PageWootters

PageWootters, also known as the Page-Wootters mechanism, is a framework in quantum theory for describing time as an emergent property of correlations within a composite quantum system. Proposed by Don N. Page and William K. Wootters in 1983, it addresses the problem of time by treating time as relational rather than external.

In the setup, the universe is partitioned into a clock subsystem C and a system of interest S, with a joint Hilbert space H = H_C ⊗ H_S. The total Hamiltonian is H_total = H_C ⊗ I_S + I_C ⊗ H_S, and the global state |Ψ⟩ is taken to be stationary, satisfying H_total |Ψ⟩ = 0 (a clocked, energy-constrained description). The clock is assumed to have a continuous spectrum and to be entangled with the system.

The conditional state of S given a clock reading t is defined as |ψ_S(t)⟩ ∝ ⟨t|Ψ⟩, where |t⟩ is a clock eigenstate with reading t. In this construction, the conditional states evolve unitarily with the system Hamiltonian: |ψ_S(t)⟩ ≈ e^{-i H_S t/ħ} |ψ_S(0)⟩. Thus, time emerges from internal correlations rather than an external parameter.

Interpretation and implications: the mechanism treats time as relational and shows how conventional quantum dynamics can arise from a timeless global description. It has influenced discussions on the problem of time in quantum gravity and relational quantum mechanics, though it relies on idealized clock models and faces challenges in modeling realistic clocks and interactions.

Limitations and critiques: dependence on an accurate, isolated clock; questions about the exact conditions under which standard dynamics are recovered; and debates about interpretational status within timeless or relational frameworks.

Applications and influence: used in conceptual studies of quantum cosmology, gravity, and quantum information to illustrate relational time and the role of entanglement between clock and system.