calculating a position field from the time integral of the velocity field, and finally calculating a quantum trajectory from the expectation value of the position field. The quantum trajectories are of course spin dependent, and the theory can be validated by checking that Pauli's exclusion principle is obeyed for a collection of fermions. Classical forces The force exerted by one mass on another and the force exerted by one charge on another are strikingly similar. Both fall off as the square of the distance between the bodies. Both are proportional to the product of properties of the bodies, mass in the case of gravitation and charge in the case of electrostatics. They also have a striking difference. Two masses attract each other, while two like charges repel each other. In both cases, the bodies appear to act on each other over a distance. The concept of field was invented to mediate the interaction among bodies thus eliminating the need for action at a distance. The gravitational force is mediated by the gravitational field and the Coulomb force is mediated by the electromagnetic field. Gravitational force The gravitational force on a mass exerted by another mass is where is the gravitational constant, is the distance between the masses, and is the unit vector from mass to mass . The force can also be written where is the gravitational field described by the field equation where is the mass density at each point in space. Coulomb force The electrostatic Coulomb force on a charge exerted by a charge is (SI units) where is the vacuum permittivity, is the separation of the two charges, and is a unit vector in the direction from charge to charge . The Coulomb force can also be written in terms of an electrostatic field: where being the charge density at each point in space. Virtual-particle exchange In perturbation theory, forces are generated by the exchange of virtual particles. The mechanics of virtual-particle exchange is best described with the path integral formulation of quantum mechanics. There are insights that can be obtained, however, without going into the machinery of path integrals, such as why classical gravitational and electrostatic forces fall off as the inverse square of the distance between bodies. Path-integral formulation of virtual-particle exchange A virtual particle is created by a disturbance to the vacuum state, and the virtual particle is destroyed when it is absorbed back