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There has been a recent revival of interest in the notion of a 'trajectory' of a quantum particle. In this paper, we detail the relationship between Dirac's ideas, Feynman paths and the Bohm approach. The key to the relationship is the weak value of the momentum which Feynman calls a transition probability amplitude. With this identification we are...
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... still let us consider a small volume surrounding the midpoint X. At this point there is a spray arriving and a spray leaving a volume ∆V(X) as shown in Figure 1. To see how the local momenta behave at the midpoint X, we will use the real part of S (x, x ) defined by ...
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... see how this unexpected result also emerges from a different perspective, let us consider the process in Figure 1 which we regard as an image of an ensemble of actual individual quantum processes. We are interested in finding the average behaviour of the momentum, P X , at the point X. ...
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There has been a recent revival of interest in the notion of a `trajectory' of a quantum particle. In this paper we detail the relationship between Dirac's ideas, Feynman paths and the Bohm approach. The key to the relationship is the weak value of the momentum which Feynman calls a transition probability amplitude. With this identification we are...
Citations
... cf. Flack and Hiley [21]. Note that the minimum (10) can also be interpreted as E ψ (p 2 O,j ). ...
Given a normalized state-vector , we define the conditional expectation of a Hermitian operator A with respect to a strongly commuting family of self-adjoint operators B as the best approximation, in the operator mean square norm associated to , of A by a real-valued function of A fundamental example is the conditional expectation of the momentum operator P given the position operator X , which is found to be the Bohm momentum. After developing the Bohm theory from this point of view we treat conditional expectations with respect to general B , which we apply to non-relativistic spin 1/2-particles. We derive the dynamics of the conditional expectations of momentum and spin with respect to position and the third spin component. These dynamics can be interpreted in terms of classical continuum mechanics as a two-component fluid whose components carry intrinsic angular momentum. Interpreting the joint spectrum of the conditioning operators as a space of beables, we can introduce a classical-stochastic particle dynamics on this space which is compatible with the time-evolution of the Born probability, by combining the de Broglie-Bohm guidance condition with a Markov jump process, following an idea of J. Bell. This results in a new Bohm-type model for particles with spin. A basic problem is that such auxiliary particle dynamics are far from unique. We finally examine the relation of our conditional expectations with the conditional expectations of the theory of -algebras and, as an application, derive a general evolution equation for conditional expectations for operators acting on finite dimensional Hilbert spaces. Two appendices re-interpret the classical Bohm model as an integrable constrained Hamiltonian system, and provide the details of the two-fluid interpretation.
... In this way, the continuous "trajectories" calculated by Philippidis, Dewdney and Hiley [13] appear as energy flow lines analogous to those shown in Berry [11,12] for optical flow lines. Furthermore, it can be shown that the quantum "trajectories" are an average of an ensemble of Feynman paths [50]. This then provides us with an intuitive image of the Heisenberg equation of motion (Equation (11)). ...
What is striking about de Broglie’s foundational work on wave–particle dualism is the role played by pseudo-Riemannian geometry in his early thinking. While exploring a fully covariant description of the Klein–Gordon equation, he was led to the revolutionary idea that a variable rest mass was essential. DeWitt later explained that in order to obtain a covariant quantum Hamiltonian, one must supplement the classical Hamiltonian with an additional energy ℏ2Q from which the quantum potential emerges, a potential that Berry has recently shown also arises in classical wave optics. In this paper, we show how these ideas emerge from an essentially geometric structure in which the information normally carried by the wave function is contained within the algebraic description of the geometry itself, within an element of a minimal left ideal. We establish the fundamental importance of conformal symmetry, in which rescaling of the rest mass plays a vital role. Thus, we have the basis for a radically new theory of quantum phenomena based on the process of mass-energy flow.
... The connection between p W, j (x, t ) and v j (x, t ) is approached elsewhere [52][53][54][55][56][57][58][59], while more attention has recently been paid to the meaning of u j (x, t ) [60][61][62][63][64][65][66][67][68]. We distinguish in this paper three types of expectation values: (i) â , with the "hat," for the operatorâ; (ii) a W , with subscript "W," for 033168-3 the weak value a W ; and (iii) a for the value a obtained by postprocessing the weak value. ...
According to both Bohmian and stochastic quantum mechanics, the standard quantum mechanical kinetic energy can be understood as consisting of two hidden-variable components. One component is associated with the current (or Bohmian) velocity, while the other is associated with the osmotic velocity (or quantum potential), and they are identified with the phase and the amplitude, respectively, of the wave function. These two components are experimentally accessible through the real and imaginary parts of the weak value of the momentum postselected in position. In this paper, a kinetic energy equipartition is presented as a signature of quantum thermalization in closed systems. This means that the expectation value of the standard kinetic energy is equally shared between the expectation values of the squares of these two hidden-variable components. Such components cannot be reached from expectation values linked to typical Hermitian operators. To illustrate these concepts, numerical results for the nonequilibrium dynamics of a few-particle harmonic trap under random disorder are presented. Furthermore, the advantages of using the center-of-mass frame of reference for dealing with systems containing many indistinguishable particles are also discussed.
