10. Emergent Time: The Flow of Time in a Frozen Universe (Page–Wootters)

“There is no flow of time outside the universe.” This shocking statement is at the heart of the ‘Problem of Time’ that arises in the study of quantum gravity. If we consider the entire universe as a single quantum state \(|\Psi\rangle\), this state cannot change over time because there is no external observer (according to the creator theory in Chapter 3). \((H_{total}|\Psi\rangle = 0\), Wheeler–DeWitt equation).

Then, where does the dynamic ‘flow of time’ that we experience come from? The Page–Wootters (PW) mechanism provides a beautiful answer to this. Time is not a fundamental background of the universe, but rather an emergent phenomenon that appears relatively when we take a part of the universe as a clock and observe the rest.

1. Fundamental Concepts

  • Global Static State: Assume that there is a global state vector \(|\Psi\rangle\) that describes the entire universe (system S + everything else C). This state is constrained by the total Hamiltonian \(H = H_S + H_C\): \(H|\Psi\rangle = 0\).

    • This can be interpreted as meaning “the total energy of the universe is zero,” and from outside, this universe would appear as an eternal static state with no changes. 🌌

  • Conditioning and the Birth of Internal Time: The flow of time emerges within ‘measurement’ and ‘relation’. We observe a part of the universe, the clock (C), pointing to a specific time ‘\(t\)’ and describe the state of the system (S) under this condition.

    • That is, we ask, “If the clock is in state \(|t\rangle_C\), what state \(|\psi_S(t)\rangle\) is the system in?”
    • As learned in Chapter 2 about entanglement, \(|\Psi\rangle\) can be viewed as an entangled state between C and S (\(|\Psi\rangle = \sum_t |t\rangle_C \otimes |\psi(t)\rangle_S\)). This ‘conditioning’ process is the key to extracting dynamic internal time from the static global state.

    💡 Detailed Explanation: How to See a Movie in a Single Photograph 🖼️➡️🎬

    Imagine a single large photograph of many dancers, each performing different static poses. This photograph itself is an unchanging ‘global static state’ (\(|\Psi\rangle\)).

    Now, focus on the arm angle (\(t\)) of one dancer (the clock C). We ask, “When the arm angle is 30 degrees (\(|t_1\rangle_C\)), what movement (\(|\psi(t_1)\rangle_S\)) is another dancer (system S) performing?” We then ask, “What movement occurs when the arm angle is 60 degrees?”

    If there is a strong correlation between the arm angle of the clock dancer and the movement of the system dancer, we can arrange the system’s movements in order based on the clock’s arm angle, reconstructing the ‘flow of time’ as if it were a movie.

    The PW mechanism shows that the quantum entanglement learned in Chapter 2 plays this role of correlation, demonstrating how the Wheeler–DeWitt equation, a movie, is created within the single photograph of the global state.

  • Emergence of Schrödinger’s Equation: Remarkably, the conditioned state of the system \(|\psi_S(t)\rangle\) depending on the clock’s state as described above evolves according to the well-known Schrödinger equation \(i\hbar \frac{\partial}{\partial t}|\psi_S(t)\rangle = H_S |\psi_S(t)\rangle\). In other words, for an internal observer, a perfectly standard quantum mechanical world unfolds. Time is not given from the outside, but rather emerges from the entanglement relation between the system and the clock.

  • Experimental Verification: This is not merely a philosophical idea. In 2013, using entangled photon pairs (Type 1 entanglement), researchers successfully experimentally implemented this mechanism by taking one photon as the ‘clock’ and the other as the ‘system’. 🔬 By observing that the system photon’s state evolved according to the predicted Schrödinger equation when conditioned on the clock photon’s state in a ‘frozen’ entangled state, the concept of relational time was experimentally demonstrated in the laboratory.

