When the vacuum becomes a prism
Renormalization is sometimes seen as a formal trick. Yet it uncovers a new layer of reality with real physical effects. One of them is the birefringence of the vacuum in a strong electromagnetic field, which follows from the Lagrangian of Euler and Heisenberg [1]. In this project you will see that quantum fluctuations make the vacuum an optically anisotropic medium.
In classical electrodynamics the Lagrangian of the field is \[L_0 = \frac{1}{8\pi}(\boldsymbol{E}^2 - \boldsymbol{B}^2).\] Quantum corrections due to virtual electron–positron pairs lead to nonlinear effects. At low energies the effective Lagrangian (the Euler–Heisenberg Lagrangian) takes the form \[L_{\text{eff}} = L_0 + \frac{e^4}{360\pi^2 m_e^4} \left[(\boldsymbol{E}^2 - \boldsymbol{B}^2)^2 + 7(\boldsymbol{E} \cdot \boldsymbol{B})^2\right].\]
Derive the expressions for the displacement and the magnetizing field: \[\boldsymbol{D} = 4\pi \frac{\partial L_{\text{eff}}}{\partial \boldsymbol{E}}, \quad \boldsymbol{H} = -4\pi \frac{\partial L_{\text{eff}}}{\partial \boldsymbol{B}}.\]
Compare the constitutive relations you obtain with those of the vacuum. What has changed?
Linearize the equations, assuming that the field consists of a background \(\boldsymbol{E}_0, \boldsymbol{B}_0\) and a small perturbation \(\delta \boldsymbol{E}, \delta \boldsymbol{B}\). Obtain the permittivity and permeability tensors of the medium.
Write Maxwell’s equations for the wave perturbation. Find the dispersion relation for a wave propagating along the \(x\) axis, assuming that the constant magnetic field is directed along the \(z\) axis: \[\boldsymbol{B}_0 = B_0 \boldsymbol{e}_z.\]
Consider two cases:
the electric field of the wave is parallel to \(\boldsymbol{B}_0\) (polarization along the \(z\) axis);
the electric field of the wave is perpendicular to \(\boldsymbol{B}_0\) (polarization along the \(y\) axis).
✓ Find the refractive indices \(n_\parallel\) and \(n_\perp\) and their difference \[\Delta n = n_\parallel - n_\perp.\]
Propose an experiment to observe vacuum birefringence. Which effect could be used: interference, rotation of the polarization, a phase shift? What fields and what sensitivity are needed?
References
1 Heisenberg, Werner, Euler, Hans, Consequences of Dirac’s theory of the positron, Zeitschrift für Physik 98 714–732 (1936) https://doi.org/10.1007/BF01343663