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{{Short description|Physical interaction in post-classical physics}}
{{Use American English|date = February 2019}}
'''Static force fields''' are fields, such as a simple [[Electric field|electric]], [[Magnetic field|magnetic]] or [[gravitational field]]s, that exist without excitations. The [[Perturbation theory (quantum mechanics)|most common approximation method]] that physicists use for [[Scattering theory|scattering calculations]] can be interpreted as static forces arising from the interactions between two bodies mediated by '''[[virtual particle]]s''', particles that exist for only a short time determined by the [[uncertainty principle]].<ref>{{cite journal|last1=Jaeger|first1=Gregg|title=Are virtual particles less real?| journal=Entropy |volume=21 |issue=2|page=141|date=2019|doi=10.3390/e21020141|pmid=33266857 |pmc=7514619 |bibcode=2019Entrp..21..141J|doi-access=free}}</ref> The virtual particles, also known as [[force carrier]]s, are [[boson]]s, with different bosons associated with each force.<ref name="Zee">{{cite book | first = A. | last = Zee | title=Quantum Field Theory in a Nutshell| publisher= Princeton University| year=2003 | isbn=0-691-01019-6}}</ref>{{rp|pp=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 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.
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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 (physics)|spin]] 0, 1, and 2 fields. [[Pion]]s, [[photon]]s, and [[graviton]]s fall into these respective categories.
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==Classical forces==
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In both cases, the bodies appear to act on each other over a distance. The concept of [[Field (physics)|field]] was invented to mediate the interaction among bodies thus eliminating the need for [[Action at a distance (physics)|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 [[Newton's law of universal gravitation|gravitational force]] on a mass <math> m </math> exerted by another mass <math> M </math> is
<math display="block">\mathbf{F} = - G \frac{m M}{r^2} \, \hat\mathbf{r} = m \mathbf{g} \left ( \mathbf{r} \right ),</math>
where {{math|''G''}} is the [[Newtonian constant of gravitation]], {{mvar|r}} is the distance between the masses, and <math> \hat\mathbf{r} </math> is the [[unit vector]] from mass <math> M </math> to mass <math> m </math>.
The force can also be written
<math display="block">\mathbf{F} = m \mathbf{g} \left ( \mathbf{r} \right ),</math>
where <math> \mathbf{g} \left ( \mathbf{r} \right ) </math> is the [[gravitational field]] described by the field equation
<math display="block">\nabla\cdot \mathbf{g} = -4\pi G\rho_m, </math>
where <math>\rho_m</math> is the [[density|mass density]] at each point in space.
=== Coulomb force ===
The electrostatic [[Coulomb force]] on a charge <math> q </math> exerted by a charge <math> Q </math> is ([[SI units]])
<math display="block">\mathbf{F} = \frac{1}{4\pi\varepsilon_0}\frac{q Q}{r^2}\mathbf{\hat{r}},</math>
where <math> \varepsilon_0 </math> is the [[vacuum permittivity]], <math>r</math> is the separation of the two charges, and <math>\mathbf{\hat{r}}</math> is a [[unit vector]] in the direction from charge <math> Q </math> to charge <math> q </math>.
The Coulomb force can also be written in terms of an [[electrostatic field]]:
<math display="block">\mathbf{F} = q \mathbf{E} \left ( \mathbf{r} \right ),</math>
where
<math display="block"> \nabla \cdot \mathbf{E} = \frac {\rho_q}{\varepsilon _0};</math>
<math>\rho_q</math> being the [[density|charge density]] at each point in space.
