User:Jostylr/Bohmian Mechanics Draft

The de Broglie-Bohm theory is a quantum theory describing the motion of point particles. At all times, the particles have definite positions. These positions change according to the guiding equation. The velocity of any one particle may depend on the simultaneous positions of any or all of the other particles in the universe with the connection being made by the standard quantum mechanical wavefunction. The velocities arise from the gradient of the wavefunction.

Schrödinger's equation completes the dynamical specification of this theory as it gives the evolution of the wavefunction. That evolution does not depend on the position of the particles.

This theory gives rise to a measurement formalism, analogous to thermodynamics for classical mechanics, which yields the standard quantum formalism generally associated with the Copenhagen interpretation. The measurement problem is resolved by this theory since the state of the system is independent of any observers. The familiar wavefunction collapse of standard quantum mechanics emerges from an analysis of subsystems and the quantum equilibrium hypothesis.

This theory is deterministic though relativistic extensions of this theory tend to lose this property. It handles spin and curved spaces without difficulty. It can be modified to handle quantum field theory. Bell's theorem was inspired by Bell discovering the work of David Bohm and wondering if the obvious non-locality of the theory could be removed; Bell's work demonstrated that it cannot be. Given the EPR paradox, this implied that a weak form of nonlocality was a fact of nature.

The theory has a number of equivalent mathematical formulations and names.

The ontology
This theory is a theory of particles moving in space-time. Each particle has a definite position $$x$$ at every time $$t$$. This is the starting point of the theory. The correspondence to our experiences is made by the identification of where the particles are and where we witness them to be, as in classical mechanics.

The difference between de Broglie-Bohm theory and classical mechanics is not in the ontology, but in the dynamics. In classical mechanics, the acceleration of the particles are given by forces. In de Broglie-Bohm theory, the velocities of the particles are given by the wavefunction.

In what follows below, we will give the setup for one particle moving in $$\mathbb{R}^3$$ followed by the setup for $$N$$ particles moving in 3 dimensions. In the first instance, configuration space and real space are the same while in the second, real space is still $$\mathbb{R}^3$$, but configuration space becomes $$\mathbb{R}^{3N}$$. While the particle positions themselves are in real space, the velocity field and wavefunction are on configuration space which is how particles are entangled with each other in this theory.

Extensions to this theory include spin and more complicated configuration spaces.

We use variations of $$\mathbf{Q}$$ for particle positions while $$\psi$$ represents the complex-valued wavefunction on configuration space.

Guiding equation
For a single particle moving in $$\mathbb{R}^3$$, the particle's velocity is given


 * $$\frac{d \mathbf{Q}}{dt} (t) = \frac{\hbar}{m} Im \left(\frac{\nabla \psi}{\psi} \right) (\mathbf{Q}, t)$$.

For many particles, we label them as $$\mathbf{Q}_k$$ for the $$k$$th particle and their velocities are given by


 * $$\frac{d \mathbf{Q}_k}{dt} (t) = \frac{\hbar}{m} Im \left(\frac{\nabla_k \psi}{\psi} \right) (\mathbf{Q}_1, \mathbf{Q}_2, \ldots, \mathbf{Q}_N, t)$$.

The key fact to notice is that this velocity field depends on the actual positions of all of the $$N$$ particles in the universe. As explained below, in most experimental situations, the influence of all of those particles can be encapsulated into an effective wavefunction for a subsystem of the universe.

Schrödinger's equation
The one particle Schrödinger equation governs the time evolution of a complex-valued wavefunction on $$\mathbb{R}^3$$. The equation represents a quantized version of the total energy of a classical system evolving under a real-valued potential function $$V$$ on $$\mathbb{R}^3$$:


 * $$i\hbar\frac{\partial}{\partial t}\psi=-\frac{\hbar^2}{2m}\nabla^2\psi + V\psi$$

For many particles, the equation is the same except that $$\psi$$ and $$V$$ are now on configuration space, $$\mathbb{R}^{3N}$$.


