Temporal aspects of quantum theory and emergent classicality

Questions concerning the nature of time have fascinated scientists and thinkers from all areas of life for millenia. The physicists conception of time first took fully concrete form with the development of Newtonian mechanics in the eighteenth century and the depth and subtleties involved with our concepts of time became apparent with the advent of relativity at the beginning of the twentieth century. Quantum theory emerged in the 1920s as the fundamental theory of matter and at its heart lies Schrodinger's famous equation, describing the evolution, in time, of the wave function representing the wave-like properties of atoms. Quantum theory was immediately spectacularly successful in all its applications. Yet an issue that was not addressed concerns the status of the time parameter in the Schrodinger equation. It appears in a way which is different to other variables, such as position and momentum. These variables (which are mathematically represented by operators) obey the uncertainty principle, which limits the degree to which they may be specified. Yet time appears in the Schrodinger equation as an essentially classical quantity, corresponding to the time measured by a classical clock described by Newtonian mechanics, seemingly unrestricted by the uncertainty principle. This is a peculiarly hybrid state of affairs and a very surprising one for a supposedly fundamental theory. The question of clarifying the nature of time in quantum theory has become a topic of considerable interest in recent years.This question is a particularly exciting one since a study of time in quantum theory leads us to a deeper understanding of what time actually is.

One particular notion of time one would like to define in quantum theory is the arrival time: what is the probability that a particle in a given quantum state arrives at a certain point in space during a given time interval? The peculiar status of time in quantum theory means that methods outside the usual quantum-mechanical toolbox have to be used to answer such questions. Many such methods are indirect. For example, by measuring the amount of probability in a spatial region at two different times one can deduce the flux of probability leaving that spatial region during the given time interval, from which one can define the arrival time at a point. But indirect methods such as this have interesting problems. For example, the arrival time probability defined through the flux, classically positive, can come out to be negative for certain quantum states. This curious and little-investigated non-classical phenomenon is called backflow and haunts many attempts to define time in quantum theory.

More elaborate means of defining the arrival time involve repeated position measurements at short time intervals, checking to see if the particle is still there. Yet these methods may also suffer from interesting problems. A basic
property of any quantum system is that measurement disturbs it. If measurements are made too frequently, it is disturbed so much that there is nothing left to measure! This non-classical effect is called the quantum Zeno effect and is another phenemonon that gets in the way of attempts to define time in quantum theory.

The proposed research addresses the definition of time in quantum theory in a variety of contexts, and also addresses the associated problems that arise. The backflow effect will be investigated in detail. This turns out to have some
interesting relationships to a novel form of the Bell inequalities which involve measurements distributed in time. The quantum Zeno effect will also be investigated. In particular, an interesting question is how this highly non-classical effect goes away in the classical limit. These and related questions may have interesting experimental consequences and will also shed light on the nature of time itself.

The proposed research is likely to have impact across a number of academic areas including quantum cosmology, the foundations of quantum theory, the emergence of classical behaviour from quantum theory, and time in quantum theory. The specific beneficiaries in these areas are listed in the beneficiaries section.

The last twenty years have seen a phenomenal amount of activity in the area of experimental tests of foundational questions in quantum theory, with tests of Bell's inequalities perhaps foremost amongst these. In addition to the drive to confirm theoretical predictions of fascinating quantum phenomena, these developments are likely to have significant technological consequences, for example, in the construction of more powerful computers. Quantum information scientists therefore devote much effort to exploring and exploiting unusual quantum effects. The backflow effect is a
little-studied effect, although may as we hope to show, be related to the temporal Bell inequalities. The study of both the backflow effect and temporal Bell inequalities might contribute to this drive to exploit unusual quantum effects and may in the long run have interesting technological consequences.

The proposed work on the classical limit of the quantum Zeno effect is part of a more general study of the role of decoherence, the destruction of interference, in quantum phenomena, an area of considerable theoretical and practical interest. For example, decoherence destroys the ability of a quantum computer to do its job properly, so a thorough understanding of this process is important technologically. Furtheremore, the reduction of quantum mechanical reflection due to decoherence is little-studied theoretically and there is, so far, little experimental work on this area. Theoretical and experimental work in this area are likely to lead to interesting new results such as realistic tests of decoherence theory.

The proposed work on the dwell time is similar in many ways to problems encountered in quantum cosmology -- in both areas one encounters operators characterizing time which commute with the Hamiltonian -- so there will be interesting impact in that area. Indeed, some of the PIs recent work has already made progress in this area and one can anticipate a fruitful cross-fertilization between quantum cosmology and studies of time in non-relativistic quantum mechanics.

Aside from these more practical benefits, research on the nature of time has a major impact on education,
outreach and culture more generally. Students and the scientifically literate public are deeply fascinated by
fundamental science and many artists and writers draw inspiration from it. All good teachers know that student learning is significantly enhanced when students are inspired and society generally profits as a result.