Bohmian Research Articles

Local Theories and Bohmian Mechanics

One of the last develoonents in the research for extending the scope of the quantum theory is the recently appearing work on the Bohmian Mechanics. The motivation for an extension is provided by the conclusions of the EPR paradoxon and the famous alternative concerning the physical reality. Discussed are some properties of Bohmian Mechanics concerning the self-consistency of the theory.

The effect of time-dilation on Bell experiments in the retrocausal Brans model

The possibility of using retrocausality to obtain a fundamentally relativistic account of the Bell correlations has gained increasing recognition in recent years. It is not known, however, the extent to which these models can make use of their relativistic properties to account for relativistic effects on entangled systems. We consider here a hypothetical relativistic Bell experiment, where one of the wings experiences time-dilation effects. We show that the retrocausal Brans model ( Found. Phys. , 49 (2), 2019) can be easily generalized to analyse this experiment, and that it predicts less separation of eigenpackets in the wing experiencing the time-dilation. This causes the particle distribution patterns on the photographic plates to differ between the wings—an experimentally testable prediction of the model. We discuss the difficulties faced by other hidden variable models in describing this experiment, and their natural resolution in our model due to its relativistic properties. In particular, we discuss how a ψ -epistemic interpretation may resolve several difficulties encountered in relativistic generalizations of de Broglie–Bohm theory and objective state reduction models. Lastly, we argue that it is not clear at present, due to technical difficulties, if our prediction is reproduced by quantum field theory. We conclude that if it is, then the retrocausal Brans model predicts the same result with great simplicity in comparison. If not, the model can be experimentally tested.

Quantum-Phase-Field: From de Broglie–Bohm Double-Solution Program to Doublon Networks

Abstract Different forms of linear and non-linear field equations, so-called ‘phase-field’ equations, are studied in relation to the de Broglie–Bohm double-solution program. This defines a framework in which elementary particles are described by localised non-linear wave solutions moving by the guidance of a pilot wave, defined by the solution of a Schrödinger-type equation. First, we consider the phase-field order parameter as the phase for the linear pilot wave, second as the pilot wave itself, and third as a moving soliton interpreted as a massive particle. In the last case, we introduce the equation for a superwave, the amplitude of which can be considered as a particle moving in accordance to the de Broglie–Bohm theory. Lax pairs for the coupled problems are constructed in order to discover possible non-linear equations that can describe the moving particle and to propose a framework for investigating coupled solutions. Finally, doublons in 1 + 1 dimensions are constructed as self-similar solutions of a non-linear phase-field equation forming a finite space object. Vacuum quantum oscillations within the doublon determine the evolution of the coupled system. Applying a conservation constraint and using general symmetry considerations, the doublons are arranged as a network in 1 + 1 + 2 dimensions, where nodes are interpreted as elementary particles. A canonical procedure is proposed to treat charge and electromagnetic exchange.

Randomness? What Randomness?

This is a review of the issue of randomness in quantum mechanics, with special emphasis on its ambiguity; for example, randomness has different antipodal relationships to determinism, computability, and compressibility. Following a (Wittgensteinian) philosophical discussion of randomness in general, I argue that deterministic interpretations of quantum mechanics (like Bohmian mechanics or ’t Hooft’s Cellular Automaton interpretation) are strictly speaking incompatible with the Born rule. I also stress the role of outliers, i.e. measurement outcomes that are not 1-random. Although these occur with low (or even zero) probability, their very existence implies that the no-signaling principle used in proofs of randomness of outcomes of quantum-mechanical measurements (and of the safety of quantum cryptography) should be reinterpreted statistically, like the second law of thermodynamics. In three appendices I discuss the Born rule and its status in both single and repeated experiments, review the notion of 1-randomness (or algorithmic randomness) that in various guises was investigated by Kolmogorov and others and treat Bell’s (Physics 1:195–200, 1964) Theorem and the Free Will Theorem with their implications for randomness.

Nano-refraction Analysis for Advanced Materials: A Theoretical Roadmap for Quantum Computers

