Relativistic Spinning Particle without Grassmann Variables and the Dirac Equation

Abstract

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.

1. Discussion: Nonrelativistic Spin

Starting from the classical works [1–6], a lot of efforts have been spent in attempts to understand behaviour of a particle with spin on the base of semiclassical mechanical models [7–24].

In the course of canonical quantization of a given classical theory, one associates Hermitian operators with classical variables. Let 𝑧^𝛼 stands for the basic phase-space variables that describe the classical system, and {𝑧^𝛼,𝑧^𝛽} is the corresponding classical bracket. (It is the Poisson (Dirac) bracket in the theory without (with) second-class constraints.) According to the Dirac quantization paradigm [3], the operators 𝑧^𝛼̂ must be chosen to obey the quantization rule𝑧^𝛼̂,𝑧^𝛽̂∓=𝑖ℏ𝑧^𝛼,𝑧^𝛽||_𝑧→𝑧̂.(1.1) In this equation, we take commutator (anticommutator) of the operators for the antisymmetric (symmetric) classical bracket. Antisymmetric (symmetric) classical bracket arises in the classical mechanics of even (odd = Grassmann) variables.

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 (1.1) for the known left-hand side. Components of the nonrelativistic spin operator 𝑆_𝑖=(ℏ/2)𝜎^𝑖 (𝜎^𝑖 are the Pauli matrices (1.5)) form a simple algebra with respect to commutator𝑆^𝑖,𝑆_𝑗₋=𝑖ℏ𝜖_𝑖𝑗𝑘𝑆_𝑘,(1.2) as well as to anticommutator𝑆^𝑖,𝑆_𝑗₊=ℏ2²𝛿^𝑖𝑗.(1.3) So, the operators can be produced starting from a classical model based on either even or odd spin-space variables.

In their pioneer work [13, 14], Berezin and Marinov have constructed the model based on the odd variables and showed that it gives very economic scheme for semiclassical description of both nonrelativistic and relativistic spin. Their prescription can be shortly resumed as follows. For nonrelativistic spin, the noninteracting Lagrangian reads (𝑚/2)(𝑥̇_𝑖)²+(𝑖/2)𝜉_𝑖𝜉̇_𝑖, where the spin inner space is constructed from vector-like Grassmann variables 𝜉_𝑖, 𝜉_𝑖𝜉_𝑗=−𝜉_𝑗𝜉_𝑖. Since the Lagrangian is linear on 𝜉̇_𝑖, their conjugate momenta coincide with 𝜉, 𝜋_𝑖=𝜕𝐿/𝜕𝜉̇_𝑖=𝑖𝜉_𝑖. The relations represent the Dirac second-class constraints and are taken into account by transition from the odd Poisson bracket to the Dirac one, the latter reads𝜉_𝑖,𝜉_𝑗_DB=𝑖𝛿_𝑖𝑗.(1.4) Dealing with the Dirac bracket, one can resolve the constraints, excluding the momenta from consideration. So, there are only three spin variables 𝜉_𝑖 with the desired brackets (1.4). According to (1.1), (1.4), and (1.3), canonical quantization is performed replacing the variables by the spin operators 𝑆_𝑖 proportional to the Pauli 𝜎-matrices, 𝑆_𝑖=(ℏ/2)𝜎^𝑖, [𝜎^𝑖,𝜎^𝑗]₊=2𝛿^𝑖𝑗,𝜎¹=⎛⎜⎜⎝0110⎞⎟⎟⎠,𝜎²=⎛⎜⎜⎝0𝑖−0𝑖⎞⎟⎟⎠,𝜎³=⎛⎜⎜⎝10−01⎞⎟⎟⎠,(1.5) acting on two-dimensional spinor space Ψ_𝛼(𝑡,𝑥)⃗. Canonical quantization of the particle on an external electromagnetic background leads to the Pauli equation𝑖ℏ𝜕Ψ𝜕𝑡=21𝑚𝑝^𝑖̂−𝑒𝑐𝐴^𝑖²−𝑒𝐴₀−𝑒ℏ2𝑚𝑐𝐵^𝑖𝜎^𝑖Ψ.(1.6) It has been denoted that 𝑝_𝑖=−𝑖ℏ(𝜕/𝜕𝑥_𝑖), (𝐴₀,𝐴_𝑖) is the four-vector potential of electromagnetic field, and the magnetic field is 𝐵_𝑖=𝜖_𝑖𝑗𝑘𝜕_𝑗𝐴_𝑘, where 𝜖_𝑖𝑗𝑘 represents the totally antisymmetric tensor with 𝜖₁₂₃=1. Relativistic spin is described in a similar way [13, 14].