== Virtual-particle exchange ==
In perturbation theory, forces are generated by the exchange of [[virtual particle]]s. 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
====
Using [[natural units]], <math> \hbar = c = 1 </math>, the probability amplitude for the creation, propagation, and destruction of a virtual particle is given, in the [[path integral formulation]] by
<math display="block"> Z \equiv
\langle 0 | \exp\left ( -i \hat H T \right ) |0 \rangle
= \exp\left ( -i E T \right )
=
= \exp\left ( i W \right )
</math>
where <math> \hat H </math> is the [[Hamiltonian operator]], <math> T </math> is elapsed time, <math> E </math> is the energy change due to the disturbance, <math> W = - E T </math> is the change in action due to the disturbance, <math> \varphi </math> is the field of the virtual particle, the integral is over all paths, and the classical [[Action (physics)|action]] is given by
<math display="block">\mathcal{S} [\varphi] = \int \mathrm{d}^4x\; {\mathcal{L} [\varphi (x)]\,} </math>
Here, the [[spacetime]] metric is given by
<math display="block">\eta_{\mu\nu} = \begin{pmatrix}
1 & 0 & 0 & 0 \\
0 &
0 & 0 & 0 & -1
\end{pmatrix}.</math>
The path integral often can be converted to the form
<math display="block"> Z = \int \exp\left[ i \int d^4x \left ( \frac 1 2 \varphi \hat O \varphi + J \varphi \right) \right ] D\varphi</math>
where <math> \hat O </math> is a differential operator with <math> \varphi </math> and <math> J </math> 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
<math display="block"> Z \propto \exp\left( i W\left ( J \right )\right)</math>
where
<math display="block"> W\left ( J \right ) = -\frac{1}{2} \iint d^4x \; d^4y \; J\left ( x \right ) D\left ( x-y \right ) J\left ( y \right )</math>
is the change in the action due to the disturbances and the [[propagator]] <math> D\left ( x-y \right ) </math> is the solution of
<math display="block">\hat O D\left ( x - y \right ) = \delta^4 \left ( x - y \right ).</math>
==== 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
<math display="block"> J(x) = \left( J_1 +J_2,0,0,0 \right)</math>
<math display="block">\begin{align}
J_1 &= a_1 \delta^3\left (
J_2 &= a_2 \delta^3\left ( \mathbf x - \mathbf x_2 \right )
\end{align}</math>
where the delta functions are in space, the disturbances are located at <math> \mathbf x_1 </math> and <math> \mathbf x_2 </math>, and the coefficients <math> a_1 </math> and <math> a_2 </math> are the strengths of the disturbances.
If we neglect self-interactions of the disturbances then W becomes
<math display="block"> W\left ( J \right ) = - \iint d^4x \; d^4y \; 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 ),</math>
which can be written
<math display="block"> W\left ( J \right ) =
- T a_1 a_2\int \frac{d^3k}{(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 ).
</math>
Finally, the change in energy due to the static disturbances of the vacuum is
<math display="block"> E = - \frac{W}{T} = a_1 a_2\int \frac{d^3k}{(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 ).</math>
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 [[
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
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
== Selected examples ==
===
Consider the [[Spin (physics)|spin]]-0 Lagrangian density<ref
<math display="block">
\mathcal{L} [\varphi (x)]
= \frac{1}{2} \left [ \left ( \partial \varphi \right )^2 -m^2 \varphi^2 \right ].</math>
The equation of motion for this Lagrangian is the [[Klein–Gordon equation]]
<math display="block"> \partial^2 \varphi + m^2 \varphi =0.</math>
If we add a disturbance the probability amplitude becomes
<math display="block"> Z =
\int D\varphi \; \exp \left \{ i \int d^4\mathbf{x}\; \left [ \frac{1}{2} \left ( \left ( \partial \varphi \right )^2 - m^2\varphi^2 \right ) + J\varphi \right ] \right \}.</math>
If we integrate by parts and neglect boundary terms at infinity the probability amplitude becomes
