Advances in Mathematical Physics

Advances in Mathematical Physics / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 680367 |

A. A. Deriglazov, "Relativistic Spinning Particle without Grassmann Variables and the Dirac Equation", Advances in Mathematical Physics, vol. 2011, Article ID 680367, 10 pages, 2011.

Relativistic Spinning Particle without Grassmann Variables and the Dirac Equation

Academic Editor: Stephen Anco
Received31 Mar 2011
Revised30 Jul 2011
Accepted23 Aug 2011
Published23 Oct 2011


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 [1ā€“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 [7ā€“24].

In the course of canonical quantization of a given classical theory, one associates Hermitian operators with classical variables. Let š‘§š›¼ stands for the basic phase-space variables that describe the classical system, and {š‘§š›¼,š‘§š›½} 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 Ģ‚š‘§š›¼ must be chosen to obey the quantization ruleī€ŗĢ‚š‘§š›¼,Ģ‚š‘§š›½ī€»ī€½š‘§āˆ“=š‘–ā„š›¼,š‘§š›½ī€¾||š‘§ā†’.Ģ‚š‘§(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 īš‘†š‘–=(ā„/2)šœŽš‘– (šœŽš‘– are the Pauli matrices (1.5)) form a simple algebra with respect to commutatorī‚ƒīš‘†š‘–,īš‘†š‘—ī‚„āˆ’=š‘–ā„šœ–š‘–š‘—š‘˜īš‘†š‘˜,(1.2) as well as to anticommutatorī‚ƒīš‘†š‘–,īš‘†š‘—ī‚„+=ā„22š›æš‘–š‘—.(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 (š‘š/2)(Ģ‡š‘„š‘–)2+(š‘–/2)šœ‰š‘–Ģ‡šœ‰š‘–, where the spin inner space is constructed from vector-like Grassmann variables šœ‰š‘–, šœ‰š‘–šœ‰š‘—=āˆ’šœ‰š‘—šœ‰š‘–. Since the Lagrangian is linear on Ģ‡šœ‰š‘–, their conjugate momenta coincide with šœ‰, šœ‹š‘–Ģ‡šœ‰=šœ•šæ/šœ•š‘–=š‘–šœ‰š‘–. 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 readsī€½šœ‰š‘–,šœ‰š‘—ī€¾DB=š‘–š›æš‘–š‘—.(1.4) Dealing with the Dirac bracket, one can resolve the constraints, excluding the momenta from consideration. So, there are only three spin variables šœ‰š‘– 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 īš‘†š‘– proportional to the Pauli šœŽ-matrices, īš‘†š‘–=(ā„/2)šœŽš‘–, [šœŽš‘–,šœŽš‘—]+=2š›æš‘–š‘—,šœŽ1=āŽ›āŽœāŽœāŽāŽžāŽŸāŽŸāŽ 0110,šœŽ2=āŽ›āŽœāŽœāŽāŽžāŽŸāŽŸāŽ 0āˆ’š‘–š‘–0,šœŽ3=āŽ›āŽœāŽœāŽāŽžāŽŸāŽŸāŽ 100āˆ’1,(1.5) acting on two-dimensional spinor space ĪØš›¼(š‘”,āƒ—š‘„). Canonical quantization of the particle on an external electromagnetic background leads to the Pauli equationš‘–ā„šœ•ĪØ=ī‚µ1šœ•š‘”ī‚€2š‘šĢ‚š‘š‘–āˆ’š‘’š‘š“š‘–ī‚2āˆ’š‘’š“0āˆ’š‘’ā„šµ2š‘šš‘š‘–šœŽš‘–ī‚¶ĪØ.(1.6) It has been denoted that š‘š‘–=āˆ’š‘–ā„(šœ•/šœ•š‘„š‘–), (š“0,š“š‘–) is the four-vector potential of electromagnetic field, and the magnetic field is šµš‘–=šœ–š‘–š‘—š‘˜šœ•š‘—š“š‘˜, where šœ–š‘–š‘—š‘˜ represents the totally antisymmetric tensor with šœ–123=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 [7ā€“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 Lagrangianš‘ššæ=2ī€·Ģ‡š‘„š‘–ī€ø2+š‘’š‘š“š‘–Ģ‡š‘„š‘–+š‘’š“0+1ī‚€2š‘”Ģ‡šœ”š‘–āˆ’š‘’šœ–š‘šš‘š‘–š‘—š‘˜šœ”š‘—šµš‘˜ī‚2+3š‘”ā„28š‘Ž2+1šœ™ī‚€ī€·šœ”š‘–ī€ø2āˆ’š‘Ž2ī‚.(1.7) The configuration-space variables are š‘„š‘–(š‘”), šœ”š‘–(š‘”), š‘”(š‘”), šœ™(š‘”). Here š‘„š‘– represents the spatial coordinates of the particle with the mass š‘š and the charge š‘’, šœ”š‘– are the spin-space coordinates, š‘”, šœ™ are the auxiliary variables and š‘Ž=const. Second and third terms in (1.7) represent minimal interaction with the vector potential š“0, š“š‘– 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 šœ”š‘–, šœ‹š‘–. The surface can be parameterized by the angular-momentum coordinates š‘†š‘–=šœ–š‘–š‘—š‘˜šœ”š‘—šœ‹š‘˜, subject to the condition š‘†2=3ā„2/4. They obey the classical brackets {š‘†š‘–,š‘†š‘—}=šœ–š‘–š‘—š‘˜š‘†š‘˜. Hence we quantize them according the rule š‘†š‘–ā†’īš‘†š‘–.

