De Broglie–Bohm theory

The de Broglie -Bohm theory or Bohmian mechanics is - depending on the definition of terms - an interpretation or an alternative formalism of quantum mechanics. It reproduces all predictions of the ( non-relativistic ) quantum mechanics, but as an interpretation conceived a radically deviates from the Copenhagen interpretation understanding of reality as the basis. The Bohmian mechanics is a deterministic theory and enables a simple solution to the measurement problem of quantum mechanics, that is, the act of measurement or observation plays no distinguished role (for measurement problem see also the articles on Schrödinger's cat). As with most interpretations of quantum mechanics is no way to experimentally distinguish between the Bohmian mechanics and standard quantum mechanics, that is, Bohmian mechanics and quantum mechanics found in all experimentally verifiable situations the same predictions.

• 3.1 Solution of the quantum mechanical measurement problem
• 3.2 Status of observables and contextuality
• 3.3 nonlocality
• 3.4 determinism
• 3.5 Complementarity superfluous

• 4.1 The quantum potential

• 7.1 Working Groups
• 7.2 launches

History

The Bohmian mechanics was developed in the 1920s by the French physicist Louis de Broglie. De Broglie described it as " theory of the guide shaft " ( theory de l' onde pilote, Eng. Pilot wave theory ). However, these received only limited attention and fell into oblivion. Without de Broglie to know works, developed in the 1950s, the American physicist David Bohm, an equivalent version of this theory. Bohm described the theory later than ontological or causal interpretation of quantum mechanics.

Since the 1970s, the Irish physicist John Stewart Bell was one of the few prominent physicists who have stood up for the Bohmian mechanics.

Since the 1990s, it comes down to this area again to increased research activity, for example with a research group at the University of Munich ( Detlef Dürr ), Rutgers University in New Jersey ( Sheldon Goldstein ) and the University of Genoa ( Nino Zanghì ). Also the name of Bohmian Mechanics (English Bohmian mechanics ) was coined by this working group. This designation may accuse you that it omits the role of de Broglie. Given their history seems the name " De Broglie -Bohm theory " reasonable. The following two terms are used interchangeably.

Formalism

The basic idea of ​​de Broglie Bohm theory is not to be described a system by the shaft () function only, but by the pair of wave function and the Teilchenorten () of the respective objects (electrons, atoms, etc.). The trajectories of the particles are the so-called hidden parameter theory. The Bohmian mechanics is thus defined by two basic equations: one is the usual Schrödinger equation of quantum mechanics:

And by the equation of motion ( " guide equation ") for the Teilchenorte:

Here, the phase of the wave function referred to in the polar representation, ie

Designates the mass of the i- th particle and Nablaoperator applied to the coordinate of the i-th particle.

Figuratively speaking, "derives " or "leads" the wave function ie the motion of the particles. Within this theory, the quantum objects thus move to continuous ( and deterministic ) paths. This movement is clearly defined course only when initial conditions are specified. Note that the guidance equation is a first order differential equation, ie, already defines the specification of the Teilchenorte at a time when the movement. In contrast, set in classical mechanics only the position and velocity (or momentum) the movement clearly. All predictions of quantum mechanics can be just then reproduced by the de Broglie -Bohm theory, if one chooses as the initial condition, the so-called quantum equilibrium distribution, that is, the validity of the following " quantum equilibrium hypothesis " assumes.

Quantum equilibrium hypothesis

The quantum equilibrium hypothesis is that the spatial distribution of a system described by the wave function is.

The distribution is the so-called quantum equilibrium distribution.

Due to the quantum equilibrium hypothesis, the Heisenberg uncertainty principle is not violated in the Bohmian mechanics. In contrast to the usual quantum mechanics, the probability statements of Bohmian mechanics, however, only our ignorance of the specific initial conditions are owed.