<math display="block"> Z =
\int D\varphi \; \exp \left \{ i \int d^4x\; \left [ -\frac{1}{2}\varphi \left ( \partial^2 + m^2\right )\varphi + J\varphi \right ] \right \}.</math>
With the amplitude in this form it can be seen that the propagator is the solution of
<math display="block"> -\left ( \partial^2 + m^2\right ) D\left ( x-y \right ) = \delta^4\left ( x-y \right ).</math>
From this it can be seen that
<math display="block">D\left ( k \right )\mid_{k_0=0} \; = \; -\frac{1}{ k^2 + m^2}.</math>
The energy due to the static disturbances becomes (see {{slink|Common integrals in quantum field theory#Yukawa Potential: The Coulomb potential with mass}})
<math display="block">E =-\frac{a_1 a_2}{4 \pi r} \exp \left ( -m r \right )</math>
with
<math display="block">r^2 = \left (\mathbf x_1 - \mathbf x_2 \right )^2</math>
which is attractive and has a range of <math display="block">\frac{1}{m}.</math>
[[Hideki Yukawa|Yukawa]] proposed that this field describes the force between two [[nucleon]]s 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 ===
====
Consider the [[Spin (physics)|spin]]-1 [[Proca action|Proca Lagrangian]] with a disturbance<ref name="Zee"/>{{rp|pp=30–31}}
<math display="block">\mathcal{L} [\varphi (x)] = -\frac{1}{4} F_{\mu \nu} F^{\mu \nu} + \frac{1}{2} m^2 A_{\mu} A^{\mu} + A_{\mu} J^{\mu}</math>
where
<math display="block"> F_{\mu \nu} = \partial_{\mu} A_{\nu} - \partial_{\nu} A_{\mu},</math>
charge is conserved
<math display="block"> \partial_{\mu} J^{\mu} = 0,</math>
and we choose the [[Lorenz gauge]]
<math display="block"> \partial_{\mu} A^{\mu} = 0.</math>
Moreover, we assume that there is only a time-like component <math>J^0 </math> 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
<math display="block">\begin{align}
-\frac{1}{4} \int d^4x F_{\mu \nu}F^{\mu \nu}
&= -\frac{1
&=
&= \frac{1}{2}\int d^4x \; A^{\mu} \left( \eta_{\mu \nu} \partial^{2} \right) A^{\nu},
\end{align}</math>
which implies
<math display="block"> \eta_{\mu \alpha} \left ( \partial^2 + m^2\right ) D^{\alpha \nu}\left ( x-y \right ) = \delta_{\mu}^{\nu} \delta^4\left ( x-y \right )</math>
and
<math display="block">D_{\mu \nu}\left ( k \right )\mid_{k_0=0} \; = \; \eta_{\mu \nu}\frac{1}{- k^2 + m^2}.</math>
This yields
<math display="block">D\left( k \right)\mid_{k_0=0}\; = \; \frac{1}{\mathbf k^2 + m^2}</math>
for the [[timelike]] propagator and
<math display="block">E = + \frac{a_1 a_2}{4 \pi r} \exp \left( -m r \right)</math>
which has the opposite sign to the Yukawa case.
In the limit of zero [[photon]] mass, the Lagrangian reduces to the Lagrangian for [[electromagnetism]]
<math display="block">E = \frac{a_1 a_2}{4 \pi r}.</math>
Therefore the energy reduces to the potential energy for the Coulomb force and the coefficients <math>a_1 </math> and <math>a_2 </math> 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 wave]]s is<ref name="Chen">{{cite book | first = Francis F. | last = Chen | title=Introduction to Plasma Physics| publisher= Plenum Press| year=1974 | isbn=0-306-30755-3}}</ref>{{rp|pp=75–82}}
<math display="block">\omega^2 = \omega_p^2 + \gamma\left( \omega \right) \frac{T_\text{e}}{m} \mathbf k^2.</math>
where <math>\omega </math> is the angular frequency of the wave,
<math display="block">\omega_p^2 = \frac{4\pi n e^2}{m}</math>
is the [[plasma frequency]], <math>e </math> is the magnitude of the [[electron charge]], <math>m </math> is the [[electron mass]], <math>T_\text{e} </math> is the electron [[temperature]] (the [[Boltzmann constant]] equal to one), and <math>\gamma\left( \omega \right) </math> 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 <math>\gamma\left( \omega \right) </math> is equal to three. At low frequencies, the compression is an [[isothermal process]] and <math>\gamma\left( \omega \right) </math> is equal to one. [[Retarded potential|Retardation]] effects have been neglected in obtaining the plasma-wave dispersion relation.
