The idea that time may emerge from quantum entanglement originated from a mechanism proposed in 1983 by Don Page and William Wootters to solve the, so called, “problem of time” that arises in the context of canonical quantization of gravity. The Page and Wootter (PaW) theory consists in dividing the total Hilbert space into two sub-systems and assigning one of it to time. The “flow of time” then consists simply in the entanglement between the quantum degree of freedom of time and the rest of the system, a correlation present in a global, time-independent state. In this framework we do not consider time as an abstract, external coordinate, but as “what is shown on a clock”, where the clock is some physical system that is taken as time reference.
We reviewed PaW theory focusing on which systems can be considered as good clocks. We have therefore extended the mechanism so that we can consider clock hamiltonians with discrete energy spectrum and unequally-spaced energy levels (the only constraint prevents us in using Hamiltonians with energy eigenstates for which the ratios of the energy differences are not rational). In this framework time is described by a POVM and we demonstrate that the introduced POVM’s provide a consistent dynamical evolution for the rest of the system even if they are they are not orthogonal, and therefore partially un-distinguishables.
We also notice that in the particular case in which the clock’s Hamiltonian is chosen to be equally spaced in energy spectrum, is possible to construct an Hermitian operator T that is the complement of the clock’s Hamiltonian. It is well known the Pauli objection regarding the existence of a time operator: time and energy must have the same spectrum since conjugate operator are unitarily equivalent, but this is clearly not always true because normal Hamiltonians have lower bounded spectrum. In the PaW mechanism this objection is overcome and it is possible considering the operator T as a time operator for each generic Hamiltonian of the rest of the system.