Lagrangian for Frenkel electron and position's non-commutativity due to spin

Abstract: We construct relativistic-invariant spinning-particle Lagrangian without auxiliary variables. Spin is considered as a composed quantity constructed on the base of non-Grassmann vector-like variable. The variational problem guarantees both fixed value of spin and Frenkel condition on spin-tensor. Taking into account the Frenkel condition, we obtain, inevitably, relativistic corrections to the algebra of position variables: their classical brackets became noncommutative, with the "parameter of non-commutativity" proportional to the spin-tensor. This leads to a number of interesting consequences in quantum theory. We construct the relativistic quantum mechanics in canonical formalism (in physical-time parametrization) and in covariant formalism (in arbitrary parametrization). We show how state-vectors and operators of covariant formulation can be used to compute mean values of physical operators of position and spin. This proves relativistic covariance of canonical formalism. Various candidates for position and spin operators of an electron acquire clear meaning and interpretation in the Lagrangian model of Frenkel electron. We also establish the relation between Frenkel electron and positive-energy sector of Dirac equation, this allowed us to turn to the long-standing problem on spin and position operators of Dirac theory. Contrary to widely assumed opinion, our results argue in favor of Pryce's (d)-type operators. This implies that effects of non-commutativity could be presented at the Compton wave length, in contrast to conventional expectations at the Planck length. At last, we present the manifestly covariant form of spin and position operators of Dirac equation.