 * $$i\hbar\frac{\partial}{\partial t}\psi=-\sum_{k=1}^{N}\frac{\hbar^2}{2m_k}\nabla_k^2\psi + V\psi$$

This is the same wavefunction of conventional quantum mechanics.

Quantum Equilibrium Hypothesis
In Bohm's original papers [Bohm 1952], he discusses how de Broglie-Bohm theory gives rise to the usual measurement results of quantum mechanics. The key idea is that this is true if the positions of the particles satisfy the statistical distribution given by $$|\psi|^2$$. And that distribution is guaranteed to be true for all time under the guiding equation if the initial distribution of the particles satisfies $$|\psi|^2$$.

For a given experiment, we can postulate this as being true and verify experimentally that it does indeed hold true, as it does. But, as argued in Dürr et al, one needs to argue that this distribution for subsystems is typical. They argue that $$|\psi|^2$$ by virtue of its equivariance under the dynamical evolution of the system, is the appropriate measure of typicality for initial conditions of the positions of the particles. And because of that typicality, just in the same fashion that we expect entropy to increase, we should expect the universe to be in quantum equilibrium. And if it is, then the subsystems governed by their own effective Schrödinger equation will have the particles distributed according to $$|\psi|^2$$.

Note that with this argument, it seems that the wavefunction is what gives rise to the non-equilibrium of classical physics.

Spin
To incorporate spin, the wavefunction becomes complex-vector valued. The guiding equation is modified by taking local inner products on the value space to reduce the complex-vectors to complex numbers. The Schrödinger equation is modified by adding a Pauli-Spin term.


 * $$\frac{d \mathbf{Q}_k}{dt} (t) = \frac{\hbar}{m} Im \left(\frac{(\psi,\nabla_k \psi)}{(\psi,\psi)} \right) (\mathbf{Q}_1, \mathbf{Q}_2, \ldots, \mathbf{Q}_N, t)$$


 * $$i\hbar\frac{\partial}{\partial t}\psi=-\sum_{k=1}^{N}\frac{\hbar^2}{2m_k}\nabla_k^2\psi + V\psi + \sum_{k=1}^{N} \mu_i \mathbf{S}^{(k)}\cdot \mathbf{B}(\mathbf{q}_k)$$

where $$\mathbf{S}^{(k)}$$ is the appropriate spin operator acting on the $$k$$th particle's value space and $$\mathbf{B}$$ is the magnetic field in $$\mathbb{R}^{3}$$ (all other functions are fully on configuration space).

For an example of the value space of the wavefunction, a system consisting of two spin 1/2 particle and one spin 1 particle has its wavefunctions satisfying $$\psi: \mathbb{R}^{9}\times \mathbb{R} \to \mathbb{C}^{2}\otimes \mathbb{C}^{2} \otimes \mathbb{C}^{3}$$. That is, its value space is a 12 dimensional space.

Curved space
To extend de Broglie-Bohm theory to curved space (Riemannian manifolds in mathematical parlance), one simply notes that all of the elements of these equations make sense, such as gradients and Laplacians. Thus, we use equations that have the same form as above. Topological and boundary conditions may apply in supplementing the evolution of Schrödinger's equation.

For a de Broglie-Bohm theory on curved space with spin, the spin space becomes a vector bundle over configuration space and the potential in Schrödinger's equation becomes a local self-adjoint operator acting on that space.

Quantum Field Theory
In Dürr et al, the authors describe an extension of de Broglie-Bohm theory for handling creation and annihilation operators. The basic idea is that configuration space becomes the (disjoint) space of all possible configurations of any number of particles. For part of the time, the system evolves deterministically under the guiding equation with a fixed number of particles. But under a stochastic process, particles may be created and annihilated. The distribution of creation events is dictated by the wavefunction. The wavefunction itself is evolving at all times over the full multi-particle configuration space.

