Search Results (44)

We study in detail the critical points of Bohmian flow, both in the inertial frame of reference (Y-points) and in the frames centered at the moving nodal points of the guiding wavefunction (X-points), and analyze their role in the onset of chaos in a system of two entangled qubits. We find the distances between these critical points and a moving Bohmian particle at varying levels of entanglement, with particular emphasis on the times at which chaos arises. Then, we find why some trajectories are ordered, without any chaos. Finally, we examine numerically how the Lyapunov Characteristic Number ($LCN$) depends on the degree of quantum entanglement. Our results indicate that increasing entanglement reduces the convergence time of the finite-time $LCN$ of the chaotic trajectories toward its final positive value. Full article

The POVM theorem is a central result in Bohmian mechanics, grounding the measurement formalism of standard quantum mechanics in a statistical analysis based on the quantum equilibrium hypothesis (the Born rule for Bohmian particle positions). It states that the outcome statistics of an experiment are described by a positive operator-valued measure (POVM) acting on the Hilbert space of the measured system. In light of recent debates about the scope and status of this result, we provide a systematic presentation of the POVM theorem and its underlying assumptions with a focus on their conceptual foundations and physical justifications. We conclude with a brief discussion of the scope of the POVM theorem—especially the sense in which it does (and does not) place limits on what is “measurable” in Bohmian mechanics. Full article

In this work, we present a new theoretical approach to interpreting and reproducing quantum mechanics using trajectory-guided wavelets. Inspired by the 1925 work of Louis de Broglie, we demonstrate that pulses composed of a difference between a delayed wave and an advanced wave (known as antisymmetric waves) are capable of following quantum trajectories predicted by the de Broglie–Bohm theory (also known as Bohmian mechanics). Our theory reproduces the main results of orthodox quantum mechanics and unlike Bohmian theory, is local in the Bell sense. We show that this is linked to the superdeterminism and past–future (anti)symmetry of our theory. Full article

In the present paper, we study both classical and quantum Hénon–Heiles systems. In particular, we make a comparison between the classical and quantum trajectories of integrable and nonintegrable Hénon–Heiles Hamiltonians. From a classical standpoint, we study both theoretically and numerically the form of invariant curves in the Poincaré surfaces of section for several values of the coupling parameter in the integrable case and compare them with those in the nonintegrable case. Then, we examine the corresponding Bohmian trajectories, and we find that they are chaotic in both cases, but with chaos emerging at different times. Full article

In this work, we analyze recent proposals by Das and Dürr (DD) to measure the arrival time distributions of quantum particles within the framework of de Broglie Bohm theory (or Bohmian mechanics). We also analyze the criticisms made by Goldstein Tumulka and Zanghì (GTZ) of these same proposals, and show that each protagonist is both right and wrong. In detail, we show that DD’s predictions are indeed measurable in principle, but that they will not lead to violations of the no-signalling theorem used in Bell’s theorem, in contradiction with some of Das and Maudlin’s hopes. Full article

A precise interpretation of the universe wave function is forbidden in the spirit of the Copenhagen School since a precise notion of measure operation cannot be satisfactorily defined. Here, we propose a Bohmian interpretation of the isotropic universe quantum dynamics, in which the Hamilton–Jacobi equation is restated by including quantum corrections, which lead to a classical trajectory containing effects of order ${\hslash }^2$. This solution is then used to determine the spectrum of gauge-invariant quantum fluctuations living on the obtained background model. The analysis is performed adopting the wave function approach to describe the fluctuation dynamics, which gives a time-dependent harmonic oscillator for each Fourier mode and whose frequency is affected by the ${\hslash }^2$ corrections. The properties of the emerging spectrum are discussed, outlining the modification induced with respect to the scale-invariant result, and the hierarchy of the spectral index running is discussed. Full article

Two distinct measures of information, connected respectively to the amplitude and phase of the wave function of a particle, are proposed. There are relations between the time derivatives of these two measures and their gradients on the configuration space, which are equivalent to the wave equation. The information related to the amplitude measures the strength of the potential coupling of the particle (which is itself aspatial) with each volume of its configuration space, i.e., its tendency to participate in an interaction localized in a region of ordinary physical space corresponding to that volume. The information connected to the phase is that required to obtain the time evolution of the particle as a persistent entity starting from a random succession of bits. It can be considered as the information provided by conservation principles. The meaning of the so-called “quantum potential” in this context is briefly discussed. Full article

Starting from the dynamics of a bouncing ball in classical and quantum regime, we have suggested in a previous paper to add an arbitrary function of time to the standard expression of the probability current in quantum mechanics. In this paper, we suggest a way to determine this function: imposing a suitable normalization condition. The application of our proposal to the case of the harmonic oscillator is discussed. Full article

In this work, we review and extend a version of the old attempt made by Louis de Broglie for interpreting quantum mechanics in realistic terms, namely, the double solution. In this theory, quantum particles are localized waves, i.e., solitons, that are solutions of relativistic nonlinear field equations. The theory that we present here is the natural extension of this old work and relies on a strong time-symmetry requiring the presence of advanced and retarded waves converging on particles. Using this method, we are able to justify wave–particle duality and to explain the violations of Bell’s inequalities. Moreover, the theory recovers the predictions of the pilot-wave theory of de Broglie and Bohm, often known as Bohmian mechanics. As a direct consequence, we reinterpret the nonlocal action-at-a-distance in the pilot-wave theory. In the double solution developed here, there is fundamentally no action-at-a-distance but the theory requires a form of superdeterminism driven by time-symmetry. Full article

