Path integrals appear to offer natural and intuitively appealing methods for defining quantum-mechanical amplitudes for questions involving spacetime regions. For example, the amplitude for entering a spatial region during a given time interval is typically defined by summing over all paths between given initial and final points but restricting them to pass through the region at any time. We argue that there is, however, under very general conditions, a significant complication in such constructions. This is the fact that the concrete implementation of the restrictions on paths over an interval of time corresponds, in an operator language, to sharp monitoring at every moment of time in the given time interval. Such processes suffer from the quantum Zeno effect – the continual monitoring of a quantum system in a Hilbert subspace prevents its state from leaving that subspace. As a consequence, path integral amplitudes defined in this seemingly obvious way have physically and intuitively unreasonable properties and in particular, no sensible classical limit. In this paper we describe this frequently-occurring but little-appreciated phenomenon in some detail, showing clearly the connection with the quantum Zeno effect. We then show that it may be avoided by implementing the restriction on paths in the path integral in a "softer" way. The resulting amplitudes then involve a new coarse graining parameter, which may be taken to be a timescale , describing the softening of the restrictions on the paths. We argue that the complications arising from the Zeno effect are then negligible as long as >> 1/ E , where E is the energy scale of the incoming state. Our criticisms of path integral constructions largely apply to approaches to quantum theory such as the decoherent histories approach or quantum measure theory, which do not specifically involve measurements. We address some criticisms of our approach by Sokolovksi, concerning the relevance of our results to measurement-based models.
Path integrals appear to offer natural and intuitively appealing methods for defining quantum-mechanical amplitudes for questions involving spacetime regions. For example, the amplitude for entering a spatial region during a given time interval is typically defined by summing over all paths between given initial and final points but restricting them to pass through the region at any time. We argue that there is, however, under very general conditions, a significant complication in such constructions. This is the fact that the concrete implementation of the restrictions on paths over an interval of time corresponds, in an operator language, to sharp monitoring at every moment of time in the given time interval. Such processes suffer from the quantum Zeno effect – the continual monitoring of a quantum system in a Hilbert subspace prevents its state from leaving that subspace. As a consequence, path integral amplitudes defined in this seemingly obvious way have physically and intuitively unreasonable properties and in particular, no sensible classical limit. In this paper we describe this frequently-occurring but little-appreciated phenomenon in some detail, showing clearly the connection with the quantum Zeno effect. We then show that it may be avoided by implementing the restriction on paths in the path integral in a "softer" way. The resulting amplitudes then involve a new coarse graining parameter, which may be taken to be a timescale , describing the softening of the restrictions on the paths. We argue that the complications arising from the Zeno effect are then negligible as long as >> 1/ E , where E is the energy scale of the incoming state. Our criticisms of path integral constructions largely apply to approaches to quantum theory such as the decoherent histories approach or quantum measure theory, which do not specifically involve measurements. We address some criticisms of our approach by Sokolovksi, concerning the relevance of our results to measurement-based models.
Path integrals appear to offer natural and intuitively appealing methods for defining quantum-mechanical amplitudes for questions involving spacetime regions. For example, the amplitude for entering a spatial region during a given time interval is typically defined by summing over all paths between given initial and final points but restricting them to pass through the region at any time. We argue that there is, however, under very general conditions, a significant complication in such constructions. This is the fact that the concrete implementation of the restrictions on paths over an interval of time corresponds, in an operator language, to sharp monitoring at every moment of time in the given time interval. Such processes suffer from the quantum Zeno effect – the continual monitoring of a quantum system in a Hilbert subspace prevents its state from leaving that subspace. As a consequence, path integral amplitudes defined in this seemingly obvious way have physically and intuitively unreasonable properties and in particular, no sensible classical limit. In this paper we describe this frequently-occurring but little-appreciated phenomenon in some detail, showing clearly the connection with the quantum Zeno effect. We then show that it may be avoided by implementing the restriction on paths in the path integral in a "softer" way. The resulting amplitudes then involve a new coarse graining parameter, which may be taken to be a timescale , describing the softening of the restrictions on the paths. We argue that the complications arising from the Zeno effect are then negligible as long as >> 1/E, where E is the energy scale of the incoming state. Our criticisms of path integral constructions largely apply to approaches to quantum theory such as the decoherent histories approach or quantum measure theory, which do not specifically involve measurements. We address some criticisms of our approach by Sokolovksi, concerning the relevance of our results to measurement-based models.