The quantum mechanical continuity equation

Ensures that a once - divided system retains this property. This still remains as to why this distribution should be available at any time. To answer this question, various approaches exist. Obviously, it is not satisfactory, this fact very specific initial conditions (about the Universe) attributable. Physically intuitive it if you could specify a dynamic mechanism that explains how ( as many as possible ) initial conditions approach the quantum equilibrium would be. That approach is followed about Valentini, who argues lead a larger class of initial conditions to an approximate quantum equilibrium distribution as a result of the Bohmian dynamics.

The question of whether many or few initial conditions are compatible with the quantum equilibrium hypothesis, of course, presupposes a measure with which these quantities can be measured. Dürr et al. chose this freedom to the starting point. These authors choose a level at which almost all initial conditions are tolerated by a special weighting to the quantum equilibrium hypothesis, and argue why this measure is natural. So they justify why a hypothetical bohmsches universe is in quantum equilibrium. The main result of this work now is to ( i) define the concept of the wave function of a subsystem, and to show ( ii ), that these subsystems meet the quantum equilibrium hypothesis. In this sense, according to Dürr quantum equilibrium hypothesis does not postulate, but a consequence of Bohmian mechanics.

Properties of the Bohm trajectories

As mentioned, sets an initial condition each Bohmian trajectory clearly identified as the lead equation ( 2) is a first order differential equation. This has the consequence that the particle trajectories in the configuration space can not intersect. In the one -particle case, the movement is so free from overlapping place in the spatial domain. In this way one can for simple systems without numerical simulation make a qualitative picture of particle motion.

Figure 1 shows the simulation of some trajectories in the double slit. The property of no overlapping together with the symmetry of the arrangement ensures that the orbits the center plane can not intersect. This figure also illustrates that the Bohmian trajectories extend completely non-classical. They have changes in direction, although the range is field-free behind the gap in the classical sense. In this sense applies at the level of individual tracks neither energy nor momentum conservation.

In the case of real wave functions, the situation is even simpler. Since the phase of the wave function here disappears, the particle - dispersed locations to rest. This situation exists, for example the ground state of the hydrogen atom or the energy eigenstates of the harmonic oscillator.

Spin in Bohmian mechanics

It is instructive to consider how the de Broglie -Bohm theory describes the spin. Here are various approaches, but an obvious possibility is not attributable to the spin of the particles, but only as a property of the wave function (or the Pauli spinor ) interpreted.

Specifically, one starts with the Schrödinger equation on the Pauli equation. From the wave function is a 2-component spinor. There is - similar to the description spinless particles - a stream:

Here, the vector potential and the Spinorindex. The guiding equation is analogous to the spinless case:

Even without the mathematical details to survey, the following point should be clear: the spinning property is not associated with the particles, ie the object on the Bohmian trajectory, and the configuration space remains the same as in the case of spinless objects. In particular, no " hidden variables " for the spin is introduced. The usual way of speaking is that the spin " contextualized " is (see below).

Key Features

Solution of the quantum mechanical measurement problem

The single most important property of the de Broglie -Bohm theory is that it solves the measurement problem of quantum mechanics and that, within this theory, the measurement problem does not even occur. Reminder: The measurement problem in quantum mechanics is in the core is to interpret macroscopic superpositions of different states. These occur in the quantum mechanical treatment of the measurement on a completely natural way, although each actually performed measurement always a defined result (that is not described by a superposition ) has.

To clarify this contradiction, a special state change has been postulated in the act of measurement by John von Neumann, the so-called collapse of the wave function. But this is less a solution than an admission of the measurement problem dar. Finally, it remains unclear what interaction has the rank of a measurement and how this mechanism is physically to understand.

In Bohmian mechanics, however, there is a simple mechanism that characterizes the component of the wave function corresponding to the actual measurement result: It is the particle location, which has reached a branch of the wave function in a continuous motion. In other words, various measurement results are distinguished by different configurations in the de Broglie Bohm theory.

