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{{Use American English|date = February 2019}}
{{Short description|Physical interaction in post-classical physics}}
'''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|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 |
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 Yukawa potential: The force between two nucleons in an atomic nucleus===
Consider the [[Spin (physics)|spin]]-0 Lagrangian density<ref
<math display="block">
\mathcal{L} [\varphi (x)]
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====The Coulomb potential in a vacuum====
Consider the [[Spin (physics)|spin]]-1 [[Proca action|Proca Lagrangian]] with a disturbance<ref
<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>
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=====Plasma waves=====
The [[dispersion relation]] for [[plasma wave]]s is<ref name="Chen">{{cite book |
<math display="block">\omega^2 = \omega_p^2 + \gamma\left( \omega \right) \frac{T_e}{m} \vec k^2.</math>
where <math>\omega </math> is the angular frequency of the wave,
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[[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>\theta = \frac{\pi}{4}</math> or <math> \frac{\mathit l}{\mathit l'} = 1 </math>. The highest energy plotted is for <math>\theta = 0.90\frac{\pi}{4}</math>. Lengths are in units of <math>r_{\mathit l \mathit l'}</math>.]]
[[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> { \mathit l = \mathit l'} </math> or <math> \frac{ \mathit l}{\mathit l^{*}} = \frac{1}{2} </math>. 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 <math> { \mathit l \ne \mathit l'} </math>, the total energy is higher than the case when <math> { \mathit l = \mathit l'} </math> for a given value of <math> { \mathit l^{*}} </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_{\mathit l} = \sqrt{\mathit l}\;r_B\; \; \; \mathit l=0,1,2, \ldots</math>
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======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}
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====Darwin interaction in a plasma====
In a plasma, the [[dispersion relation]] for an [[electromagnetic wave]] is<ref
<math display="block">k_0^2 = \omega_p^2 +\vec k^2,</math>
which implies
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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
<math display="block"> -k_0^2 +\vec 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
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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
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
==References==
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