We are now in the beginning of quantum supremacy to build Quantum Computers (Q.C.). So, a theoretical roadmap is required. It seems experimental works in Q.C. is ahead, in comparison with theory. In this study, Nano-Refraction (N.R.), as basic concept in developing Quantum Photonic Computers (Q.P.C.), is defined and discussed. N.R., is quite different from classical macroscopic refraction. It plays major role in photon deflection, when it travels in few atomic layers from the boundary between two transparent materials. Specifically, when we’re looking for a theoretically intuitive explanation of how advanced materials work in atomic scales in atto-second regime. To explain N.R., mathematically, Quantum Electrodynamics (QED) may help. But QED, only can describe how N.R. occurs in Hilbert space and does not explain why it happens in Real space? Why attosecond optical pulses are squeezed in kilometers long optical fibers coiled in first reported development of Q.C. in 2020. Also, “Duality” has no reply in this area. For answering why?, in this paper, we’ll use Quantum Photonic (Q.P.) theoretical analysis. Q.P., is based on Bohmian mechanics with intuition physics belief and “Causality”. Bohm theory in shadow of Quantum Mechanics (Q.M.) has been mostly ignored and even boycotted during last 70 years. Q.P. corpuscular viewpoint of light, estimated both physically and mathematically with enough precision, that flight route of photons at first few molecular surfaces in boundary region is not refracted suddenly. As can be observed macroscopically in Snell’s refraction equation. But instead, N.R. is happened gradually in attosecond regime (figure 2). Finally, it asymptotes to macroscopic refraction, in large space-time scales according to “correspondence” principle. Errors in our theoretical analysis using Montecarlo time domain simulation, compared with experiments, always are much less than or at most five percent.

“It from Bit” and Quantum Mechanics

John Archibald Wheeler is one of the staunchest advocates of the idea that information is more fundamental than anything else in physics. In this paper, we examine the status of this idea summarized in Wheeler’s slogan ‘it from bit’ in the context of Bohmian Mechanics and spontaneous collapse models. We will argue that any question about the status of ‘it from bit’ crucially depends on the particular interpretation of these theories one favors.

Bohmian mechanics of Klein–Gordon equation via quantum metric and mass

The causal stochastic interpretation of relativistic quantum mechanics has the problems of superluminal velocities, motion backward in time and the incorrect non-relativistic limit. In this paper, according to the original ideas of de Broglie, Bohm and Takabayasi, we introduce simultaneously a quantum mass and a quantum metric of a curved spacetime to obtain a correct relativistic theory free of mentioned problems.

Indeterminism in Physics, Classical Chaos and Bohmian Mechanics: Are Real Numbers Really Real?

It is usual to identify initial conditions of classical dynamical systems with mathematical real numbers. However, almost all real numbers contain an infinite amount of information. I argue that a finite volume of space can’t contain more than a finite amount of information, hence that the mathematical real numbers are not physically relevant. Moreover, a better terminology for the so-called real numbers is “random numbers”, as their series of bits are truly random. I propose an alternative classical mechanics, which is empirically equivalent to classical mechanics, but uses only finite-information numbers. This alternative classical mechanics is non-deterministic, despite the use of deterministic equations, in a way similar to quantum theory. Interestingly, both alternative classical mechanics and quantum theories can be supplemented by additional variables in such a way that the supplemented theory is deterministic. Most physicists straightforwardly supplement classical theory with real numbers to which they attribute physical existence, while most physicists reject Bohmian mechanics as supplemented quantum theory, arguing that Bohmian positions have no physical reality.

Lorentz-Invariant, Retrocausal, and Deterministic Hidden Variables

We review several no-go theorems attributed to Gisin and Hardy, Conway and Kochen purporting the impossibility of Lorentz-invariant deterministic hidden-variable model for explaining quantum nonlocality. Those theorems claim that the only known solution to escape the conclusions is either to accept a preferred reference frame or to abandon the hidden-variable program altogether. Here we present a different alternative based on a foliation dependent framework adapted to deterministic hidden variables. We analyse the impact of such an approach on Bohmian mechanics and show that retrocausation (that is future influencing the past) necessarily comes out without time-loop paradox.

Gravitational lensing by wormholes supported by electromagnetic, scalar, and quantum effects

Wormholes are one of the most interesting topological features in spacetime, offering a rat run between two vastly separated regions of the universe. In this paper, we study the deflection angle of light by wormholes, which are supported by electric charge, magnetic charge, and scalar fields in the weak field limit approximation. To this end, we apply new geometric methods -- the Gauss-Bonnet theorem and the optical geometry -- to compute the deflection angles. We also verify our findings by using the well-known geodesics method. There exists a similarity between the charge and the quantum corrections on a black hole solution, which has been recently discussed in the context of the relativistic Bohmian quantum mechanics. By replacing classical geodesics with Bohmian trajectories, we introduce a new wormhole solution, whose having matter sources and anisotropic pressure supported by Bohmian quantum effects. The problem of fulfillment of the energy conditions of the Morris-Thorne traversable wormhole is also discussed.