Exploiting nonlocality
Valentini has extended the de Broglie-Bohm theory to include signal nonlocality that would allow entanglement to be used as a stand-alone communication channel without a secondary classical "key" signal to "unlock" the message encoded in the entanglement. This violates orthodox quantum theory but it has the virtue that it makes the parallel universes of the chaotic inflation theory observable in principle.

Unlike de Broglie-Bohm theory, Valentini's theory has the wavefunction evolution also depend on the onotological variables. This introduces an instability, a feedback loop that pushes the hidden variables out of "sub-quantal heat death". The resulting theory becomes nonlinear and non-unitary.

Relativity
While Bell's theorem does suggest a certain amount of tension between quantum theory and relativity, it is possible to formulate models demonstrating the potential for relativistic theories. There has been work in developing nonlocal versions of de Broglie-Bohm theory. See Bohm and Hiley: The Undivided Universe, and, , and references therein. A quantum field theory treatment is given in and. Another approach is given in the work of Dürr et al in which they use Bohm-Dirac models and a Lorentz-invariant foliation of space-time.

Results
Below are some highlights of the results that arise out of an analysis of de Broglie-Bohm theory. Experimental results agree with all of the standard predictions of quantum mechanics in so far as the latter has predictions. However, while standard quantum mechanics is limited to discussing experiments with human observers, de Broglie-Bohm theory is a theory which governs the dynamics of a system without the intervention of outside observers (p. 117 in Bell ).

The basis for agreement with standard quantum mechanics is that the particles are distributed according to $$|\psi|^2$$. This is a statement of observer ignorance, but it can be proven that for a universe governed by this theory, this will typically be the case. There is apparent collapse of the wave function governing subsystems of the universe, but there is no collapse of the universal wavefunction.

Two-slit experiment


The double-slit experiment is an illustration of wave-particle duality. In it, a beam of particles (such as photons) travels through a barrier with two slits removed. If one puts a detector screen on the other side, the pattern of detected particles shows interference fringes characteristic of waves; however, the detector screen responds to particles. The system exhibits behaviour of both waves (interference patterns) and particles (dots on the screen).

If we modify this experiment so that one slit is closed, no interference pattern is observed. Thus, the state of both slits affects the final results. We can also arrange to have a minimally invasive detector at one of the slits to see which slit the particle went through. When we do that, the interference pattern disappears.

The Copenhagen interpretation requires that the particles are not localised in space until they are detected, and travel through both slits. If one slit has a detector on it, then the wavefunction collapses due to that detection. This is the reason that having a well-timed collapse of the wavefunction is crucial.

In de Broglie-Bohm theory, the wavefunction travels through both slits. The initial positions of the particles are not known to the observer, but they are guided by the wavefunction. Each particle that is detected has traveled through exactly one of the slits. The interference pattern on the screen is built up out of many particles traveling along the interference pattern. The wave interferes with itself and that is translated to the particles via the guiding equation. The particles have a single, well-defined trajectory and, in particular, they are localized at all times.

To explain the behavior when the particle is detected to go through one slit, one needs to appreciate the role of the conditional wavefunction and how it gives rise to the collapse of the wavefunction; this is explained below. The basic idea is that the environment registering the detection effectively separates the two wave packets in configuration space.

Measuring spin and polarization
It is not possible to measure the spin or polarization of a particle directly; instead, the component in one direction is measured; the outcome from a single particle may be 1, meaning that the particle is aligned with the measuring apparatus, or -1, meaning that it is aligned the opposite way. For an ensemble of particles, if we expect the particles to be aligned, the results are all 1. If we expect them to be aligned oppositely, the results are all -1. For other alignments, we expect some results to be 1 and some to be -1 with a probability that depends on the expected alignment. For a full explanation of this, see the Stern-Gerlach Experiment.

