Two primitives for time: Relational time (Page–Wootters) and process time (actualization)

Abstract

In canonical quantum gravity, the Hamiltonian constraint Ĥ|Ψ⟩ = 0 deprives the universe of any explicit temporal evolution: this is the "problem of time". The Page–Wootters (PaW) mechanism answers it by reconstructing a relational time: from a stationary global state, the evolution of a subsystem appears as a family of conditional states relative to an internal clock ("evolution without evolution"). The Time–Actualization (TA) theory answers the same question with an opposite primitive: time is the process of actualization of quantum potentialities into real events, with a primitive arrow — whose dynamical law remains an open problem — and objective events. We compare the two frameworks by carefully separating the operational equivalence (the same conditional probabilities within a given protocol — the two coincide in the effective regime) from the ontological divergence, which bears on three axes: the status of events, the origin of irreversibility, and the dependence on the choice of clock. We argue that TA and PaW are not extensions of one another, but two distinct primitives — the conditional state and the ontic event — and we remain cautious about the metaphysical status of PaW, whose eternalist reading, common though it is, is itself debated.