# Euler–Heisenberg Lagrangian

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In physics, the Euler–Heisenberg Lagrangian describes the non-linear dynamics of electromagnetic fields in vacuum. It was first obtained by Werner Heisenberg and Hans Heinrich Euler[1] in 1936. By treating the vacuum as a medium, it predicts rates of quantum electrodynamics (QED) light interaction processes.[clarification needed]

## Physics

It takes into account vacuum polarization to one loop, and is valid for electromagnetic fields that change slowly compared to the inverse electron mass:

${\displaystyle {\mathcal {L}}=-{\mathcal {F}}-{\frac {1}{8\pi ^{2}}}\int _{0}^{\infty }\exp \left(-m^{2}s\right)\left[(es)^{2}{\frac {\operatorname {Re} \cosh \left(es{\sqrt {2\left({\mathcal {F}}+i{\mathcal {G}}\right)}}\right)}{\operatorname {Im} \cosh \left(es{\sqrt {2\left({\mathcal {F}}+i{\mathcal {G}}\right)}}\right)}}{\mathcal {G}}-{\frac {2}{3}}(es)^{2}{\mathcal {F}}-1\right]{\frac {ds}{s^{3}}}}$

Here m is the electron mass, e the electron charge, ${\displaystyle {\mathcal {F}}={\frac {1}{2}}\left(\mathbf {B} ^{2}-\mathbf {E} ^{2}\right)}$, and ${\displaystyle {\mathcal {G}}=\mathbf {E} \cdot \mathbf {B} }$.

In the weak field limit, this becomes: ${\displaystyle {\mathcal {L}}={\frac {1}{2}}\left(\mathbf {E} ^{2}-\mathbf {B} ^{2}\right)+{\frac {2\alpha ^{2}}{45m^{4}}}\left[\left(\mathbf {E} ^{2}-\mathbf {B} ^{2}\right)^{2}+7\left(\mathbf {E} \cdot \mathbf {B} \right)^{2}\right]}$

It describes photon-photon scattering in QED; Robert Karplus and Maurice Neuman calculated the full amplitude,[2] which is very small and has not been seen.

## Experiments

Delbrück scattering of gamma rays was observed in 1953 by Robert Wilson.[3] Photon splitting in strong magnetic fields was measured in 2002.[4]

PVLAS is searching for vacuum polarization of laser beams crossing magnetic fields to detect effects from axion dark matter. No signal has been found and searches continue. OSQAR at CERN is also studying vacuum birefringence.

A team of astronomers from Italy, Poland, and the U.K. has reported in 2016[5] observations of the light emitted by a neutron star (pulsar RX J1856.5−3754). The star is surrounded by a very strong magnetic field (1013G), and one expects birefringence from the vacuum polarization described by the Euler–Heisenberg Lagrangian. A degree of polarization of about 16% was measured and was claimed to be "large enough to support the presence of vacuum birefringence, as predicted by QED".

## References

1. ^ W. Heisenberg and H. Euler, Folgerungen aus der Diracschen Theorie des Positrons Z. Phys. 98, 714 (1936).
2. ^ R. Karplus and M. Neuman, “The Scattering of Light by Light”, Phys. Rev. 83, 776 (1951).
3. ^ Sh. Zh. Akhmadaliev et al, “Delbrück scattering at energies of 140-450 MeV”, Phys. Rev. C 58, 2844 (1998).
4. ^ Sh. Zh. Akhmadaliev et al, “ Experimental investigation of high-energy photon splitting in atomic fields”, Phys. Rev. Lett. 89, 061802 (2002).
5. ^ R. P. Mignani, V. Testa, D. González Caniulef, R. Taverna, R. Turolla, S. Zane, and K. Wu, "Evidence for vacuum birefringence from the first optical-polarimetry measurement of the isolated neutron star RX J1856.5−3754", Month. Not. Roy. Astron. Soc. 465 (2017) 492, published online 02 Nov 2016.