Static force fields are fields, such as a simpleelectric,magnetic orgravitational fields, that exist without excitations. Themost common approximation method that physicists use forscattering calculations can be interpreted as static forces arising from the interactions between two bodies mediated byvirtual particles, particles that exist for only a short time determined by theuncertainty principle.^[1] The virtual particles, also known asforce carriers, arebosons, with different bosons associated with each force.^{[2]: 16–37}

The virtual-particle description of static forces is capable of identifying the spatial form of the forces, such as the inverse-square behavior inNewton's law of universal gravitation and inCoulomb's law. It is also able to predict whether the forces are attractive or repulsive for like bodies.

Thepath integral formulation is the natural language for describing force carriers. This article uses the path integral formulation to describe the force carriers forspin 0, 1, and 2 fields.Pions,photons, andgravitons 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 asperturbation 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 bindingquarks intonucleons 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. Similarly, forbound states the method fails.^[4] 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.^{[citation needed]}

Use of the "force-mediating particle" picture (FMPP) is unnecessary innonrelativistic 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-perturbativerelativistic 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 thatPauli's exclusion principle is obeyed for a collection offermions.

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 offield was invented to mediate the interaction among bodies thus eliminating the need foraction at a distance. The gravitational force is mediated by thegravitational field and the Coulomb force is mediated by theelectromagnetic field.

Thegravitational force on a mass${\displaystyle m}$ exerted by another mass${\displaystyle M}$ is$${\displaystyle \mathbf{F}=-G\frac{mM}{r^2}\hat{\mathbf{r}}=m\mathbf{g}\left(\mathbf{r}\right),}$$whereG is theNewtonian constant of gravitation,r is the distance between the masses, and${\displaystyle \hat{\mathbf{r}}}$ is theunit vector from mass${\displaystyle M}$ to mass${\displaystyle m}$.

The force can also be written$${\displaystyle \mathbf{F}=m\mathbf{g}\left(\mathbf{r}\right),}$$where${\displaystyle \mathbf{g}\left(\mathbf{r}\right)}$ is thegravitational field described by the field equation$${\displaystyle \mathrm{\nabla }\cdot \mathbf{g}=-4\pi G{\rho }_m,}$$where${\displaystyle {\rho }_m}$ is themass density at each point in space.

The electrostaticCoulomb force on a charge${\displaystyle q}$ exerted by a charge${\displaystyle Q}$ is (SI units)$${\displaystyle \mathbf{F}=\frac{1}{4\pi {\epsilon }_0}\frac{qQ}{r^2}\hat{\mathbf{r}},}$$where${\displaystyle {\epsilon }_0}$ is thevacuum permittivity,${\displaystyle r}$ is the separation of the two charges, and${\displaystyle \hat{\mathbf{r}}}$ is aunit vector in the direction from charge${\displaystyle Q}$ to charge${\displaystyle q}$.

The Coulomb force can also be written in terms of anelectrostatic field:$${\displaystyle \mathbf{F}=q\mathbf{E}\left(\mathbf{r}\right),}$$where$${\displaystyle \mathrm{\nabla }\cdot \mathbf{E}=\frac{{\rho }_q}{{\epsilon }_0};}$$${\displaystyle {\rho }_q}$ being thecharge density at each point in space.

In perturbation theory, forces are generated by the exchange ofvirtual particles. The mechanics of virtual-particle exchange is best described with thepath 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.

A virtual particle is created by a disturbance to thevacuum 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.

Usingnatural units,${\displaystyle \hslash =c=1}$, the probability amplitude for the creation, propagation, and destruction of a virtual particle is given, in thepath integral formulation by$${\displaystyle Z\equiv ⟨0|\exp \left(-i\hat{H}T\right)|0⟩=\exp \left(-iET\right)=\int D\phi \exp \left(i\mathcal{S}[\phi ]\right)=\exp \left(iW\right)}$$where${\displaystyle \hat{H}}$ is theHamiltonian operator,${\displaystyle T}$ is elapsed time,${\displaystyle E}$ is the energy change due to the disturbance,${\displaystyle W=-ET}$ is the change in action due to the disturbance,${\displaystyle \phi }$ is the field of the virtual particle, the integral is over all paths, and the classicalaction is given by$${\displaystyle \mathcal{S}[\phi ]=\int {\mathrm{d}}^{4}x\mathcal{L}[\phi (x)]}$$where${\displaystyle \mathcal{L}[\phi (x)]}$ is theLagrangian density.

