11. The Thermal Time Hypothesis: How the State Defines Its Own Time

So far, we have learned from Chapter 3 (Generator) that the Hamiltonian (\(H\)) generates the flow of time (\(U(t) = e^{-iHt/\hbar}\)), and from Chapter 9 (Page-Wootters) we explored how ‘internal time’ emerges through entanglement in a universe without time (\(H|\Psi\rangle=0\)).

This chapter addresses the third and most radical approach to the ‘problem of time’. It is the “Thermal Time Hypothesis”. This hypothesis claims that the flow of time does not originate from the Hamiltonian or an external clock, but is instead intrinsic to the state (State) itself.

This remarkable idea emerged from the meeting of quantum field theory (QFT) and pure mathematics (algebra theory). To understand this theory, we will examine how a physical ‘state’ generates its own unique ‘dynamics’ and why it is related to ‘thermal’ phenomena.

1. Fundamental Concepts

Operator Algebra (Operator Algebra: C* / von Neumann) Instead of treating the ‘operators’ learned in Chapter 1 individually, this mathematical framework treats the entire ‘set’ of all measurable operators as a single algebra (Algebra) structure.
- **C*-algebra:** The most general and stable ‘operating system’ that deals with physical observables (\(A=A^\dagger\)) and their sums, products, and norms (\(\|A\|\)).
- von Neumann algebra: A more refined structure than C*-algebra, specialized in dealing with ‘measurement’ and ‘subsystems’, and the standard language for handling local observations in QFT.
KMS State (KMS State) The thermal equilibrium state \(\rho = e^{-\beta H} / Z\) (Gibbs state) learned in Chapter 5 requires the Hamiltonian \(H\) to be defined. However, in general relativity or quantum cosmology, the concept of a “global Hamiltonian” itself is ambiguous. The KMS state is a generalized method to mathematically define a thermal equilibrium state without \(H\).
💡 Thermal equilibrium without a Hamiltonian?
There are two ways to say “the room is hot.”
1. “Boiler (\(H\)) is operating at 100°C.” (Gibbs state)
2. “By looking at the statistical distribution (correlation) of air molecules in the room, it shows a thermal distribution corresponding to 100°C.” (KMS state)
The KMS condition extracts the complex correlations of the Gibbs state (\(\omega(A(t)B)\) and \(\omega(B A(t+i\beta))\)) and serves as a ‘thermal certificate’ that authenticates “this state is a thermal equilibrium state at temperature \(\beta\)” even without \(H\).
Tomita-Takesaki Modular Theory (Tomita-Takesaki Modular Theory) This theory is an extremely surprising theorem discovered in mid-20th century pure mathematics. “Given any (von Neumann) algebra \(\mathcal{M}\) and a ‘faithful’ state \(\omega\) defined on it, the state \(\omega\) generates a unique dynamics (flow) \(\sigma_t^\omega\) by itself.” > 💡 The state is the clock > > This is not a metaphor but a mathematical fact. The Page-Wooters (PW) mechanism in Chapter 9 creates time through entanglement with an external clock (C). > > However, the Tomita-Takesaki (T-T) theory goes deeper. No external clock is needed. > 1. Given any physical system (algebra \(\mathcal{M}\)) and > 2. a specific state of that system (state \(\omega\), e.g., vacuum state, thermal state), > > the T-T theory automatically finds a time evolution operator called ‘modular flow’ \(\sigma_t^\omega\), which is uniquely determined mathematically from this \((\mathcal{M}, \omega)\) pair. It is as if the state \(\omega\) itself says, “This is my unique flow of time.” The generator of this flow is called the modular Hamiltonian (\(K_\omega \sim -\ln \rho\)).
Thermal Time Hypothesis In the 1990s, Alain Connes and Carlo Rovelli proposed that this mathematical ‘flow’ from T-T theory could be a physical entity.
Hypothesis: In a fundamental theory without a pre-given background time (like general relativity), the physical flow of time experienced by a system is none other than the modular flow \(\sigma_t^\omega\) generated by the system’s (KMS) state \(\omega\).
Justification: In standard QFT, computing the modular flow \(\sigma_t^\omega\) of a Gibbs thermal equilibrium state (KMS state) surprisingly matches exactly with the standard time evolution we know, \(e^{iHt/\hbar}(\cdot)e^{-iHt/\hbar}\).
Conclusion: ‘Thermal time’ is not a new time but a redefinition of the time we know from the perspective of the state.

