2\pi }{\hslash }\int ⟨T_{kk}⟩dA$$

Physical Meaning:

• Modular time provides a definition of “time arrow”
• Relative entropy is monotonic along modular time
• This is consistent with the second law of thermodynamics

graph TB
    MOD["Modular Flow<br/>σ_t"] --> REL["Relative Entropy<br/>S(ρ₁||ρ₂)"]
    REL --> MONO["Monotonicity<br/>dS/dt ≥ 0"]
    MONO --> ARROW["Time Arrow<br/>t → +∞"]

    MONO --> QNEC["QNEC<br/>S'' ≥ ∫⟨T_kk⟩"]
    QNEC --> EIN["Einstein Equations<br/>R_kk = 8πG⟨T_kk⟩"]

    style MOD fill:#e1f5ff
    style MONO fill:#fff4e1,stroke:#ff6b6b,stroke-width:3px
    style ARROW fill:#ffe1e1
    style EIN fill:#e1ffe1

📊 Connection to Unified Time Scale

Modular Time ↔ Geometric Time

Theorem: Under appropriate conditions (Rindler wedge, accelerating observer, etc.):

$$t_{\text{mod}}=\beta t_{\text{geo}}$$

where $\beta$ is determined by KMS temperature.

Unruh Effect:

$$\beta =\frac{2\pi }{a},T=\frac{a}{2\pi }$$

Therefore:

$$t_{\text{mod}}=2\pi \frac{t_{\text{Rindler}}}{a}$$

Modular Time ↔ Scattering Time

In AdS/CFT correspondence:

• Modular Hamiltonian $K_{\text{CFT}}$ of boundary CFT
• Quasi-local energy $E_{\text{bulk}}$ of bulk

Correspondence:

$$K_{\text{CFT}}\leftrightarrow E_{\text{bulk}}$$

Time Correspondence:

$$t_{\text{mod}}^{\text{CFT}}\leftrightarrow t_{\text{geo}}^{\text{bulk}}$$

Through JLMS equivalence.

Unified Scale

Time Scale Equivalence Class:

$$[T]\sim \{\tau ,t_K,N,\lambda ,u,v,\eta ,{\omega }^{-1},z,t_{\text{mod}}\}$$

Position of Modular Time:

• Connected to geometric time through KMS condition
• Connected to scattering time through boundary correspondence
• Connected to entropy evolution through relative entropy

Theoretical Consistency: The logic of each part is mutually closed.

graph TB
    MOD["Modular Time<br/>t_mod"] --> KMS["KMS Condition<br/>β = 2π/a"]
    KMS --> GEO["Geometric Time<br/>t_geo ~ t_mod/β"]

    MOD --> JLMS["JLMS Equivalence<br/>K_CFT ~ E_bulk"]
    JLMS --> SCAT["Scattering Time<br/>∫tr Q dω"]

    MOD --> ENTROPY["Relative Entropy<br/>S(ρ₁||ρ₂)"]
    ENTROPY --> QNEC["QNEC<br/>S'' ≥ ∫⟨T_kk⟩"]

    GEO --> UNIFIED["Unified Scale<br/>[T]"]
    SCAT --> UNIFIED
    QNEC --> UNIFIED

    style MOD fill:#fff4e1,stroke:#ff6b6b,stroke-width:4px
    style UNIFIED fill:#e1ffe1,stroke:#ff6b6b,stroke-width:4px

🎓 Profound Significance

1. Emergence of Time

Traditional View: Time is an external parameter

Modular View: Time may emerge from the entanglement structure of quantum states.

Argument:

1. Pure state → no modular flow → no time
2. Entangled state → non-trivial modular flow → time emerges
3. Inference: Entanglement may be one of the origins of time

2. Gravity as Thermodynamics

Jacobson’s Argument (1995):

• Generalized entropy $S_{\text{gen}}=A/(4G\hslash )+S_{\text{out}}$
• Relative entropy monotonicity → QNEC
• QNEC → Einstein equations

Modular Perspective:

• Modular Hamiltonian $K\sim \int T_{kk}$
• Relative entropy evolves along $K$
• Monotonicity → energy conditions → gravitational equations

Conclusion: Gravity can be viewed as the geometric projection of modular flow.

3. Quantum Error Correction and Time

Almheiri et al. (2015): Time evolution can be viewed as quantum error correction code

• Code subspace: physical states
• Modular flow: time evolution
• Entanglement wedge reconstruction: error correction recovery

Perspective: Time structure is closely related to entanglement encoding structure.

🤔 Exercises

1. Conceptual Understanding:

   • Why is the modular flow of pure states trivial?
   • What is the physical meaning of the KMS condition?
   • What is the core argument of the thermal time hypothesis?

2. Calculation Exercises:

   • Canonical ensemble $\omega (A)=\text{tr}(e^{-\beta H}A)/Z$, calculate ${\sigma }_t^{\omega }(A)$
   • Unruh temperature $T=a/(2\pi )$, calculate the temperature for acceleration $a=1g$
   • Half-space modular Hamiltonian $K=2\pi \int T_{00}xdx$, verify boost generator

3. Physical Applications:

   • How does a Rindler observer understand the Unruh effect through modular flow?
   • What is the modular flow near a black hole horizon?
   • How does boundary modular flow correspond to bulk time in AdS/CFT?

4. Advanced Thinking:

   • What is the role of modular flow in quantum gravity?
   • What is the relationship between relative entropy monotonicity and causality?
   • How to derive Einstein equations from modular flow?

Navigation:

• Previous: 05-geometric-times_en.md - Geometric Times
• Next: 07-cosmological-redshift_en.md - Cosmological Redshift
• Overview: 00-time-overview_en.md - Unified Time Overview
• GLS Theory: unified-time-scale-geometry.md
• References:
  • Connes & Rovelli, “Von Neumann algebra automorphisms and time–thermodynamics relation” (1994)
  • Bisognano & Wichmann, “On the Duality Condition for Quantum Fields” (1976)
  • Tomita-Takesaki theory: Takesaki, “Theory of Operator Algebras” (2002)