... In other words, while p W,j (x, t) is linked to the hermitian operatorp j , no hermitian operators can be linked to v j (x, t) and u j (x, t). The connection among p W,j (x, t) and v j (x, t) is approached elsewhere [49][50][51][52][53][54][55][56], while more attention has recently been paid to the meaning of u j (x, t) [57][58][59][60][61][62][63][64][65]. We distinguish in the paper three types of expectation values: (i) â with "hat" for the operatorâ; (ii) a W with subindex "W" for the weak value a W ; (iii) a for the value a obtained by post-processing the weak value. ...
The Orthodox kinetic energy has, in fact, two hidden-variable components: one linked to the current (or Bohmian) velocity, and another linked to the osmotic velocity (or quantum potential), and which are respectively identified with phase and amplitude of the wavefunction. Inspired by Bohmian and Stochastic quantum mechanics, we address what happens to each of these two velocity components when the Orthodox kinetic energy thermalizes in closed systems, and how the pertinent weak values yield experimental information about them. We show that, after thermalization, the expectation values of both the (squared) current and osmotic velocities approach the same stationary value, that is, each of the Bohmian kinetic and quantum potential energies approaches half of the Orthodox kinetic energy. Such a `kinetic energy equipartition' is a novel signature of quantum thermalization that can empirically be tested in the laboratory, following a well-defined operational protocol as given by the expectation values of (squared) real and imaginary parts of the local-in-position weak value of the momentum, which are respectively related to the current and osmotic velocities. Thus, the kinetic energy equipartion presented here is independent on any ontological status given to these hidden variables, and it could be used as a novel element to characterize quantum thermalization in the laboratory, beyond the traditional use of expectation values linked to Hermitian operators. Numerical results for the nonequilibrium dynamics of a few-particle harmonic trap under random disorder are presented as illustration. And the advantages in using the center-of-mass frame of reference for dealing with systems with many indistinguishable particles are also discussed.
... So the idea that a single quantum object moves along a trajectory ought to be seen as a hypothesis which has not been empirically verified. However, by making use of measurements of weak values it is possible to measure average trajectories (see Flack and Hiley 2018). ...
Consciousness and quantum mechanics are two mysteries in our times. A careful and thorough examination of possible connections between them may help unravel these two mysteries. On the one hand, an analysis of the conscious mind and psychophysical connection seems indispensable in understanding quantum mechanics and solving the notorious measurement problem. On the other hand, it seems that in the end quantum mechanics, the most fundamental theory of the physical world, will be relevant to understanding consciousness and even solving the mind-body problem when assuming a naturalist view. This book is the first volume which provides a comprehensive review and thorough analysis of intriguing conjectures about the connection between consciousness and quantum mechanics. Written by leading experts in this research field, this book will be of value to students and researchers working on the foundations of quantum mechanics and philosophy of mind.
... In this work, we consider momentum conditionally averaged on positions following Moyal's [26] and Sonego's [38] arguments. Also, it was previously realized by many researchers thatp are the real, measurable part of the weak value of the momenta [32,76,77]. Therefore, the conditionally averaged momenta act as an effective momenta of a system. ...
... It being equal to the conditionally averaged momentum also shows that it is an effective object. This was also realized in certain other studies [38] and its relation to the weak measurements of Aharonov et al. [72] has been established before [32,76,77]. ...
The hydrodynamic interpretation of quantum mechanics treats a system of particles in an effective manner which allows one to study the system in a statistical fashion. In this work, we investigate squeezed coherent states within the hydrodynamic interpretation. The Hamiltonian operator in question is time dependent, n-dimensional and in quadratic order. We start by deriving a phase space Wigner probability distribution and an associated equilibrium entropy for the squeezed coherent states. Then, we decompose the joint phase space distribution into two portions: A marginal position distribution and a momentum distribution that is conditioned on the post-selection of positions. Our conditionally averaged momenta are shown to be equal to the Bohm’s momenta whose connection to the weak measurements is already known. We also keep track of the corresponding classical system evolution by identifying shear, magnification and rotation components of the symplectic phase space dynamics. This allows us to pinpoint which portion of the underlying classical motion appears in which quantum statistical concept. We show that our probability distributions satisfy the Fokker–Planck equations exactly and they can be used to decompose the equilibrium entropy into the missing information in positions and in momenta as in the Sackur–Tetrode entropy of the classical kinetic theory. Eventually, we define a quantum pressure, a quantum temperature and a quantum internal energy which are related to each other in the same fashion as in the classical kinetic theory. We show that the quantum potential incorporates the kinetic part of the internal energy and the fluctuations around it. This allows us to suggest a quantum conditional virial relation. In the end, we show that the kinetic internal energy is linked to the fractional Fourier transformer part of the underlying classical dynamics similar to the case where the energy of a quantum oscillator is linked to its Maslov index.
