Pilot wave theory

In theoretical physics, the pilot wave theory, also known as Bohmian mechanics, was the first known example of a hidden-variable theory, presented by Louis de Broglie in 1927. Its more modern version, the de Broglie–Bohm theory, interprets quantum mechanics as a deterministic theory, avoiding troublesome notions such as wave–particle duality, instantaneous wave function collapse, and the paradox of Schrödinger's cat. To solve these problems, the theory is inherently nonlocal.
The de Broglie–Bohm pilot wave theory is one of several interpretations of (non-relativistic) quantum mechanics.
An extension to the relativistic case has been developed since the 1990s.
History.
Louis de Broglie's early results on the pilot wave theory were presented in his thesis (1924) in the context of atomic orbitals where the waves are stationary. Early attempts to develop a general formulation for the dynamics of these guiding waves in terms of a relativistic wave equation were unsuccessful until in 1926 Schrödinger developed his non-relativistic wave equation, and further suggested that since the equation described waves in configuration space the particle picture should be abandoned. Shortly thereafter, Max Born suggested that the wave function of Schrödinger's wave equation represents the probability density of finding a particle. Following these results, de Broglie developed the dynamical equations for his pilot wave theory. Initially, de Broglie proposed a "double solution" approach, in which the quantum object consists of a physical wave ("u"-wave) in real space which has a spherical singular region that gives rise to particle-like behaviour; in this initial form of his theory he did not have to postulate the existence of a quantum particle. He later formulated it as a theory in which a particle is accompanied by a pilot wave.
De Broglie presented the pilot wave theory at the 1927 Solvay Conference. However, Wolfgang Pauli raised an objection to it at the conference, saying that it did not deal properly with the case of inelastic scattering. De Broglie was not able to find a response to this objection, and he abandoned the pilot-wave approach. Unlike David Bohm years later, de Broglie did not complete his theory to encompass the many-particle case. The many-particle case shows mathematically that the energy dissipation in inelastic scattering could be distributed to the surrounding field structure by a yet-unknown mechanism of the theory of hidden variables.
In 1932, John von Neumann published a book, part of which claimed to prove that all hidden variable theories were impossible. This result was found to be flawed by Grete Hermann three years later, though this went unnoticed by the physics community for over fifty years.
In 1952, David Bohm, dissatisfied with the prevailing orthodoxy, rediscovered de Broglie's pilot wave theory. Bohm developed pilot wave theory into what is now called the de Broglie–Bohm theory. The de Broglie–Bohm theory itself might have gone unnoticed by most physicists, if it had not been championed by John Bell, who also countered the objections to it. In 1987, John Bell rediscovered Grete Hermann's work, and thus showed the physics community that Pauli's and von Neumann's objections "only" showed that the pilot wave theory did not have locality. 
Yves Couder and co-workers in 2010 reported a macroscopic pilot wave system in the form of "walking droplets". This system was said to exhibit behavior of a pilot wave, heretofore considered to be reserved to microscopic phenomena. However, more careful fluid dynamics experiments have been carried out since 2015 by two American groups and one Danish team led by Tomas Bohr (grandson of Niels Bohr). These new experiments have not replicated the results of the 2010 experiment as of 2018.
The pilot wave theory.
Principles.
The pilot wave theory is a hidden-variable theory. Consequently:
The positions of the particles are considered to be the hidden variables. The observer not only doesn't know the precise value of these variables of the quantum system considered, and cannot know them precisely because any measurement disturbs them. On the other hand, one (the observer) is defined not by the wave function of one's atoms, but by the atoms' positions. So what one sees around oneself are also the positions of nearby things, not their wave functions. 
A collection of particles has an associated matter wave, which evolves according to the Schrödinger equation. Each particle follows a deterministic trajectory, which is guided by the wave function; collectively, the density of the particles conforms to the magnitude of the wave function. The wave function is not influenced by the particle and can exist also as an empty wave function.
