Pilot wave theory

(a) A walker in a circular corral. Trajectories of increasing length are colour-coded according to the droplet’s local speed (b) The probability distribution of the walker’s position corresponds roughly to the amplitude of the corral’s Faraday wave mode.[19]

The pilot wave theory is a hidden-variable theory. Consequently:

• the theory has realism (meaning that its concepts exist independently of the observer);
• the theory has determinism.

The positions of the particles are considered to be the hidden variables. The observer doesn't know the precise values of these variables; they cannot know them precisely because any measurement disturbs them. On the other hand, the observer is defined not by the wave function of their own 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.[20]

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.[21]

Pilot wave theory is based on Hamilton–Jacobi dynamics,[29] rather than Lagrangian or Hamiltonian dynamics. Using the Hamilton–Jacobi equation

${\displaystyle H\left(\,{\vec {x}}\,,\;{\vec {\nabla }}_{\!x}\,S\,,\;t\,\right)+{\partial S \over \partial t}\left(\,{\vec {x}},\,t\,\right)=0}$

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 ${\displaystyle \rho ({\vec {x}},t)}$ is known. Probability must be conserved, i.e. ${\displaystyle \int \rho \,\mathrm {d} ^{3}{\vec {x}}=1}$ for each ${\displaystyle t}$. Therefore, it must satisfy the continuity equation

${\displaystyle {\frac {\,\partial \rho \,}{\partial t}}=-{\vec {\nabla }}\cdot (\rho \,{\vec {v}})\qquad \qquad (1)}$

where ${\displaystyle \,{\vec {v}}({\vec {x}},t)\,}$ is the velocity of the particle.

In the Hamilton–Jacobi formulation of classical mechanics, velocity is given by ${\displaystyle \;{\vec {v}}({\vec {x}},t)={\frac {1}{\,m\,}}\,{\vec {\nabla }}_{\!x}S({\vec {x}},\,t)\;}$ where ${\displaystyle \,S({\vec {x}},t)\,}$ is a solution of the Hamilton-Jacobi equation

${\displaystyle -{\frac {\partial S}{\partial t}}={\frac {\;\left|\,\nabla S\,\right|^{2}\,}{2m}}+{\tilde {V}}\qquad \qquad (2)}$

${\displaystyle \,(1)\,}$ and ${\displaystyle \,(2)\,}$ can be combined into a single complex equation by introducing the complex function ${\displaystyle \;\psi ={\sqrt {\rho \,}}\,e^{\frac {\,i\,S\,}{\hbar }}\;,}$ then the two equations are equivalent to

${\displaystyle i\,\hbar \,{\frac {\,\partial \psi \,}{\partial t}}=\left(-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+{\tilde {V}}-Q\right)\psi \quad }$

with

${\displaystyle \;Q=-{\frac {\;\hbar ^{2}\,}{\,2m\,}}{\frac {\nabla ^{2}{\sqrt {\rho \,}}}{\sqrt {\rho \,}}}~.}$

The time-dependent Schrödinger equation is obtained if we start with ${\displaystyle \;{\tilde {V}}=V+Q\;,}$ the usual potential with an extra quantum potential ${\displaystyle Q}$. 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.

Note this potential is the same one that appears in the Madelung equations, a classical analog of the Schrödinger equation.

The matter wave of de Broglie is described by the time-dependent Schrödinger equation:

${\displaystyle i\,\hbar \,{\frac {\,\partial \psi \,}{\partial t}}=\left(-{\frac {\hbar ^{2}}{\,2m\,}}\nabla ^{2}+V\right)\psi \quad }$

The complex wave function can be represented as:

${\displaystyle \psi ={\sqrt {\rho \,}}\;\exp \left({\frac {i\,S}{\hbar }}\right)~}$

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 density ${\displaystyle \,\rho \,:}$[15]

${\displaystyle {\frac {\,\partial \rho \,}{\,\partial t\,}}+{\vec {\nabla }}\cdot \left(\rho \,{\vec {v}}\right)=0~,}$

where the velocity field is determined by the “guidance equation”

${\displaystyle {\vec {v}}\left(\,{\vec {r}},\,t\,\right)={\frac {1}{\,m\,}}\,{\vec {\nabla }}S\left(\,{\vec {r}},\,t\,\right)~.}$

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 ${\displaystyle \;\rho =|\psi |^{2}~.}$ 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 S:

${\displaystyle -{\frac {\partial S}{\partial t}}={\frac {\;\left|\,{\vec {\nabla }}S\,\right|^{2}\,}{\,2m\,}}+V+Q~,}$

where Q is the quantum potential defined by

${\displaystyle Q=-{\frac {\hbar ^{2}}{\,2m\,}}{\frac {\nabla ^{2}{\sqrt {\rho \,}}}{\sqrt {\rho \,}}}~.}$

If we choose to neglect Q, our equation is reduced to the Hamilton–Jacobi equation of a classical point particle.[lower-alpha 1] 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

${\displaystyle m\,{\frac {d}{dt}}\,{\vec {v}}=-{\vec {\nabla }}(V+Q)~,}$

where the hydrodynamic time derivative is defined as

${\displaystyle {\frac {d}{dt}}={\frac {\partial }{\,\partial t\,}}+{\vec {v}}\cdot {\vec {\nabla }}~.}$

The Schrödinger equation for the many-body wave function ${\displaystyle \psi ({\vec {r}}_{1},{\vec {r}}_{2},\cdots ,t)}$ is given by

${\displaystyle i\hbar {\frac {\partial \psi }{\partial t}}=\left(-{\frac {\hbar ^{2}}{2}}\sum _{i=1}^{N}{\frac {\nabla _{i}^{2}}{m_{i}}}+V(\mathbf {r} _{1},\mathbf {r} _{2},\cdots \mathbf {r} _{N})\right)\psi }$

The complex wave function can be represented as:

${\displaystyle \psi ={\sqrt {\rho \,}}\;\exp \left({\frac {i\,S}{\hbar }}\right)}$

The pilot wave guides the motion of the particles. The guidance equation for the jth particle is:

${\displaystyle {\vec {v}}_{j}={\frac {\nabla _{j}S}{m_{j}}}\;.}$

The velocity of the jth particle explicitly depends on the positions of the other particles. This means that the theory is nonlocal.

Lucien Hardy[30] and John Stewart Bell[20] 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,[31] and not associated with a particle. The same concept was called ghost waves (or "Gespensterfelder", ghost fields) by Albert Einstein.[31] The empty wave function notion has been discussed controversially.[32][33][34] In contrast, the many-worlds interpretation of quantum mechanics does not call for empty wave functions.[20]