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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 [[
The force can also be written
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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]])
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<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 particle’s field.
====
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
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<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
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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|two line charges embedded in a plasma or electron gas]], [[#Coulomb potential between two current loops embedded in a magnetic field|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|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 Hall effect|fractional quantum numbers]].
== Selected examples ==
===
Consider the [[Spin (physics)|spin]]-0 Lagrangian density<ref name="Zee"/>{{rp|pp=21–29}}
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[[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}}
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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{
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>
For low frequencies, the dispersion relation becomes
<math display="block"> \mathbf k^2 + \mathbf
where
<math display="block">
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 +
In fact, if the retardation effects are not neglected, then the dispersion relation is
<math display="block"> -k_0^2 + k^2 +
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 ( -
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 ( -
where the inverse of the Thomas–Fermi screening length is
<math display="block">
and <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 + \
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 electron gas
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<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
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\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
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The density fluctuation is then
<math display="block"> \delta n = \frac{e \varphi}{\hbar \omega_c
where <math>
[[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
The propagator is then
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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>\ell = \ell'</math>. Lengths are in units are in <math>r_\ell</math>, and the energy is in units of <math display="inline"> \frac{e^2}{L_B}</math>. Here <math>r = r_{12}</math>. Note that there are local minima for large values of <math> k_{B}</math>.]]
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The interaction energy for <math> \ell = \ell'</math> is given in Figure 1 for various values of <math>k_B r_\ell</math>. The energy for two different values is given in Figure 2.
====== 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
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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 name="Ezewa"/>{{rp|189}}
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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
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\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\{
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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
In a plasma, the [[dispersion relation]] for an [[electromagnetic wave]] is<ref name="Chen"/>{{rp|pp=100–103}} (<math>c = 1</math>)
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\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 ====
=====
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
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= 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}}
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\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,
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\frac{1}{2}\;\left( \frac{1}{mr}\right) . </math>
===== Relation to the quantum Hall effect =====
The screening [[wavenumber]] can be written ([[Gaussian units]])
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<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>
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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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