Spacetime Coarse Grainings and the Problem of Time in the Decoherent Histories Approach to Quantum Theory

Wallden, Petros

doi:10.1007/s10773-007-9592-y

Cited by 9 publications

(11 citation statements)

References 68 publications

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“…so is unitary in the Hilbert space of states with support only in x > 0 [27,28]. Differently put, an incoming wave packet evolving according to the restricted propagator undergoes total reflection, so never crosses x = 0.…”

Section: B Complex Potentialsmentioning

confidence: 99%

Arrival times, complex potentials, and decoherent histories

Halliwell

Yearsley

2009

Phys. Rev. A

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We carry out a decoherent-histories analysis of the arrival-time problem, taking advantage of a recently demonstrated connection between time-ordered strings of projection operators and evolution in the presence of a complex potential of step-function form. We concentrate on the limit of a weak potential, in which the resulting arrival-time distribution function is closely related to the quantum-mechanical current. We first consider the analogous classical arrival-time problem involving an absorbing potential, and this sheds some light on certain aspects of the quantum case. We use the path-decomposition expansion to give a derivation of the standard arrival-time distribution defined using a complex potential. This derivation is then used in the decoherent-histories analysis to obtain very simple and plausible expressions for the class operators ͑describing the amplitudes for crossing the origin during intervals of time͒. We show that decoherence of histories is obtained for a wide class of initial states ͑such as simple wave packets and superpositions of wave packets͒. We find that the decoherent-histories approach gives results with a sensible classical limit that are fully compatible with standard results on the arrival-time problem. We also find some interesting connections between backflow and decoherence.

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Section: B Complex Potentialsmentioning

confidence: 99%

Arrival times, complex potentials, and decoherent histories

Halliwell

Yearsley

2009

Phys. Rev. A

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“…(1.2) is required to have zero support in the region ∆ at every moment of time between t 1 and t 2 . Evolution with this propagator therefore suffers from the quantum Zeno effect -the fact that continual monitoring of a quantum system in a Hilbert subspace prevents it from leaving that subspace [57][58][59][60][61][62][63][64][65]. The consequence is that the amplitude Eq.…”

Section: Introductionmentioning

confidence: 99%

Pitfalls of path integrals: Amplitudes for spacetime regions and the quantum Zeno effect

Halliwell

Yearsley

2012

Phys. Rev. D

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Path integrals appear to offer natural and intuitively appealing methods for defining quantummechanical 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 ǫ, 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 ǫ ≫ 1/E, where E is the energy scale of the incoming state.

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“…It is of interest to explore the properties of this amplitude for a range of values of the time spacing ǫ. It is known that as ǫ → 0, we approach the Zeno limit, in which the state becomes entirely confined to the Hilbert subspace of states with support only in x > 0, so that an incoming wave packet from the right is totally reflected [11][12][13]. However, it is of greater physical interest to explore the regime of non-zero ǫ, in which the system is monitored sufficiently well to get some idea of whether the particle is in x > 0, yet not monitored so much that an incoming state is significantly reflected at x = 0.…”

Section: Introductionmentioning

confidence: 99%

On the relationship between complex potentials and strings of projection operators

Halliwell¹,

Yearsley²

2010

J. Phys. A: Math. Theor.

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It is of interest in a variety of contexts, and in particular in the arrival time problem, to consider the quantum state obtained through unitary evolution of an initial state regularly interspersed with periodic projections onto the positive x-axis (pulsed measurements). Echanobe, del Campo and Muga have given a compelling but heuristic argument that the state thus obtained is approximately equivalent to the state obtained by evolving in the presence of a certain complex potential of stepfunction form. In this paper, with the help of the path decomposition expansion of the associated propagators, we give a detailed derivation of this approximate equivalence. The propagator for the complex potential is known so the bulk of the derivation consists of an approximate evaluation of the propagator for the free particle interspersed with periodic position projections. This approximate equivalence may be used to show that to produce significant reflection, the projections must act at time spacing less than /E, where E is the energy scale of the initial state.

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Spacetime Coarse Grainings and the Problem of Time in the Decoherent Histories Approach to Quantum Theory

Cited by 9 publications

References 68 publications

Arrival times, complex potentials, and decoherent histories

Arrival times, complex potentials, and decoherent histories

Pitfalls of path integrals: Amplitudes for spacetime regions and the quantum Zeno effect

On the relationship between complex potentials and strings of projection operators

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