Local gauge and magnetic translation groups
Wojciech Florek
Abstract
The magnetic translation group was introduced as a set of operators T(R)=[-iR.(p-eA/c)/]. However,these operators commute with the Hamiltonian for an electron in a periodic potential and a uniform magnetic field if the vector potential A (the gauge) is chosen in a symmetric way. It is showed that a local gauge field AR(r) on a crystal lattice leads to operators, which commute with the Hamiltonian for any (global) gauge field A=A(r). Such choice of the local gauge determines afactor system ω(R,R')= T(R)T(R') T(R+R')-1, which depends on a global gauge only. Moreover, for any potential A a commutator T(R)T(R')T(R)-1T(R')-1 depends only on the magnetic field and not on the gauge.
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