The model leads to reasonable picture both on classical and quantum levels. The classical dynamics is governed by the Lagrangian equationsš‘šĢˆš‘„š‘–=š‘’šøš‘–+š‘’š‘šœ–š‘–š‘—š‘˜Ģ‡š‘„š‘—šµš‘˜āˆ’š‘’š‘†š‘šš‘š‘˜šœ•š‘–šµš‘˜,Ģ‡š‘†(1.8)š‘–=š‘’šœ–š‘šš‘š‘–š‘—š‘˜š‘†š‘—šµš‘˜.(1.9) It has been denoted that šø=āˆ’(1/š‘)(šœ•š“/šœ•š‘”)+āˆ‡š“0. Since š‘†2ā‰ˆā„2, the š‘†-term disappears from (1.8) in the classical limit ā„ā†’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 īš‘†š‘– (1.2) either into the Frenkel tensor Ī¦šœ‡šœˆ, Ī¦šœ‡šœˆš‘¢šœˆ=0, or into the Bargmann-Michel-Telegdi four-vector (The conditions Ī¦šœ‡šœˆš‘¢šœˆ=0 and š‘†šœ‡š‘¢šœ‡=0 guarantee that in the rest frame survive only three components of these quantities, which implies the right nonrelativistic limit). š‘†šœ‡, š‘†šœ‡š‘¢šœ‡=0, where š‘¢šœ‡ 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 īš‘†š‘– into the š‘†š‘‚(3,2) angular-momentum tensor īšæš“šµ of five-dimensional Minkowski space š“=(šœ‡,5)=(0,š‘–,5)=(0,1,2,3,5), šœ‚š“šµ=(āˆ’+++āˆ’).

In the passage from nonrelativistic to relativistic spin, we replace the Pauli equation by the Dirac oneī€·Ģ‚š‘šœ‡Ī“šœ‡ī€øĪØ+š‘šš‘(š‘„šœ‡)=0,(2.1) where Ģ‚š‘šœ‡=āˆ’š‘–ā„šœ•šœ‡. We use the representation with Hermitian Ī“0 and anti-Hermitian Ī“š‘–Ī“0=āŽ›āŽœāŽœāŽāŽžāŽŸāŽŸāŽ 100āˆ’1,Ī“š‘–=āŽ›āŽœāŽœāŽ0šœŽš‘–āˆ’šœŽš‘–0āŽžāŽŸāŽŸāŽ ,(2.2) then [Ī“šœ‡,Ī“šœˆ]+=āˆ’2šœ‚šœ‡šœˆ, šœ‚šœ‡šœˆ=(āˆ’+++), and Ī“0Ī“š‘–, Ī“0 are the Dirac matrices [3] š›¼š‘–, š›½š›¼š‘–=āŽ›āŽœāŽœāŽ0šœŽš‘–šœŽš‘–0āŽžāŽŸāŽŸāŽ āŽ›āŽœāŽœāŽāŽžāŽŸāŽŸāŽ ,š›½=100āˆ’1.(2.3) We take the classical counterparts of the operators Ģ‚š‘„šœ‡ and Ģ‚š‘šœ‡=āˆ’š‘–ā„šœ•šœ‡ in the standard way, which are š‘„šœ‡, š‘šœˆ, with the Poisson brackets {š‘„šœ‡,š‘šœˆ}PB=šœ‚šœ‡šœˆ.

