This article generalizes the conditional probability interpretation of time in which time evolution is realized through entanglement between a clock and a system of interest. This formalism is based upon conditioning a solution to the Wheeler-DeWitt equation on a subsystem of the Universe, serving as a clock, being in a state corresponding to a time $t$. Doing so assigns a conditional state to the rest of the Universe $|\psi_S(t)\rangle$, referred to as the system. We demonstrate that when the total Hamiltonian appearing in the Wheeler-DeWitt equation contains an interaction term coupling the clock and system, the conditional state $|\psi_S(t)\rangle$ satisfies a time-nonlocal Schrödinger equation in which the system Hamiltonian is replaced with a self-adjoint integral operator. This time-nonlocal Schrödinger equation is solved perturbatively and three examples of clock-system interactions are examined. One example considered supposes that the clock and system interact via Newtonian gravity, which leads to the system's Hamiltonian developing corrections on the order of $G/c^4$ and inversely proportional to the distance between the clock and system.
Such a reformulation of quantum theory was put forward by Don Page and William Wootters in 1982, which begins by partitioning the Universe into two parts: a clock and a system comprised of everything else. Then any statement we would ordinarily make regarding the time dependence of the system takes the form, “The quantum state of the system is $\psi(t)$, if a measurement of the clock reads the time $t$”. The crucial difference in this formulation of quantum theory is that it makes no reference to an external background time; instead, time appears as an operational concept defined in terms of the measurement of a physical clock. An entangled state is assigned to the clock and system and the usual dynamics of quantum theory, as described by the Schrödinger equation, emerge from quantum correlations between the clock and system.
This article explores the effects of when the Universe cannot be partitioned cleanly into two parts, that is, when the clock and system interact. In some situations we should expect such an interaction given that gravity couples to everything! It is shown that such clock-system interactions lead to a more general quantum dynamics described by a time-nonlocal modification to the Schrödinger equation. This temporal nonlocality implies that the way in which the system changes in time depends not just on its current state, but also on its state in the future and the past. Three different clock-system interactions are examined to explore the effects of this temporal nonlocality. The first interaction considered leads to the system’s Hamiltonian developing a particular time dependence. The second example supposes that the clock and system interact via Newtonian gravity, which leads to corrections to the system’s Hamiltonian at order $G / c^4$ and is inversely proportional to the distance separating the clock and system. The third interaction gives an example of a qubit clock interacting with another qubit.
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