Here, thespacetime metric is given by

The path integral often can be converted to the form$${\displaystyle Z=\int \exp \left[i\int d^{4}x\left(\frac{1}{2}\phi \hat{O}\phi +J\phi \right)\right]D\phi }$$where${\displaystyle \hat{O}}$ is a differential operator with${\displaystyle \phi }$ and${\displaystyle J}$ functions ofspacetime. 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 (seeCommon integrals in quantum field theory § Integrals with differential operators in the argument)$${\displaystyle Z\propto \exp \left(iW\left(J\right)\right)}$$where$${\displaystyle W\left(J\right)=-\frac{1}{2}\iint d^{4}x{d}^{4}yJ\left(x\right)D\left(x-y\right)J\left(y\right)}$$is the change in the action due to the disturbances and thepropagator${\displaystyle D\left(x-y\right)}$ is the solution of$${\displaystyle \hat{O}D\left(x-y\right)={\delta }^4\left(x-y\right).}$$

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$${\displaystyle J(x)=\left(J_1+J_2,0,0,0\right)}$$where the delta functions are in space, the disturbances are located at${\displaystyle {\mathbf{x}}_1}$ and${\displaystyle {\mathbf{x}}_2}$, and the coefficients${\displaystyle a_1}$ and${\displaystyle a_2}$ are the strengths of the disturbances.

If we neglect self-interactions of the disturbances then W becomes$${\displaystyle W\left(J\right)=-\iint d^{4}x{d}^{4}y{J}_1\left(x\right)\frac{1}{2}\left[D\left(x-y\right)+D\left(y-x\right)\right]J_2\left(y\right),}$$

which can be written$${\displaystyle W\left(J\right)=-T{a}_1{a}_2\int \frac{d^{3}k}{(2\pi {)}^3}D\left(k\right){\mid }_{k_0=0}\exp \left(i\mathbf{k}\cdot \left({\mathbf{x}}_1-{\mathbf{x}}_2\right)\right).}$$

Here${\displaystyle D\left(k\right)}$ is the Fourier transform of$${\displaystyle \frac{1}{2}\left[D\left(x-y\right)+D\left(y-x\right)\right].}$$

Finally, the change in energy due to the static disturbances of the vacuum is$${\displaystyle E=-\frac{W}{T}=a_1{a}_2\int \frac{d^{3}k}{(2\pi {)}^3}D\left(k\right){\mid }_{k_0=0}\exp \left(i\mathbf{k}\cdot \left({\mathbf{x}}_1-{\mathbf{x}}_2\right)\right).}$$

If this quantity is negative, the force is attractive. If it is positive, the force is repulsive.

Examples of static, motionless, interacting currents are theYukawa potential, theCoulomb potential in a vacuum, and theCoulomb 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 interactionin a vacuum andin 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 themagnetic 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 asfractional quantum numbers.

Consider thespin-0 Lagrangian density^{[2]: 21–29}$${\displaystyle \mathcal{L}[\phi (x)]=\frac{1}{2}\left[{\left(\mathrm{\partial }\phi \right)}^2-m^2{\phi }^2\right].}$$

The equation of motion for this Lagrangian is theKlein–Gordon equation$${\displaystyle {\mathrm{\partial }}^2\phi +m^2\phi =0.}$$

If we add a disturbance the probability amplitude becomes$${\displaystyle Z=\int D\phi \exp \left\{i\int d^4\mathbf{x}\left[\frac{1}{2}\left({\left(\mathrm{\partial }\phi \right)}^2-m^2{\phi }^2\right)+J\phi \right]\right\}.}$$

If we integrate by parts and neglect boundary terms at infinity the probability amplitude becomes$${\displaystyle Z=\int D\phi \exp \left\{i\int d^{4}x\left[-\frac{1}{2}\phi \left({\mathrm{\partial }}^2+m^2\right)\phi +J\phi \right]\right\}.}$$

With the amplitude in this form it can be seen that the propagator is the solution of$${\displaystyle -\left({\mathrm{\partial }}^2+m^2\right)D\left(x-y\right)={\delta }^4\left(x-y\right).}$$

From this it can be seen that$${\displaystyle D\left(k\right){\mid }_{k_0=0}=-\frac{1}{k^2+m^2}.}$$

The energy due to the static disturbances becomes (seeCommon integrals in quantum field theory § Yukawa Potential: The Coulomb potential with mass)$${\displaystyle E=-\frac{a_1{a}_2}{4\pi r}\exp \left(-mr\right)}$$with$${\displaystyle r^2={\left({\mathbf{x}}_1-{\mathbf{x}}_2\right)}^2}$$which is attractive and has a range of$${\displaystyle \frac{1}{m}.}$$

Yukawa proposed that this field describes the force between twonucleons in an atomic nucleus. It allowed him to predict both the range and the mass of the particle, now known as thepion, associated with this field.

