Gauge-invariant description of a Dirac electron moving in a noncommutative gauge field background

Abstract We provide a gauge-invariant description of the quantum dynamics of an electron in a constant noncommutative (NC) gauge field background using the Dirac equation in an NC plane. We show that the physical force imparted on the electron is the standard Lorentz force but with modified NC magnetic fields that pick up two different first-order corrections in the relativistic and nonrelativistic (NR) domains. Taking the NR limit of the NC modified Dirac equation we show that noncommutativity affects the cyclotron frequency but leaves the Hall conductivity and cyclotron motion trajectory unaffected to first order in the NC parameter. The hyperfine splitting of hydrogen atom (H-atom) spectrum also shows a first-order correction.