We introduce a trajectory-based decomposition of quantum variances within Bohmian mechanics. By extending the weak actual value to a field on configuration space, we prove, under strong regularity conditions for stationary bound states, that the standard quantum variance splits into two non-negative terms: the ensemble variance of weak actual value and a quantum termarising from phase-amplitude coupling. For momentum, linking variance-level fluctuations to the average quantum potential. The decomposition fails to provide a physical interpretation for spin, reinforcing the Bohmian tenet that only position is fundamental. The work provides a formal tool for analyzing quantum fluctuations and clarifies the interpretative limits of such a trajectory-based approach.
We introduce a trajectory-based decomposition of quantum variances within Bohmian mechanics. By extending the weak actual value to a field on configuration space, we prove, under strong regularity conditions for stationary bound states, that the standard quantum variance splits into two non-negative terms: the ensemble variance of weak actual value and a quantum termarising from phase-amplitude coupling. For momentum, linking variance-level fluctuations to the average quantum potential. The decomposition fails to provide a physical interpretation for spin, reinforcing the Bohmian tenet that only position is fundamental. The work provides a formal tool for analyzing quantum fluctuations and clarifies the interpretative limits of such a trajectory-based approach.
The ontological aspect of Bohmian mechanics, as a hidden-variable theory that provides us with an objective description of a quantum world without observers, is widely known. Yet its practicality is getting more and more acceptance and relevance, for it has proven to be an efficient and useful resource to tackle, explore, describe and explain such phenomena. This practical aspect emerges precisely when the pragmatic application of the formalism prevails over any other interpretational question, still a matter of debate and controversy. In this regard, the purpose here is to show and discuss how Bohmian mechanics emphasizes in a natural manner a series of dynamical features difficult to find out through other quantum approaches. This arises from the fact that Bohmian mechanics allows us to establish a direct link between the dynamics exhibited by quantum systems and the local variations of the quantum phase associated with their state. To illustrate these facts, simple models of two physically insightful quantum phenomena have been chosen, namely, the dispersion of a free Gaussian wave packet and Young-type two-slit interference. As it is shown, the outcomes from their analysis render a novel, alternative understanding of the dynamics displayed by these quantum phenomena in terms of the underlying local velocity field that connects the probability density with the quantum flux. This field, nothing but the so-called guidance condition in standard Bohmian mechanics, thus acquires a prominent role to understand quantum dynamics, as the mechanism responsible for such dynamics. This goes beyond the passive role typically assigned to this field in Bohmian mechanics, where traditionally trajectories and quantum potentials have received more attention instead.
We systematically analyze Holland's local expectation values within Bohmian mechanics, referring to them as weak actual values to emphasize their connection with weak measurement theory. We derive the exact time evolution equation for these quantities along a Bohmian trajectory and formally establish their correspondence with the real part of the weak value under position postselection. The explanatory power of this framework is demonstrated by revisiting a recent waveguide experiment: we show that the measured quantity corresponds to the imaginary part of the momentum weak value, which reflects the spatial decay of the wave field, not the particle velocity. This cleanly clarifies the distinct physical roles of the real and imaginary parts of weak values.
Bohmian mechanics (BM) is a popular interpretation of quantum mechanics (QM) in which particles have real positions. The velocity of a point x in configuration space is defined as the standard probability current j ( x ) divided by the probability density P ( x ). However, this 'standard' j is in fact only one of infinitely many that transform correctly and satisfy . In this paper, I show that a particular j is singled out if one requires that j be determined experimentally as a weak value , using a technique that would make sense to a physicist with no knowledge of QM. This 'naively observable' j seems the most natural way to define j operationally. Moreover, I show that this operationally defined j equals the standard j , so, assuming , one obtains the dynamics of BM. It follows that the possible Bohmian paths are naively observable from a large enough ensemble. Furthermore, this justification for the Bohmian law of motion singles out x as the hidden variable, because (for example) the analogously defined momentum current is in general incompatible with the evolution of the momentum distribution. Finally I discuss how, in this setting, the usual quantum probabilities can be motivated from a Bayesian standpoint, via the principle of indifference.
We analyze finite-sample statistics of Bohmian trajectories for single spinless and spin-1/2 particles. Equivariance ensures agreement with ∣ψ∣2 in the quantum equilibrium limit, yet experiments and simulations necessarily use finite ensembles. We show that in regular flows (e.g., wavepackets or low-mode superpositions of eigenstates of harmonic oscillators) sample means and/or variances over modest N are consistent with Born-rule moments. In contrast, degenerate superpositions of 3D oscillators with nodal barriers and chaotic Bohmian dynamics exhibit sensitive dependence on initial conditions and complex flow partitioning, which can yield noticeable finite-sample deviations in the mean and variance. For the spin-1/2 particle, both convective and Pauli currents conserve ∣ψ∣2, but they are associated with different velocity fields and thus might yield different finite-sample trajectory statistics. These findings calibrate the interpretation of trajectory-based uncertainty and provide practical guidance for numerical Bohmian simulations of spin and transport, without challenging the equivalence to orthodox quantum mechanics in the quantum equilibrium ensemble.
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.
Bohmian mechanics, also referred to as the de Broglie-Bohm pilot-wave theory, represents a deterministic and nonlocal interpretation of quantum mechanics. Central to this framework is a description of quantum motion in terms of particle trajectories, whose reconciliation with relativity remains a fundamental challenge. Although relativistic guidance laws have been proposed, their predicted dynamics has remained experimentally untested. Here we use weak measurements in a single-photon interferometer to jointly reconstruct the photon energy and momentum weak values and the associated relativistic velocity field. The reconstructed average trajectories agree with theoretical predictions. Near destructive interference, the flow exhibits local reconstructed velocities exceeding the speed of light and negative values of the effective squared-mass parameter. Our results establish an experimental basis for investigating the physical meaning of quantum trajectories and their role in describing motion in relativistic spacetime.