Consider thespin-1Proca Lagrangian with a disturbance^{[2]: 30–31}

$${\displaystyle \mathcal{L}[\phi (x)]=-\frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu }+\frac{1}{2}{m}^2{A}_{\mu }{A}^{\mu }+A_{\mu }{J}^{\mu }}$$where$${\displaystyle F_{\mu \nu }={\mathrm{\partial }}_{\mu }{A}_{\nu }-{\mathrm{\partial }}_{\nu }{A}_{\mu },}$$charge is conserved$${\displaystyle {\mathrm{\partial }}_{\mu }{J}^{\mu }=0,}$$and we choose theLorenz gauge$${\displaystyle {\mathrm{\partial }}_{\mu }{A}^{\mu }=0.}$$

Moreover, we assume that there is only a time-like component${\displaystyle J^0}$ 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 thatwhich implies$${\displaystyle {\eta }_{\mu \alpha }\left({\mathrm{\partial }}^2+m^2\right)D^{\alpha \nu }\left(x-y\right)={\delta }_{\mu }^{\nu }{\delta }^4\left(x-y\right)}$$and$${\displaystyle D_{\mu \nu }\left(k\right){\mid }_{k_0=0}={\eta }_{\mu \nu }\frac{1}{-k^2+m^2}.}$$

This yields$${\displaystyle D\left(k\right){\mid }_{k_0=0}=\frac{1}{{\mathbf{k}}^2+m^2}}$$for thetimelike propagator and$${\displaystyle E=+\frac{a_1{a}_2}{4\pi r}\exp \left(-mr\right)}$$which has the opposite sign to the Yukawa case.

In the limit of zerophoton mass, the Lagrangian reduces to the Lagrangian forelectromagnetism$${\displaystyle E=\frac{a_1{a}_2}{4\pi r}.}$$

Therefore the energy reduces to the potential energy for the Coulomb force and the coefficients${\displaystyle a_1}$ and${\displaystyle a_2}$ are proportional to the electric charge. Unlike the Yukawa case, like bodies, in this electrostatic case, repel each other.

Thedispersion relation forplasma waves is^{[5]: 75–82}$${\displaystyle {\omega }^2={\omega }_p^2+\gamma \left(\omega \right)\frac{T_{\text{e}}}{m}{\mathbf{k}}^2.}$$where${\displaystyle \omega }$ is the angular frequency of the wave,$${\displaystyle {\omega }_p^2=\frac{4\pi n{e}^2}{m}}$$is theplasma frequency,${\displaystyle e}$ is the magnitude of theelectron charge,${\displaystyle m}$ is theelectron mass,${\displaystyle T_{\text{e}}}$ is the electrontemperature (theBoltzmann constant equal to one), and${\displaystyle \gamma \left(\omega \right)}$ 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 anadiabatic process and${\displaystyle \gamma \left(\omega \right)}$ is equal to three. At low frequencies, the compression is anisothermal process and${\displaystyle \gamma \left(\omega \right)}$ is equal to one.Retardation effects have been neglected in obtaining the plasma-wave dispersion relation.

For low frequencies, the dispersion relation becomes$${\displaystyle {\mathbf{k}}^2+{\mathbf{k}}_{\text{D}}^2=0}$$where$${\displaystyle k_{\text{D}}^2=\frac{4\pi n{e}^2}{T_e}}$$is the Debye number, which is the inverse of theDebye length. This suggests that the propagator is$${\displaystyle D\left(k\right){\mid }_{k_0=0}=\frac{1}{k^2+k_{\text{D}}^2}.}$$

In fact, if the retardation effects are not neglected, then the dispersion relation is$${\displaystyle -k_0^2+k^2+k_{\text{D}}^2-\frac{m}{T_{\text{e}}}{k}_0^2=0,}$$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$${\displaystyle E=\frac{a_1{a}_2}{4\pi r}\exp \left(-k_{\text{D}}r\right).}$$The Coulomb potential is screened on length scales of a Debye length.

In a quantumelectron gas, plasma waves are known asplasmons. Debye screening is replaced withThomas–Fermi screening to yield^[6]$${\displaystyle E=\frac{a_1{a}_2}{4\pi r}\exp \left(-k_{\text{s}}r\right)}$$where the inverse of the Thomas–Fermi screening length is$${\displaystyle k_{\text{s}}^2=\frac{6\pi n{e}^2}{{\epsilon }_{\text{F}}}}$$and${\displaystyle {\epsilon }_{\text{F}}}$ is theFermi energy${\epsilon }_{\text{F}}=\frac{{\hslash }^2}{2m}{\left(3{\pi }^{2}n\right)}^{2/3}.$

This expression can be derived from thechemical potential for an electron gas and fromPoisson's equation. The chemical potential for an electron gas near equilibrium is constant and given by$${\displaystyle \mu =-e\phi +{\epsilon }_{\text{F}}}$$where${\displaystyle \phi }$ is theelectric 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 theplasma wave.

