A modification of the projective construction of quantum states for field theories

Kijowski, Jerzy; Okołów, Andrzej

doi:10.1063/1.4989550

Cited by 26 publications

(41 citation statements)

References 27 publications

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“…This is exactly the time-of-arrival probability density obtained by several authors via different approaches [8,[30][31][32][33] and is in excellent agreement with numerical "quantum jump" time-of-flight simulations [18]. However, the models employed have faced problems even in the free particle case.…”

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confidence: 86%

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Space-time-symmetric extension of nonrelativistic quantum mechanics

Dias

Parisio

2017

Phys. Rev. A

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In quantum theory we refer to the probability of finding a particle between positions x and x + dx at the instant t, although we have no capacity of predicting exactly when the detection occurs. In this work, first we present an extended non-relativistic quantum formalism where space and time play equivalent roles. It leads to the probability of finding a particle between x and x + dx during [t,t + dt]. Then, we find a Schrödinger-like equation for a "mirror" wave function φ(t, x) associated with the probability of measuring the system between t and t + dt, given that detection occurs at x. In this framework, it is shown that energy measurements of a stationary state display a non-zero dispersion, and that energy-time uncertainty arises from first principles. We show that a central result on arrival time, obtained through approaches that resort to ad hoc assumptions, is a natural, built-in part of the formalism presented here. In Schrödinger quantum mechanics (QM) there is a clear asymmetry between time and space. Time is a continuous parameter that can be chosen with arbitrary precision and used to label the solution of the wave equation. In contrast, the position of a particle is seen as an operator, and therefore its value under a measurement is inherently probabilistic. It is common to hear that this asymmetry is due to the non-relativistic character of the Schrödinger equation (SE). Although partially correct, this argument is largely insufficient to justify all the disparity between space and time in the formalism of QM.A clear illustration is as follows. In a position measurement, ψ(x, t) = x|ψ(t) gives the probability amplitude of finding the particle within [x, x + dx], given that the time of detection is t. Would it not be equally reasonable, even in the non-relativistic domain, to ask about the probability of measuring the particle between x and x + dx, and t and t + dt? In this broader scenario, inquiring about the state of a particle at a given time t (as we often do), should make as much sense as asking about the state of that particle in a given position x (which we never do). In addition, if symmetry is to hold at this level, then there should exist a "mirror" wave function φ(t, x) = t|φ(x) , where x is a continuous parameter and t is the eigenvalue of an observable. If the location of particle becomes a physical reality only when a measurement is made, then it is a tenable position to expect that time should emerge in the same way. To earnestly consider these issues is the main goal of this manuscript.Time has been addressed in different contexts in QM . Common to several of these works is the attempt to remain within the borders of the standard theory. However, the solution to the arrival-time problem is considered by several authors to lay outside the framework of QM. It concerns the arrival of a particle in a spatially localized apparatus, where a time operator may be defined so that the relation [T ,Ĥ] = i is satisfied, and * corresponding author: eduardodias@df.ufpe.br † parisio@df.ufpe.br ...

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confidence: 86%

“…In Refs. [30,32], e. g., Eq. (14) was obtained from the Schrödinger current density, which is not positive definite.…”

mentioning

confidence: 99%

Space-time-symmetric extension of nonrelativistic quantum mechanics

Dias

Parisio

2017

Phys. Rev. A

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“…Another context in which the disformal symmetry could be of use is in the formulation of purely affine theories of gravitation (see e.g. [63][64][65][66][67][68][69][70][71][72][73]). Since it is easier to construct kinetic terms for matter fields with the help of a metric, purely affine theories could make use of disformal symmetry in order to introduce a fiducial metric.…”

Section: A Two Examples Of Disformal Weyl Structuresmentioning

confidence: 99%

A generalized Weyl structure with arbitrary non-metricity

et al. 2019

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A Weyl structure is usually defined by an equivalence class of pairs (g, ω) related by Weyl transformations, which preserve the relation ∇g = ω ⊗ g, where g and ω denote the metric tensor and a 1-form field. An equivalent way of defining such a structure is as an equivalence class of conformally related metrics with a unique affine connection Γω, which is invariant under Weyl transformations. In a standard Weyl structure, this unique connection is assumed to be torsion-free and have vectorial non-metricity. This second view allows us to present two different generalizations of standard Weyl structures. The first one relies on conformal symmetry while allowing for a general non-metricity tensor, and the other comes from extending the symmetry to arbitrary (disformal) transformations of the metric.1 The usual definition of the non-metricity tensor is Q = ∇g.

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“…Note that we do not make any contact with the ideal arrival time distribution of Kijowksi [20]. This is not surprising since the Kijowski distribution is always non-negative, yet here,…”

Section: Relation To Standard Arrival Time Distribution Resultsmentioning

confidence: 79%

“…But the failure to satisfy the sum rule suggests that such a probability formula has in general significant dependence on the measurement procedure employed to determine it and indeed specific models exhibit exactly this property. (It is however, sometimes possible to extract the ideal arrival time distribution proposed by Kijowski [20] from a particular measurement process [21]).…”

Section: Introductionmentioning

confidence: 99%

Quasiprobability for the arrival-time problem with links to backflow and the Leggett-Garg inequalities

Halliwell¹,

Beck²,

Lee³

et al. 2019

Phys. Rev. A

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The arrival time problem for the free particle in one dimension may be formulated as the problem of determining a joint probability for the particle being found on opposite sides of the x-axis at two different times. We explore this problem using a two-time quasi-probability linear in the projection operators, a natural counterpart of the corresponding classical problem. We show that it can be measured either indirectly, by measuring its moments in different experiments, or directly, in a single experiment using a pair of sequential measurements in which the first measurement is weak (or more generally, ambiguous). We argue that when positive, it corresponds to a measurementindependent arrival time probability. For small time intervals it coincides approximately with the time-averaged current, in agreement with semiclassical expectations. The quasi-probability can be negative and we exhibit a number of situations in which this is the case. We interpret these situations as the presence of "quantumness", in which the arrival time probability is not properly defined in a measurement-independent manner. Backflow states, in which the current flows in the direction opposite to the momentum, are shown to provide an interesting class of examples such situations. We also show that the quasi-probability is closely linked to a set of two-time Leggett-Garg inequalities, which test for macroscopic realism.represents the amplitude for the history in which the particle lies in x < 0 at times t 1 , t 2 , · · · t n−1 and is in x > 0 at t n , where P + (t) denotes the projection operator in the Heisenberg picture. For sufficiently close spacing of the times, this object is then a plausible candidate for the amplitude for the particle to make a left-right crossing of the origin, for the first time, during the time interval [t n−1 , t n ]. The probability for the crossing is then the norm of this state.Note that in an expression of the form Eq.(1.1), one would expect the Zeno effect [11] to come into play for sufficiently frequent measurements. This is indeed the case -it becomes significant when the time interval between projectors is smaller than /E, where E is the energy scale of the incoming packet [12,13].

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A modification of the projective construction of quantum states for field theories

Cited by 26 publications

References 27 publications

Space-time-symmetric extension of nonrelativistic quantum mechanics

Space-time-symmetric extension of nonrelativistic quantum mechanics

A generalized Weyl structure with arbitrary non-metricity

Quasiprobability for the arrival-time problem with links to backflow and the Leggett-Garg inequalities

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