In de Broglie-Bohm theory, the results of a spin experiment cannot be analyzed without some knowledge of the experimental setup. It is possible to modify the setup so that the trajectory of the particle is unaffected, but that the particle with one setup registers as spin up while in the other setup it registers as spin down. That is to say, the particle's spin is not an intrinsic property of the particle. This is related to naive realism about operators.

Measurements, the quantum formalism, and observer independence
De Broglie-Bohm theory gives the same results as quantum mechanics. It treats the wavefunction as a fundamental object in the theory as the wavefunction describes how the particles move. This means that no experiment can distinguish between the two theories. This section outlines the ideas as to how the standard quantum formalism arises out of quantum mechanics. References include Bohm's original 1952 paper and Dürr et al.

Collapse of the wavefunction
De Broglie-Bohm theory is a theory that applies to the whole universe. That is, there is a single wavefunction governing the motion of all of the particles in the universe according to the guiding equation. Theoretically, the motion of one particle depends on the positions of all of the other particles in the universe. In some situations, such as in experimental systems, we can represent the system itself in terms of a de Broglie-Bohm theory in which the wavefunction of the system is obtained by conditioning on the environment of the system. Thus, the system can be analyzed with Schrödinger's equation and the guiding equation, with an initial $$|\psi|^2$$ distribution for the particles in the system.

It requires a special setup for the conditional wavefunction of a system to obey a quantum evolution. When a system interacts with its environment, such as through a measurement, then the conditional wavefunction of the system evolves in a different way. The evolution of the universal wavefunction can become such that the wavefunction of the system appears to be in a superposition of distinct states. But if the environment has recorded the results of the experiment, then using the actual Bohmian configuration of the environment to condition on, the conditional wavefunction collapses to just one alternative, the one corresponding with the measurement results.

Collapse of the universal wavefunction never occurs in de Broglie-Bohm theory. Its entire evolution is governed by Schrödinger's equation and the particles' evolutions are governed by the guiding equation. Collapse only occurs in a phenomenological way for systems that seem to follow their own Schrödinger's equation. As this is an effective description of the system, it is a matter of choice as to what to define the experimental system to include and this will affect when "collapse" occurs.

Operators as observables
In the standard quantum formalism, measuring observables is generally thought of as measuring operators on the Hilbert space. For example, measuring position is considered to be a measurement of the position operator. This is taken to be an axiom of standard quantum mechanics. This axiom can be understood and deduced from de Broglie-Bohm theory. A major point of the analysis is that many of the measurements of the observables do not correspond to properties of the particles; they are measurements of the wavefunction.

In the history of de Broglie-Bohm theory, the proponents have often had to deal with claims that this theory is impossible. Such arguments are generally based on inappropriate analysis of operators as observables. If one believes that the spin measurements is indeed measuring the spin of a particle that existed prior to the measurement, then one does reach contradictions. De Broglie-Bohm theory deals with this by noting that spin is not a feature of the particle, but rather that of the wavefunction. As such, it only has a definite outcome once the experimental apparatus is chosen. Once that is taken into account, the impossibility theorems become irrelevant.

There have also been claims that experiments reject the Bohm trajectories in favor of the standard QM lines. As shown in and, such experiments cited above only disprove a misinterpretation of the de Broglie-Bohm theory, not the theory itself.

There are also objections to this theory based on what it says about particular situations usually involving eigenstates of an operator. For example, the ground state of hydrogen is a real wavefunction. According to the guiding equation, this means that the electron is at rest when in this state. Nevertheless, it is distributed according to $$|\psi|^2$$ and no contradiction to experimental results is possible to detect.

Operators as observables leads many to believe that many operators are equivalent. De Broglie-Bohm theory, from this perspective, chooses the position observable as a favored observable rather than, say, the momentum observable. Again, the link to the position observable is a consequence of the dynamics. The motivation for de Broglie-Bohm theory is to describe a system of particles. This implies that the goal of the theory is to describe the positions of those particles at all times. Other observables do not have this compelling ontological status. Having definite positions explains having definite results such as flashes on a detector screen. Other observables would not lead to that conclusion, but there need not be any problem in defining a mathematical theory for other observables; see Hyman et al for an exploration of the fact that a probability density and probability current can be defined for any set of commuting operators.