For low frequencies, the dispersion relation becomes
<math display="block"> \mathbf k^2 + \mathbf k_\text{D}^2 = 0</math>
where
<math display="block"> k_\text{D}^2= \frac{4\pi n e^2}{T_e}</math>
is the Debye number, which is the inverse of the [[Debye length]]. This suggests that the propagator is
<math display="block">D\left ( k \right )\mid_{k_0=0} \; = \; \frac{1}{ k^2 + k_\text{D}^2}.</math>
In fact, if the retardation effects are not neglected, then the dispersion relation is
<math display="block"> -k_0^2 + k^2 + k_\text{D}^2 -\frac{m}{T_\text{e}} k_0^2 = 0,</math>
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
<math display="block">E = \frac{a_1 a_2}{4 \pi r} \exp \left ( -k_\text{D} r \right ).</math>
The Coulomb potential is screened on length scales of a Debye length.
===== Plasmons =====
In a quantum [[Free electron model|electron gas]], plasma waves are known as [[plasmon]]s. Debye screening is replaced with [[Thomas–Fermi screening]] to yield<ref>{{cite book | author=C. Kittel | title=[[Introduction to Solid State Physics]]|edition=Fifth | publisher= John Wiley and Sons| year=1976 | isbn=0-471-49024-5}} pp. 296-299.</ref>
<math display="block">E = \frac{a_1 a_2}{4 \pi r} \exp \left ( -k_\text{s} r \right )</math>
where the inverse of the Thomas–Fermi screening length is
<math display="block"> k_\text{s}^2 = \frac{6\pi n e^2}{\varepsilon_\text{F}}</math>
and <math>\varepsilon_\text{F}</math> is the [[Fermi energy]] <math display="inline">\varepsilon_\text{F} = \frac{\hbar^2}{2m} \left( {3 \pi^2 n} \right)^{2/3} .</math>
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
<math display="block"> \mu = -e\varphi + \varepsilon_\text{F}</math>
where <math>\varphi</math> 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
<math display="block"> J_1\left( x\right) = \frac{a_1}{L_B} \frac{1}{2 \pi r} \delta^2\left( r \right)</math>
where <math>r</math> is the distance in the ''xy''-plane from the line of charge, <math>L_B</math> 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
<math display="block">D\left ( k \right )\mid_{k_0=0}\; = \; \frac{1}{\mathbf k^2 + k_{Ds}^2}</math>
where <math>k_{Ds} </math> is either the inverse [[Debye–Hückel equation|Debye–Hückel screening length]] or the inverse [[Thomas–Fermi screening]] length.
The interaction energy is
<math display="block"> E = \left( \frac{a_1\, a_2}{2 \pi L_B}\right) \int_0^{\infty} \frac{k\,dk}{k^2 + k_{Ds}^2} \mathcal J_0 ( kr_{12} )
= \left( \frac{a_1\, a_2}{2 \pi L_B}\right) K_0 \left( k_{Ds} r_{12} \right)</math>
where <math> \mathcal J_n ( x ) </math> and <math> K_0 ( x )</math> are [[Bessel function]]s and <math> r_{12}</math> is the distance between the two line charges. In obtaining the interaction energy we made use of the integrals (see {{slink|Common integrals in quantum field theory#Integration of the cylindrical propagator with mass}})
<math display="block"> \int_0^{2 \pi} \frac{d\varphi}{2 \pi} \exp\left( i p \cos\left( \varphi \right) \right) = \mathcal J_0 ( p ) </math>
and
<math display="block"> \int_0^{\infty} \frac{k\,dk}{k^2 + m^2} \mathcal J_0 (kr) = K_0 (mr).</math>
For <math> k_{Ds} r_{12} \ll 1</math>, we have
<math display="block">K_0 \left( k_{Ds} r_{12} \right) \to -\ln \left(\frac{k_{Ds} r_{12}}{2}\right) + 0.5772.</math>
==== 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
<math display="block"> J_1\left( x\right) = \frac{a_1}{L_b} \frac{1}{2 \pi r} \delta^2{\left( r - r_{B1}\right)}</math>
where <math>r</math> is the distance from the [[guiding center]], <math>L_B</math> is the width of the material in the direction of the magnetic field
<math display="block"> r_{B1} = \frac{\sqrt{4 \pi} m_1 v_1}{a_1 B} = \sqrt{\frac{2 \hbar}{m_1 \omega_c}} </math>
where the [[cyclotron frequency]] is ([[Gaussian units]])
<math display="block">\omega_c = \frac{a_1 B}{\sqrt{4 \pi} m_1 c} </math>
and
<math display="block"> v_1 = \sqrt {\frac{2 \hbar \omega_c}{m_1}} </math>
