ar X iv : h ep - t h / 07 01 06 2 v 1 8 J an 2 00 7 Comments on spin operators and spin - polarization states of 2 + 1 fermions

In this brief article we discuss spin polarization operators and spin polarization states of 2 + 1 massive Dirac fermions and find a convenient representation by the help of 4-spinors for their description. We stress that in particular the use of such a representation allows us to introduce the conserved covariant spin operator in the 2 + 1 field theory. Another advantage of this representation is related to the pseudoclassical limit of the theory. Indeed, quantization of the pseudoclassical model of a spinning particle in 2 + 1 dimensions leads to the 4-spinor representation as the adequate realization of the operator algebra, where the corresponding operator of a first-class constraint, which cannot be gauged out by imposing the gauge condition, is just the covariant operator previously introduced in the quantum theory. I. The 2 + 1 spinor field theory [1] has attracted in recent years great attention due to various reasons, e.g., because of nontrivial topological properties, and due to a possibility of the existence of particles with fractional spins and exotic statistics (anyons), having probably applications to fractional Hall effect, high-T c superconductivity and so on [2]. In many practical situations the quantum behavior of spin 1/2 fermions (from now on simply called fermions) in 2 + 1 dimensions can be described by the corresponding Dirac equation with an external electromagnetic field. The main difference between the relativistic quantum mechanics of fermions in 3 + 1 and in 2 + 1 dimensions is related to the different description of spin polarization states. It is well know that in 3 + 1 dim. there exist two massive spin 1/2 fermions, the electron and its corresponding antiparticle, i.e., the positron. Both the electron and the positron have two spin polarization states. In 2 + 1 dim. there exist four massive fermions: two different types of electrons and two corresponding positrons. In contrast to the situation in 3 + 1 dim., each particle in 2 + 1 dim. has only one polarization state. We recall that constructing the covariant and conserved spin operators for the 3 + 1 Dirac equation in an external field is an important problem as regards finding exact solutions of this equation and specificating the spin polarization states [3]. Here, there do not exist universal covariant conserved spin operators which relate to any external field, for each specific configuration of the external field one has to determine such …