Abstract

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 |ψS(t)⟩, 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 |ψS(t)⟩ 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/c4 and inversely proportional to the distance between the clock and system.

Highlights

  • In quantum theory, time enters through its appearance as a classical parameter in the Schrödinger equation, as opposed to other physical quantities, such as position or momentum, which are associated with self-adjoint operators and treated dynamically

  • We conclude that the coupling between the clock and and system described by the interaction Hamiltonian in Eq (24) has no effect on the relational dynamics between the clock and system, and the conditional state |ψS(t) satisfies the usual Schrödinger equation, which in this case is i d dt

  • In the case of an interaction between the clock and system, we find the conditional state of the system |ψS(t) satisfies a modified time-nonlocal Schrödinger equation

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Summary

Introduction

The canonical quantization of gravity leads to the Wheeler-DeWitt equation: physical states are annihilated by the Hamiltonian of the theory. A necessary requirement for any quantum theory of gravity is to answer this question and explain how the familiar Schrödinger equation comes about from the Wheeler-DeWitt equation. As introduced by Page and Wootters [3,4,5,6], the conditional probability interpretation defines the state of a system at a time t as a solution to the Wheeler-DeWitt equation conditioned on a subsystem of the Universe, serving as a clock, to be in a state corresponding to the time t. We find that taking into account a possible clock-system interaction within the conditional probability interpretation of time results in a time-nonlocal modification to the Schrödinger equation.

The Hamiltonian constraint in classical and quantum mechanics
The conditional probability interpretation
The modified Schrödinger equation
Solving the modified Schrödinger equation
Interactions leading to time-dependent system Hamiltonians
Gravitationally interacting clocks and systems
Qubit clock and system
Conclusions and outlook
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