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 exerted by another mass is
where is the Newtonian constant of gravitation, 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 into the vacuum state by another disturbance. The disturbances are imagined to be due to bodies that interact with the virtual particle’s field.
Probability amplitude Using natural units, , the probability amplitude for the creation, propagation, and destruction of a virtual particle is given, in the path integral formulation by
where is the Hamiltonian operator, is elapsed time, is the energy change due to the disturbance, is the change in action due to the disturbance, is the field of the virtual particle, the integral is over all paths, and the classical action is given by
where is the Lagrangian density.
Here, the spacetime metric is given by
The path integral often can be converted to the form
where is a differential operator with and functions of spacetime. The first term in the argument represents the free particle and the second term represents the disturbance to the field from an external source such as a charge or a mass.
The integral can be written (see )
where
is the change in the action due to the disturbances and the propagator is the solution of
Energy of interaction
We assume that there are two point disturbances representing two bodies and that the disturbances are motionless and constant in time. The disturbances can be written
where the delta functions are in space, the disturbances are located at and , and the coefficients and are the strengths of the disturbances.
If we neglect self-interactions of the disturbances then W becomes
which can be written
Here is the Fourier transform of
Finally, the change in energy due to the static disturbances of the vacuum is
If this quantity is negative, the force is attractive. If it is positive, the force is repulsive.
Examples of static, motionless, interacting currents are the Yukawa potential, the Coulomb potential in a vacuum, and the Coulomb potential in a simple plasma or electron gas.
The expression for the interaction energy can be generalized to the situation in which the point particles are moving, but the motion is slow compared with the speed of light. Examples are the Darwin interaction in a vacuum and in a plasma.
Finally, the expression for the interaction energy can be generalized to situations in which the disturbances are not point particles, but are possibly line charges, tubes of charges, or current vortices. Examples include: two line charges embedded in a plasma or electron gas, Coulomb potential between two current loops embedded in a magnetic field, and the magnetic interaction between current loops in a simple plasma or electron gas. As seen from the Coulomb interaction between tubes of charge example, shown below, these more complicated geometries can lead to such exotic phenomena as fractional quantum numbers.
Selected examples
Yukawa potential: the force between two nucleons in an atomic nucleus
Consider the spin-0 Lagrangian density
The equation of motion for this Lagrangian is the Klein–Gordon equation
If we add a disturbance the probability amplitude becomes
If we integrate by parts and neglect boundary terms at infinity the probability amplitude becomes
With the amplitude in this form it can be seen that the propagator is the solution of
From this it can be seen that
The energy due to the static disturbances becomes (see )
with
which is attractive and has a range of
Yukawa proposed that this field describes the force between two nucleons in an atomic nucleus. It allowed him to predict both the range and the mass of the particle, now known as the pion, associated with this field.
Electrostatics
Coulomb potential in vacuum Consider the spin-1 Proca Lagrangian with a disturbance
where
charge is conserved
and we choose the Lorenz gauge
Moreover, we assume that there is only a time-like component to the disturbance. In ordinary language, this means that there is a charge at the points of disturbance, but there are no electric currents.
If we follow the same procedure as we did with the Yukawa potential we find that
which implies
and
This yields
for the timelike propagator and
which has the opposite sign to the Yukawa case.
In the limit of zero photon mass, the Lagrangian reduces to the Lagrangian for electromagnetism
Therefore the energy reduces to the potential energy for the Coulomb force and the coefficients and are proportional to the electric charge. Unlike the Yukawa case, like bodies, in this electrostatic case, repel each other.
Coulomb potential in a simple plasma or electron gas
Plasma waves
The dispersion relation for plasma waves is
where is the angular frequency of the wave,
is the plasma frequency, is the magnitude of the electron charge, is the electron mass, is the electron temperature (the Boltzmann constant equal to one), and is a factor that varies with frequency from one to three. At high frequencies, on the order of the plasma frequency, the compression of the electron fluid is an adiabatic process and is equal to three. At low frequencies, the compression is an isothermal process and is equal to one. Retardation effects have been neglected in obtaining the plasma-wave dispersion relation.
For low frequencies, the dispersion relation becomes
where
is the Debye number, which is the inverse of the Debye length. This suggests that the propagator is
In fact, if the retardation effects are not neglected, then the dispersion relation is
which does indeed yield the guessed propagator. This propagator is the same as the massive Coulomb propagator with the mass equal to the inverse Debye length. The interaction energy is therefore
The Coulomb potential is screened on length scales of a Debye length.
Plasmons
In a quantum electron gas, plasma waves are known as plasmons. Debye screening is replaced with Thomas–Fermi screening to yield pp. 296-299.
where the inverse of the Thomas–Fermi screening length is
and is the Fermi energy
This expression can be derived from the chemical potential for an electron gas and from Poisson's equation. The chemical potential for an electron gas near equilibrium is constant and given by
where is the electric potential. Linearizing the Fermi energy to first order in the density fluctuation and combining with Poisson's equation yields the screening length. The force carrier is the quantum version of the plasma wave.
Two line charges embedded in a plasma or electron gas
We consider a line of charge with axis in the z direction embedded in an electron gas
where is the distance in the xy-plane from the line of charge, is the width of the material in the z direction. The superscript 2 indicates that the Dirac delta function is in two dimensions. The propagator is
where is either the inverse Debye–Hückel screening length or the inverse Thomas–Fermi screening length.
The interaction energy is