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Finally, the change in energy due to the static disturbances of the vacuum is
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<math> E =
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The energy due to the static disturbances becomes (see [[Common integrals in quantum field theory#Yukawa Potential: The Coulomb potential with mass|Common integrals in quantum field theory]])
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<math>
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with
:<math>r^2 = \left (\vec x_1 - \vec x_2 \right )^2</math>▼
▲\left (\vec x_1 - \vec x_2 \right )^2
which is attractive and has a range of
:<math>{1 \over 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.
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In the limit of zero [[photon]] mass, the Lagrangian reduces to the Lagrangian for [[electromagnetism]]
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<math>
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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
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<math>
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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>
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<math>
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The interaction energy is
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:<math>
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In this geometry, the interaction energy can be written
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:<math>
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and the interaction energy becomes
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:<math>
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where <math>\mathit l</math> is the angular momentum [[quantum number]]. When <math>\mathit l=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>\mathit l^{ }_{ }>0 </math> and <math>\mathit l^{\prime} \ge \mathit l^{ }_{ } </math> interact, and we assume the charge densities are delta functions at radius <math>r_{\mathit l}</math>, then the interaction energy is
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:<math>
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If we scale the lengths as <math> r_{\mathit l \mathit l^{\prime}} </math>, then the interaction energy becomes
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:<math>
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The interaction energy becomes
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:<math>
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which reduces to
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<math> E =
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Here <math>\omega_p</math> is the [[plasma frequency]]. The interaction energy is therefore
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<math> E =
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The interaction energy becomes, for like currents,
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<math>
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In the limit that the distance between current loops is small,
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<math>
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For cases of interest in the quantum Hall effect, <math>\mu</math> is small. In that case the interaction energy is
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<math>
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and
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<math>
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