Additive K-theory

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

According to, 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 ${\ell }_B=\sqrt{\hslash /eB}$, which makes them mathematically similar to the position and momentum operators $Q=q$ and $P=-i\hslash \frac{d}{dq}$ 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 $(p{\text{'}}_x,p{\text{'}}_y)$ do not commute.

References

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