Physics in ultra-strong magnetic fields

In magnetic fields stronger than B_Q≡m_e²c³/ℏe=4.4×10¹³ Gauss, an electron's Landau excitation energy exceeds its rest energy. I review the physics of this strange regime and some of its implications for the crusts and magnetospheres of neutron stars. In particular, I describe how ultra-strong fields • render the vacuum birefringent and capable of distorting and magnifying images (``magnetic lensing'') • change the self-energy of electrons: as B increases they are first slightly lighter than m_e, then slightly heavier; • cause photons to rapidly split and merge with each other; • distort atoms into long, thin cylinders and molecules into strong, polymer-like chains; • enhance the pair density in thermal pair-photon gases; • strongly suppress photon-electron scattering, and • drive the vacuum itself unstable, at extremely large B. In a concluding section, I discuss the spindown of ultra-magnetized neutron stars and recent soft gamma repeater observations. .