2. Symbols and Key Relations

  • Total State Space and Hamiltonian Constraint:
    • Total Hilbert space: \(\mathcal{H} = \mathcal{H}_C \otimes \mathcal{H}_S\) (clock ⊗ system)
    • Global state: \(|\Psi\rangle \in \mathcal{H}\)
    • Wheeler-DeWitt Equation: \((H_C + H_S)|\Psi\rangle = 0\)
  • Ideal Clock Conditions:
    • Clock state (time basis): \(|t\rangle_C\)
    • The clock’s Hamiltonian \(H_C\) must act as the generator of time translation, as learned in Chapter 3. \(H_C |t\rangle_C \approx i\hbar \frac{\partial}{\partial t} |t\rangle_C\)
  • Conditional State and Probability:
    • System state at time \(t\) (including normalization): \[|\psi_S(t)\rangle = \frac{{}_C\langle t | \Psi \rangle}{\sqrt{p(t)}}\]
    • Probability of observing time \(t\) (similar to a partial trajectory): \[p(t) = \| {}_C\langle t | \Psi \rangle \|^2 = \langle \Psi | (|t\rangle\langle t|_C \otimes \mathbf{1}_S) | \Psi \rangle\]
  • Derivation of Schrödinger’s Equation: > 💡 Exchange of Time: Dynamics of the Clock Become Dynamics of the System
    >
    > The derivation is surprisingly simple. Take the inner product of the clock state \(\langle t|_C\) with both sides of the global constraint equation.
    >
    > \({}_C\langle t | (H_C + H_S) | \Psi \rangle = 0\)
    >
    > \({}_C\langle t | H_C | \Psi \rangle + {}_C\langle t | H_S | \Psi \rangle = 0\)
    >
    > Applying the ideal clock condition \(H_C|t\rangle_C \approx i\hbar \partial_t|t\rangle_C\) to the first term (exactly \(\langle t|H_C = -i\hbar \partial_t \langle t|\)),
    >
    > \(-i\hbar \frac{\partial}{\partial t} ({}_C\langle t | \Psi \rangle) = -H_S ({}_C\langle t | \Psi \rangle)\)
    >
    > Multiplying both sides by \(-1\) and substituting the conditional state definition \(|\psi_S(t)\rangle \propto {}_C\langle t|\Psi\rangle\),
    >
    > \[i\hbar \frac{\partial}{\partial t} |\psi_S(t)\rangle = H_S |\psi_S(t)\rangle\]
    >
    > The flow of time that the clock’s Hamiltonian \(H_C\) was supposed to generate has been magically transformed into the flow of time generated by the system’s Hamiltonian \(H_S\) due to the constraint.

3. Easy Example (Examples with Deeper Insight)

  • Example 1: Ticking, a Clock That Only Goes Twice (Discrete Time)
    • Scenario: Imagine a very simple universe where the clock can only be in two states, \(|t_1\rangle_C\) and \(|t_2\rangle_C\). The global state is entangled as follows:
      \[|\Psi\rangle = \frac{1}{\sqrt{2}} \left( |t_1\rangle_C \otimes |\text{initial state}\rangle_S + |t_2\rangle_C \otimes |\text{later state}\rangle_S \right)\]
    • Conditioning:
      • “If the clock is at \(t_1\)?” \(\implies |\psi_S(t_1)\rangle = |\text{initial state}\rangle_S\)
      • “If the clock is at \(t_2\)?” \(\implies |\psi_S(t_2)\rangle = |\text{later state}\rangle_S\)
    • Interpretation: From the static entangled state \(|\Psi\rangle\), a single step of “temporal evolution” naturally emerges, transitioning from the “initial state” to the “later state.” If the relation \(|\text{later state}\rangle_S \approx e^{-iH_S(t_2-t_1)/\hbar} |\text{initial state}\rangle_S\) holds, this discrete evolution follows the Schrödinger equation.
  • Example 2: Photon Entanglement Experiment (2013)
    • Setup: Generate two photons A, B with entangled polarization. (Type 1 ‘clean’ entanglement)
      • Clock (C): Photon A. The path length through which photon A passes is precisely adjusted to encode ‘time’ \(t\). (e.g., rotate a polarizer at angle \(t\))
      • System (S): Photon B. The polarization state of photon B is exactly \(|\psi_S(t)\rangle\).
    • Measurement: Only when measuring photon A (the ‘clock’) at a specific angle \(t\), measure photon B (polarization tomography).
    • Result: By changing the angle \(t\) of the clock and performing conditional measurements, it is observed that the polarization state of photon B rotates following the Schrödinger equation depending on \(t\). This clearly demonstrates that dynamic time evolution emerges from a static entangled state.