We consider a line of charge with axis in thez direction embedded in an electron gas$${\displaystyle J_1\left(x\right)=\frac{a_1}{L_B}\frac{1}{2\pi r}{\delta }^2\left(r\right)}$$where${\displaystyle r}$ is the distance in thexy-plane from the line of charge,${\displaystyle L_B}$ is the width of the material in the z direction. The superscript 2 indicates that theDirac delta function is in two dimensions. The propagator is$${\displaystyle D\left(k\right){\mid }_{k_0=0}=\frac{1}{{\mathbf{k}}^2+k_{Ds}^2}}$$where${\displaystyle k_{Ds}}$ is either the inverseDebye–Hückel screening length or the inverseThomas–Fermi screening length.

The interaction energy is$${\displaystyle E=\left(\frac{a_1{a}_2}{2\pi L_B}\right){\int }_0^{\mathrm{\infty }}\frac{kdk}{k^2+k_{Ds}^2}{\mathcal{J}}_0(k{r}_{12})=\left(\frac{a_1{a}_2}{2\pi L_B}\right)K_0\left(k_{Ds}{r}_{12}\right)}$$where${\displaystyle {\mathcal{J}}_n(x)}$ and${\displaystyle K_0(x)}$ areBessel functions and${\displaystyle r_{12}}$ is the distance between the two line charges. In obtaining the interaction energy we made use of the integrals (seeCommon integrals in quantum field theory § Integration of the cylindrical propagator with mass)$${\displaystyle {\int }_0^{2\pi }\frac{d\phi }{2\pi }\exp \left(ip\cos \left(\phi \right)\right)={\mathcal{J}}_0(p)}$$and$${\displaystyle {\int }_0^{\mathrm{\infty }}\frac{kdk}{k^2+m^2}{\mathcal{J}}_0(kr)=K_0(mr).}$$

For${\displaystyle k_{Ds}{r}_{12}\ll 1}$, we have$${\displaystyle K_0\left(k_{Ds}{r}_{12}\right)\to -\ln \left(\frac{k_{Ds}{r}_{12}}{2}\right)+\mathrm{0.5772.}}$$

We consider a charge density in tube with axis along a magnetic field embedded in an electron gas$${\displaystyle J_1\left(x\right)=\frac{a_1}{L_b}\frac{1}{2\pi r}{\delta }^2\left(r-r_{B1}\right)}$$where${\displaystyle r}$ is the distance from theguiding center,${\displaystyle L_B}$ is the width of the material in the direction of the magnetic field$${\displaystyle r_{B1}=\frac{\sqrt{4\pi }{m}_1{v}_1}{a_{1}B}=\sqrt{\frac{2\hslash }{m_1{\omega }_c}}}$$where thecyclotron frequency is (Gaussian units)$${\displaystyle {\omega }_c=\frac{a_{1}B}{\sqrt{4\pi }{m}_{1}c}}$$and$${\displaystyle v_1=\sqrt{\frac{2\hslash {\omega }_c}{m_1}}}$$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 betweenLandau levels in the quantum treatment of a charged particle in a magnetic field.

In this geometry, the interaction energy can be written$${\displaystyle E=\left(\frac{a_1{a}_2}{2\pi L_B}\right){\int }_0^{\mathrm{\infty }}kdkD\left(k\right){\mid }_{k_0=k_B=0}{\mathcal{J}}_0\left(k{r}_{B1}\right){\mathcal{J}}_0\left(k{r}_{B2}\right){\mathcal{J}}_0\left(k{r}_{12}\right)}$$where${\displaystyle r_{12}}$ is the distance between the centers of the current loops and${\displaystyle {\mathcal{J}}_n(x)}$ is aBessel function of the first kind. In obtaining the interaction energy we made use of the integral$${\displaystyle {\int }_0^{2\pi }\frac{d\phi }{2\pi }\exp \left(ip\cos (\phi )\right)={\mathcal{J}}_0(p).}$$

Thechemical potential near equilibrium, is given by$${\displaystyle \mu =-e\phi +N\hslash {\omega }_c=N_0\hslash {\omega }_c}$$where${\displaystyle -e\phi }$ is thepotential energy of an electron in anelectric potential and${\displaystyle N_0}$ and${\displaystyle N}$ 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$${\displaystyle \delta n=\frac{e\phi }{\hslash {\omega }_c{A}_{\text{M}}{L}_B}}$$where${\displaystyle A_{\text{M}}}$ is the area of the material in the plane perpendicular to the magnetic field.