2. Symbols and Key Relations

C* / von Neumann Algebras (\(\mathcal{A}, \mathcal{M}\)):
- **C*-identity:** \(\|A^*A\| = \|A\|^2\) (ensures stability of the algebra)
- von Neumann: A C*-algebra closed in the weak operator topology.
States (\(\omega\)):
- \(\omega\) is a linear function (functional) from the algebra \(\mathcal{A}\) to the complex numbers \(\mathbb{C}\), satisfying \(\omega(A^*A) \ge 0\) (positivity) and \(\omega(\mathbf{1}) = 1\) (normalization). (Generalization of \(\omega(A) = \mathrm{Tr}(\rho A)\) from Chapter 2.)
KMS Condition (Kubo-Martin-Schwinger):
- A state \(\omega\) being a \((\beta, \alpha_t)\)-KMS state with respect to dynamics \(\alpha_t\) (time evolution group) means that,
- for all \(A, B \in \mathcal{A}\), the function \(F(t) = \omega(A \, \alpha_t(B))\) is analytically continuable into the strip of the complex plane (\(0 < \mathrm{Im}(z) < \beta\)) and satisfies the following on the boundary: \[F(t + i\beta) = \omega(\alpha_t(B) A)\]
- Intuition: This condition connects the order (BA) at complex time \(t + i\beta\) with the order (AB) at time \(t\), representing thermal correlations.
Tomita-Takesaki (Modular Theory):
- Given a faithful state \(\omega\), there exists a unique modular operator \(\Delta_\omega\) and modular conjugation \(J_\omega\).
- Modular Flow: \(\sigma_t^\omega (A) = \Delta_\omega^{it} A \Delta_\omega^{-it}\)
- Core Theorem: The state \(\omega\) is always a \((\beta=1)\)-KMS state with respect to its modular flow \(\sigma_t^\omega\). (That is, every state is an equilibrium state at temperature 1 with respect to its own unique ‘thermal time’.)
Thermal Time Hypothesis:
- Physical time \(t_{\text{phys}}\) = modular time \(\tau\).
- \(H_{\text{phys}} = K_\omega / \beta\) (the physical Hamiltonian is the modular Hamiltonian divided by temperature)

3. Easy Examples (Examples with Deeper Insight)

Example 1: Finite-Dimensional Gibbs State (KMS Verification)
- Situation: \(n \times n\) matrix algebra (\(\mathcal{A}=M_n(\mathbb{C})\)), Hamiltonian \(H\), time evolution \(\alpha_t(A) = e^{iHt} A e^{-iHt}\).
- State: Gibbs state \(\rho_\beta = e^{-\beta H} / Z\), \(\omega(A) = \mathrm{Tr}(\rho_\beta A)\).
- Result: This \(\omega\) exactly satisfies the \((\beta, \alpha_t)\)-KMS condition. This shows that the KMS condition properly generalizes Gibbs states.
Example 2: Maximal Mixed State (Trivial Time)
- Situation: In Example 1, \(\rho = \mathbf{1}/n\) (infinite temperature, \(\beta=0\)). This is a ‘tracial state’.
- Modular Flow: The modular operator for this state is \(\Delta_\omega = \mathbf{1}\).
- Result: \(\sigma_t^\omega(A) = \mathbf{1}^{it} A \mathbf{1}^{-it} = A\). The modular flow is trivial. That is, time does not evolve.
- Interpretation: The maximally mixed (disordered) state defines no unique direction of time.
Example 3: Unruh Effect and Thermal Time
- Situation: (Conceptual explanation for understanding)
- Inertial Frame Observer: Sees the entire algebra \(\mathcal{M}\) of Minkowski spacetime. The state \(\omega_{\text{Mink}}\) of this observer is ‘vacuum’.
- Accelerated Frame Observer (Rindler): Due to the light speed limit, this observer can only see part of spacetime (Rindler wedge) \(\mathcal{M}_R\).
- T-T Theory Application: The state \(\omega_R\) obtained by restricting the entire vacuum state \(\omega_{\text{Mink}}\) to the Rindler algebra \(\mathcal{M}_R\) is no longer a ‘trivial’ state.
- Surprising Result: The modular flow \(\sigma_t^{\omega_R}\) of this \(\omega_R\) exactly matches the flow of proper time of the accelerated observer.
- Interpretation: According to T-T theory, \(\omega_R\) must be a KMS state with respect to its modular flow. This means that the accelerated observer perceives vacuum as a thermal bath (Unruh effect). At this time, ‘thermal time’ is exactly the accelerated observer’s ‘physical time’.

