Static force fields are fields, such as a simple electric, magnetic or gravitational fields, that exist without excitations. The most common approximation method that physicists use for scattering calculations can be interpreted as static forces arising from the interactions between two bodies mediated by virtual particles, particles that exist for only a short time determined by the uncertainty principle. The virtual particles, also known as force carriers, are bosons, with different bosons associated with each force. The virtual-particle description of static forces is capable of identifying the spatial form of the forces, such as the inverse-square behavior in Newton's law of universal gravitation and in Coulomb's law. It is also able to predict whether the forces are attractive or repulsive for like bodies. The path integral formulation is the natural language for describing force carriers. This article uses the path integral formulation to describe the force carriers for spin 0, 1, and 2 fields. Pions, photons, and gravitons fall into these respective categories. There are limits to the validity of the virtual particle picture. The virtual-particle formulation is derived from a method known as perturbation theory which is an approximation assuming interactions are not too strong, and was intended for scattering problems, not bound states such as atoms. For the strong force binding quarks into nucleons at low energies, perturbation theory has never been shown to yield results in accord with experiments, thus, the validity of the "force-mediating particle" picture is questionable. Similarly, for bound states the method fails. In these cases, the physical interpretation must be re-examined. As an example, the calculations of atomic structure in atomic physics or of molecular structure in quantum chemistry could not easily be repeated, if at all, using the "force-mediating particle" picture. Use of the "force-mediating particle" picture (FMPP) is unnecessary in nonrelativistic quantum mechanics, and Coulomb's law is used as given in atomic physics and quantum chemistry to calculate both bound and scattering states. A non-perturbative relativistic quantum theory, in which Lorentz invariance is preserved, is achievable by evaluating Coulomb's law as a 4-space interaction using the 3-space position vector of a reference electron obeying Dirac's equation and the quantum trajectory of a second electron which depends only on the scaled time. The quantum trajectory of each electron in an ensemble is inferred from the Dirac current for each electron by setting it equal to a velocity field times a quantum density, 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 m {\displaystyle m} exerted by another mass M {\displaystyle M} is
F = − G m M r 2 r ^ = m g ( r ) , {\displaystyle \mathbf {F} =-G{\frac {mM}{r^{2}}}\,{\hat {\mathbf {r} }}=m\mathbf {g} \left(\mathbf {r} \right),}
where G is the Newtonian constant of gravitation, r is the distance between the masses, and r ^ {\displaystyle {\hat {\mathbf {r} }}} is the unit vector from mass M {\displaystyle M} to mass m {\displaystyle m} . The force can also be written
F = m g ( r ) , {\displaystyle \mathbf {F} =m\mathbf {g} \left(\mathbf {r} \right),}
where g ( r ) {\displaystyle \mathbf {g} \left(\mathbf {r} \right)} is the gravitational field described by the field equation
∇ ⋅ g = − 4 π G ρ m , {\displaystyle \nabla \cdot \mathbf {g} =-4\pi G\rho _{m},}
where ρ m {\displaystyle \rho _{m}} is the mass density at each point in space.
Coulomb force The electrostatic Coulomb force on a charge q {\displaystyle q} exerted by a charge Q {\displaystyle Q} is (SI units)
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