Newton's law for Bloch electrons, Klein factors and deviations from canonical commutation relations

Abstract

The acceleration theorem for Bloch electrons in a homogenous external field is usually presented using quasiclassical arguments. In quantum mechanical versions the Heisenberg equations of motion for an operator $\hat {\vec k}(t)$ are presented mostly without properly defining this operator. This leads to the surprising fact that the generally accepted version of the theorem is incorrect for the most natural definition of $\hat {\vec k}$. This operator is shown not to obey canonical commutation relations with the position operator. A similar result is shown for the phase operators defined via the Klein factors which take care of the change of particle number in the bosonization of the field operator in the description of interacting fermions in one dimension. The phase operators are also shown not to obey canonical commutation relations with the corresponding particle number operators. Implications of this fact are discussed for Tomonaga-Luttinger type models.