Status of observables and contextuality

A radical innovation of the de Broglie -Bohm theory is their reinterpretation of the observables concept of quantum mechanics represents the usual quantum mechanics identifies all observables with Hermitian operators acting on the Hilbert space of states. The Nichtvertauschen of these operators is interpreted as an expression of the radical novelty of quantum mechanics.

The de Broglie -Bohm theory proposes here a different path. It records the location and explicitly describes him by real coordinates and the particle velocity by a real vector field (on configuration space ). All other sizes ( spin, energy, momentum, etc.) have only a derivative status. The reason for this is simple: When performing an experiment for "measuring " of, eg, the spin component of the output is ( as with any other event) defined by the wave function and the initial location. So there is no measurement literally take, i.e., it is determined not an intrinsic property that is independent of the measurement. The somewhat unfortunate way of speaking is that these quantities ( ie about the spin) are contextualized, ie, the measured value depends on the context of the measurement setup and the initial location. Specifically about cases can be constructed in which are to systems that are described by the same wave function different spin components " measured " by different starting locations. This property is the key to why the Cooking- Specker theorem does not affect the consistency of the de Broglie -Bohm theory in the rest. The Hermitian operators of the usual quantum mechanics play in the Bohmian mechanics no fundamental role, but act as mathematical objects that encode probability distributions (see ).

Nonlocality

Since the wave function is defined on the configuration space (where N the number of particles ), the guide link, in principle the movement equation of individual particles with the location of all the other at the same time. In this way, space-like separate objects can influence each other, ie this form of interaction occurs with over- the speed of light, even instantaneous. This mechanism explains the Bohmian mechanics the EPR effect and the violation of Bell 's inequality. Due to the quantum equilibrium hypothesis, however, a signal transmission using these correlations is not possible. This form of " Einstein locality" is so well respected.

If the Mehrteilchenzustand is not crossed, ie, factored into the shares of the individual parts, decouple the equations of motion of Bohmian mechanics and the corresponding subsystems evolve independently.

Results such as the already mentioned violation of Bell 's inequalities or the " Free Will Theorem" by John Horton Conway and Simon cooking show that there can be no completions or formulations of quantum mechanics that are local and deterministic.

Determinism

The de Broglie -Bohm theory describes the quantum phenomena deterministic, that is, all changes of state are completely determined by the initial conditions ( wave function and configuration). All probability statements are only the ignorance of the specific starting locations owed.

In contrast, the claimed principle of randomness quantum phenomena in the conventional view, as in the act of measurement.

However, it must be emphasized that due to the quantum equilibrium hypothesis, the ignorance of the initial conditions in the de Broglie -Bohm theory is in principle and thus the descriptive content of the two theories is the same. In philosophical terminology is from the ontological indeterminacy of quantum physics an " epistemic " uncertainty in the de Broglie -Bohm theory.

Complementarity superfluous

The concept of complementarity was introduced to justify the common use of conflicting in the strict sense types of description in quantum mechanics. For example, wave and particle are complementary to each other usually believed. So you would think that they complement each other and to determine the scope must be considered when using it.

In Bohmian mechanics, wave and particle, for example, electrons, however, are a simple consequence of the fact that to describe them both a particle property is used (namely the place ) and a wave-like size (the wave function). Figure 1 illustrates the simulation of some trajectories in the double slit experiment and illustrates this point particularly clear.

The de Broglie -Bohm theory allows - like any other theory also - various equivalent representations. Our previous presentation for example, has placed no value on the so-called quantum potential, making it the reading of Bell followed, which was further developed by Dürr, among others. In many representations of the de Broglie -Bohm theory, the quantum potential is, however, highlighted as the key feature. For this reason, it should also be mentioned at this point here ( other differences between different schools of de Broglie Bohm theory concerning the status of the observables and the wave function and the derivation of the quantum equilibrium).