We study order, chaos and ergodicity in the Bohmian trajectories of a 2D quantum harmonic oscillator. We first present all the possible types (chaotic, ordered) of Bohmian trajectories in wavefunctions made of superpositions of two and three energy eigenstates of the oscillator. There is no chaos in the case of two terms and in some cases of three terms. Then, we show the different geometries of nodal points in bipartite Bohmian systems of entangled qubits. Finally, we study multinodal wavefunctions and find that a large number of nodal points does not always imply the dominance of chaos. We show that, in some cases, the Born distribution is dominated by ordered trajectories, something that has a significant impact on the accessibility of Born’s rule $P={\left|\mathrm{\Psi }\right|}^2$ by initial distributions of Bohmian particles with $P_0\ne {\left|{\mathrm{\Psi }}_0\right|}^2$. Full article

We study the role of unstable points in the Bohmian flow of a 2d system composed of two non-interacting harmonic oscillators. In particular, we study the unstable points in the inertial frame of reference as well as in the frame of reference of the moving nodal points, in cases with 1, 2 and multiple nodal points. Then, we find the contributions of the ordered and chaotic trajectories in the Born distribution, and when the latter is accessible by an initial particle distribution which does not satisfy Born’s rule. Full article

We consider the concept of velocity fields, taken from Bohmian mechanics, to investigate the dynamical effects of entanglement in bipartite realizations of Young’s two-slit experiment. In particular, by comparing the behavior exhibited by factorizable two-slit states (cat-type state analogs in the position representation) with the dynamics exhibited by a continuous-variable Bell-type maximally entangled state, we find that, while the velocity fields associated with each particle in the separable scenario are well-defined and act separately on each subspace, in the entangled case there is a strong deformation in the total space that prevents this behavior. Consequently, the trajectories for each subsystem are not constrained any longer to remain confined within the corresponding subspace; rather, they exhibit seemingly wandering behavior across the total space. In this way, within the subspace associated with each particle (that is, when we trace over the other subsystem), not only interference features are washed out, but also the so-called Bohmian non-crossing rule (i.e., particle trajectories are allowed to get across the same point at the same time). Full article

We have implemented quantum modeling mainly based on Bohmian mechanics to study time series that contain strong coupling between their events. Compared to time series with normal densities, such time series are associated with rare events. Hence, employing Gaussian statistics drastically underestimates the occurrence of their rare events. The central objective of this study was to investigate the effects of rare events in the probability densities of time series from the point of view of quantum measurements. For this purpose, we first model the non-Gaussian behavior of time series using the multifractal random walk (MRW) approach. Then, we examine the role of the key parameter of MRW, $\lambda$, which controls the degree of non-Gaussianity, in quantum potentials derived for time series. Our Bohmian quantum analysis shows that the derived potential takes some negative values in high frequencies (its mean values), then substantially increases, and the value drops again for rare events. Thus, rare events can generate a potential barrier in the high-frequency region of the quantum potential, and the effect of such a barrier becomes prominent when the system transverses it. Finally, as an example of applying the quantum potential beyond the microscopic world, we compute quantum potentials for the S&P financial market time series to verify the presence of rare events in the non-Gaussian densities and demonstrate deviation from the Gaussian case. Full article

Shortly after Schrödinger’s wave mechanics in terms of complex wave functions was published, Madelung formulated this theory in terms of two real hydrodynamic-like equations. This version is also the formal basis of Bohmian mechanics, albeit with a different ontological interpretation. A point of criticism raised by Pauli against Bohmian mechanics is its missing symmetry between position and momentum that is present in classical phase space as well as in the quantum mechanical position and momentum representations. Both Madelung’s quantum hydrodynamics formulation and Bohmian mechanics are usually expressed only in position space. Recently, with the use of complex quantities, we were able to provide a hydrodynamic formulation also in momentum space. In this paper, we extend this formalism to include dissipative systems with broken time-reversal symmetry. In classical Hamiltonian mechanics and conventional quantum mechanics, closed systems with reversible time-evolution are usually considered. Extending the discussion to include open systems with dissipation, another form of symmetry is broken, that under time-reversal. There are different ways of describing such systems; for instance, Langevin and Fokker–Planck-type equations are commonly used in classical physics. We now investigate how these aspects can be incorporated into our complex hydrodynamic description and what modifications occur in the corresponding equations, not only in position, but particularly in momentum space. Full article

Within the framework of quantum mechanics, the wave function squared describes the probability density of particles. In this article, another description of the wave function is given which embeds quantum mechanics into the traditional fields of physics, thus making new interpretations dispensable. The new concept is based on the idea that each microscopic particle with non-vanishing rest mass is accompanied by a matter wave, which is formed by adjusting the phases of the vacuum fluctuations in the vicinity of the vibrating particle. The vibrations of the particle and wave are phase-coupled. Particles move on continuous approximately classical trajectories. By the phase coupling mechanism, the particle transfers the information on its kinematics and thus also on the external potential to the wave. The space dependence of the escorting wave turns out to be equal to the wave function. The new concept fundamentally differs from the pilot wave concept of Bohmian mechanics. Full article