Magnetic translation

Magnetic translations are naturally defined operators acting on wave function on a two-dimensional particle in a magnetic field.

According to,^[1] the motion of an electron in a magnetic field on a plane is described by the following four variables: guiding center coordinates $(X,Y)$ and the relative coordinates $(R_x,R_y)$.

The guiding center coordinates are independent of the relative coordinates and, when quantized, satisfy
$[X,Y]=-i \ell_B^2$,
where $\ell_B=\sqrt{\hbar/eB}$, which makes them mathematically similar to the position and momentum operators $Q =q$ and $P=-i\hbar \frac{d}{dq}$ in one-dimensional quantum mechanics.

Much like acting on a wave function $f(q)$ of a one-dimensional quantum particle by the operators $e^{iaP}$ and $e^{ibQ}$ 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
$e^{i(p_x X + p_y Y)},$
for any pair of numbers $(p_x, p_y)$.

The magnetic translation operators corresponding to two different pairs $(p_x,p_y)$ and $(p'_x,p'_y)$ do not commute.

References

1. Z.Ezawa. Quantum Hall Effect, 2nd ed, World Scientific. Chapter 28