Path integrals appear to offer natural and intuitively appealing methods for defining quantum-mechanical amplitudes for questions involving spacetime regions. For example, the amplitude for entering a spatial region during a given time interval is typically defined by summing over all paths between given initial and final points but restricting them to pass through the region at any time. We argue that there is, however, under very general conditions, a significant complication in such constructions. This is the fact that the concrete implementation of the restrictions on paths over an interval of time corresponds, in an operator language, to sharp monitoring at every moment of time in the given time interval. Such processes suffer from the quantum Zeno effect---the continual monitoring of a quantum system in a Hilbert subspace prevents its state from leaving that subspace. As a consequence, path integral amplitudes defined in this seemingly obvious way have physically and intuitively unreasonable properties and, in particular, no sensible classical limit. In this paper we describe this frequently occurring but little-appreciated phenomenon in some detail, showing clearly the connection with the quantum Zeno effect. We then show that it may be avoided by implementing the restriction on paths in the path integral in a ``softer'' way. The resulting amplitudes then involve a new coarse-graining parameter, which may be taken to be a time scale ϵ, describing the softening of the restrictions on the paths. We argue that the complications arising from the Zeno effect are then negligible as long as $ϵ\ensuremath{\gg}1/E$, where E is the energy scale of the incoming state.
We consider a point particle in one dimension initially confined to a finite spatial region whose state is frequently monitored by projection operators onto that region. In the limit of infinitely frequent monitoring, the state never escapes from the region---this is the Zeno effect. In the corresponding classical problem, by contrast, the state diffuses out of the region with the frequent monitoring simply removing probability. The aim of this paper is to show how the Zeno effect disappears in the classical limit in this and similar examples. We give a general argument showing that the Zeno effect is suppressed in the presence of a decoherence mechanism which suppresses interference between histories. We show how this works explicitly in two examples involving projections onto a one-dimensional subspace and identify the key time scales for the process. We extend this understanding to our main problem of interest, the case of a particle in a spatial region, by coupling it to a decohering environment. Smoothed projectors are required to give the problem proper definition and this implies the existence of a momentum cutoff and minimum length scale. We show that the escape rate from the region approaches the classically expected result, and hence the Zeno effect is suppressed, as long as the environmentally induced fluctuations in momentum are sufficiently large. We establish the time scale on which an arbitrary initial state develops sufficiently large fluctuations to satisfy this condition. We link our results to earlier work on the $\ensuremath{\hbar}\ensuremath{\rightarrow}0$ limit of the Zeno effect. We illustrate our results by plotting the probability flux lines for the density matrix (which are equivalent to Bohm trajectories in the pure-state case). These illustrate both the Zeno and anti-Zeno effects very clearly, and their suppression. Our results are closely related to our earlier paper [Phys. Rev. A 88, 022128 (2013)], demonstrating the suppression of quantum-mechanical reflection by decoherence.
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.
The quantum Zeno effect arises due to frequent observation. 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, and 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.
It is accepted that among the ways through which a quantum phenomenon decoheres and becomes a classical one is what is termed in the literature the Zeno effect. This effect, named after the ancient Greek philosopher Zeno of Elea (born about 485 B.C), were used in 1977 to analytically predict that an initial quantum state may be preserved in time by merely repeating a large number of times, in a finite total time, the experiment of checking its state. Since then this effect has been experimentally validated and has become an established physical fact. It has been argued by Simonius that the Zeno effect must be related not only to quantum phenomena but also to many macroscopic and classical effects. Thus, since it operates in both quantum and classical regimes it must cause to a more generalized kind of decoherence than the restricted one that ``classicalizes'' a quantum phenomenon. We show that this generalized decoherence, {\it obtained as a result of dense measurement}, not only gives rise to new phenomena that are demonstrated through new responses of the densely interacted-upon system but also may physically {\it establish} them. For that matter we have found and established the analogous {\it space Zeno effect} which leads to the necessity of an ensemble of related observers (systems) for the remarked physical validation of new phenomena. As will be shown in Chapters 3-5 of this work the new phenomena (new responses of the system) that result from the space Zeno effect may be of an unexpected nature. We use quantum field theory in addition to the more conventional methods of analysis and also corroborate our analytical findings by numerical simulations.