Hidden variables
De Broglie-Bohm theory is often referred to as a "hidden variable" theory. The idea is that particle positions are not directly observable. And that is based on the fact that the wavefunction does not depend on the positions of the particles. There is no analogue of Newton's third law in this theory. Since particles cannot influence the wavefunction it would seem naively that they are superfluous and unobservable.

But as demonstrated by flashes on the screen and the definiteness of results of experiments, the particles are what humans, more or less, are observing. While the exact details of how humans process such information and what is based on is beyond the scope of this theory, the basic idea of any particle ontology is that if the particles in the theory appear where they seem to be from human observations, then it is considered a successful prediction.

Heisenberg's uncertainty principle
The Heisenberg uncertainty principle states that when two complementary measurements are made, there is a limit to the product of their accuracy. As an example, if one measures the position with an accuracy of $$\Delta x$$, and the momentum with an accuracy of $$\Delta p$$, then $$\Delta x\Delta p\gtrsim h.$$ If we make further measurements in order to get more information, we disturb the system and change the trajectory into a new one depending on the measurement setup; therefore, the measurement results are still subject to Heisenberg's uncertainty relation.

In de Broglie-Bohm theory, there is no theoretical uncertainty in the position and momentum of a particle. Each particle has a well defined trajectory. Observers have limited knowledge as to what this trajectory is (and thus of the position and momentum). It is the knowledge of the particle's trajectory that is subject to the uncertainty relation. What we know about the particle in the present is described by the same wavefunction that other interpretations use, so the uncertainty relation can be derived in the same way as for other interpretations of quantum mechanics.

To put the statement differently, the particles' positions are only known statistically. As in classical mechanics, successive observations of the particles' positions refine the initial conditions. Thus, with succeeding observations, the initial conditions become more and more restricted. This formalism is consistent with the normal use of the Schrödinger equation.

For the derivation of the uncertainty relation, see Heisenberg uncertainty principle, noting that it describes it from the viewpoint of the Copenhagen interpretation.

Quantum entanglement, Einstein-Podolsky-Rosen paradox, Bell's theorem, and nonlocality
De Broglie-Bohm theory has highlighted the issue of nonlocality: it inspired John Stewart Bell to prove his now-famous theorem, which in turn led to the Bell test experiments. These showed that all theories of quantum mechanics must address nonlocality.

In the Einstein-Podolsky-Rosen paradox, the authors point out that it is possible to create pairs of particles with quantum states that are mirror-images of each other; these particles are now described as entangled. They describe a thought-experiment showing that either quantum mechanics is an incomplete theory or that it has nonlocality.

John Bell then described Bell's theorem (see p. 14 in Bell ), in which he asserts that all hidden-variable theories (including the Bohm interpretation) have nonlocality. Bell went further, arguing that quantum mechanics itself is nonlocal and that this cannot be avoided by appealing to any alternative interpretation (p. 196 in Bell ): "It is known that with Bohm's example of EPR correlations, involving particles with spin, there is an irreducible nonlocality."

Alain Aspect took this further by creating Bell test experiments, which realize the thought experiments on which Bell's theorem is based. He was able to show experimentally that Bell's results hold.

In the Bell test experiment, entangled pairs of particles are created; the particles are separated, traveling to remote measuring apparatus. The orientation of the measuring apparatus can be changed while the particles are in flight, demonstrating the apparent non-locality of the effect. The apparatus makes a statistical detection of the orientation of the particles. Bell asserts that the results at each detector depend on the orientation of both detectors.

