Lagrangian treatment of magnetic dielectrics

A general Lagrangian-based long-wavelength theory of ordered magnetic dielectric crystals (ferromagnetic, antiferromagnetic, and ferrimagnetic) is formulated. Our classical treatment of intrinsic spin uses the anticommuting Grassmann algebra G₃ developed by Berezin and Marinov and by Casalbuoni. The Grassmann formulation of classical spin gives by the Dirac quantization procedure the usual nonrelativistic spin-1/2 quantum theory. The treatment begins at the microscopic level before a long-wavelength limit is performed to obtain a macroscopic theory so as to incorporate all long-wavelength modes of motion (acoustic, optic, electromagnetic, and spin) and their interaction to all orders of nonlinearity. The crystals can have any symmetry, anisotropy, and structural complexity. The equations of motion for all the modes are obtained and the energy, momentum, pseudomomentum, angular momentum, and spin conservation laws are found. The magnetizations arising from spin and from the motion of bound charge are found to enter the energy conservation law distinguishably. This theory should be particularly useful for the study of magneto-optical phenomena.