Euler–Heisenberg Lagrangian

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.

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, $\mathcal{F}=\frac{1}{2}\left(\mathbf{B}^2 - \mathbf{E}^2\right)$, and $\mathcal{G}=\mathbf{E}\cdot\mathbf{B}$.

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

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.

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, “Delbr¨uck 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).