4. Practice Problems

(KMS Verification): Show directly by calculation that the finite-dimensional Gibbs state \(\omega(A)=\mathrm{Tr}(\rho_\beta A)\) of Example 1 satisfies the KMS condition \(F(t+i\beta) = \omega(\alpha_t(B)A)\). (Hint: cyclicity of \(\mathrm{Tr}\), \(e^{iH(t+i\beta)} = e^{iHt}e^{-\beta H}\))
(Modular Flow): Show that the modular flow \(\sigma_t^\omega(A)=\Delta_\omega^{it}A\Delta_\omega^{-it}\) satisfies the 1-parameter group property \(\sigma_{t+s}^\omega = \sigma_t^\omega \circ \sigma_s^\omega\).
(Trivial Flow): Show that \(\Delta_\omega = \mathbf{1}\) for the trace state \(\omega(A)=\tfrac{1}{n}\mathrm{Tr}(A)\) of Example 2, and conclude that \(\sigma_t^\omega = \mathrm{id}\) (identity transformation).
(KMS and Detailed Balance): Conceptually explain how the KMS condition relates to the ‘detailed balance’ principle in statistical mechanics.
(PW vs Thermal Time): Both the Page-Wooters mechanism in Chapter 9 and the thermal time hypothesis in Chapter 10 deal with the ‘problem of time’. Compare and explain the fundamental differences between these two approaches (e.g., necessity of a clock).
(State Dependence): If the state of the system changes from \(\omega\) to \(\omega'\) (at a different temperature), discuss how the ‘thermal time flow’ would change (e.g., speed).
(Type III Algebra): (Advanced) The local algebras of QFT are mostly ‘Type III’ von Neumann algebras. These algebras do not mathematically admit a ‘trace’. Explain why this fact makes modular theory essential for QFT.
(Euclidean Path Integral): The ‘complex time’ periodicity (\(t+i\beta\)) of the KMS condition is deeply related to identifying imaginary time \(\tau\) as \(\tau \sim \tau+\beta\) in the Euclidean path integral of Chapter 4. Explain this relationship.

5. Exercise Solutions

\(F(t+i\beta) = \mathrm{Tr}(\rho_\beta A \, e^{iH(t+i\beta)} B e^{-iH(t+i\beta)}) = \mathrm{Tr}(\rho_\beta A e^{iHt} e^{-\beta H} B e^{\beta H} e^{-iHt})\). \(\rho_\beta e^{\beta H} = (e^{-\beta H}/Z) e^{\beta H} = \mathbf{1}/Z\) and, using the cyclicity of \(\mathrm{Tr}\), move \(e^{-\beta H}\) to the end. \(\dots = \mathrm{Tr}(e^{-\beta H} e^{-iHt} A e^{iHt} B)/Z = \mathrm{Tr}(\rho_\beta \, \alpha_t(B) A) = \omega(\alpha_t(B)A)\).
\(\sigma_t^\omega(\sigma_s^\omega(A)) = \sigma_t^\omega(\Delta_\omega^{is} A \Delta_\omega^{-is}) = \Delta_\omega^{it} (\Delta_\omega^{is} A \Delta_\omega^{-is}) \Delta_\omega^{-it}\) \(= (\Delta_\omega^{it}\Delta_\omega^{is}) A (\Delta_\omega^{-is}\Delta_\omega^{-it}) = \Delta_\omega^{i(t+s)} A \Delta_\omega^{-i(t+s)} = \sigma_{t+s}^\omega(A)\).
Trace states satisfy \(\omega(AB) = \omega(BA)\), so the operator \(S\) in T-T theory becomes \(S=J\) (isometry operator). Therefore, \(\Delta_\omega = S^\dagger S = J^\dagger J = \mathbf{1}\). Since \(\Delta_\omega=\mathbf{1}\), \(\sigma_t^\omega(A) = \mathbf{1}^{it} A \mathbf{1}^{-it} = A\).
The KMS condition links the transition rate \((A \to B)\) at time \(t\) with the transition rate \((B \to A)\) at time \(t+i\beta\), generalizing the detailed balance condition that forward and backward flows balance in thermal equilibrium.
PW: Time emerges from the ‘entanglement relation’ between the external clock (C) and the system (S). A clock is necessary. Thermal time: Time is an intrinsic property of the ‘state itself’ (modular flow). No external clock is needed.
When the state changes from \(\omega \to \omega'\), the modular operator \(\Delta_\omega \to \Delta_{\omega'}\) changes. This implies that the flow of ‘thermal time’ \(\sigma_t^{\omega'}\) differs from \(\sigma_t^\omega\). (E.g., if temperature changes (\(\beta \to \beta'\)), the ‘speed’ of time flow can be interpreted as changing). This suggests that time is ‘observer/state dependent’.
Without ‘trace’, defining states as \(\mathrm{Tr}(\rho A)\) or entropy (\(S = -\mathrm{Tr}(\rho \ln \rho)\)) becomes impossible. Modular theory provides the only mathematical tool that allows defining states (\(\omega\)), dynamics (\(\sigma_t^\omega\)), relative entropy, etc., without trace.
In the Euclidean path integral, the partition function \(Z = \mathrm{Tr}(e^{-\beta H})\) is computed by integrating imaginary time from \(0\) to \(\beta\) and imposing periodic boundary conditions (period \(\beta\)). The periodicity of KMS condition in \(t+i\beta\) exactly matches the periodicity of imaginary time, and both represent the same mathematical structure for thermal equilibrium at temperature \(\beta\).

