[Paper Review] Arrows of Time and Initial and Final Conditions in the Quantum Mechanics of Closed Systems Like the Universe

James B. Hartle|arXiv (Cornell University)|Feb 17, 2020

Quantum Mechanics and Applications18 references4 citations

TL;DR

This paper proposes that arrows of time in a closed quantum universe emerge not from initial conditions alone, but from time-asymmetric boundary conditions—specifically, a pure initial state and a final state of indifference (ρF ∝ I). Using the decoherent histories formulation of quantum mechanics, it shows that decoherence requires time asymmetry between initial and final conditions, and that the final state of indifference allows the finest-grained consistent description of the universe, resolving a key tension between causality and quantum predictability.

ABSTRACT

A model quantum cosmology is used to illustrate how arrows of time emerge in a universe governed by a time-neutral dynamical theory constrained by time asymmetric initial and final boundary conditions represented by initial and final density matrices. In a quantum universe universe arrows of time are described by the probabilities of appropriately coarse grained sets of histories of quantities like entropy that grow or decay. We show that the requirement of that these sets of histories decohere implies two things: (1) A time asymmetry between initial and final conditions that is a basis for arrows ot time. (2) How a final state of indifference that is represented by a final density matrix proportional to the unit density matrix is consistent with causality, and allows a finer-grained description of the model universe in terms of decoherent histories than any other final state.

Motivation & Objective

• To explain how arrows of time—such as thermodynamic, electromagnetic, and cosmological—can emerge in a time-neutral quantum universe governed by consistent dynamics.
• To investigate the role of both initial and final boundary conditions in determining the directionality of time in quantum cosmology.
• To clarify how decoherence in the consistent histories framework constrains the form of final conditions, especially the physically preferred state ρF ∝ I.
• To assess whether a final condition of indifference is consistent with causality and provides a more detailed description of the universe than other final states.

Proposed method

• The paper employs the consistent (decoherent) histories formulation of quantum mechanics to describe alternative time-ordered histories of physical quantities like entropy or retarded radiation.
• It models the universe as a closed system with a time-neutral dynamical theory H and imposes initial (ρI) and final (ρF) density matrix boundary conditions.
• Decoherence is enforced by requiring negligible quantum interference between histories in a coarse-grained set, ensuring probabilities obey classical probability rules.
• The analysis focuses on the implications of setting ρF proportional to the identity matrix (a state of indifference) and compares it to other possible final states.
• It examines the consistency of such a final condition with causality and its capacity to allow a finer-grained description of the universe through more decoherent history sets.
• The framework is applied to a model quantum cosmology to derive constraints on ρF and ρI, particularly under the assumption of a pure initial state.

Experimental results

Research questions

• RQ1 Can arrows of time emerge in a time-neutral quantum universe without assuming a special initial condition alone?
• RQ2 What constraints does the requirement of decoherence impose on the form of the final boundary condition ρF?
• RQ3 Is a final condition of indifference (ρF ∝ I) consistent with causality in quantum mechanics?
• RQ4 Does a final state of indifference allow a more detailed, finer-grained description of the universe than any other final state?
• RQ5 How do initial and final conditions jointly determine the direction of time in a closed quantum system?

Key findings

• Decoherence in a closed quantum system requires time asymmetry between initial and final boundary conditions; a pure initial state ρI cannot be paired with a pure final state ρF, as this would violate decoherence.
• A final condition of indifference, represented by ρF proportional to the identity matrix, is consistent with causality and allows a finer-grained description of the universe than any other final state.
• The final state ρF ∝ I maximizes the number of decoherent history sets, enabling the most detailed consistent quantum description of the universe’s evolution.
• The emergence of arrows of time is not solely due to special initial conditions but arises from the interplay between time-neutral dynamics and time-asymmetric boundary conditions.
• The requirement of decoherence implies that the final condition must differ from the initial condition, providing a dynamical basis for the observed time asymmetry.
• The analysis suggests that the final condition ρF ∝ I is theoretically preferred, though its physical validity must ultimately be tested against large-scale cosmological observations.