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
<math display="block"> E =
\left( \frac{a_1\, a_2}{2 \pi L_B}\right) \int_0^{\infty} {k\;dk \;} D\left( k \right) \mid_{k_0=k_B=0}
\mathcal J_0 \left ( kr_{B1} \right) \mathcal J_0 \left ( kr_{B2} \right) \mathcal J_0 \left ( kr_{12} \right)</math>
where <math>r_{12}</math> is the distance between the centers of the current loops and <math> \mathcal J_n ( x ) </math> is a [[Bessel function]] of the first kind. In obtaining the interaction energy we made use of the integral
<math display="block">
\int_0^{2 \pi} \frac{d\varphi}{2 \pi} \exp\left( i p \cos(\varphi) \right) = \mathcal J_0 ( p ) . </math>
===== Electric field due to a density perturbation =====
The [[chemical potential]] near equilibrium, is given by
<math display="block"> \mu = -e\varphi + N\hbar \omega_c = N_0\hbar \omega_c</math>
where <math>-e\varphi</math> is the [[potential energy]] of an electron in an [[electric potential]] and <math>N_0</math> and <math>N</math> 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
<math display="block"> \delta n = \frac{e \varphi}{\hbar \omega_c A_\text{M} L_B}</math>
where <math> A_\text{M} </math> is the area of the material in the plane perpendicular to the magnetic field.
[[Poisson's equation]] yields
<math display="block">\left( k^2 + k_B^2 \right) \varphi = 0</math>
where
<math display="block"> k_B^2 = \frac{4 \pi e^2}{\hbar \omega_c A_\text{M} L_B}.</math>
The propagator is then
<math display="block"> D\left( k \right) \mid_{k_0=k_B=0} = \frac{1}{k^2 + k_B^2}</math>
and the interaction energy becomes
<math display="block"> E =
\left( \frac{a_1\, a_2}{2 \pi L_B}\right) \int_0^{\infty} \frac{k\;dk \;}{k^2 + k_B^2}
\mathcal J_0 \left ( kr_{B1} \right) \mathcal J_0 \left ( kr_{B2} \right) \mathcal J_0 \left ( kr_{12} \right)
=
\left( \frac{
\mathcal J_0^2 \left ( k \right) \mathcal J_0 \left ( k\frac{r_{12}}{r_B} \right)
</math>
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 ======
[[Image:100927c Angular momentum 11.jpg|thumb|250px|right|Figure 1. Interaction energy vs. ''r'' for angular momentum states of value one. The curves are identical to these for any values of <math>
[[Image:100927 Angular momentum 15.jpg|thumb|250px|right|Figure 2. Interaction energy vs. r for angular momentum states of value one and five.]]
[[Image:101005 energy by theta.jpg|thumb|250px|right|Figure 3. Interaction energy vs. r for various values of theta. The lowest energy is for <math display="inline">\theta = \frac{\pi
[[Image:101011 energy picture.jpg|thumb|250px|right|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 <math>
Unlike classical currents, quantum current loops can have various values of the [[Larmor radius]] for a given energy.<ref name="Ezewa">{{cite book |
<math display="block">r_{\ell} = \sqrt{\ell}\;r_B\; \; \; \ell=0,1,2, \ldots</math>
where <math>\ell</math> is the angular momentum [[quantum number]]. When <math> \ell = 1</math> we recover the classical situation in which the electron orbits the magnetic field at the [[Larmor radius]]. If currents of two angular momentum <math> \ell > 0 </math> and <math>\ell' \ge \ell </math> interact, and we assume the charge densities are delta functions at radius <math>r_{\ell}</math>, then the interaction energy is
<math display="block"> E =
\left( \frac{2 e^2}{L_B}\right) \int_0^{\infty} \frac{k\;dk \;}{k^2 + k_B^2 r_{\ell}^2}
\;\mathcal J_0 \left ( k \right) \;\mathcal J_0 \left ( \sqrt{\frac{\ell'}{\ell}} \;k \right) \;\mathcal J_0 \left ( k \frac{r_{12}}{r_{\ell}} \right).</math>
The interaction energy for <math> \
====== Quasiparticles ======
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
<math display="block">r_{12} = r_{\ell \ell'} = \sqrt{\ell + \ell'} \; r_B.</math>
This suggests that the pair of particles that are bound and separated by a distance <math>r_{\ell \ell'} </math> act as a single [[quasiparticle]] with angular momentum <math> \ell + \ell'</math>.