The quantum potential

In Bohm's presentation of the theory in 1952 (as well as the views of other authors, 1993), the novelty of the de Broglie -Bohm theory in the creation of an additional potential term is seen. Substituting the Polar representation of the Schrödinger equation and separates the real and imaginary parts, one is led to the following equations:

Of expression (3) is just the continuity equation of Quantum Mechanics. Equation (4) corresponds to the classical Hamilton-Jacobi equation for the effect. Here, however, occurs in addition to the kinetic and the potential energy term on an additional term, the so-called quantum well:

The classical Hamilton -Jacobi theory is a reformulation of the Newtonian (or Hamiltonian ) mechanics. The Hamilton -Jacobi equation is first order (but non- linear). The speed (or pulse) is defined by the condition. This just corresponds to the reference equation of the de Broglie -Bohm theory.

With the help of the quantum potential can ultimately the Bohmian equation of motion a Newtonian impression be given:

Since the particle is, however, already completely determined by the guiding equation, one can dispense with the derivation of the analogy with the Hamilton -Jacobi theory and the additional potential term. The leadership equation can be motivated, for example, directly from symmetry considerations. Due to the quantum potential formulation it will be also invited to the misconception that the de Broglie -Bohm theory is essentially the classical mechanics with an additional potential term. Basically, however, the preference for one or another formulation of the theory is a matter of taste. In addition, each formulation can have a meaningful application. For instance, particularly easy-to- formulate the quantum version, the problem of the conventional limit of de Broglie Bohm theory.

Criticism

The de Broglie -Bohm theory is only represented by a small minority of physicists. This is, however, only partly explicit criticism of this theory, but primarily because the de Broglie -Bohm theory is a contribution to questions of interpretation in physics. At these discussions, most scientists do not participate.

The criticism is voiced in the de Broglie -Bohm theory, can be divided into different groups. Approximately one can accuse the theory that distinguishes the spatial domain and that the wave function acts on the Teilchenorte, but not vice versa. In addition, it can seem unsatisfactory that the de Broglie -Bohm theory, the world populated with "empty" wave functions, ie those components that do not contain the particle and have no more influence on the particle dynamics due to decoherence should. Instead it to see a criticism, these properties are sometimes regarded as a remarkable new features of the description of nature.

Often, the nonlocality of the de Broglie -Bohm theory is put forward as an objection. This can reciprocate at different levels. Firstly, there is the face of the violation of Bell inequalities between numerous physicists that claim as well as the " non-locality " of the usual quantum mechanics. At least the EPR effect between space-like distant objects is an experimental fact. This discussion suffers from the fact that you can give to the concept of non-locality or non - separability numerous meanings. For example, the " signal -locality ", ie no signal propagation with superluminal velocity, respected both by quantum mechanics as well as the de Broglie -Bohm theory.

It can be argued that the allegation of non-locality is a non- relativistic theory such as Bohmian mechanics does not apply. The charge of non-locality is usually associated with the doubt of being able to specify a satisfactory relativistic ( and quantum field theory ) generalization of the de Broglie -Bohm theory. This objection, the generalization ability of the de Broglie -Bohm theory in question and look for a more substantive discussion of the theory, or require their further development. In particular, the quantum equilibrium condition and the requirement of Lorentz covariance seem to contradict each other, ie in a " bohmartigen " relativistic theory must be introduced a preferred reference frame. However, there are " bohmartige " models of the Dirac theory in which this excellent reference system is no experimental effect and all the statistical predictions of relativistic quantum mechanics can be reproduced.

Likewise, there are different approaches to " bohmartigen " quantum field theory. While some maintain that " Teilchenontologie " the non-relativistic formulation, perform other fields as hidden parameters. This approach, however, been successful only for bosonic fields. The further development in this field is expected to play a major role in the reception of the de Broglie -Bohm theory. A more detailed discussion of criticism of the de Broglie -Bohm theory can be found in a 2004 article by Passon. An overview of the different approaches to quantum field theoretical generalization of the de Broglie -Bohm theory gives Struyve.