Local Gauge and Magnetic Translation Groups
Tle magnetic translation group was introduced as a set of operators T(R) = exp{—iR.(p-cA/c)/h]. However, these operators commute with the Hamiltonian for an electron in a periodic potential and a uniform magnetic field if the vector potential Α (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 _ Α (r). Such choice of the local gauge determines a factor system ω(11, R') = T(R)T(R')T(R + R') -1 , which depends on a global gauge only. Moreover, for any potential Α a commutator Τ(R)T(R')T(R) -1 T(R') -1 depends only on the magnetic field and not on the gauge. PACS numbers: 02.20.-a, 71.45.—d, 11.15.Ha, 05.50.+q