Physics in Ultra-strong Magnetic Fields

In magnetic fields stronger than B_Q = 4.4 X 10^13 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 magneto- spheres 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 recent observations of the spindown histories of soft gamma repeaters and anomalous X-ray pulsars. The magnetar model gives a promising framework for understanding these data.