Uehling potential

In quantum electrodynamics, the Uehling potential describes the interaction potential between two electric charges which, in addition to the classical Coulomb potential, contains an extra term responsible for the electric polarization of the vacuum. This potential was found by Uehling in 1935.[1][2]

The Uehling potential is given by

${\displaystyle V(r)={\frac {-e^{2}}{4\pi r}}\left(1+{\frac {e^{2}}{6\pi ^{2}}}\int _{1}^{\infty }dx\,e^{-2rm_{\text{e}}x}{\frac {2x^{2}+1}{2x^{4}}}{\sqrt {x^{2}-1}}\right),}$

from where it is apparent that this potential is indeed a refinement of the classical Coulomb potential. Here ${\displaystyle m_{\text{e}}}$ is the electron mass, ${\displaystyle e}$ is its charge measured at large distances. If ${\displaystyle r\gg 1/m}$, this potential simplifies to

${\displaystyle V(r)\approx {\frac {-e^{2}}{4\pi r}}\left(1+{\frac {e^{2}}{16\pi ^{3/2}}}{\frac {1}{(m_{\text{e}}r)^{3/2}}}e^{-2m_{\text{e}}r}\right),}$

while for ${\displaystyle r\ll 1/m}$ we have

${\displaystyle V(r)\approx {\frac {-e^{2}}{4\pi r}}\left(1+{\frac {e^{2}}{6\pi ^{2}}}\left(\log {\frac {1}{m_{\text{e}}r}}-\gamma -{\frac {5}{6}}\right)\right),}$

where ${\displaystyle \gamma }$ is the Euler–Mascheroni constant.

Properties

It was recently demonstrated that the above integral in the expression of ${\displaystyle V(r)}$ can be evaluated in closed form by using the modified Bessel functions of the second kind ${\displaystyle K_{0}(z)}$ and its successive integrals.[3]

See also

• QED vacuum
• Virtual particles

References

1. Uehling, E. A. (1935). "Polarization Effects in the Positron Theory". Physical Review. 48: 55–63. doi:10.1103/physrev.48.55.
2. Schwartz, M. D. (2013). "16". Quantum Field Theory and the Standard Model. Cambridge University Press. ISBN 978-1-107-03473-0.
3. Frolov, A. E.; Wardlaw, D. M. (2012). "Analytical formula for the Uehling potential". The European Physical Journal B. 85. arXiv:1110.3433. doi:10.1140/epjb/e2012-30408-4.

Further reading

• More on the vacuum polarization in QED, see section 7.5 of M.E. Peskin and D.V. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley, 1995.
• Both the exact form and the ${\displaystyle r\gg 1/m}$, ${\displaystyle r\ll 1/m}$ approximations are proved in details in the 114th section of V. B. Berestetskii, E. M. Lifshitz, L. P. Pitaevskii, Quantum Electrodynamics, Butterworth-Heinemann, 1982.