Static forces and virtual-particle exchange

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.

This article uses the path integral formulation to describe the force carriers for spin 0, 1, and 2 fields.

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,[3] thus, the validity of the "force-mediating particle" picture is questionable.

[citation needed] 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 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 concept of field was invented to mediate the interaction among bodies thus eliminating the need for action at a distance.

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.

, the probability amplitude for the creation, propagation, and destruction of a virtual particle is given, in the path integral formulation by

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 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.

Linearizing the Fermi energy to first order in the density fluctuation and combining with Poisson's equation yields the screening length.

We consider a charge density in tube with axis along a magnetic field embedded in an electron gas

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.

are the number of particles in the electron gas in the absence of and in the presence of an electrostatic potential, respectively.

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.

[7]: 187–190  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.

we recover the classical situation in which the electron orbits the magnetic field at the Larmor radius.

For large values of angular momentum, the energy can have local minima at distances other than zero and infinity.

The lowest energy states for odd total angular momentum occur when

Also, as with the delta function charges, the energy at the minimum increases as the ratio of angular momenta varies from one.

If the expectation value of the interaction energy is taken over a Laughlin wavefunction, these series are also preserved.

Consider a tube of current rotating in a magnetic field embedded in a simple plasma or electron gas.

A current in a plasma confined to the plane perpendicular to the magnetic field generates an extraordinary wave.

[5]: 110–112  This wave generates Hall currents that interact and modify the electromagnetic field.

In the limit of small graviton mass, we recover the inverse-square behavior of Newton's Law.

[2]: 32–37 Unlike the electrostatic case, however, taking the small-mass limit of the boson does not yield the correct result.

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 . Here . 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 or . 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 or . 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 .