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Journal of Physics Conference Series

Decoherent histories approach to tunneling times and its implication

Norifumi Yamada

Published May 1, 2007• 0 citations
Publisher

Abstract

Decoherent histories approach to quantum mechanics assigns probabilities to events that are not restricted to a single moment of time, provided that the decoherence condition holds for the set of events. The condition is however hardly satisfied in general, which has been considered to limit the usefulness of the histories approach. The present paper applies the histories approach to the tunneling time problem to show that, although the decoherence condition does not hold, the histories approach is still useful in predicting the central tendency and the dispersion of resident times. We consider such quantities that would become the average and the standard deviation of resident times if the decoherence condition held. Since the condition does not actually hold, they are not regarded as the true average and the true standard deviation. Nevertheless, a quantum clock approach allows us to interpret these quantities as a central value and a dispersion of resident times within the time resolution of the clock.

DOI
10.1088/1742-6596/67/1/012042

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Decoherent histories approach to quantum mechanics assigns probabilities to events that are not restricted to a single moment of time, provided that the decoherence condition holds for the set of events. The condition is however hardly satisfied in general, which has been considered to limit the usefulness of the histories approach. The present paper applies the histories approach to the tunneling time problem to show that, although the decoherence condition does not hold, the histories approach is still useful in predicting the central tendency and the dispersion of resident times. We consider such quantities that would become the average and the standard deviation of resident times if the decoherence condition held. Since the condition does not actually hold, they are not regarded as the true average and the true standard deviation. Nevertheless, a quantum clock approach allows us to interpret these quantities as a central value and a dispersion of resident times within the time resolution of the clock.
We study a formulation of quantum mechanics in which the central notion is that of a quantum-mechanical history---a sequence of events at a succession of times. The primary aim is to identify sets of ``decoherent'' (or ``consistent'') histories for the system. These are quantum-mechanical histories suffering negligible interference with each other, and, therefore, to which probabilities may be assigned. These histories may be found for a given system using the so-called decoherence functional. When the decoherence functional is exactly diagonal, probabilities may be assigned to the histories, and all probability sum rules are satisfied exactly. We propose a condition for approximate decoherence, and argue that it implies that most probability sum rules will be satisfied to approximately the same degree. We also derive an inequality bounding the size of the off-diagonal terms of the decoherence functional. We calculate the decoherence functional for some simple one-dimensional systems, with a variety of initial states. For these systems, we explore the extent to which decoherence is produced using two different types of coarse graining. The first type of coarse graining involves imprecise specification of the particle's position. The second involves coupling the particle to a thermal bath of harmonic oscillators and ignoring the details of the bath (the Caldeira-Leggett model). We argue that both types of coarse graining are necessary in general. We explicitly exhibit the degree of decoherence as a function of the temperature of the bath, and of the width to within which the particle's position is specified. We study the diagonal elements of the decoherence functional, representing the probabilities for the possible histories of the system. To the extent that the histories decohere, we show that the probability distributions are peaked about the classical histories of the system, with the distribution of their initial positions and momenta given by a smeared version of the Wigner function. We discuss this result in connection with earlier uses of the Wigner function in this context. We find that there is a certain amount of tension between the demands of decoherence and peaking about classical paths.
The decoherent (or consistent) histories approach is applied to the arrival time problem. Since the decoherence condition does not hold for arrival time, a probability distribution of arrival times cannot be defined. Instead, a quasidistribution, which is not positive definite, can be defined in the histories approach. It is shown that well‐known Kijowski's distribution, which is positive definite and often claimed to be the correct distribution of arrival times, is obtained from the quasidistribution when the momentum wave function has certain properties. This suggests that the histories approach still holds some predictive power even when the decoherence condition does not hold.
What is the probability of a particle entering a given region of space at any time between t1{t}_{1}t1​ and t2?{t}_{2}?t2​? Standard quantum theory assigns probabilities to alternatives at a fixed moment of time and is not immediately suited to questions of this type. We use the decoherent histories approach to quantum theory to compute the probability of a nonrelativistic particle crossing x=0x=0x=0 during an interval of time. For a system consisting of a single nonrelativistic particle, histories coarse grained according to whether or not they pass through spacetime regions are generally not decoherent, except for very special initial states, and thus probabilities cannot be assigned. Decoherence may, however, be achieved by coupling the particle to an environment consisting of a set of harmonic oscillators in a thermal bath. Probabilities for spacetime coarse grainings are thus calculated by considering restricted density operator propagators of the quantum Brownian motion model. We also show how to achieve decoherence by replicating the system NNN times and then projecting onto the number density of particles that cross during a given time interval, and this gives an alternative expression for the crossing probability. The latter approach shows that the relative frequency for histories is approximately decoherent for sufficiently large N,N,N, a result related to the Finkelstein-Graham-Hartle theorem.
The quantum Zeno effect arises due to frequent observations. That implies the existence of some experimenter and its interaction with the system. In this contribution, we examine what happens for a closed system if one considers a quantum Zeno type of question, namely what is the probability of a system, remaining always in a particular subspace. This has implications to the arrival time problem that is also discussed. We employ the decoherent histories approach to quantum theory, as this is the better developed formulation of closed system quantum mechanics, in particular, dealing with questions that involve time in a non-trivial way. We get a very restrictive decoherence condition, that implies that even if we do introduce an environment, there will be very few cases that we can assign probabilities to these histories, but in those cases the quantum Zeno effect is still present.
A pedagogical introduction is given to the quantum mechanics of closed systems, most generally the universe as a whole. Quantum mechanics aims at predicting the probabilities of alternative coarse-grained time histories of a closed system. Not every set of alternative coarse-grained histories that can be described may be consistently assigned probabilities because of quantum mechanical interference between individual histories of the set. In “Copenhagen ” quantum mechanics, probabilities can be assigned to histories of a subsystem that have been “measured”. In the quantum mechanics of closed systems, containing both observer and observed, probabilities are assigned to those sets of alternative histories for which there is negligible interference between individual histories as a consequence of the system’s initial condition and dynamics. Such sets of histories are said to decohere. We define decoherence for closed systems in the simplified case when quantum gravity can be neglected and the initial state is pure. Typical mechanisms of decoherence that are widespread in our universe are illustrated. Copenhagen quantum mechanics is an approximation to the more general quantum framework of closed subsystems. It is appropriate when there is an approximately isolated subsystem that is a participant in a measurement situation in which (among other things) the decoherence of alternative registrations of the apparatus can be idealized as exact. Since the quantum mechanics of closed systems does not posit the existence of the quasiclassical domain of everyday experience, the domain of the approximate aplicability of classical physics must be explained. We describe how a quasiclassical domain described by averages of densities of approximately conserved quantities could be an emergent feature of an initial condition of the universe that implies the approximate classical behavior of spacetime on accessible scales.