The problem here is that the Grassmann classical mechanics represents a rather formal mathematical construction. It leads to certain difficulties [13, 14, 17] in attempts to use it for description the spin effects on the semiclassical level, before the quantization. Hence it would be interesting to describe spin on a base of usual variables. While the problem has a long history (see [7–15] and references therein), there appears to be no wholly satisfactory solution to date. It seems to be surprisingly difficult [15] to construct, in a systematic way, a consistent model that would lead to the Dirac equation in the the course of canonical quantization. It is the aim of this work to construct an example of mechanical model for the Dirac equation.

To describe the nonrelativistic spin by commuting variables, we need to construct a mechanical model which implies the commutator (even) operator algebra (1.2) instead of the anticommutator one (1.3). It has been achieved in the recent work [18] starting from the Lagrangian𝐿=𝑚2𝑥̇^𝑖²+𝑒𝑐𝐴^𝑖𝑥̇_𝑖+𝑒𝐴₀+12𝑔𝜔̇^𝑖−𝑒𝑚𝑐𝜖^𝑖𝑗𝑘𝜔_𝑗𝐵_𝑘²+3𝑔ℏ²8𝑎²+1𝜙𝜔^𝑖²−𝑎².(1.7) The configuration-space variables are 𝑥_𝑖(𝑡), 𝜔_𝑖(𝑡), 𝑔(𝑡), 𝜙(𝑡). Here 𝑥_𝑖 represents the spatial coordinates of the particle with the mass 𝑚 and the charge 𝑒, 𝜔_𝑖 are the spin-space coordinates, 𝑔, 𝜙 are the auxiliary variables and 𝑎=const. Second and third terms in (1.7) represent minimal interaction with the vector potential 𝐴₀, 𝐴_𝑖 of an external electromagnetic field, while the fourth term contains interaction of spin with a magnetic field. At the end, it produces the Pauli term in quantum mechanical Hamiltonian.

The Dirac constraints presented in the model imply [18] that spin lives on two-dimensional surface of six-dimensional spin phase space 𝜔_𝑖, 𝜋_𝑖. The surface can be parameterized by the angular-momentum coordinates 𝑆_𝑖=𝜖_𝑖𝑗𝑘𝜔_𝑗𝜋_𝑘, subject to the condition 𝑆²=3ℏ²/4. They obey the classical brackets {𝑆_𝑖,𝑆_𝑗}=𝜖_𝑖𝑗𝑘𝑆_𝑘. Hence we quantize them according the rule 𝑆_𝑖→𝑆_𝑖.

The model leads to reasonable picture both on classical and quantum levels. The classical dynamics is governed by the Lagrangian equations𝑚𝑥̈_𝑖=𝑒𝐸_𝑖+𝑒𝑐𝜖^𝑖𝑗𝑘𝑥̇_𝑗𝐵_𝑘−𝑒𝑚𝑐𝑆^𝑘𝑆̇𝜕_𝑖𝐵_𝑘,(1.8)_𝑖=𝑒𝑚𝑐𝜖^𝑖𝑗𝑘𝑆_𝑗𝐵_𝑘.(1.9) It has been denoted that 𝐸=−(1/𝑐)(𝜕𝐴/𝜕𝑡)+∇𝐴₀. Since 𝑆²≈ℏ², the 𝑆-term disappears from (1.8) in the classical limit ℏ→0. Then (1.8) reproduces the classical motion on an external electromagnetic field. Notice also that in absence of interaction, the spinning particle does not experience an undesirable Zitterbewegung. Equation (1.9) describes the classical spin precession in an external magnetic field. On the other hand, canonical quantization of the model immediately produces the Pauli equation (1.6).

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.

