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![2: The dwell time problem, defined using a model clock. The clock runs while the particle is in the region [−L, L].](profile/James-Yearsley/publication/51948888/figure/fig12/AS:669548697092125@1536644257922/The-dwell-time-problem-defined-using-a-model-clock-The-clock-runs-while-the-particle.png)
2: The dwell time problem, defined using a model clock. The clock runs while the particle is in the region [−L, L].
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We consider a number of aspects of the problem of defining time observables
in quantum theory. Time observables are interesting quantities in quantum
theory because they often cannot be associated with self-adjoint operators.
Their definition therefore touches on foundational issues in quantum theory.
Various operational approaches to defining time...
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The conformational change of biological macromolecule is investigated from the point of quantum transition. A quantum theory on protein folding is proposed. Compared with other dynamical variables such as mobile electrons, chemical bonds and stretching-bending vibrations the molecular torsion has the lowest energy and can be looked as the slow vari...
Citations
... The most popular time-of-arrival operator seems to be one developed by Kijowski in 1974 [52]. It and variations on it have seen much use since then [11,10,64,9,75,2,82,55,84,40,53,20,83]. We used Kijowski as a starting point in our previous paper: it gives reasonable results in simple cases. ...
Does the Heisenberg uncertainty principle (HUP) apply along the time dimension in the same way it applies along the three space dimensions? Relativity says it should; current practice says no. With recent advances in measurement at the attosecond scale it is now possible to decide this question experimentally.
The most direct test is to measure the time-of-arrival of a quantum particle: if the HUP applies in time, then the dispersion in the time-of-arrival will be measurably increased.
We develop an appropriate metric of time-of-arrival in the standard case; extend this to include the case where there is uncertainty in time; then compare. There is – as expected – increased uncertainty in the time-of-arrival if the HUP applies along the time axis. The results are fully constrained by Lorentz covariance, therefore uniquely defined, therefore falsifiable.
So we have an experimental question on our hands. Any definite resolution would have significant implications with respect to the role of time in quantum mechanics and relativity. A positive result would also have significant practical applications in the areas of quantum communication, attosecond physics (e.g. protein folding), and quantum computing.
... The most popular time-of-arrival operator seems to be one developed by Kijowski in 1974 [52]. It and variations on it have seen much use since then [11,10,64,9,75,2,82,55,84,40,53,20,83]. We used Kijowski as a starting point in our previous paper: it gives reasonable results in simple cases. ...
Does the Heisenberg uncertainty principle (HUP) apply along the time dimension in the same way it applies along the three space dimensions? Relativity says it should; current practice says no. With recent advances in measurement at the attosecond scale it is now possible to decide this question experimentally. The most direct test is to measure the time-of-arrival of a quantum particle: if the HUP applies in time, then the dispersion in the time-of-arrival will be measurably increased. We develop an appropriate metric of time-of-arrival in the standard case; extend this to include the case where there is uncertainty in time; then compare. There is -- as expected -- increased uncertainty in the time-of-arrival if the HUP applies along the time axis. The results are fully constrained by Lorentz covariance, therefore uniquely defined, therefore falsifiable. And therefore we have an experimental question on our hands. Any definite resolution would have significant implications with respect to the role of time in quantum mechanics and relativity. A positive result would also have significant practical applications in the areas of quantum communication, attosecond physics (e.g. protein folding), and quantum computing.
... The Schrödinger Eqn. and the standard postulates of quantum mechanics [12] do not give a ready-made recipe for calculating these statistics. There is no textbook quantum operators or wave function associated with the first passage time measurements (see [13][14][15]for related historical accounts). Actually, time is a non-quantum ingredient of quantum mechanics and is treated as a object detached from the probabilistic interpretation inherent to non-classical reality. ...
Even after decades of research the problem of first passage time statistics for quantum dynamics remains a challenging topic of fundamental and practical importance. Using a projective measurement approach, with a sampling time $\tau$, we obtain the statistics of first detection events for quantum dynamics on a lattice, with the detector located at the origin. A quantum renewal equation for a first detection wave function, in terms of which the first detection probability can be calculated, is derived. This formula gives the relation between first detection statistics and the solution of the corresponding Schr\"odinger equation in the absence of measurement. We demonstrate our results with tight binding quantum walk models. We examine a closed system, i.e. a ring, and reveal the intricate influence of the sampling time $\tau$ on the statistics of detection, discussing the quantum Zeno effect, half dark states, revivals and optimal detection. The initial condition modifies the statistics of a quantum walk on a finite ring in surprising ways. In some cases the average detection time is independent of the sampling time while in others the average exhibits multiple divergences as the sampling time is modified. For an unbounded one dimensional quantum walk the probability of first detection decays like $(\mbox{time})^{(-3)}$ with superimposed oscillations, with exceptional behavior when the sampling period $\tau$ times the tunnelling rate $\gamma$ is a multiple of $\pi/2$. The amplitude of the power law decay is suppressed as $\tau\to 0$ due to the Zeno effect. Our work presented here, is an extended version of Friedman et al. arXiv:1603.02046 [cond-mat.stat-mech], and it predicts rich physical behaviors compared with classical Brownian motion, for which the first passage probability density decays monotonically like $(\mbox{time})^{-3/2}$, as elucidated by Schr\"odinger in $1915$.