If we scale the lengths as <math> r_{\ell \ell'} </math>, then the interaction energy becomes
<math display="block"> E = \frac{2 e^2}{L_B} \int_0^{\infty} \frac{k\,dk}{k^2 + k_B^2 r_{\ell \ell'}^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'}} \right)}</math>
where
<math display="block">\tan \theta = \sqrt{\frac{\ell}{\ell'}}.</math>
The value of the <math> r_{12} </math> at which the energy is minimum, <math>r_{12} = r_{\ell \ell'} </math>, is independent of the ratio <math display="inline"> \tan \theta = \sqrt{{\ell}/{\ell'}}</math>. However the value of the energy at the minimum depends on the ratio. The lowest energy minimum occurs when
<math display="block"> \frac{\ell}{\ell'} = 1.</math>
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)
<math display="block"> \ell = \ell' = 1</math>
or
<math display="block"> \frac{\ell}{\ell^*} = \frac{1}{2} </math>
where the total angular momentum is written as
<math display="block"> \ell^* = \ell + \ell'. </math>
When the total angular momentum is odd, the minima cannot occur for <math> \ell = \ell' . </math> The lowest energy states for odd total angular momentum occur when
<math display="block"> \frac{\ell}{\ell^*} = \; \frac{\ell^*\pm 1}{2\ell^*}</math>
or
<math display="block">\frac{\ell}{\ell^*} = \frac{1}{3}, \frac{2}{5}, \frac{3}{7}, \text{etc.,} </math>
and
<math display="block">\frac{\ell}{\ell^*} = \frac{2}{3}, \frac{3}{5}, \frac{4}{7}, \text{etc.,} </math>
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<ref
<math display="block">
\frac{1}{\pi r_B^2 L_B}
\frac{1}{n!}
\left( \frac{r}{r_B} \right)^{2l}
\exp \left( -\frac{r
The interaction energy becomes
<math display="block"> E =
\left( \frac{2 e^2}{L_B}\right) \int_0^{\infty} \frac{k\;dk \;}{k^2 + k_B^2 r_{B}^2}
\; M {\left ( \ell + 1, 1, -\frac{k^2}{4} \right)} \;M {\left ( \ell' + 1, 1, -\frac{k^2}{4} \right)} \;\mathcal J_0 {\left ( k \frac{r_{12}}{r_{B}} \right)}
</math>
where <math>M</math> is a [[confluent hypergeometric function]] or [[Kummer function]]. In obtaining the interaction energy we have used the integral (see {{slink|Common integrals in quantum field theory#Integration over a magnetic wave function}})
<math display="block">
\frac{2}{n!} \int_0^{\infty} dr \; r^{2n+1} e^{-r^2} J_0(kr)
= M\left( n+1, 1, -\frac{k^2}{4}\right). </math>
As with delta function charges, the value of <math>r_{12}</math> 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
<math display="block">\frac{\ell}{\ell^*} = \frac{1}{3}, \frac{2}{5}, \frac{3}{7}, \text{etc.,} </math>
and
<math display="block">\frac{\ell}{\ell^*} = \frac{2}{3}, \frac{3}{5}, \frac{4}{7}, \text{etc.,} </math>
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.
=== Magnetostatics ===
==== 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 Lagrangian|Darwin interaction]]. To calculate this, consider the electrical currents in space generated by a moving charge
<math display="block">\mathbf J_1{\left( \mathbf x \right)} = a_1 \mathbf v_1 \delta^3 {\left( \mathbf x - \mathbf x_1 \right)}</math>
with a comparable expression for <math> \mathbf J_2 </math>.
The Fourier transform of this current is
<math display="block">\mathbf J_1{\left( \mathbf k \right)} = a_1 \mathbf v_1 \exp\left( i \mathbf k \cdot \mathbf x_1 \right).</math>
The current can be decomposed into a transverse and a longitudinal part (see [[Helmholtz decomposition]]).