📚 Note: Two ‘Times’ in Chapters 10 and 11 – Relation or Attribute?

Chapters 10 (Page-Wootters) and 11 (Thermal Time Hypothesis) present the two most prominent approaches in modern physics to the ‘problem of time.’ Rather than contradicting each other, they view the origin of ‘time’ from fundamentally different perspectives.

Chapter 10. Page-Wootters (PW): “Time is a relation.”
- In the PW theory, time does not exist externally.
- Time emerges through the ‘entanglement relation’ between a part of the universe (clock C) and the rest (system S).
- For the theory to work, it is essential to divide the universe into two parts (C and S).
- Verification: This ‘relational time’ concept was successfully verified experimentally (concept proof) using entangled photon pairs in 2013.
Chapter 11. Thermal Time Hypothesis (TTH): “Time is an attribute of a state.”
- TTH is more radical. This theory claims that an external clock or system division is not necessary.
- Time is a mathematical attribute inherent to the state (\(\omega\)) itself.
- Given any state \(\omega\), the Tomita-Takesaki theory mathematically identifies a unique ‘flow’ (i.e., modular flow (\(\sigma_t^\omega\))) uniquely defined by that state. TTH declares this mathematical flow to be physical time.
- Verification: Direct evidence for TTH is linked to the observation of the ‘Unruh Effect,’ but this remains unverified experimentally with current technology.

Chapter 10 explains time as a ‘relation’ with an ‘external clock,’ while Chapter 11 explains time as an ‘intrinsic mathematical property’ of the ‘state’ itself.

📚 Note: Time, Objectivity, and the Dual Role of Entanglement

Chapters 9 (Quantum Darwinism) and 10 (Page-Wooters) of the main text answer two of the deepest questions in quantum theory.

The Problem of Time (Chapter 9): “If the entire universe is stationary (\(H|\Psi\rangle=0\)), why do we experience the ‘flow of time’?”
The Problem of Objectivity (Chapter 8): “If the universe is filled with quantum superposition, why do we experience a ‘classical objective world’?”

These two theories are not contradictory; they are two essential pieces completing one picture. PW explains the “order of history,” while QD explains why each “frame of history” is classical.

1. Page-Wooters (PW): Creating the ‘Film Roll of History’

The PW mechanism shows how the concept (order) of “the flow of time” itself emerges.