... So the idea that a single quantum object moves along a trajectory ought to be seen as a hypothesis which has not been empirically verified. However, by making use of measurements of weak values it is possible to measure average trajectories (see Flack and Hiley 2018). ...
Researchers have suggested since the early days of quantum theory that there are strong analogies between quantum phenomena and mental phenomena and these have developed into a vibrant new field of quantum cognition during recent decades. After revisiting some early analogies by Niels Bohr and David Bohm, this paper focuses upon Bohm and Hiley’s ontological interpretation of quantum theory which suggests further analogies between quantum phenomena and biological and psychological phenomena, including the proposal that the human brain operates in some ways like a quantum measuring apparatus. After discussing these analogies I will also consider, from a quantum perspective, Hintikka’s suggestion that Kant’s notion of things in themselves can be better understood by making an analogy between our knowledge-seeking activities and an elaborate measuring apparatus.
... as a sum over classical paths [7]. In more complex situations [8][9][10][11][12][13], several Feynman paths compatible with pre-and post-selection can interfere at the position of a weakly coupled probe, and may even result in a vanishing weak value (e.g., [14]). In a generic situation, Feynman paths interfere as the system evolves, so that reconstructing a weak trajectory from observed pointers is expected to be a difficult task. ...
... Assume a probe placed at r a = (x a , z a ) is weakly coupled to the system with the interaction Hamiltonian given by Eq. (11). The pre-selected state is given by Eq. (15) ...
... A weak measurement of the momentum followed immediately by a projective position measurement leads to the velocity field associated with the Schrödinger current density. From there it is possible to reconstruct Bohmian trajectories [15,24], though as discussed elsewhere [11,25] Bohmian trajectories are not the observed quantities -contrary to the paths measured in this work that can be measured by following the wavefunction in space (through the position r a of the probes) and time (by turning on and off the interaction at the desired time t b ). Indeed, in the case examined here, a weak interaction is not followed immediately by post-selection (that would terminate the system's evolution), but it is followed by successive weak interactions, in order to sample the intermediate dynamics before detecting the particle on the screen. ...
The interference pattern produced by a quantum particle in Young's double-slit setup is attributed to the particle's wavefunction having gone through both slits. In the path integral formulation, this interference involves a superposition of paths, going through either slit, linking the source to the detection point. We show how these paths superpositions can in principle be observed by implementing a series of minimally-perturbing weak measurements between the slits and the detection plane. We further propose a simplified protocol in order to observe these "weak trajectories" with single photons.
... When a weak measurement of transverse momentum can be made with sufficient resolution at one longitudinal position, the measurements at different z positions would need to be made in order to reconstruct the momentum flow lines. Theorists collaborating with this project are currently developing ideas of how the experiment could be modified to test the 'quantum potential' [63] which arises in Bohm's treatment of the Schrödinger equation, along with the phase representing the atom's local momentum. ...
This thesis describes the construction of an atomic matter-wave interferometer, combined with a spin-state interferometer, used to weakly measure the average transverse momentum of the atoms. A velocity-tuneable cold atomic beam was constructed and characterised. By applying radiation pressure to a magneto-optical trap, the beam's velocity was selected between 1-52 ms-1. The beam was used to produce interference fringes in a matter-wave interferometer which consisted of a mutli-slit Si3Ni4 grating and a planar atom detector placed below the grating. The interference pattern was used to measure the average beam velocity and the Van der Waals coefficient between the atoms and the grating. This was the first instance of such a measurement. Below the grating a longitudinal Stern-Gerlach interferometer was constructed. The phase shift of the atom's spin, due to the Zeeman effect from the interferometer's magnetic field, was measured. The phase shift provided a measurement of the interferometer's magnetic field in the μT range. An experiment combining the two interferometers to weakly measure the atom's transverse momentum is described and modelled.
... Surprisingly, very few works have employed weak values in a path integral context. Even then, the interest was restricted to the weak measurement of Feynman paths in semiclassical systems [10][11][12], to WV of specific operators [13][14][15], or as a way to probe virtual histories [16]. ...
... FIG. 2. A quantum particle propagating in an interferometer and postselected at D displays discontinuous trajectories when observed with weak measurements [the weakly coupled probes depicted in black remain unaffected by the interaction with the particle, while the probes depicted in orange see their pointers shift after postselection; see Eq. (14)]. The Feynman paths of the particle, represented as pencils of trajectories along the arms, are continuous and propagate the interactions with the probes up to D, where superposition with the postselected state yield zero WV at E and F even when the wave function on these segments does not vanish (see text for details). ...
We connect the weak measurements framework to the path integral formulation of quantum mechanics. We show how Feynman propagators can in principle be experimentally inferred from weak value measurements. We also obtain expressions for weak values parsing unambiguously the quantum and the classical aspects of weak couplings between a system and a probe. These expressions are shown to be useful in quantum-chaos-related studies (an illustration involving quantum scars is given), and also in solving current weak-value-related controversies (we discuss the existence of discontinuous trajectories in interferometers and the issue of anomalous weak values in the classical limit).