Poisson's equation yields$${\displaystyle \left(k^2+k_B^2\right)\phi =0}$$where$${\displaystyle k_B^2=\frac{4\pi e^2}{\hslash {\omega }_c{A}_{\text{M}}{L}_B}.}$$

The propagator is then$${\displaystyle D\left(k\right){\mid }_{k_0=k_B=0}=\frac{1}{k^2+k_B^2}}$$and the interaction energy becomes$${\displaystyle E=\left(\frac{a_1{a}_2}{2\pi L_B}\right){\int }_0^{\mathrm{\infty }}\frac{kdk}{k^2+k_B^2}{\mathcal{J}}_0\left(k{r}_{B1}\right){\mathcal{J}}_0\left(k{r}_{B2}\right){\mathcal{J}}_0\left(k{r}_{12}\right)=\left(\frac{2e^2}{L_B}\right){\int }_0^{\mathrm{\infty }}\frac{kdk}{k^2+k_B^2{r}_B^2}{\mathcal{J}}_0^2\left(k\right){\mathcal{J}}_0\left(k\frac{r_{12}}{r_B}\right)}$$where in the second equality (Gaussian units) we assume that the vortices had the same energy and the electron charge.

In analogy withplasmons, theforce carrier is the quantum version of theupper hybrid oscillation which is a longitudinalplasma wave that propagates perpendicular to the magnetic field.

Unlike classical currents, quantum current loops can have various values of theLarmor radius for a given energy.^{[7]: 187–190}Landau levels, the energy states of a charged particle in the presence of a magnetic field, are multiplydegenerate. The current loops correspond toangular momentum states of the charged particle that may have the same energy. Specifically, the charge density is peaked around radii of$${\displaystyle r_{\ell }=\sqrt{\ell }{r}_B\ell =0,1,2,\dots }$$where${\displaystyle \ell }$ is the angular momentumquantum number. When${\displaystyle \ell =1}$ we recover the classical situation in which the electron orbits the magnetic field at theLarmor radius. If currents of two angular momentum${\displaystyle \ell >0}$ and${\displaystyle {\ell }^{\prime }\ge \ell }$ interact, and we assume the charge densities are delta functions at radius${\displaystyle r_{\ell }}$, then the interaction energy is$${\displaystyle E=\left(\frac{2e^2}{L_B}\right){\int }_0^{\mathrm{\infty }}\frac{kdk}{k^2+k_B^2{r}_{\ell }^2}{\mathcal{J}}_0\left(k\right){\mathcal{J}}_0\left(\sqrt{\frac{{\ell }^{\prime }}{\ell }}k\right){\mathcal{J}}_0\left(k\frac{r_{12}}{r_{\ell }}\right).}$$

The interaction energy for${\displaystyle \ell ={\ell }^{\prime }}$ is given in Figure 1 for various values of${\displaystyle k_B{r}_{\ell }}$. 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$${\displaystyle r_{12}=r_{\ell {\ell }^{\prime }}=\sqrt{\ell +{\ell }^{\prime }}{r}_B.}$$

This suggests that the pair of particles that are bound and separated by a distance${\displaystyle r_{\ell {\ell }^{\prime }}}$ act as a singlequasiparticle with angular momentum${\displaystyle \ell +{\ell }^{\prime }}$.

If we scale the lengths as${\displaystyle r_{\ell {\ell }^{\prime }}}$, then the interaction energy becomes$${\displaystyle E=\frac{2e^2}{L_B}{\int }_0^{\mathrm{\infty }}\frac{kdk}{k^2+k_B^2{r}_{\ell {\ell }^{\prime }}^2}{\mathcal{J}}_0\left(\cos \theta k\right){\mathcal{J}}_0(\sin \theta k){\mathcal{J}}_0\left(k\frac{r_{12}}{r_{\ell {\ell }^{\prime }}}\right)}$$where$${\displaystyle \tan \theta =\sqrt{\frac{\ell }{{\ell }^{\prime }}}.}$$

The value of the${\displaystyle r_{12}}$ at which the energy is minimum,${\displaystyle r_{12}=r_{\ell {\ell }^{\prime }}}$, is independent of the ratio$\tan \theta =\sqrt{\ell /{\ell }^{\prime }}$. However the value of the energy at the minimum depends on the ratio. The lowest energy minimum occurs when$${\displaystyle \frac{\ell }{{\ell }^{\prime }}=1.}$$

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)$${\displaystyle \ell ={\ell }^{\prime }=1}$$or$${\displaystyle \frac{\ell }{{\ell }^{\ast }}=\frac{1}{2}}$$where the total angular momentum is written as$${\displaystyle {\ell }^{\ast }=\ell +{\ell }^{\prime }.}$$

When the total angular momentum is odd, the minima cannot occur for${\displaystyle \ell ={\ell }^{\prime }.}$ The lowest energy states for odd total angular momentum occur when$${\displaystyle \frac{\ell }{{\ell }^{\ast }}=\frac{{\ell }^{\ast }\pm 1}{2{\ell }^{\ast }}}$$or$${\displaystyle \frac{\ell }{{\ell }^{\ast }}=\frac{1}{3},\frac{2}{5},\frac{3}{7},\text{etc.,}}$$and$${\displaystyle \frac{\ell }{{\ell }^{\ast }}=\frac{2}{3},\frac{3}{5},\frac{4}{7},\text{etc.,}}$$which also appear as series for the filling factor in thefractional quantum Hall effect.

