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 is the potential energy, is the velocity and 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 . 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 is known. Probability must be conserved, i.e. for each . Therefore, it must satisfy the continuity equation where is the velocity of the particle. In the Hamilton–Jacobi formulation of classical mechanics, velocity is given by where is a solution of the Hamilton-Jacobi equation and can be combined into a single complex equation by introducing the complex function then the two equations are equivalent to with The time-dependent Schrödinger equation is obtained if we start with the usual potential with an extra quantum potential . 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. 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: 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 where the velocity field is determined 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