The universe in the \(H|\Psi\rangle=0\) state can be metaphorically thought of as a single giant ‘film roll’ (\(|\Psi\rangle\)) where all frames are already printed.
This film roll itself is stationary, but within it, entanglement encodes the frame labeled “1:01 (\(|t_1\rangle\))” and the “state of the universe at that time (\(|\psi(t_1)\rangle\))” perfectly correlated. \[|\Psi\rangle = \sum_t |t\rangle_C \otimes |\psi(t)\rangle_S\]
“Time flows” means that we, as observers (part of the universe), experience reading the stationary film roll (\(|\Psi\rangle\)) one frame at a time in order, using a ‘clock (\(C\))’ as a reference.

2. Quantum Darwinism (QD): Sharpening ‘Each Frame’

Whereas PW provides the ‘film roll’, QD explains why each ‘frame (\(|\psi(t)\rangle\))’ of the film is not a blurry quantum superposition but a sharp classical picture.

The \(|\psi(t)\rangle_S\) frame created by PW itself consists of a ‘system’ (e.g., an apple) and ‘environment \(E\)’ (millions of photons/air molecules) surrounding the system.
As you mentioned, “you are each other’s observer,” QD explains that the environment (\(E\)) continuously ‘observes’ the system (apple).
This ‘observation’ is exactly the Decoherence learned in Chapter 5. The moment a photon hits the apple, the information that “the apple is at A” becomes entangled with the photon (measurement), and this information is immediately duplicated to millions (\(10^{20}\)) of other photons.
This overwhelming redundancy determines the state of that frame as an ‘objective fact’ that cannot be reversed.

3. The Dual Role of Entanglement: ‘Resource’ vs ‘Information Leak’

To understand these two theories, we must distinguish the two different usages of the term ‘entanglement’.

Type 1 (Clean Entanglement): Quantum ‘Resource’ (Chapter 2)
- Definition: A few particles (A, B) share an inseparable single pure state (\(|\Phi^+\rangle\)).
- Features: “All information is perfect (pure state), partial information is incomplete (mixed state).”
- Role: Quantum computers, core principle of the PW mechanism.
- Metaphor: A ciphertext that only makes sense when two pieces are combined.
Type 2 (Dirty Entanglement): ‘Information Leak’ / Decoherence (Chapters 5, 8)
- Definition: The process where a system (S) becomes entangled with an uncontrollable large environment (E).
- Features: From the system (S)’s perspective, information disappears and superposition (interference term) is destroyed.
- Role: Quantum Darwinism, emergence of classicality.
- Metaphor: Secret information being torn into millions of copies and leaking everywhere. Conclusion:
  PW explains ‘the order of time’ using Type 1 entanglement, and QD explains ‘the objectivity of each moment’ using Type 2 entanglement.

4. Experimental Verification: Reality, Not Imagination

These two theories are imaginative, but they are not mere philosophy—they provide verifiable physical predictions.

Page-Wooters (PW) Verification

Key Prediction: Even in a globally stopped state (\(H|\Psi\rangle=0\)), a system (S) conditioned on an internal clock (C) should appear to evolve according to the Schrödinger equation.
Experiment (2013, Moreva et al.):
- As mentioned, we created a ‘stopped universe’ using two entangled photons (Type 1 entanglement).
- One photon was treated as the ‘clock (C)’, and the other as the ‘system (S)’.
- Under the condition that the polarization of the clock (C) was measured at a specific angle (\(t\) role), observing the system (S) experimentally and perfectly proved that the state of S evolves exactly according to the Schrödinger equation depending on \(t\).
- This 2013 experiment remains the most representative and crucial experiment that successfully demonstrated the concept of the PW mechanism.

Quantum Darwinism (QD) Verification

Key Prediction: If QD is correct, information of the system (S) should be redundantly copied into the environment (E). Thus, even by observing a very small fragment (F) of the environment, one should be able to know the complete information of the system. (This is called the ‘Information Plateau’.)
Experiments (Since around 2010):
- Experimental teams created a system (S) and an environment (E), and split the environment into multiple fragments (\(F\)) to measure how many fragments one needs to steal to know the system’s information (\(I(S:F)\)).
- Result: As predicted, once the size of the environmental fragment (\(f\)) increased slightly, \(I(S:F)\) reached the maximum value (the total information of the system), and no further increase in information was observed even when more fragments were added—clearly revealing the ‘plateau’.
- This proves that information is redundantly duplicated in the environment, strongly supporting QD’s core claim. (Examples: photon cluster experiments, diamond NV center experiments, etc.)