On the Classical Limit of Quantum Mechanics

Contrary to the widespread belief, the problem of the emergence of classical mechanics from quantum mechanics is still open. In spite of many results of the standard approach, it is not yet clear how to explain within standard quantum mechanics the classical motion of macroscopic bodies. In this paper, we shall formulate the classical limit as a scaling limit in terms of an adimensional parameter ε. We shall take the first steps toward a comprehensive understanding of the classical limit, analyzing special cases of classical behavior in the framework of a precise formulation of quantum mechanics called Bohmian mechanics which contains in its own structure the possibility of describing real objects in an observer-independent way.     

Some Clarifications on the Relation Between Bohmian Quantum Potential and Mach’s Principle

Mach’s principle asserts that the inertial mass of a body is related to the distribution of other distant bodies. This means that in the absence of other bodies, a single body has no mass. In this case, talking about motion is not possible, because the detection of motion is possible only relative to other bodies. But in physics we are faced with situations that are not fully Machian. As in the case of general theory of relativity where geodesics exist in the absence of any matter, the motion has meaning. Another example which is the main topic of our discussion, refers to Bohmian quantum mechanics, where the inertial mass of a single particle does not vanish, but is modified. We can call such situations in which motion or mass of a single particle has meaning, pseudo-Machian situations. In this paper, we use the Machian or pseudo-Machian considerations to clarify under what circumstances and how a Machian effect leads us to Bohmian quantum mechanics. Then, we shall get the Bohmian quantum potential and its higher order terms for the Klein-Gordon particle through Machian considerations, without using any quantum mechanical postulate or operator formalism.     

Applied Bohmian mechanics

Bohmian mechanics provides an explanation of quantum phenomena in terms of point-like particles guided by wave functions. This review focuses on the use of nonrelativistic Bohmian mechanics to address practical problems, rather than on its interpretation. Although the Bohmian and standard quantum theories have different formalisms, both give exactly the same predictions for all phenomena. Fifteen years ago, the quantum chemistry community began to study the practical usefulness of Bohmian mechanics. Since then, the scientific community has mainly applied it to study the (unitary) evolution of single-particle wave functions, either by developing efficient quantum trajectory algorithms or by providing a trajectory-based explanation of complicated quantum phenomena. Here we present a large list of examples showing how the Bohmian formalism provides a useful solution in different forefront research fields for this kind of problems (where the Bohmian and the quantum hydrodynamic formalisms coincide). In addition, this work also emphasizes that the Bohmian formalism can be a useful tool in other types of (nonunitary and nonlinear) quantum problems where the influence of the environment or the nonsimulated degrees of freedom are relevant. This review contains also examples on the use of the Bohmian formalism for the many-body problem, decoherence and measurement processes. The ability of the Bohmian formalism to analyze this last type of problems for (open) quantum systems remains mainly unexplored by the scientific community. The authors of this review are convinced that the final status of the Bohmian theory among the scientific community will be greatly influenced by its potential success in those types of problems that present nonunitary and/or nonlinear quantum evolutions. A brief introduction of the Bohmian formalism and some of its extensions are presented in the last part of this review.     

Bohmian Mechanics and Quantum Information

Many recent results suggest that quantum theory is about information, and that quantum theory is best understood as arising from principles concerning information and information processing. At the same time, by far the simplest version of quantum mechanics, Bohmian mechanics, is concerned, not with information but with the behavior of an objective microscopic reality given by particles and their positions. What I would like to do here is to examine whether, and to what extent, the importance of information, observation, and the like in quantum theory can be understood from a Bohmian perspective. I would like to explore the hypothesis that the idea that information plays a special role in physics naturally emerges in a Bohmian universe.     

On the Weak Measurement of Velocity in Bohmian Mechanics

In a recent article (Wiseman in New J. Phys. 9:165, 2007), Wiseman has proposed the use of so-called weak measurements for the determination of the velocity of a quantum particle at a given position, and has shown that according to quantum mechanics the result of such a procedure is the Bohmian velocity of the particle. Although Bohmian mechanics is empirically equivalent to variants based on velocity formulas different from the Bohmian one, and although it has been proven that the velocity in Bohmian mechanics is not measurable, we argue here for the somewhat paradoxical conclusion that Wiseman’s weak measurement procedure indeed constitutes a genuine measurement of velocity in Bohmian mechanics. We reconcile the apparent contradictions and elaborate on some of the different senses of measurement at play here.     