4. Exercises

  1. (Normalization of Conditional States): Show that the norm of the conditional state \(|\psi_S(t)\rangle=\frac{{}_C\langle t|\Psi\rangle}{\sqrt{p(t)}}\) is 1, i.e., \(\langle \psi_S(t)|\psi_S(t)\rangle=1\).
  2. (Derivation of Internal Schrödinger Equation): Re-derive the process of 💡 Time Exchange from Section 2 of the text by hand.
  3. (Resolution of Time): Based on the time-energy uncertainty principle, discuss the relationship between the energy spectrum width of the clock (\(\Delta E_C\)) and the minimum distinguishable time unit (\(\Delta t\)) perceived by the internal observer.
  4. (What is a Perfect Clock?): What are the conditions for an ‘ideal clock’ required for the PW mechanism to function properly? Discuss what problems may arise if the clock’s Hamiltonian \(H_C\) interacts with the system \(H_S\) (\([H_C, H_S] \neq 0\)).
  5. (Comparison with Thermal Time Hypothesis): In one paragraph, summarize the fundamental differences in approach between the ‘Thermal Time Hypothesis’ to be learned in Chapter 11 and the ‘Page-Wootters Mechanism’ in Chapter 10. (Hint: Does an ‘external’ clock need to be present?)

5. Solutions

  1. \(\langle \psi_S(t)|\psi_S(t)\rangle = \frac{({}_C\langle t | \Psi \rangle)^\dagger ({}_C\langle t | \Psi \rangle)}{p(t)} = \frac{\langle \Psi | t \rangle_C {}_C\langle t | \Psi \rangle}{p(t)}\). The numerator is the inner product in the system space \(\mathcal{H}_S\), and is equal to \(\langle \Psi | (|t\rangle\langle t|_C \otimes \mathbf{1}_S) | \Psi \rangle\). Since this value is consistent with the definition of probability \(p(t)\), the result is \(p(t)/p(t)=1\).
  2. Refer to the 💡 Exchange of Time box in Section 2 of the main text. The key is to combine the global constraint \((H_C+H_S)|\Psi\rangle=0\) with the property of the clock \(\langle t|H_C = -i\hbar \partial_t \langle t|\).
  3. According to the \(\Delta E_C \Delta t \gtrsim \hbar\) relation, to distinguish time precisely (\(\Delta t\) small), the clock must have a very broad energy spectrum (\(\Delta E_C\) large). A clock with limited energy has fundamental limitations in time measurement.
  4. An ideal clock must (1) not interact with the system (\([H_C, H_S]=0\)), (2) change monotonically with time, and (3) have a continuous spectrum capable of distinguishing all time values. If there is an interaction, \(H_S\) can affect the clock’s scale, potentially distorting time measurement.
  5. The Thermal Time Hypothesis (Chapter 11) defines physical time as the intrinsic dynamical flow (modular flow) inherent to a specific thermal equilibrium state (KMS state), whereas the Page-Wooters Mechanism (Chapter 10) defines the dynamical time of one subsystem relatively through quantum entanglement between two subsystems (clock and system) and conditional measurements on one subsystem. That is, the former finds the origin of time in the ‘property of the state itself’, while the latter finds it in the ‘relationship between subsystems’.