... Thus for analyses in the decoherent histories framework, cf. Halliwell (2011),Yearsley (2011). For a recent time-energy uncertainty principle for an arrival time, cf.Kiukas et al. (2011). ...
First, I briefly review the different conceptions of time held by three rival
interpretations of quantum theory: the collapse of the wave-packet, the
pilot-wave interpretation, and the Everett interpretation (Section 2).
Then I turn to a much less controversial task: to expound the recent
understanding of the time-energy uncertainty principle, and indeed of
uncertainty principles in general, that has been established by such authors as
Busch, Hilgevoord and Uffink.
Although this may at first seem a narrow topic, I point out connections to
other conceptual topics about time in quantum theory: for example, the question
under what circumstances there is a time operator (Section 4.3).
... The implications of the decoherent histories framework for the foundations of physics have been widely discussed, for instance with respect to quantum measurement problem [13,14,16,21,35], the arrival time problem [20,36,37], as well as the problem of time in quantum gravity [1,9,36]. While the results of this paper are mostly formal, their foundational implications should be the subject of future work. ...
... Let α = (α 1 , ..., α n ) denote a list of diffinvariant properties and {c α } the associated exhaustive diff-invariant set of exclusive classes c α of histories. The class operator for this coarse graining is by matrix elements (37) where sum runs over all bulk configurations for which the properties α are satisfied. ...
... A decoherence functional is readily constructed from the class operator above by multiplying the restricted amplitudes (37) and summing over all possible boundary states ψ = ψ f ⊗ ψ i . ...
The decoherent histories formalism, developed by Griffiths, Gell-Mann, and Hartle [M. Gell-Mann and J. B. Hartle, “Quasiclassical coarse graining and thermodynamic entropy”, Phys. Rev. A 76, No. 2, Article No. 022104, 16 p. (2007; doi:10.1103/PhysRevA.76.022104); R. B. Griffiths, Consistent Quantum Theory. Cambridge University Press (2002; Zbl 1140.81304)] is a general framework in which to formulate a timeless, ‘generalised’ quantum theory and extract predictions from it. Recent advances in spin foam models allow for loop gravity to be cast in this framework. In this paper, I propose a decoherence functional for loop gravity and interpret existing results [E. Bianchi et al., “Cosmological constant in spinfoam cosmology”, Phys. Rev. D 83, No. 10, Article No. 104015, 4 p. (2011; doi: 10.1103/PhysRevD.83.104015); “Towards spinfoam cosmology”, Phys. Rev. D 82, No. 8, Article No. 084035, 8 p. (2010; doi:10.1103/PhysRevD.82.084035)] as showing that coarse grained histories follow quasiclassical trajectories in the appropriate limit.
... The analysis of this situation with one of the above modified propagators essentially follows from the work described in Ref.[31]. For superpositions of incoming wave packets (in either direction), the sum rules are approximately satisfied if the energy scale of the wave packet E satisfies E ≫ V 0 and the probabilities are the expected semiclassical ones, so intuitive properties are restored. ...
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 \epsilon, 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 \epsilon >> 1/ E, where E is the
energy scale of the incoming state.
This paper gives a philosophical assessment of the Montevideo interpretation of quantum theory, advocated by Gambini, Pullin and co-authors. This interpretation has the merit of linking its proposal about how to solve the measurement problem to the search for quantum gravity: namely by suggesting that quantum gravity makes for fundamental limitations on the accuracy of clocks, which imply a type of decoherence that ‘collapses the wave-packet’.
I begin (Section 2) by sketching the topics of decoherence, and quantum clocks, on which the interpretation depends. Then I expound the interpretation, from a philosopher׳s perspective (, and ). Finally, in Section 6, I argue that the interpretation, at least as developed so far, is best seen as a form of the Everett interpretation: namely with an effective or approximate branching, that is induced by environmental decoherence of the familiar kind, and by the Montevideans’ ‘temporal decoherence’.
In this paper the notion of Dirac basis will be introduced. It is the continuous pendant of the discrete basis for Hilbert spaces. The introduction of this new notion is closely related to the theory of generalized functions. Here De Graaf's theory will be employed. It is based on the triplet where X is a Hilbert space. In a well specified way any member of can be expanded with respect to a Dirac basis. Both the introduction of Dirac bases and a new interpretation of Dirac's bracket notion will lead to a mathematical rigorization of various aspects of Dirac's formalism for quantum mechanics. This rigorization goes much beyond earlier proposals.