A New Way for the Extension of Quantum Theory: Non-Bohmian Quantum Potentials

Quantum Mechanics is a good example of a successful theory. Most of atomic phenomena are described well by quantum mechanics and cases such as Lamb Shift that are not described by quantum mechanics, are described by quantum electrodynamics. Of course, at the nuclear level, because of some complications, it is not clear that we can claim the same confidence. One way of taking these complications and corrections into account seems to be a modification of the standard quantum theory. In this paper and its follow ups we consider a straightforward way of extending quantum theory. Our method is based on a Bohmian approach. We show that this approach has the essential ability for extending quantum theory, and we do this by introducing “non-Bohmian” forms for the quantum potential.     

Quantum backreaction through the Bohmian particle

A novel solution to the quantum backreaction problem in a mixed quantum-classical simulation is provided using the Bohmian interpretation of quantum mechanics. The Bohmian backreaction is unique, computationally simple, features reaction channel branching, and easily gives the full classical limit. The Bohmian quantum-classical method is illustrated by application to a model of O2 interacting with a Pt surface.     

On the Green’s function and iterative solutions of loop quantum cosmology

Here we shall find the Green’s function of the difference equation of loop quantum cosmology. To illustrate how to use it, we shall obtain an iterative solution for closed model and evaluate its corresponding Bohmian trajectory.     

Deriving relativistic Bohmian quantum potential using variational method and conformal transformations

In this paper we shall argue that conformal transformations give some new aspects to a metric and changes the physics that arises from the classical metric. It is equivalent to adding a new potential to relativistic Hamilton–Jacobi equation. We start by using conformal transformations on a metric and obtain modified geodesics. Then, we try to show that extra terms in the modified geodesics are indications of a background force. We obtain this potential by using variational method. Then, we see that this background potential is the same as the Bohmian non-local quantum potential. This approach gives a method stronger than Bohm’s original method in deriving Bohmian quantum potential. We do not use any quantum mechanical postulates in this approach.     

Bohmian Trajectories for Kerr–Newman Particles in Complex Space-Time

Complexified Liénard–Wiechert potentials simplify the mathematics of Kerr–Newman particles. Here we constrain them by fiat to move along Bohmian trajectories to see if anything interesting occurs, as their equations of motion are not known. A covariant theory due to Stueckelberg is used. This paper deviates from the traditional Bohmian interpretation of quantum mechanics since the electromagnetic interactions of Kerr–Newman particles are dictated by general relativity. A Gaussian wave function is used to produce the Bohmian trajectories, which are found to be multi-valued. A generalized analytic continuation is introduced which leads to an infinite number of trajectories. These include the entire set of Bohmian trajectories. This leads to multiple retarded times which come into play in complex space-time. If one weights these trajectories by their natural Bohmian weighting factors, then it is found that the particles do not radiate, that they are extended, and that they can have a finite electrostatic self energy, thus avoiding the usual divergence of the charged point particle. This effort does not in any way criticize or downplay the traditional Bohmian interpretation which does not assume the standard electromagnetic coupling to charged particles, but it suggests that a hybridization of Kerr–Newman particle theory with Bohmian mechanics might lead to interesting new physics, and maybe even the possibility of emergent quantum mechanics.     

Relativistic Quantum Mechanics and the Bohmian Interpretation

No Heading Conventional relativistic quantum mechanics, based on the Klein-Gordon equation, does not possess a natural probabilistic interpretation in configuration space. The Bohmian interpretation, in which probabilities play a secondary role, provides a viable interpretation of relativistic quantum mechanics. We formulate the Bohmian interpretation of many-particle wave functions in a Lorentz-covariant way. In contrast with the nonrelativistic case, the relativistic Bohmian interpretation may lead to measurable predictions on particle positions even when the conventional interpretation does not lead to such predictions.     

On the Classical Limit in Bohm’s Theory

The standard means of seeking the classical limit in Bohmian mechanics is through the imposition of vanishing quantum force and quantum potential for pure states. We argue that this approach fails, and that the Bohmian classical limit can be realized only by combining narrow wave packets, mixed states, and environmental decoherence.     