The Bohm interpretation describes this experiment as follows: to understand the evolution of these particles, we need to set up a wave equation for both particles; the orientation of the apparatus affects the wavefunction. The particles in the experiment follow the guidance of the wavefunction. It is the wavefunction that carries the faster-than-light effect of changing the orientation of the apparatus. As such, this nonlocality is present in some form in any quantum theory. An analysis of exactly what kind of nonlocality is present and how it is compatible with relativity can be found in Maudlin. Note that in Bell's work, and in more detail in Maudlin's work, it is shown that the nonlocality does not allow for signaling at speeds faster than light.

Classical limit
Bohm's formulation of de Broglie-Bohm theory in terms of a classical-looking version has the merits that the emergence of classical behavior seems to follow immediately for any situation in which the quantum potential is negligible, as noted by Bohm in 1952. Modern methods of decoherence are relevant to an analysis of this limit. See Allori et al for steps towards a rigorous analysis.

Quantum trajectory method
Work by Robert Wyatt in the early 2000s attempted to use the Bohm "particles" as an adaptive mesh that follows the actual trajectory of a quantum state in time and space. In the "quantum trajectory" method, one samples the quantum wavefunction with a mesh of quadrature points. One then evolves the quadrature points in time according to the Bohm equations of motion. At each time-step, one then re-synthesizes the wavefunction from the points, recomputes the quantum forces, and continues the calculation. (Quick-time movies of this for H+H2 reactive scattering can be found on the Wyatt group web-site at UT Austin.) This approach has been adapted, extended, and used by a number of researchers in the Chemical Physics community as a way to compute semi-classical and quasi-classical molecular dynamics. A recent (2007) issue of the Journal of Physical Chemistry A was dedicated to Prof. Wyatt and his work on "Computational Bohmian Dynamics".

Eric Bittner's group at the University of Houston has advanced a statistical variant of this approach that uses Bayesian sampling technique to sample the quantum density and compute the quantum potential on a structureless mesh of points. This technique was recently used to estimate quantum effects in the heat-capacity of small clusters Nen for n~100.

There remain difficulties using the Bohmian approach, mostly associated with the formation of singularities in the quantum potential due to nodes in the quantum wavefunction. In general, nodes forming due to interference effects lead to the case where $$\frac{1}{R}\nabla^2R\rightarrow\infty.$$ This results in an infinite force on the sample particles forcing them to move away from the node and often crossing the path of other sample points (which violates single-valuedness). Various schemes have been developed to overcome this; however, no general solution has yet emerged.

These methods, as does Bohm's Hamilton-Jacobi formulation, do not apply to situations in which the full dynamics of spin need to be taken into account.

Relation to the many worlds interpretation
For those familiar with the many worlds interpretation, de Broglie-Bohm theory is often criticized as having a superfluous particle. Talking of Bohm's approach, Everett says: Our main criticism of this view is on the grounds of simplicity - if one desires to hold the view that $\psi$ is a real field then the associated particle is superfluous since, as we have endeavored to illustrate, the pure wave theory is itself satisfactory.

Bell responded to this criticism by noting that the many worlds interpretation does contain more than just the wavefunction, namely the instrumental readings upon which the world divides itself (p. 97-98 in Bell ). From the perspective of Everett, de Broglie-Bohm theory is based on giving reality to a single branch of the wave function. As Bell argues, a compromise between the two with no vagueness is as follows: But we learn from Evertt that if we do not like these trajectories we can simply leave them out. We could just as well redistribute the configuration $(\mathbf{x}_1, \mathbf{x}_2, \ldots)$ at random (with weight $

Derivations
De Broglie-Bohm theory has been derived many times and in many ways. Below are five derivations all of which are very different and lead to different ways of understanding and extending this theory.


 * Schrödinger's equation can be derived by using Einstein's light quanta hypothesis: $$E = \hbar \omega \;$$ and de Broglie's hypothesis: $$\mathbf{p} = \hbar \mathbf{k}\;$$.