On non-linear Schrödinger equations for open quantum systems

Recently two generalized nonlinear Schr\"{o}dinger equations have been proposed by Chavanis [Eur. Phys. J. Plus 132 (2017) 286] by applying Nottale's theory of scale relativity relying on a fractal space-time to describe dissipation in quantum systems. Several existing nonlinear equations are then derived and discussed in this context leading to a continuity equation with an extra source/sink term which violates Ehrenfest theorem. An extension to describe stochastic dynamics is also carried out by including thermal fluctuations or noise of the environment. These two generalized nonlinear equations are analyzed within the Bohmian mechanics framework to describe the corresponding dissipative and stochastic dynamics in terms of quantum trajectories. Several applications of this second generalized equation which can be considered as a generalized Kostin equation have been carried out. The first application consists of the study of the position-momentum uncertainty principle in a dissiaptive dynamics. After, the so-called Brownian-Bohmian motion is investigated by calculating classical and quantum diffusion coefficients. And as a third example, transmission through a transient (time dependent) parabolic repeller is studied where the interesting phenomenon of early arrival is observed even in the stochastic dynamics although the magnitude of early arrival is reduced by friction.

Violation of the Bell’s type inequalities as a local expression of incompatibility

By filtering out the philosophic component we can be said that the EPR-paper was directed against the straightforward interpretation of the Heisenberg’s uncertainty principle or more generally the Bohr’s complementarity principle. The latter expresses contextuality of quantum measurements: dependence of measurement’s output on the complete experimental arrangement. However, Bell restructured the EPR-argument against complementarity to justify nonlocal theories with hidden variables of the Bohmian mechanics’ type. Then this Bell’s kind of nonlocality - subquantum nonlocality - was lifted to the level of quantum theory - up to the terminology “quantum nonlocality”. The aim of this short note is to explain that Bell’s test is simply a special test of local incompatibility of quantum observables, similar to interference experiments, e.g., the two-slit experiment.

Quantum Motion on Shape Space and the Gauge Dependent Emergence of Dynamics and Probability in Absolute Space and Time

Relational formulations of classical mechanics and gravity have been developed by Julian Barbour and collaborators. Crucial to these formulations is the notion of shape space. We indicate here that the metric structure of shape space allows one to straightforwardly define a quantum motion, a Bohmian mechanics, on shape space. We show how this motion gives rise to the more or less familiar theory in absolute space and time. We find that free motion on shape space, when lifted to configuration space, becomes an interacting theory. Many different lifts are possible corresponding in fact to different choices of gauges. Taking the laws of Bohmian mechanics on shape space as physically fundamental, we show how the theory can be statistically analyzed by using conditional wave functions, for subsystems of the universe, represented in terms of absolute space and time.

Schrödinger’s Paradox and Proofs of Nonlocality Using Only Perfect Correlations

We discuss proofs of nonlocality based on a generalization by Erwin Schr\"odinger of the argument of Einstein, Podolsky and Rosen. These proofs do not appeal in any way to Bell's inequalities. Indeed, one striking feature of the proofs is that they can be used to establish nonlocality solely on the basis of suitably robust perfect correlations. First we explain that Schr\"odinger's argument shows that locality and the perfect correlations between measurements of observables on spatially separated systems implies the existence of a non-contextual value-map for quantum observables; non-contextual means that the observable has a particular value before its measurement, for any given quantum system, and that any experiment "measuring this observable" will reveal that value. Then, we establish the impossibility of a non-contextual value-map for quantum observables {\it without invoking any further quantum predictions}. Combining this with Schr\"odinger's argument implies nonlocality. Finally, we illustrate how Bohmian mechanics is compatible with the impossibility of a non-contextual value-map.

Bohmian mechanics at space–time singularities. I. Timelike singularities

We develop an extension of Bohmian mechanics to a curved background space–time containing a singularity. The present paper focuses on timelike singularities. We use the naked timelike singularity of the super-critical Reissner–Nordström geometry as an example. While one could impose boundary conditions at the singularity that would prevent the particles from falling into the singularity, we are interested here in the case in which particles have positive probability to hit the singularity and get annihilated. The wish for reversibility, equivariance, and the Markov property then dictates that particles must also be created by the singularity, and indeed dictates the rate at which this must occur. That is, a stochastic law prescribes what comes out of the singularity. We specify explicit equations of a model involving an interior–boundary condition on the wave function at the singularity, which can be used also in other versions of quantum theory besides Bohmian mechanics. As the resulting theory involves particle creation and annihilation, it can be regarded as a quantum field theory, and the stochastic process for the Bohmian particles is analogous to Bell-type quantum field theories.

Determination of classical behaviour of the Earth for large quantum numbers using quantum guiding equation

For quantum systems, we expect to see the classical behaviour at the limit of large quantum numbers. Hence, we apply Bohmian approach for describing the evolution of Earth around the Sun. We obtain possible trajectories of the Earth system with different initial conditions which converge to a certain stable orbit, known as the Kepler orbit, after a given time. The trajectories are resulted from the guiding equation $p=\nabla S$ in the Bohmian mechanics, which relates the momentum of the system to the phase part of the wave function. Except at some special situations, Bohmian trajectories are not Newtonian in character. We show that the classic behaviour of the Earth can be interpreted as the consequence of the guiding equation at the limit of large quantum numbers.

