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arXiv:1504.04215 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Quantum time

Vittorio Giovannetti, Seth Lloyd, Lorenzo Maccone

Published Aug 26, 2015• 161 citations
arXivHTMLPDF

Abstract

We give a consistent quantum description of time, based on Page and Wootters's conditional probabilities mechanism, which overcomes the criticisms that were raised against similar previous proposals. In particular we show how the model allows one to reproduce the correct statistics of sequential measurements performed on a system at different times.

DOI
10.1103/physrevd.92.045033

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We give a consistent quantum description of time, based on Page and Wootters's conditional probabilities mechanism, which overcomes the criticisms that were raised against similar previous proposals. In particular we show how the model allows one to reproduce the correct statistics of sequential measurements performed on a system at different times.
The quantum measurement problem can be regarded as the tension between the two alternative dynamics prescribed by quantum mechanics: the unitary evolution of the wave function and the state-update rule (or "collapse") at the instant a measurement takes place. The notorious Wigner's friend gedankenexperiment constitutes the paradoxical scenario in which different observers (one of whom is observed by the other) describe one and the same interaction differently, one –the Friend– via state-update and the other –Wigner– unitarily. This can lead to Wigner and his friend assigning different probabilities to the outcome of the same subsequent measurement. In this paper, we apply the Page-Wootters mechanism (PWM) as a timeless description of Wigner's friend-like scenarios. We show that the standard rules to assign two-time conditional probabilities within the PWM need to be modified to deal with the Wigner's friend gedankenexperiment. We identify three main definitions of such modified rules to assign two-time conditional probabilities, all of which reduce to standard quantum theory for non-Wigner's friend scenarios. However, when applied to the Wigner's friend setup each rule assigns different conditional probabilities, potentially resolving the probability-assignment paradox in a different manner. Moreover, one rule imposes strict limits on when a joint probability distribution for the measurement outcomes of Wigner and his Friend is well-defined, which single out those cases where Wigner's measurement does not disturb the Friend's memory and such a probability has an operational meaning in terms of collectible statistics. Interestingly, the same limits guarantee that said measurement outcomes fulfill the consistency condition of the consistent histories framework.
In quantum theory it is usual one refers to the probability of finding a particle between the positions x and x + dx at time t, although the theory does not provide when detection occurs.In the primordies of the theory, Pauli presented an argument refuting the idea of introducing an operator describing the probability of the instant in which the detection occurs.Later, Page and Wootters, motivated by the fact that one is not able to observe energy eigenstates superposition, presented a non-relativistic quantum formalism in an extended space in which space and time play equivalent roles, being consistent with Pauli's inquiries.This leads to a description given by the conditional probability of finding a particle between x and x + dx in respect to the time [t, t + dt] given by another particle, called the clock particle.This dissertation consists in a review of the Page and Wootters proposal to address the time problem in quantum mechanics as well as a detailed analysis of this using recent tools introduced in the area of quantum information for the treatment of symmetries in physics.