Trajectories for the Wave Function of the Universe from a Simple Detector Model

Abstract

Inspired by Mott's (1929) analysis of particle tracks in a cloud chamber, we consider a simple model for quantum cosmology which includes, in the total Hamiltonian, model detectors registering whether or not the system, at any stage in its entire history, passes through a series of regions in configuration space. We thus derive a variety of well-defined formulas for the probabilities for trajectories associated with the solutions to the Wheeler-DeWitt equation. The probability distribution is peaked about classical trajectories in configuration space. The ``measured'' wave functions still satisfy the Wheeler-DeWitt equation, except for small corrections due to the disturbance of the measuring device. With modified boundary conditions, the measurement amplitudes essentially agree with an earlier result of Hartle derived on rather different grounds. In the special case where the system is a collection of harmonic oscillators, the interpretation of the results is aided by the introduction of ``timeless'' coherent states -- eigenstates of the Hamiltonian which are concentrated about entire classical trajectories.

37 pages, plain Tex. Second draft. Substantial revisions

References in corpus (12)

• Measurement of Time-of-Arrival in Quantum Mechanics
• A Uniqueness Theorem for Constraint Quantization
• On the Generality of Refined Algebraic Quantization
• Somewhere in the Universe: Where is the Information Stored When Histories Decohere?
• Quantum arrival time measurement and backflow effect
• Comparing Formulations of Generalized Quantum Mechanics for Reparametrization-Invariant Systems
• The time of arrival concept in quantum mechanics
• Localization of Events in Space-Time
• Arrival Times in Quantum Theory from an Irreversible Detector Model
• Universal Procedure for Enforcing Quantum Constraints
• Decoherent Histories Approach to the Arrival Time Problem
• Probability distribution of arrival times in quantum mechanics

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