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^[7]: 189$${\displaystyle \frac{1}{\pi r_B^2{L}_B}\frac{1}{n!}{\left(\frac{r}{r_B}\right)}^{2l}\exp \left(-\frac{r^2}{r_B^2}\right).}$$

The interaction energy becomes$${\displaystyle E=\left(\frac{2e^2}{L_B}\right){\int }_0^{\mathrm{\infty }}\frac{kdk}{k^2+k_B^2{r}_B^2}M\left(\ell +1,1,-\frac{k^2}{4}\right)M\left({\ell }^{\prime }+1,1,-\frac{k^2}{4}\right){\mathcal{J}}_0\left(k\frac{r_{12}}{r_B}\right)}$$where${\displaystyle M}$ is aconfluent hypergeometric function orKummer function. In obtaining the interaction energy we have used the integral (seeCommon integrals in quantum field theory § Integration over a magnetic wave function)

$${\displaystyle \frac{2}{n!}{\int }_0^{\mathrm{\infty }}dr{r}^{2n+1}{e}^{-r^2}{J}_0(kr)=M\left(n+1,1,-\frac{k^2}{4}\right).}$$

As with delta function charges, the value of${\displaystyle r_{12}}$ 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$${\displaystyle \frac{\ell }{{\ell }^{\ast }}=\frac{1}{3},\frac{2}{5},\frac{3}{7},\text{etc.,}}$$and$${\displaystyle \frac{\ell }{{\ell }^{\ast }}=\frac{2}{3},\frac{3}{5},\frac{4}{7},\text{etc.,}}$$appear as well in the case of charges spread by the wave function.

TheLaughlin wavefunction is anansatz for the quasiparticle wavefunction. If the expectation value of the interaction energy is taken over aLaughlin wavefunction, these series are also preserved.

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 theDarwin interaction. To calculate this, consider the electrical currents in space generated by a moving charge$${\displaystyle {\mathbf{J}}_1\left(\mathbf{x}\right)=a_1{\mathbf{v}}_1{\delta }^3\left(\mathbf{x}-{\mathbf{x}}_1\right)}$$with a comparable expression for${\displaystyle {\mathbf{J}}_2}$.

The Fourier transform of this current is$${\displaystyle {\mathbf{J}}_1\left(\mathbf{k}\right)=a_1{\mathbf{v}}_1\exp \left(i\mathbf{k}\cdot {\mathbf{x}}_1\right).}$$

The current can be decomposed into a transverse and a longitudinal part (seeHelmholtz decomposition).$${\displaystyle {\mathbf{J}}_1\left(\mathbf{k}\right)=a_1\left[1-\hat{\mathbf{k}}\hat{\mathbf{k}}\right]\cdot {\mathbf{v}}_1\exp \left(i\mathbf{k}\cdot {\mathbf{x}}_1\right)+a_1\left[\hat{\mathbf{k}}\hat{\mathbf{k}}\right]\cdot {\mathbf{v}}_1\exp \left(i\mathbf{k}\cdot {\mathbf{x}}_1\right).}$$

The hat indicates aunit vector. The last term disappears because$${\displaystyle \mathbf{k}\cdot \mathbf{J}=-k_0{J}^0\to 0,}$$which results from charge conservation. Here${\displaystyle k_0}$ vanishes because we are considering static forces.

With the current in this form the energy of interaction can be written$${\displaystyle E=a_1{a}_2\int \frac{d^3\mathbf{k}}{(2\pi {)}^3}D\left(k\right){\mid }_{k_0=0}{\mathbf{v}}_1\cdot \left[1-\hat{\mathbf{k}}\hat{\mathbf{k}}\right]\cdot {\mathbf{v}}_2\exp \left(i\mathbf{k}\cdot \left({\mathbf{x}}_1-{\mathbf{x}}_2\right)\right).}$$

The propagator equation for the Proca Lagrangian is$${\displaystyle {\eta }_{\mu \alpha }\left({\mathrm{\partial }}^2+m^2\right)D^{\alpha \nu }\left(x-y\right)={\delta }_{\mu }^{\nu }{\delta }^4\left(x-y\right).}$$