2. Algebraic Construction of the Relativistic Spin Space

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 𝑆^𝑖 (1.2) either into the Frenkel tensor Φ^𝜇𝜈, Φ^𝜇𝜈𝑢_𝜈=0, or into the Bargmann-Michel-Telegdi four-vector (The conditions Φ^𝜇𝜈𝑢_𝜈=0 and 𝑆^𝜇𝑢_𝜇=0 guarantee that in the rest frame survive only three components of these quantities, which implies the right nonrelativistic limit). 𝑆^𝜇, 𝑆^𝜇𝑢_𝜇=0, where 𝑢_𝜇 represents four-velocity of the particle. Unfortunately, the semiclassical models based on these schemes do not lead to a reasonable quantum theory, as they do not produce the Dirac equation through the canonical quantization. We now motivate that it can be achieved in the formulation that implies inclusion of 𝑆_𝑖 into the 𝑆𝑂(3,2) angular-momentum tensor 𝐿^𝐴𝐵 of five-dimensional Minkowski space 𝐴=(𝜇,5)=(0,𝑖,5)=(0,1,2,3,5), 𝜂^𝐴𝐵=(−+++−).

In the passage from nonrelativistic to relativistic spin, we replace the Pauli equation by the Dirac one𝑝_𝜇̂Γ^𝜇+𝑚𝑐Ψ(𝑥^𝜇)=0,(2.1) where 𝑝_𝜇̂=−𝑖ℏ𝜕_𝜇. We use the representation with Hermitian Γ⁰ and anti-Hermitian Γ^𝑖Γ⁰=⎛⎜⎜⎝10−01⎞⎟⎟⎠,Γ^𝑖=⎛⎜⎜⎝0𝜎^𝑖−𝜎^𝑖0⎞⎟⎟⎠,(2.2) then [Γ^𝜇,Γ^𝜈]₊=−2𝜂^𝜇𝜈, 𝜂^𝜇𝜈=(−+++), and Γ⁰Γ^𝑖, Γ⁰ are the Dirac matrices [3] 𝛼^𝑖, 𝛽𝛼^𝑖=⎛⎜⎜⎝0𝜎^𝑖𝜎^𝑖0⎞⎟⎟⎠,𝛽=⎛⎜⎜⎝10−01⎞⎟⎟⎠.(2.3) We take the classical counterparts of the operators 𝑥̂^𝜇 and 𝑝_𝜇̂=−𝑖ℏ𝜕_𝜇 in the standard way, which are 𝑥^𝜇, 𝑝^𝜈, with the Poisson brackets {𝑥^𝜇,𝑝^𝜈}_PB=𝜂^𝜇𝜈.

Let us discuss the classical variables that could produce the Γ-matrices. To this aim, we first study their commutators. The commutators of Γ^𝜇 do not form closed Lie algebra, but produce SO(1,3)-Lorentz generatorsΓ^𝜇,Γ^𝜈₋=−2𝑖Γ^𝜇𝜈,(2.4) where it has been denoted Γ^𝜇𝜈≡(𝑖/2)(Γ^𝜇Γ^𝜈−Γ^𝜈Γ^𝜇). The set Γ^𝜇, Γ^𝜇𝜈 form closed algebra. Besides the commutator (2.4), one hasΓΓ^𝜇𝜈,Γ^𝛼₋=2𝑖(𝜂^𝜇𝛼Γ^𝜈−𝜂^𝜈𝛼Γ^𝜇),_𝜇𝜈,Γ^𝛼𝛽₋=2𝑖𝜂^𝜇𝛼Γ^𝜈𝛽−𝜂^𝜇𝛽Γ^𝜈𝛼−𝜂^𝜈𝛼Γ^𝜇𝛽+𝜂^𝜈𝛽Γ^𝜇𝛼.(2.5) The algebra (2.4), (2.5) can be identified with the five-dimensional Lorentz algebra SO(2,3) with generators 𝐿^𝐴𝐵𝐿^𝐴𝐵,𝐿^𝐶𝐷₋=2𝑖𝜂^𝐴𝐶𝐿^𝐵𝐷−𝜂^𝐴𝐷𝐿^𝐵𝐶−𝜂^𝐵𝐶𝐿^𝐴𝐷+𝜂^𝐵𝐷𝐿^𝐴𝐶,(2.6) assuming Γ^𝜇≡𝐿^5𝜇, Γ^𝜇𝜈≡𝐿^𝜇𝜈.

