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.
The "force-mediating particle" picture (FMPP) is used because the classical two-body interaction (Coulomb's law for example), depending on six spatial dimensions, is incompatible with the Lorentz invariance of Dirac's equation. The use of the 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.
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.
The gravitational force on a mass exerted by another mass is
where G is the gravitational constant, r 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.
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:
being the charge density at each point in space.
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 field.
The 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 Common integrals in quantum field theory)
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 Yukawa Potential, The Coulomb potential in a vacuum, and 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 Darwin interaction in a vacuum and Darwin interaction 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 are Two line charges embedded in a plasma or electron gas, Coulomb potential between two current loops embedded in a magnetic field, and 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.
The 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 Common integrals in quantum field theory)
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.
The Coulomb potential in a vacuum
Consider the spin-1 Proca Lagrangian with a disturbance
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
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
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 (Boltzmann's 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
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.
In a quantum electron gas, plasma waves are known as plasmons. Debye screening is replaced with Thomas–Fermi screening to yield
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
are Bessel functions and is the distance between the two line charges. In obtaining the interaction energy we made use of the integrals (see Common integrals in quantum field theory)
For , we have
Coulomb potential between two current loops embedded in a magnetic field
Interaction energy for vortices
We consider a charge density in tube with axis along a magnetic field embedded in an electron gas
where is the distance from the guiding center, is the width of the material in the direction of the magnetic field
where the cyclotron frequency is (Gaussian units)
is the speed of the particle about the magnetic field, and B is the magnitude of the magnetic field. The speed formula comes from setting the classical kinetic energy equal to the spacing between Landau levels in the quantum treatment of a charged particle in a magnetic field.
In this geometry, the interaction energy can be written
where is the distance between the centers of the current loops and
is a Bessel function of the first kind. In obtaining the interaction energy we made use of the integral
Electric field due to a density perturbation
The chemical potential near equilibrium, is given by
where is the potential energy of an electron in an electric potential and and are the number of particles in the electron gas in the absence of and in the presence of an electrostatic potential, respectively.
The density fluctuation is then
where is the area of the material in the plane perpendicular to the magnetic field.
Poisson's equation yields
The propagator is then
and the interaction energy becomes
where in the second equality (Gaussian units) we assume that the vortices had the same energy and the electron charge.
In analogy with plasmons, the force carrier is the quantum version of the upper hybrid oscillation which is a longitudinal plasma wave that propagates perpendicular to the magnetic field.
Currents with angular momentum
Delta function currents
Figure 1. Interaction energy vs. r for angular momentum states of value one. The curves are identical to these for any values of
. Lengths are in units are in
, and the energy is in units of
. Note that there are local minima for large values of
Figure 2. Interaction energy vs. r for angular momentum states of value one and five.
Figure 3. Interaction energy vs. r for various values of theta. The lowest energy is for
. The highest energy plotted is for
. Lengths are in units of
Figure 4. Ground state energies for even and odd values of angular momenta. Energy is plotted on the vertical axis and r is plotted on the horizontal. When the total angular momentum is even, the energy minimum occurs when
. When the total angular momentum is odd, there are no integer values of angular momenta that will lie in the energy minimum. Therefore, there are two states that lie on either side of the minimum. Because
, the total energy is higher than the case when
for a given value of
Unlike classical currents, quantum current loops can have various values of the Larmor radius for a given energy. Landau levels, the energy states of a charged particle in the presence of a magnetic field, are multiply degenerate. The current loops correspond to angular momentum states of the charged particle that may have the same energy. Specifically, the charge density is peaked around radii of
where is the angular momentum quantum number. When we recover the classical situation in which the electron orbits the magnetic field at the Larmor radius. If currents of two angular momentum and interact, and we assume the charge densities are delta functions at radius , then the interaction energy is
The interaction energy for is given in Figure 1 for various values of . The energy for two different values is given in Figure 2.
For large values of angular momentum, the energy can have local minima at distances other than zero and infinity. It can be numerically verified that the minima occur at
This suggests that the pair of particles that are bound and separated by a distance act as a single quasiparticle with angular momentum .
If we scale the lengths as , then the interaction energy becomes
The value of the at which the energy is minimum, , is independent of the ratio . However the value of the energy at the minimum depends on the ratio. The lowest energy minimum occurs when
When the ratio differs from 1, then the energy minimum is higher (Figure 3). Therefore, for even values of total momentum, the lowest energy occurs when (Figure 4)
where the total angular momentum is written as
When the total angular momentum is odd, the minima cannot occur for The lowest energy states for odd total angular momentum occur when
which also appear as series for the filling factor in the fractional quantum Hall effect.