Weak Value,Quasiprobability and Bohmian Mechanics

We clarify the significance of quasiprobability (QP) in quantum mechanics that is relevant in describing physical quantities associated with a transition process. Our basic quantity is Aharonov’s weak value, from which the QP can be defined up to a certain ambiguity parameterized by a complex number. Unlike the conventional probability, the QP allows us to treat two noncommuting observables consistently, and this is utilized to embed the QP in Bohmian mechanics such that its equivalence to quantum mechanics becomes more transparent. We also show that, with the help of the QP, Bohmian mechanics can be recognized as an ontological model with a certain type of contextuality.     

On the unique mapping relationship between initial and final quantum states

In its standard formulation, quantum mechanics presents a very serious inconvenience: given a quantum system, there is no possibility at all to unambiguously (causally) connect a particular feature of its final state with some specific section of its initial state. This constitutes a practical limitation, for example, in numerical analyses of quantum systems, which often make necessary the use of some extra assistance from classical methodologies. Here it is shown how the Bohmian formulation of quantum mechanics removes the ambiguity of quantum mechanics, providing a consistent and clear answer to such a question without abandoning the quantum framework. More specifically, this formulation allows to define probability tubes, along which the enclosed probability keeps constant in time all the way through as the system evolves in configuration space. These tubes have the interesting property that once their boundary is defined at a given time, they are uniquely defined at any time. As a consequence, it is possible to determine final restricted (or partial) probabilities directly from localized sets of (Bohmian) initial conditions on the system initial state. Here, these facts are illustrated by means of two simple yet physically insightful numerical examples: tunneling transmission and grating diffraction.     

Foundations of the Quaternion Quantum Mechanics

We show that quaternion quantum mechanics has well-founded mathematical roots and can be derived from the model of the elastic continuum by French mathematician Augustin Cauchy, i.e., it can be regarded as representing the physical reality of elastic continuum. Starting from the Cauchy theory (classical balance equations for isotropic Cauchy-elastic material) and using the Hamilton quaternion algebra, we present a rigorous derivation of the quaternion form of the non- and relativistic wave equations. The family of the wave equations and the Poisson equation are a straightforward consequence of the quaternion representation of the Cauchy model of the elastic continuum. This is the most general kind of quantum mechanics possessing the same kind of calculus of assertions as conventional quantum mechanics. The problem of the Schrödinger equation, where imaginary ‘i’ should emerge, is solved. This interpretation is a serious attempt to describe the ontology of quantum mechanics, and demonstrates that, besides Bohmian mechanics, the complete ontological interpretations of quantum theory exists. The model can be generalized and falsified. To ensure this theory to be true, we specified problems, allowing exposing its falsity.     

Bohmian trajectories of Airy packets

The discovery of Berry and Balazs in 1979 that the free-particle Schrödinger equation allows a non-dispersive and accelerating Airy-packet solution has taken the folklore of quantum mechanics by surprise. Over the years, this intriguing class of wave packets has sparked enormous theoretical and experimental activities in related areas of optics and atom physics. Within the Bohmian mechanics framework, we present new features of Airy wave packet solutions to Schrödinger equation with time-dependent quadratic potentials. In particular, we provide some insights to the problem by calculating the corresponding Bohmian trajectories. It is shown that by using general space–time transformations, these trajectories can display a unique variety of cases depending upon the initial position of the individual particle in the Airy wave packet. Further, we report here a myriad of nontrivial Bohmian trajectories associated to the Airy wave packet. These new features are worth introducing to the subject’s theoretical folklore in light of the fact that the evolution of a quantum mechanical Airy wave packet governed by the Schrödinger equation is analogous to the propagation of a finite energy Airy beam satisfying the paraxial equation. Numerous experimental configurations of optics and atom physics have shown that the dynamics of Airy beams depends significantly on initial parameters and configurations of the experimental set-up.     

On the conformally coupled scalar field quantum cosmology

We propose a new initial condition for the homogeneous and isotropic quantum cosmology, where the source of the gravitational field is a conformally coupled scalar field, and the maximally symmetric hypersurfaces have positive curvature. After solving corresponding Wheeler–DeWitt equation, we obtain exact solutions in both classical and quantum levels. We propose appropriate initial condition for the wave packets which results in a complete classical and quantum correspondence. These wave packets closely follow the classical trajectories and peak on them. We also quantify this correspondence using de Broglie–Bohm interpretation of quantum mechanics. Using this proposal, the quantum potential vanishes along the Bohmian paths and the classical and Bohmian trajectories coincide with each other. We show that the model contains singularities even at the quantum level. Therefore, the resulting wave packets closely follow the classical trajectories from big-bang to big-crunch.