To reach the algebra starting from a classical-mechanics model, we introduce ten-dimensional “phase” space of the spin degrees of freedom, 𝜔^𝐴, 𝜋^𝐵, equipped with the Poisson bracket𝜔^𝐴,𝜋^𝐵_PB=𝜂^𝐴𝐵.(2.7) Then Poisson brackets of the quantities𝐽^𝐴𝐵≡2𝜔^𝐴𝜋^𝐵−𝜔^𝐵𝜋^𝐴(2.8) read𝐽^𝐴𝐵,𝐽^𝐶𝐷_PB=2𝜂^𝐴𝐶𝐽^𝐵𝐷−𝜂^𝐴𝐷𝐽^𝐵𝐶−𝜂^𝐵𝐶𝐽^𝐴𝐷+𝜂^𝐵𝐷𝐽^𝐴𝐶.(2.9) Below we use the decompositions𝐽^𝐴𝐵=𝐽^5𝜇,𝐽^𝜇𝜈=𝐽⁵⁰,𝐽^5𝑖≡𝐉⁵,𝐽^0𝑖≡𝐖,𝐽^𝑖𝑗=𝜖^𝑖𝑗𝑘𝐷^𝑘.(2.10)

The Jacobian of the transformation (𝜔^𝐴,𝜋^𝐵)→𝐽^𝐴𝐵 has rank equal seven. So, only seven among ten functions 𝐽^𝐴𝐵(𝜔,𝜋), 𝐴<𝐵, are independent quantities. They can be separated as follows. By construction, the quantities (2.8) obey the identity𝜖^{𝜇𝜈𝛼𝛽}𝐽^5𝜈𝐽_𝛼𝛽=0,⟺𝐽^𝑖𝑗=1𝐽⁵⁰𝐽^5𝑖𝐽^0𝑗−𝐽^5𝑗𝐽^0𝑖,(2.11) that is, the three-vector 𝐃 can be presented through 𝐉^𝟓, 𝐖 as𝐃=1𝐽⁵⁰𝐉⁵×𝐖.(2.12) Further, (𝜔^𝐴,𝜋^𝐵)-space can be parameterized by the coordinates 𝐽^5𝜇, 𝐽^0𝑖, 𝜔⁰, 𝜔⁵, 𝜋⁵. We can not yet quantize the variables since it would lead to the appearance of some operators 𝜔⁰, 𝜔⁵, 𝜋⁵, which are not presented in the Dirac theory and are not necessary for description of spin. To avoid the problem, we kill the variables 𝜔⁰, 𝜔⁵, 𝜋⁵, restricting our model to live on seven-dimensional surface of ten-dimensional phase space 𝜔^𝐴, 𝜋^𝐵. The only SO(2,3) quadratic invariants that can be constructed from 𝜔^𝐴,𝜋^𝐵 are 𝜔^𝐴𝜔_𝐴, 𝜔^𝐴𝜋_𝐴, 𝜋^𝐴𝜋_𝐴. We choose conventionally the surface determined by the equations𝜔𝜔^𝐴𝜔_𝐴+𝑅=0,(2.13)_𝐴𝜋_𝐴=0,𝜋^𝐴𝜋_𝐴=0,(2.14)𝑅=const>0. The quantities 𝐽^5𝜇, 𝐽^0𝑖 form a coordinate system of the spin-space surface. So, we can quantize them instead of the initial variables 𝜔^𝐴, 𝜋^𝐵.

According to (1.1), (2.6), (2.9), quantization is achieved replacing the classical variables 𝐽^5𝜇, 𝐽^𝜇𝜈 on Γ-matrices𝐽^5𝜇⟶ℏΓ^𝜇,𝐽^𝜇𝜈⟶ℏΓ^𝜇𝜈.(2.15) It implies, that the Dirac equation can be produced by the constraint (we restate that 𝐽^5𝜇≡2(𝜔⁵𝜋^𝜇−𝜔^𝜇𝜋⁵))𝑝_𝜇𝐽^5𝜇+𝑚𝑐ℏ=0.(2.16)

Summing up, to describe the relativistic spin, we need a theory that implies the Dirac constraints (2.13), (2.14), (2.16) in the Hamiltonian formulation.