<math display="block">\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).</math>
The hat indicates a [[unit vector]]. The last term disappears because
<math display="block">\mathbf k \cdot \mathbf J = -k_0 J^0 \to 0,</math>
which results from charge conservation. Here <math>k_0 </math> vanishes because we are considering static forces.
With the current in this form the energy of interaction can be written
<math display="block"> 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 ) .</math>
The propagator equation for the Proca Lagrangian is
<math display="block"> \eta_{\mu \alpha} \left ( \partial^2 + m^2\right ) D^{\alpha \nu}\left ( x-y \right ) = \delta_{\mu}^{\nu} \delta^4\left ( x-y \right ).</math>
The [[spacelike]] solution is
<math display="block">D\left ( k \right )\mid_{k_0=0}\; = \; -\frac{1}{k^2 + m^2},</math>
which yields
<math display="block"> 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 ),</math>
where <math display="inline">k = |\mathbf k|</math>. The integral evaluates to (see {{slink|Common integrals in quantum field theory#Transverse potential with mass}})
<math display="block"> E =
- \frac{1}{2} \frac{a_1 a_2}{4 \pi r} e^{ - m r} \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
</math>
which reduces to
<math display="block"> 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 </math>
in the limit of small {{mvar|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 plasma ====
In a plasma, the [[dispersion relation]] for an [[electromagnetic wave]] is<ref name="Chen"/>{{rp|pp=100–103}} (<math>c = 1</math>)
<math display="block">k_0^2 = \omega_p^2 + k^2,</math>
which implies
<math display="block">D\left ( k \right )\mid_{k_0=0}\; = \; -\frac{1}{ k^2 + \omega_p^2}.</math>
Here <math>\omega_p</math> is the [[plasma frequency]]. The interaction energy is therefore
<math display="block"> 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 \}.</math>
==== Magnetic interaction between current loops in a simple plasma or electron gas ====
===== Interaction energy =====
Consider a tube of current rotating in a magnetic field embedded in a simple [[Plasma (physics)|plasma]] or electron gas. The current, which lies in the plane perpendicular to the magnetic field, is defined as
<math display="block">\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)</math>
where
<math display="block"> r_{B1} = \frac{\sqrt{4 \pi}m_1 v_1}{a_1 B}</math>
and <math>\hat\mathbf b</math> is the unit vector in the direction of the magnetic field. Here <math>L_B</math> 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
<math display="block"> E = \left( \frac{a_1\, a_2}{2 \pi L_B}\right) v_1\, v_2\, \int_0^{\infty} {k\;dk \;} D\left( k \right) \mid_{k_0=k_B=0} \mathcal J_1 {\left ( kr_{B1} \right)} \mathcal J_1 {\left ( kr_{B2} \right)} \mathcal J_0 {\left ( kr_{12} \right)}</math>
where <math>r_{12}</math> is the distance between the centers of the current loops and <math> \mathcal J_n ( x )</math> is a [[Bessel function]] of the first kind. In obtaining the interaction energy we made use of the integrals
<math display="block">
and
<math display="block"> \int_0^{2 \pi} \frac{d\varphi}{2 \pi} \cos\left( \varphi \right) \exp\left( i p \cos\left( \varphi \right) \right)
= i\mathcal J_1 \left( p \right) . </math>
See ''{{slink|Common 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 an [[Electromagnetic electron wave#X wave|extraordinary wave]].<ref name="Chen"/>{{rp|pp=110–112}} This wave generates [[Hall current]]s that interact and modify the electromagnetic field. The [[dispersion relation]] for extraordinary waves is<ref name="Chen"/>{{rp|112}}
<math display="block"> -k_0^2 + k^2 + \omega_p^2 \frac{k_0^2 - \omega_p^2}{k_0^2- \omega_H^2} =0,</math>
which gives for the propagator
<math display="block"> D\left( k \right) \mid_{k_0=k_B=0}\;= \;-\left( \frac{1}{ k^2 + k_X^2}\right)</math>
where
<math display="block">k_X \equiv \frac{\omega_p^2}{\omega_H}</math>
in analogy with the Darwin propagator. Here, the upper hybrid frequency is given by
<math display="block"> \omega_H^2 = \omega_p^2 + \omega_c^2,</math>
the [[cyclotron frequency]] is given by ([[Gaussian units]])
<math display="block"> \omega_c = \frac{e B}{m c},</math>
and the [[plasma frequency]] ([[Gaussian units]])
<math display="block"> \omega_p^2 = \frac{4\pi n e^2}{m}.</math>
Here {{mvar|n}} is the electron density, {{math|''e''}} is the magnitude of the electron charge, and {{mvar|m}} is the electron mass.