Charge density spread over a wave function
The charge density is not actually concentrated in a delta function. The charge is spread over a wave function. In that case the electron density is
The interaction energy becomes
where is a confluent hypergeometric function or Kummer function. In obtaining the interaction energy we have used the integral (see Common integrals in quantum field theory)
As with delta function charges, the value of in which the energy is a local minimum only depends on the total angular momentum, not on the angular momenta of the individual currents. Also, as with the delta function charges, the energy at the minimum increases as the ratio of angular momenta varies from one. Therefore, the series
appear as well in the case of charges spread by the wave function.
The Laughlin wavefunction is an ansatz for the quasiparticle wavefunction. If the expectation value of the interaction energy is taken over a Laughlin wavefunction, these series are also preserved.
Darwin interaction in a vacuum
A charged moving particle can generate a magnetic field that affects the motion of another charged particle. The static version of this effect is called the Darwin interaction. To calculate this, consider the electrical currents in space generated by a moving charge
with a comparable expression for .
The Fourier transform of this current is
The current can be decomposed into a transverse and a longitudinal part (see Helmholtz decomposition).
The hat indicates a unit vector. The last term disappears because
which results from charge conservation. Here vanishes because we are considering static forces.
With the current in this form the energy of interaction can be written
The propagator equation for the Proca Lagrangian is
The spacelike solution is
which evaluates to (see Common integrals in quantum field theory)
which reduces to
in the limit of small m. The interaction energy is the negative of the interaction Lagrangian. For two like particles traveling in the same direction, the interaction is attractive, which is the opposite of the Coulomb interaction.
Darwin interaction in a plasma
In a plasma, the dispersion relation for an electromagnetic wave is ()
Here is the plasma frequency. The interaction energy is therefore
Magnetic interaction between current loops in a simple plasma or electron gas
The interaction energy
Consider a tube of current rotating in a magnetic field embedded in a simple plasma or electron gas. The current, which lies in the plane perpendicular to the magnetic field, is defined as
and is the unit vector in the direction of the magnetic field. Here indicates the dimension of the material in the direction of the magnetic field. The transverse current, perpendicular to the wave vector, drives the transverse wave.
The energy of interaction is
where is the distance between the centers of the current loops and
is a Bessel function of the first kind. In obtaining the interaction energy we made use of the integrals
See Common integrals in quantum field theory.
A current in a plasma confined to the plane perpendicular to the magnetic field generates an extraordinary wave. This wave generates Hall currents that interact and modify the electromagnetic field. The dispersion relation for extraordinary waves is
which gives for the propagator
in analogy with the Darwin propagator. Here, the upper hybrid frequency is given by
the cyclotron frequency is given by (Gaussian units)
and the plasma frequency (Gaussian units)
Here n is the electron density, e is the magnitude of the electron charge, and m is the electron mass.
The interaction energy becomes, for like currents,
Limit of small distance between current loops
In the limit that the distance between current loops is small,
and I and K are modified Bessel functions. we have assumed that the two currents have the same charge and speed.
We have made use of the integral (see Common integrals in quantum field theory)
For small mr the integral becomes
For large mr the integral becomes
Relation to the quantum Hall effect
The screening wavenumber can be written (Gaussian units)
where is the fine-structure constant and the filling factor is
and N is the number of electrons in the material and A is the area of the material perpendicular to the magnetic field. This parameter is important in the quantum Hall effect and the fractional quantum Hall effect. The filling factor is the fraction of occupied Landau states at the ground state energy.
For cases of interest in the quantum Hall effect, is small. In that case the interaction energy is
where (Gaussian units)
is the interaction energy for zero filling factor. We have set the classical kinetic energy to the quantum energy
A gravitational disturbance is generated by the stress–energy tensor ; consequently, the Lagrangian for the gravitational field is spin-2. If the disturbances are at rest, then the only component of the stress–energy tensor that persists is the component. If we use the same trick of giving the graviton some mass and then taking the mass to zero at the end of the calculation the propagator becomes
which is once again attractive rather than repulsive. The coefficients are proportional to the masses of the disturbances. In the limit of small graviton mass, we recover the inverse-square behavior of Newton's Law.
Unlike the electrostatic case, however, taking the small-mass limit of the boson does not yield the correct result. A more rigorous treatment yields a factor of one in the energy rather than 4/3.