3. Dynamical Realization

One possible dynamical realization of the construction presented above is given by the following 𝑑=4 Poincare-invariant Lagrangian𝐿=−12𝑒₂𝑥̇^𝜇+𝑒₃𝜔^𝜇²−𝑒₃𝜔⁵²−𝜎𝑚𝑐ℏ2𝜔⁵+1𝜎𝑥̇^𝜇+𝑒₃𝜔^𝜇𝜔̇_𝜇−𝑒₃𝜔⁵𝜔̇⁵−𝑒₄𝜔^𝐴𝜔_𝐴+𝑅,(3.1) written on the configuration space 𝑥^𝜇, 𝜔^𝜇, 𝜔⁵, 𝑒_𝑖, 𝜎, where 𝑒_𝑖, 𝜎 are the auxiliary variables. The variables 𝜔⁵, 𝑒_𝑖, 𝜎 are scalars under the Poincare transformations. The remaining variables transform according to the rule𝑥′^𝜇=Λ^𝜇𝜈𝑥^𝜈+𝑎^𝜇,𝜔′^𝜇=Λ^𝜇𝜈𝜔^𝜈.(3.2) Local symmetries of the theory form the two-parameter group composed by the reparametrizations𝛿𝑥^𝜇=𝛼𝑥̇^𝜇,𝛿𝜔^𝐴=𝛼𝜔̇^𝐴,𝛿𝑒_𝑖=𝛼𝑒_𝑖^.,𝛿𝜎=(𝛼𝜎)^.,(3.3) as well as by the local transformations with the parameter 𝜖(𝜏) (below we have denoted 𝛽≡𝑒₄̇𝜖+(1/2)𝑒₄𝜖̇)𝛿𝛿𝑥𝑒^𝜇₃=0,𝛿𝜔^𝐴=𝛽𝜔^𝐴,𝛿𝜎=𝛽𝜎,𝛿𝑒₂=0,=−𝛽𝑒₃+𝑒²𝜎𝛽̇,𝛿𝑒⁴=−2𝑒₄𝛽−𝑒²𝛽̇2𝜎²^⋅.(3.4) The local symmetries guarantee appearance of the first-class constraints (2.13), (2.16).

Curiously enough, the action can be rewritten in almost five-dimensional form. Indeed, after the change (𝑥^𝜇,𝜎,𝑒₃)→(𝑥̃^𝜇,𝑥̃⁵,𝑒₃̃), where 𝑥̃^𝜇=𝑥^𝜇−(𝑒₂/𝜎)𝜔^𝜇, 𝑥̃⁵=−(𝑒₂/𝜎)𝜔⁵, 𝑒₃̃=𝑒₃+(𝑒₂/𝜎), it reads𝐿=−12𝑒₂𝐷𝑥̃^𝐴²+𝑥̃⁵²2𝑒₂𝜔⁵²𝜔̇^𝐴²+𝑒²𝑚𝑐ℏ𝑥̃⁵−𝑒₄𝜔^𝐴𝜔_𝐴+𝑅,(3.5) where the covariant derivative is 𝐷𝑥̃^𝐴=𝑥̇̃^𝐴+𝑒₃̃𝜔^𝐴 (The change is an example of conversion of the second-class constraints in the Lagrangian formulation [25]).

Canonical Quantization
In the Hamiltonian formalism, the action implies the desired constraints (2.13), (2.14), (2.16). The constraints (2.13), (2.14) can be taken into account by transition from the Poisson to the Dirac bracket, and after that they are omitted from the consideration [17, 26]. The first-class constraint (2.16) is imposed on the state vector and produces the Dirac equation. In the result, canonical quantization of the model leads to the desired quantum picture.
We now discuss some properties of the classical theory and confirm that they are in correspondence with semiclassical limit [3, 27, 28] of the Dirac equation.