Thespacelike solution is$${\displaystyle D\left(k\right){\mid }_{k_0=0}=-\frac{1}{k^2+m^2},}$$which yields$${\displaystyle E=-a_1{a}_2\int \frac{d^3\mathbf{k}}{(2\pi {)}^3}\frac{{\mathbf{v}}_1\cdot \left[1-\hat{\mathbf{k}}\hat{\mathbf{k}}\right]\cdot {\mathbf{v}}_2}{k^2+m^2}\exp \left(i\mathbf{k}\cdot \left({\mathbf{x}}_1-{\mathbf{x}}_2\right)\right),}$$where$k=|\mathbf{k}|$. The integral evaluates to (seeCommon integrals in quantum field theory § Transverse potential with mass)$${\displaystyle E=-\frac{1}{2}\frac{a_1{a}_2}{4\pi r}{e}^{-mr}\left\{\frac{2}{{\left(mr\right)}^2}\left(e^{mr}-1\right)-\frac{2}{mr}\right\}{\mathbf{v}}_1\cdot \left[1+\hat{\mathbf{r}}\hat{\mathbf{r}}\right]\cdot {\mathbf{v}}_2}$$which reduces to$${\displaystyle E=-\frac{1}{2}\frac{a_1{a}_2}{4\pi r}{\mathbf{v}}_1\cdot \left[1+\hat{\mathbf{r}}\hat{\mathbf{r}}\right]\cdot {\mathbf{v}}_2}$$in the limit of smallm. 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.

In a plasma, thedispersion relation for anelectromagnetic wave is^{[5]: 100–103} (${\displaystyle c=1}$)$${\displaystyle k_0^2={\omega }_p^2+k^2,}$$which implies$${\displaystyle D\left(k\right){\mid }_{k_0=0}=-\frac{1}{k^2+{\omega }_p^2}.}$$

Here${\displaystyle {\omega }_p}$ is theplasma frequency. The interaction energy is therefore$${\displaystyle E=-\frac{1}{2}\frac{a_1{a}_2}{4\pi r}{\mathbf{v}}_1\cdot \left[1+\hat{\mathbf{r}}\hat{\mathbf{r}}\right]\cdot {\mathbf{v}}_2{e}^{-{\omega }_{p}r}\left\{\frac{2}{{\left({\omega }_{p}r\right)}^2}\left(e^{{\omega }_{p}r}-1\right)-\frac{2}{{\omega }_{p}r}\right\}.}$$

Consider a tube of current rotating in a magnetic field embedded in a simpleplasma or electron gas. The current, which lies in the plane perpendicular to the magnetic field, is defined as$${\displaystyle {\mathbf{J}}_1(\mathbf{x})=a_1{v}_1\frac{1}{2\pi r{L}_B}{\delta }^2\left(r-r_{B1}\right)\left(\hat{\mathbf{b}}\times \hat{\mathbf{r}}\right)}$$where$${\displaystyle r_{B1}=\frac{\sqrt{4\pi }{m}_1{v}_1}{a_{1}B}}$$and${\displaystyle \hat{\mathbf{b}}}$ is the unit vector in the direction of the magnetic field. Here${\displaystyle L_B}$ indicates the dimension of the material in the direction of the magnetic field. The transverse current, perpendicular to thewave vector, drives thetransverse wave.

The energy of interaction is$${\displaystyle E=\left(\frac{a_1{a}_2}{2\pi L_B}\right)v_1{v}_2{\int }_0^{\mathrm{\infty }}kdkD\left(k\right){\mid }_{k_0=k_B=0}{\mathcal{J}}_1\left(k{r}_{B1}\right){\mathcal{J}}_1\left(k{r}_{B2}\right){\mathcal{J}}_0\left(k{r}_{12}\right)}$$where${\displaystyle r_{12}}$ is the distance between the centers of the current loops and${\displaystyle {\mathcal{J}}_n(x)}$ is aBessel function of the first kind. In obtaining the interaction energy we made use of the integrals$${\displaystyle {\int }_0^{2\pi }\frac{d\phi }{2\pi }\exp \left(ip\cos \left(\phi \right)\right)={\mathcal{J}}_0\left(p\right)}$$and$${\displaystyle {\int }_0^{2\pi }\frac{d\phi }{2\pi }\cos \left(\phi \right)\exp \left(ip\cos \left(\phi \right)\right)=i{\mathcal{J}}_1\left(p\right).}$$

SeeCommon integrals in quantum field theory § Angular integration in cylindrical coordinates.

