Magnetic translation

Magnetic translations are naturally defined operators acting on wave function on a two-dimensional particle in a magnetic field.
The motion of an electron in a magnetic field on a plane is described by the following four variables: guiding center coordinates formula_1 and the relative coordinates formula_2.
The guiding center coordinates are independent of the relative coordinates and, when quantized, satisfy
formula_3, 
where formula_4, which makes them mathematically similar to the position and momentum operators formula_5 and formula_6 in one-dimensional quantum mechanics.
Much like acting on a wave function formula_7 of a one-dimensional quantum particle by the operators formula_8 and formula_9 generate the shift of momentum or position of the particle, for the quantum particle in 2D in magnetic field one considers the magnetic translation operators
formula_10 
for any pair of numbers formula_11.
The magnetic translation operators corresponding to two different pairs formula_12 and formula_13 do not commute.