Equations of Motion
The auxiliary variables 𝑒_𝑖, 𝜎 can be omitted from consideration after the partial fixation of a gauge. After that, Hamiltonian of the model reads 𝐻=12𝜋^𝐴𝜋_𝐴+12𝑝^𝜇𝐽^5𝜇+𝑚𝑐ℏ.(3.6) Since 𝜔^𝐴, 𝜋^𝐴 are not 𝜖-invariant variables, their equations of motion have no much sense. So, we write the equations of motion for 𝜖-invariant quantities 𝑥^𝜇, 𝑝_𝜇, 𝐽^5𝜇, 𝐽^𝜇𝜈𝑥̇^𝜇=12𝐽^5𝜇𝐽̇,(3.7)_5𝜇𝐽̇_𝜇𝜈=−𝐽^𝜇𝜈𝑝_𝜈,=𝑝𝑝^𝜇̇_𝜇𝐽^5𝜈−𝑝^𝜈𝐽^5𝜇,(3.8)=0.(3.9) They imply 𝑥̈^𝜇=−12𝐽^𝜇𝜈𝑝_𝜈.(3.10) In three-dimensional notations, the equation (3.8) read 𝐽̇⁵⁰=−(𝐖𝐖𝐩̇)=,𝑝₀𝐉̇⁵=−𝑝⁰𝐖+𝐃×𝐩,𝐉⁵−𝐽⁵⁰𝐩.(3.11)

Relativistic Invariance
While the canonical momentum of 𝑥^𝜇 is given by 𝑝^𝜇, the mechanical momentum, according to (3.7), coincides with the variables that turn into the Γ-matrices in quantum theory, (1/2)𝐽^5𝜇. Due to the constraints (2.13), (2.14), 𝐽^5𝜇 obeys (𝐽^5𝜇/𝜋⁵)²=−4𝑅, which is analogy of 𝑝²=−𝑚²𝑐² of the spinless particle. As a consequence, 𝑥^𝑖(𝑡) cannot exceed the speed of light, (𝑑𝑥^𝑖/𝑑𝑡)²=𝑐²(𝑥̇^𝑖/𝑥̇⁰)²=𝑐²(1−((𝜋⁵)²4𝑅/(𝐽⁵⁰)²))<𝑐². Equations (3.10), (3.11) mean that both 𝑥^𝜇-particle and the variables 𝐖, 𝐉^𝟓 experience the Zitterbewegung in noninteracting theory.

Center-of-Charge Rest Frame
Identifying the variables 𝑥^𝜇 with position of the charge, (3.7) implies that the rest frame is characterized by the conditions 𝐽⁵⁰=const,𝐉⁵=0.(3.12) According to (3.12), (2.12), only 𝐖 survives in the nonrelativistic limit.

The Variables Free of Zitterbewegung
The quantity (center-of-mass coordinate [7]) 𝑥̃^𝜇=𝑥^𝜇+(1/2𝑝²)𝐽^𝜇𝜈𝑝_𝜈 obeys 𝑥̇̃^𝜇=−(𝑚𝑐ℏ/2𝑝²)𝑝^𝜇, so, it has the free dynamics 𝑥̈̃^𝜇=0. Note also that 𝑝^𝜇 represents the mechanical momentum of 𝑥̃^𝜇-particle.
As the classical four-dimensional spin vector, let us take 𝑆^𝜇=𝜖^{𝜇𝜈𝛼𝛽}𝑝_𝜈𝐽_𝛼𝛽. It has no precession in the free theory, 𝑆̇^𝜇=0. In the rest frame, it reduces to 𝐒⁰=0, 𝐒=𝐩×𝐖.

Comparison with the Barut-Zanghi (BZ) Model
The BZ spinning particle [16] is widely used [19–24] for semiclassical analysis of spin effects. Starting from the even variable 𝑧_𝛼, where 𝛼=1,2,3,4 is SO(1,3)-spinor index, Barut and Zanghi have constructed the spin-tensor according to 𝑆_𝜇𝜈=(1/4)𝑖𝑧𝛾_𝜇𝜈𝑧. We point out that (3.7)–(3.9) of our model coincide with those of BZ-model, identifying 𝐽^5𝜇↔𝑣^𝜇, 𝐽^𝜇𝜈↔𝑆^𝜇𝜈. Besides, our model implies the equations (𝐽^5𝜇/𝜋⁵)²=−4𝑅,𝑝_𝜇𝐽^5𝜇+𝑚𝑐ℏ=0. The first equation guarantees that the center of charge cannot exceed the speed of light. The second equation implies the Dirac equation. (In the BZ theory [16], the mass of the spinning particle is not fixed from the model.)

Acknowledgment

This work has been supported by the Brazilian foundation FAPEMIG.