A current in a plasma confined to the plane perpendicular to the magnetic field generates anextraordinary wave.^{[5]: 110–112} This wave generatesHall currents that interact and modify the electromagnetic field. Thedispersion relation for extraordinary waves is^[5]: 112$${\displaystyle -k_0^2+k^2+{\omega }_p^2\frac{k_0^2-{\omega }_p^2}{k_0^2-{\omega }_H^2}=0,}$$which gives for the propagator$${\displaystyle D\left(k\right){\mid }_{k_0=k_B=0}=-\left(\frac{1}{k^2+k_X^2}\right)}$$where$${\displaystyle k_X\equiv \frac{{\omega }_p^2}{{\omega }_H}}$$in analogy with the Darwin propagator. Here, the upper hybrid frequency is given by$${\displaystyle {\omega }_H^2={\omega }_p^2+{\omega }_c^2,}$$thecyclotron frequency is given by (Gaussian units)$${\displaystyle {\omega }_c=\frac{eB}{mc},}$$and theplasma frequency (Gaussian units)$${\displaystyle {\omega }_p^2=\frac{4\pi n{e}^2}{m}.}$$

Heren is the electron density,e is the magnitude of the electron charge, andm is the electron mass.

The interaction energy becomes, for like currents,$${\displaystyle E=-\left(\frac{a^2}{2\pi L_B}\right)v^2{\int }_0^{\mathrm{\infty }}\frac{kdk}{k^2+k_X^2}{\mathcal{J}}_1^2\left(k{r}_B\right){\mathcal{J}}_0\left(k{r}_{12}\right)}$$

In the limit that the distance between current loops is small,$${\displaystyle E=-E_0{I}_1\left(\mu \right)K_1\left(\mu \right)}$$where$${\displaystyle E_0=\left(\frac{a^2}{2\pi L_B}\right)v^2}$$and$${\displaystyle \mu =\frac{{\omega }_p^2{r}_B}{{\omega }_H}=k_X{r}_B}$$andI andK are modified Bessel functions. we have assumed that the two currents have the same charge and speed.

We have made use of the integral (seeCommon integrals in quantum field theory § Integration of the cylindrical propagator with mass)$${\displaystyle {\int }_o^{\mathrm{\infty }}\frac{kdk}{k^2+m^2}{\mathcal{J}}_1^2\left(kr\right)=I_1\left(mr\right)K_1\left(mr\right).}$$

For smallmr the integral becomes$${\displaystyle I_1\left(mr\right)K_1\left(mr\right)\to \frac{1}{2}\left[1-\frac{1}{8}{\left(mr\right)}^2\right].}$$

For largemr the integral becomes$${\displaystyle I_1\left(mr\right)K_1\left(mr\right)\to \frac{1}{2}\left(\frac{1}{mr}\right).}$$

The screeningwavenumber can be written (Gaussian units)$${\displaystyle \mu =\frac{{\omega }_p^2{r}_B}{{\omega }_{H}c}=\left(\frac{2e^2{r}_B}{L_B\hslash c}\right)\frac{\nu }{\sqrt{1+\frac{{\omega }_p^2}{{\omega }_c^2}}}=2\alpha \left(\frac{r_B}{L_B}\right)\left(\frac{1}{\sqrt{1+\frac{{\omega }_p^2}{{\omega }_c^2}}}\right)\nu }$$where${\displaystyle \alpha }$ is thefine-structure constant and the filling factor is$${\displaystyle \nu =\frac{2\pi N\hslash c}{eBA}}$$andN is the number of electrons in the material andA is the area of the material perpendicular to the magnetic field. This parameter is important in thequantum Hall effect and thefractional quantum Hall effect. The filling factor is the fraction of occupiedLandau states at the ground state energy.

For cases of interest in the quantum Hall effect,${\displaystyle \mu }$ is small. In that case the interaction energy is$${\displaystyle E=-\frac{E_0}{2}\left[1-\frac{1}{8}{\mu }^2\right]}$$where (Gaussian units)$${\displaystyle E_0=4\pi \frac{e^2}{L_B}\frac{v^2}{c^2}=8\pi \frac{e^2}{L_B}\left(\frac{\hslash {\omega }_c}{m{c}^2}\right)}$$is the interaction energy for zero filling factor. We have set the classical kinetic energy to the quantum energy$${\displaystyle \frac{1}{2}m{v}^2=\hslash {\omega }_c.}$$

A gravitational disturbance is generated by thestress–energy tensor${\displaystyle T^{\mu \nu }}$; consequently, the Lagrangian for the gravitational field isspin-2. If the disturbances are at rest, then the only component of the stress–energy tensor that persists is the${\displaystyle 00}$ component. If we use the same trick of giving thegraviton some mass and then taking the mass to zero at the end of the calculation the propagator becomes$${\displaystyle D\left(k\right){\mid }_{k_0=0}=-\frac{4}{3}\frac{1}{k^2+m^2}}$$and$${\displaystyle E=-\frac{4}{3}\frac{a_1{a}_2}{4\pi r}\exp \left(-mr\right),}$$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.^{[2]: 32–37}

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.^[2]: 35

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