^{1}

We present the relativistic particle model without Grassmann variables which, being canonically quantized, leads to the Dirac equation. Classical dynamics of the model is in correspondence with the dynamics of mean values of the corresponding operators in the Dirac theory. Classical equations for the spin tensor are the same as those of the Barut-Zanghi model of spinning particle.

Starting from the classical works [

In the course of canonical quantization of a given classical theory, one associates Hermitian operators with classical variables. Let

Since the quantum theory of spin is known (it is given by the Pauli (Dirac) equation for nonrelativistic (relativistic) case), search for the corresponding semiclassical model represents the inverse task to those of canonical quantization: we look for the classical-mechanics system whose classical bracket obeys (

In their pioneer work [

The problem here is that the Grassmann classical mechanics represents a rather formal mathematical construction. It leads to certain difficulties [

To describe the nonrelativistic spin by commuting variables, we need to construct a mechanical model which implies the commutator (even) operator algebra (

The Dirac constraints presented in the model imply [

The model leads to reasonable picture both on classical and quantum levels. The classical dynamics is governed by the Lagrangian equations

Below, we generalize this scheme to the relativistic case, taking angular-momentum variables as the basic coordinates of the spin space. On this base, we construct the relativistic-invariant classical mechanics that produces the Dirac equation after the canonical quantization, and briefly discuss its classical dynamics.

We start from the model-independent construction of the relativistic-spin space. Relativistic equation for the spin precession can be obtained including the three-dimensional spin vector

In the passage from nonrelativistic to relativistic spin, we replace the Pauli equation by the Dirac one

Let us discuss the classical variables that could produce the

To reach the algebra starting from a classical-mechanics model, we introduce ten-dimensional “phase” space of the spin degrees of freedom,

The Jacobian of the transformation

According to (

Summing up, to describe the relativistic spin, we need a theory that implies the Dirac constraints (

One possible dynamical realization of the construction presented above is given by the following

Curiously enough, the action can be rewritten in almost five-dimensional form. Indeed, after the change

In the Hamiltonian formalism, the action implies the desired constraints (

We now discuss some properties of the classical theory and confirm that they are in correspondence with semiclassical limit [

The auxiliary variables

While the canonical momentum of

Identifying the variables

The quantity (center-of-mass coordinate [

As the classical four-dimensional spin vector, let us take

The BZ spinning particle [

This work has been supported by the Brazilian foundation FAPEMIG.