The interaction energy becomes, for like currents,
<math display="block"> E =
- \left( \frac{a^2}{2 \pi L_B}\right) v^2\, \int_0^{\infty} \frac{k\;dk}{ k^2 + k_X^2}
\mathcal J_1^2 \left ( kr_{B} \right) \mathcal J_0 \left ( kr_{12} \right)</math>
===== Limit of small distance between current loops =====
In the limit that the distance between current loops is small,
<math display="block"> E = - E_0 \; I_1 {\left( \mu \right)} K_1 {\left( \mu \right)}</math>
where
<math display="block"> E_0 = \left( \frac{a^2}{2 \pi L_B}\right) v^2</math>
and
<math display="block">\mu =\frac{\omega_p^2 r_B}{\omega_H}= k_X \;r_B</math>
and {{math|''I''}} and {{math|''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 {{slink|Common integrals in quantum field theory#Integration of the cylindrical propagator with mass}})
<math display="block">\int_o^{\infty} \frac{k\; dk}{k^2 +m^2} \mathcal J_1^2 \left( kr \right)
= I_1 \left( mr \right)K_1 \left( mr \right) . </math>
For small {{math|''mr''}} the integral becomes
<math display="block">
I_1 {\left( mr \right)} K_1 {\left( mr \right)}
\to
\frac{1}{2}\left[ 1- \frac{1}{8}\left( mr \right)^2 \right] . </math>
For large {{math|''mr''}} the integral becomes
<math display="block">
I_1 \left( mr \right)K_1 \left( mr \right)
\rightarrow
\frac{1
===== Relation to the quantum Hall effect =====
The screening [[wavenumber]] can be written ([[Gaussian units]])
<math display="block"> \mu = \frac{\omega_p^2 r_B}{\omega_H c}
= \left( \frac{2e^2r_B}{L_B \hbar 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</math>
where <math>\alpha</math> is the [[fine-structure constant]] and the filling factor is
<math display="block"> \nu = \frac{2\pi N \hbar c}{eBA}</math>
and {{mvar|N}} is the number of electrons in the material and {{mvar|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 levels|Landau states]] at the ground state energy.
For cases of interest in the quantum Hall effect, <math>\mu</math> is small. In that case the interaction energy is
<math display="block"> E = - \frac{E_0}{2} \left[ 1- \frac{1}{8}\mu^2\right]</math>
where ([[Gaussian units]])
<math display="block"> E_0 = {4\pi}\frac{e^2}{L_B}\frac{v^2}{c^2}
= {8\pi}\frac{e^2}{L_B}\left( \frac{\hbar \omega_c}{m c^2}\right)</math>
is the interaction energy for zero filling factor. We have set the classical kinetic energy to the quantum energy
<math display="block">\frac{1}{2} m v^2 = \hbar \omega_c.</math>
=== Gravitation ===
A gravitational disturbance is generated by the [[stress–energy tensor]] <math> T^{\mu \nu} </math>; consequently, the Lagrangian for the gravitational field is [[Spin (physics)|spin]]-2. If the disturbances are at rest, then the only component of the stress–energy tensor that persists is the <math> 00 </math> 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
<math display="block">D\left ( k \right )\mid_{k_0=0}\; = \; - \frac{4}{3} \frac{1}{ k^2 + m^2}</math>
and
<math display="block">E = -\frac{4}{3}\frac{a_1 a_2}{4 \pi r} \exp \left ( -m r \right ),</math>
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.<ref name="Zee"/>{{rp|pp=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.<ref name="Zee"/>{{rp|35}}
== References ==
{{reflist}}
[[Category:Quantum field theory]]
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