 * The guiding equation can be derived in a similar fashion. We assume a plane wave: $$\psi(\mathbf{x},t) = Ae^{i(\mathbf{k}\cdot\mathbf{x}- \omega t)}$$. Notice that $$i\mathbf{k}= \nabla\psi /\psi$$. Assuming that $$\mathbf{p} = m \mathbf{v}$$ for the particle's actual velocity, we have that $$\mathbf{v}= \frac{-i \hbar}{m} \left(\frac{\nabla\psi}{\psi}\right) $$. In greater generality, that equation would be complex-valued. Taking the real part of that vector is a reasonable guess and, because of the $$i$$ factor, this translates into the imaginary part. Thus, we have the guiding equation.


 * Notice that this derivation does not use Schrödinger's equation.


 * Preserving the density under the time evolution is another method of derivation. This is the method that Bell cites. It is this method which generalizes to many possible alternative theories. The starting point is the continuity equation $$-\frac{\partial \rho}{\partial t} = \nabla \cdot (\rho v^{\psi})$$ for the density $$\rho=|\psi|^2$$. This equation describes a probability flow along a current. We take the velocity field associated with this current as the velocity field whose integral curves yield the motion of the particle.


 * A method applicable for particles without spin is to do a polar decomposition of the wavefunction and transform Schrödinger's equation into two coupled equations: the continuity equation from above and the Hamilton-Jacobi equation. This is the method used by Bohm in 1952. The decomposition and equations are as follows:


 * Decomposition: $$\psi(\mathbf{x},t) = R(\mathbf{x},t)e^{i S(\mathbf{x},t) / \hbar}.$$ Note $$R^2(\mathbf{x},t)$$ corresponds to the probability density $$\rho (\mathbf{x},t) = |\psi (\mathbf{x},t)|^2$$.


 * Continuity Equation: $$-\frac{\partial \rho(\mathbf{x},t)}{\partial t} = \nabla \cdot \left(\rho (\mathbf{x},t)\frac{\nabla S(\mathbf{x},t)}{m}\right)$$


 * Hamilton-Jacobi Equation: $$\frac{\partial S(\mathbf{x},t)}{\partial t} = -\left[ V + \frac{1}{2m}(\nabla S(\mathbf{x},t))^2 -\frac{\hbar ^2}{2m} \frac{\nabla ^2R(\mathbf{x},t)}{R(\mathbf{x},t)} \right]. $$


 * The Hamilton-Jacobi equation is the equation derived from a Newtonian system with potential $$V -\frac{\hbar ^2}{2m} \frac{\nabla ^2 R}{R}$$ and velocity field $$\frac{\nabla S}{m}.$$ The potential $$V$$ is the classical potential that appears in Schrödinger's equation and the other term involving $$R$$ is the quantum potential, terminology introduced by Bohm.


 * This leads to viewing the quantum theory as particles moving under the classical force modified by a quantum force. However, unlike standard Newtonian mechanics, the initial velocity field is already specified by $$\frac{\nabla S}{m}$$ which is a symptom of this being a first-order theory, not a second-order theory.


 * A fourth derivation was given by Dürr et al. In their derivation, they derive the velocity field by demanding the appropriate transformation properties given by the various symmetries that Schrödinger's equation satisfies, once the wavefunction is suitably transformed. The guiding equation is what emerges from that analysis.


 * A fifth derivation, given by Dürr et al is appropriate for generalization to quantum field theory and the Dirac equation. The idea is that a velocity field can also be understood as a first order differential operator acting on functions. Thus, if we know how it acts on functions, we know what it is. Then given the Hamiltonian operator $$H$$, the equation to satisfy for all functions $$f$$ (with associated multiplication operator $$\hat{f}$$) is


 * $$(v(f))(q) = \mathrm{Re} \frac{(\psi, \frac{i}{\hbar} [H,\hat f] \psi)}{(\psi,\psi)}(q)$$ where $$(v,w)$$ is the local Hermitian inner product on the value space of the wavefunction.


 * This formulation allows for stochastic theories such as the creation and annihilation of particles.