A proposed mechanism for mind-brain interaction using extended Bohmian quantum mechanics in Avicenna's monotheistic perspective

In quantum approaches to consciousness, the authors try to propose a model and mechanism for the mind-brain interaction using modern physics and some quantum concepts which do not exist in the classical physics. The independent effect of mind on the brain has been one of the challenging issues in the history of science and philosophy. In some recent mind-brain interaction models, the direct influence of mind on matter is either not accepted (as in Stapp's model) or not clear, and there have not been any clear mechanism for it (as in Penrose-Hameroff's model or in Eccles's model). In this manuscript we propose a model and mechanism for mind's effect on the matter using an extended Bohmian quantum mechanics and Avicenna's ideas. We show that mind and mental states can affect brain's activity without any violation of physical laws. This is a mathematical and descriptive model which shows the possibility of providing a causal model for mind's effect on matter. It is shown that this model guarantees the realistic philosophical constraints and respects the laws of nature. In addition, it is shown that it is in agreement with the Libet style experimental results and parapsychological data.To propose this model, we obtained a modified (non-unitary) Schrödinger equation via second quantization method which affects the particle through a modified quantum potential and a new term in the continuity equation. At the second quantized level, which is equivalent to quantum field theory level (QFT), we can use the path integral formalism of Feynman. We show that there are three methods to extend Bohmian QM via path integral formalism, which has different interpretations. By numerical simulation of trajectories in the two-slits experiment, we show their differences and choose one of these methods for our mind-brain model which can be the basis for explaining some phenomena which are not possible to explain in the standard Bohmian QM.

Quantum cosmology of a Hořava–Lifshitz model coupled to radiation

In the present paper, we canonically quantize a homogeneous and isotropic Hořava–Lifshitz cosmological model, with constant positive spatial sections and coupled to radiation. We consider the projectable version of that gravitational theory without the detailed balance condition. We use the Arnowitt–Deser–Misner (ADM) formalism to write the gravitational Hamiltonian of the model and the Schutz variational formalism to write the perfect fluid Hamiltonian. We find the Wheeler–DeWitt equation for the model, which depends on several parameters. We study the case in which parameter values are chosen so that the solutions to the Wheeler–DeWitt equation are bounded. Initially, we solve it using the Many Worlds interpretation. Using wave packets computed with the solutions to the Wheeler–DeWitt equation, we obtain the scalar factor expected value [Formula: see text]. We show that this quantity oscillates between finite maximum and minimum values and never vanishes. Such result indicates that the model is free from singularities at the quantum level. We reinforce this indication by showing that by subtracting one standard deviation unit from the expected value [Formula: see text], the latter remains positive. Then, we use the DeBroglie–Bohm interpretation. Initially, we compute the Bohm’s trajectories for the scale factor and show that they never vanish. Then, we show that each trajectory agrees with the corresponding [Formula: see text]. Finally, we compute the quantum potential, which helps understanding why the scale factor never vanishes.

Bohmian Trajectories for Hamiltonians with Interior–Boundary Conditions

Recently, there has been progress in developing interior-boundary conditions (IBCs) as a technique of avoiding the problem of ultraviolet divergence in non-relativistic quantum field theories while treating space as a continuum and electrons as point particles. An IBC can be expressed in the particle-position representation of a Fock vector $\psi$ as a condition on the values of $\psi$ on the set of collision configurations, and the corresponding Hamiltonian is defined on a domain of vectors satisfying this condition. We describe here how Bohmian mechanics can be extended to this type of Hamiltonian. In fact, part of the development of IBCs was inspired by the Bohmian picture. Particle creation and annihilation correspond to jumps in configuration space; the annihilation is deterministic and occurs when two particles (of the appropriate species) meet, whereas the creation is stochastic and occurs at a rate dictated by the demand for the equivariance of the $|\psi|^2$ distribution, time reversal symmetry, and the Markov property. The process is closely related to processes known as Bell-type quantum field theories.

Causal interpretation of quantum mechanics: The Schrodinger equation as a condition of stability

This work presents a further development of causal interpretation of quantum mechanics. Existence of ‘non-local hidden variables’ like a physical field is taken as an assumption. Under this assumption, on the basis of the causal approach it is proven that Schrödinger equation is a necessary condition for stability of the motion of a particle. For the case of a hydrogen atom this approach gives a mathematical base with which to suggest that (1) the electron’s spin in an atom is precessing; (2) the energy of the precessional motion on Bohr orbits satisfies the Rydberg formula, and (3) structures are formed in the physical vacuum which stabilize the motion of an electron on Bohr orbits.