The theory brings to light nonlocality that is implicit in the non-relativistic formulation of quantum mechanics and uses it to satisfy Bell's theorem. These nonlocal effects can be shown to be compatible with the no-communication theorem, which prevents use of them for faster-than-light communication, and so is empirically compatible with relativity.
Mathematical foundations.
To derive the de Broglie–Bohm pilot-wave for an electron, the quantum Lagrangian
where formula_2 is the potential energy, formula_3 is the velocity and formula_4 is the potential associated with the quantum force (the particle being pushed by the wave function), is integrated along precisely one path (the one the electron actually follows). This leads to the following formula for the Bohm propagator:
This propagator allows to track the electron precisely over time under the influence of the quantum potential formula_4.
Derivation of the Schrödinger equation.
Pilot wave theory is based on Hamilton–Jacobi dynamics, rather than Lagrangian or Hamiltonian dynamics. Using the Hamilton–Jacobi equation
it is possible to derive the Schrödinger equation:
Consider a classical particle – the position of which is not known with certainty. We must deal with it statistically, so only the probability density formula_8 is known. Probability must be conserved, i.e. formula_9 for each formula_10. Therefore, it must satisfy the continuity equation
where formula_12 is the velocity of the particle.
In the Hamilton–Jacobi formulation of classical mechanics, velocity is given by formula_13 where formula_14 is a solution of the Hamilton-Jacobi equation
formula_16 and formula_17 can be combined into a single complex equation by introducing the complex function formula_18, then the two equations are equivalent to
The time dependent Schrödinger equation is obtained if we start with formula_21, the usual potential with an extra quantum potential formula_4. The quantum potential is the potential of the quantum force, which is proportional (in approximation) to the curvature of the amplitude of the wave function.
Mathematical formulation for a single particle.
The matter wave of de Broglie is described by the time-dependent Schrödinger equation:
The complex wave function can be represented as:
formula_24
By plugging this into the Schrödinger equation, one can derive two new equations for the real variables. The first is the continuity equation for the probability densityformula_25:
where the velocity field is defined by the guidance equation
According to pilot wave theory, the point particle and the matter wave are both real and distinct physical entities (unlike standard quantum mechanics, where particles and waves are considered to be the same entities, connected by wave–particle duality). 
The pilot wave guides the motion of the point particles as described by the guidance equation.
Ordinary quantum mechanics and pilot wave theory are based on the same partial differential equation. The main difference is that in ordinary quantum mechanics, the Schrödinger equation is connected to reality by the Born postulate, which states that the probability density of the particle's position is given by formula_28. Pilot wave theory considers the guidance equation to be the fundamental law, and sees the Born rule as a derived concept.
The second equation is a modified Hamilton–Jacobi equation for the action formula_29:
where Q is the quantum potential defined by
By neglecting Q, our equation is reduced to the Hamilton–Jacobi equation of a classical point particle. (Strictly speaking, this is only a semiclassical limit , because the superposition principle still holds and one needs a decoherence mechanism to get rid of it. Interaction with the environment can provide this mechanism.) So, the quantum potential is responsible for all the mysterious effects of quantum mechanics.
One can also combine the modified Hamilton–Jacobi equation with the guidance equation to derive a quasi-Newtonian equation of motion
where the hydrodynamic time derivative is defined as
Mathematical formulation for multiple particles.
The Schrödinger equation for the many-body wave function formula_34 is given by
The complex wave function can be represented as:
The pilot wave guides the motion of the particles. The guidance equation for the jth particle is:
The velocity of the jth particle explicitly depends on the positions of the other particles.
This means that the theory is nonlocal.
Empty wave function.
Lucien Hardy and John Stewart Bell have emphasized that in the de Broglie–Bohm picture of quantum mechanics there can exist empty waves, represented by wave functions propagating in space and time but not carrying energy or momentum, and not associated with a particle. The same concept was called "ghost waves" (or "Gespensterfelder", "ghost fields") by Albert Einstein. The empty wave function notion has been discussed controversially. In contrast, the many-worlds interpretation of quantum mechanics does not call for empty wave functions.