The preferred-time problem in the conditional interpretation of time

Abstract

In timeless formulations of quantum theory, the Page-Wootters proposal (PW) recovers time and dynamics as relative clock-and-world states. By interpreting the total timeless system as a clock entangled with the world, the states having a definite clock reading appear to recover the temporal states of the world as relative states, and dynamics emerges. But an entangled state admits infinitely many decompositions as a superposition of product states, each decomposition corresponding to a different but equally valid time operator of the same clock system. Different valid choices of the time operator for the same clock system lead to different physical histories, where the world states in a history are superpositions of the world states from another history. Fixing one operator as "the" clock time would import the temporal meaning that the Page-Wootters proposal is meant to explain in the first place. Therefore, the interpretation of PW that the world state is passively conditioned on the clock state cannot hold. The timeless state is resolved into time-dependent states by the intrinsic pointer observables of the world, without having to outsource the role of the time operator to a separate clock. Fortunately, the formalism of PW remains valid, provided that it is interpreted in terms of intrinsic pointer observables of the world itself.