History
De Broglie-Bohm theory has a history of different formulations and names. In this section, each stage is given a name and a main reference.

Pilot-wave theory
This was the theory which de Broglie presented at the 1927 Solvay Conference. At the conference, Wolfgang Pauli pointed out that it did not deal properly with the case of inelastic scattering. De Broglie was persuaded by this argument, and abandoned this theory. Later, in 1932, John von Neumann published a paper, claiming to prove that all hidden-variable theories are impossible. This clearly applied to de Broglie's theories.

This stage applies to many spin-less particles, and is deterministic, but lacks an adequate theory of measurement. An analysis of de Broglie's presentation is given in Bacciagaluppi et al.

Around this time Madelung also developed a hydrodynamic version of Schrödinger's equation which is the basis for the density current derivation of de Broglie-Bohm theory.

De Broglie-Bohm Theory
This was described by Bohm's original papers 'A Suggested Interpretation of the Quantum Theory in Terms of "Hidden Variables" I and II' [Bohm 1952]. It extended the original Pilot Wave Theory to incorporate a consistent theory of measurement, and to address a criticism of Pauli that de Broglie did not properly respond to; it is taken to be deterministic (though Bohm hinted in the original papers that there should be disturbances to this, in the way Brownian motion disturbs Newtonian mechanics). This stage is known as the de Broglie-Bohm Theory in Bell's work [Bell 1987] and is the basis for 'The Quantum Theory of Motion' [Holland 1993].

This stage applies to multiple particles, and is deterministic.

The de Broglie-Bohm theory is an example of a hidden variables theory. Bohm originally hoped that hidden variables could provide a local, causal, objective description that would resolve or eliminate many of the paradoxes of quantum mechanics, such as Schrödinger's cat, the measurement problem and the collapse of the wavefunction. However, Bell's theorem complicates this hope, as it demonstrates that there can be no local theory that is compatible with the predictions of quantum mechanics. The Bohmian interpretation is causal but not local.

Bohm became dissatisfied with the conventional interpretation of quantum mechanics, pointing out that, although it requires one to give up "the possibility of even conceiving what might determine the behaviour of an individual system at a quantum level", it does not prove that this requirement is necessary. Indeed, it was his understanding, given 44to him by Einstein, of the flawed nature of von Neumann's proof that inspired his papers.

Bohm's paper was largely ignored by other physicists; it was not supported by Albert Einstein (who was also dissatisfied with the prevailing orthodoxy and had discussed Bohm's ideas with him before publication). Bohm eventually abandoned it.

The cause was taken up by John Bell. In "Speakable and Unspeakable in Quantum Mechanics" [Bell 1987], several of the papers refer to hidden variables theories (which include Bohm's). Bell showed that Pauli's and von Neumann's objections amounted to showing that hidden variables theories are nonlocal, and that nonlocality is a feature of all quantum mechanical systems.

Bohmian Mechanics
This term is used to describe the same theory, but with an emphasis on the notion of current flow. In particular, it is often used to include most of the further extensions past the spin-less version of Bohm. While de Broglie-Bohm theory has Lagrangians and Hamilton-Jacobi equations as a primary focus and backdrop, with the icon of the quantum potential, Bohmian mechanics considers the continuity equation as primary and has the guiding equation as its icon. They are mathematically equivalent in so far as the Hamilton-Jacobi formulation applies, i.e., spin-less particles. The papers of Dürr et al popularized the term.

All of non-relativistic quantum mechanics can be fully accounted for in this theory.

Causal Interpretation and Ontological Interpretation
Bohm developed his original ideas, calling them the Causal Interpretation. Later he felt that causal sounded too much like deterministic and preferred to call his theory the Ontological Interpretation. The main reference is 'The Undivided Universe' [Bohm, Hiley 1993].

This stage covers work by Bohm and in collaboration with Vigier and Hiley. Bohm is clear that this theory is non-deterministic (the work with Hiley includes a stochastic theory)