Line integral convolution: Difference between revisions

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In physics, the '''Euler–Heisenberg Lagrangian''' describes the non-linear dynamics of electromagnetic fields in vacuum. It takes into account vacuum polarization to one loop, and it is valid for electromagnetic fields that change slowly compared to the inverse electron mass. It was first obtained by [[Werner Heisenberg]] and [[Hans Heinrich Euler]],<ref>W. Heisenberg and H. Euler, ''Folgerungen aus der Diracschen Theorie des Positrons'' Z. Phys. '''98''', 714 (1936).</ref> and can be expressed as:
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:<math>\mathcal{L} =-\mathcal{F} -\frac{1}{8\pi^{2}}\int_{0}^{\infty}\frac{ds}{s^{3}}\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]</math>
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Here m is the electron mass, e the electron charge,
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:<math>\mathcal{F}=\frac{1}{2}\left(\mathbf{B}^2 - \mathbf{E}^2\right)</math>,
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:<math>\mathcal{G}=\mathbf{E}\cdot\mathbf{B}</math>
 
In the weak field limit, this becomes:
 
:<math>\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]</math>
 
==References==
<references/>
 
{{DEFAULTSORT:Euler-Heisenberg Lagrangian}}
[[Category:Quantum electrodynamics]]
 
 
{{electromagnetism-stub}}
{{quantum-stub}}

Latest revision as of 20:45, 17 November 2014

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