Let us discuss the classical variables that could produce the Ī“-matrices. To this aim, we first study their commutators. The commutators of Ī“šœ‡ do not form closed Lie algebra, but produce SO(1,3)-Lorentz generatorsī€ŗĪ“šœ‡,Ī“šœˆī€»āˆ’=āˆ’2š‘–Ī“šœ‡šœˆ,(2.4) where it has been denoted Ī“šœ‡šœˆā‰”(š‘–/2)(Ī“šœ‡Ī“šœˆāˆ’Ī“šœˆĪ“šœ‡). The set Ī“šœ‡, Ī“šœ‡šœˆ form closed algebra. Besides the commutator (2.4), one hasī€ŗĪ“šœ‡šœˆ,Ī“š›¼ī€»āˆ’=2š‘–(šœ‚šœ‡š›¼Ī“šœˆāˆ’šœ‚šœˆš›¼Ī“šœ‡ī€ŗĪ“),šœ‡šœˆ,Ī“š›¼š›½ī€»āˆ’ī€·šœ‚=2š‘–šœ‡š›¼Ī“šœˆš›½āˆ’šœ‚šœ‡š›½Ī“šœˆš›¼āˆ’šœ‚šœˆš›¼Ī“šœ‡š›½+šœ‚šœˆš›½Ī“šœ‡š›¼ī€ø.(2.5) The algebra (2.4), (2.5) can be identified with the five-dimensional Lorentz algebra SO(2,3) with generators īšæš“šµī‚ƒīšæš“šµ,īšæš¶š·ī‚„āˆ’ī‚€šœ‚=2š‘–š“š¶īšæšµš·āˆ’šœ‚š“š·īšæšµš¶āˆ’šœ‚šµš¶īšæš“š·+šœ‚šµš·īšæš“š¶ī‚,(2.6) assuming Ī“šœ‡ā‰”īšæ5šœ‡, Ī“šœ‡šœˆā‰”īšæšœ‡šœˆ.

To reach the algebra starting from a classical-mechanics model, we introduce ten-dimensional ā€œphaseā€ space of the spin degrees of freedom, šœ”š“, šœ‹šµ, equipped with the Poisson bracketī€½šœ”š“,šœ‹šµī€¾PB=šœ‚š“šµ.(2.7) Then Poisson brackets of the quantitiesš½š“šµī€·šœ”ā‰”2š“šœ‹šµāˆ’šœ”šµšœ‹š“ī€ø(2.8) readī€½š½š“šµ,š½š¶š·ī€¾PBī€·šœ‚=2š“š¶š½šµš·āˆ’šœ‚š“š·š½šµš¶āˆ’šœ‚šµš¶š½š“š·+šœ‚šµš·š½š“š¶ī€ø.(2.9) Below we use the decompositionsš½š“šµ=ī€·š½5šœ‡,š½šœ‡šœˆī€ø=ī€·š½50,š½5š‘–ā‰”š‰5,š½0š‘–ā‰”š–,š½š‘–š‘—=šœ–š‘–š‘—š‘˜š·š‘˜ī€ø.(2.10)

The Jacobian of the transformation (šœ”š“,šœ‹šµ)ā†’š½š“šµ has rank equal seven. So, only seven among ten functions š½š“šµ(šœ”,šœ‹), š“<šµ, are independent quantities. They can be separated as follows. By construction, the quantities (2.8) obey the identityšœ–šœ‡šœˆš›¼š›½š½5šœˆš½š›¼š›½=0,āŸŗš½š‘–š‘—=1š½50ī€·š½5š‘–š½0š‘—āˆ’š½5š‘—š½0š‘–ī€ø,(2.11) that is, the three-vector šƒ can be presented through š‰šŸ“, š– as1šƒ=š½50š‰5Ɨš–.(2.12) Further, (šœ”š“,šœ‹šµ)-space can be parameterized by the coordinates š½5šœ‡, š½0š‘–, šœ”0, šœ”5, šœ‹5. We can not yet quantize the variables since it would lead to the appearance of some operators īšœ”0, īšœ”5, īšœ‹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 šœ”0, šœ”5, šœ‹5, restricting our model to live on seven-dimensional surface of ten-dimensional phase space šœ”š“, šœ‹šµ. The only SO(2,3) quadratic invariants that can be constructed from šœ”š“,šœ‹šµ are šœ”š“šœ”š“, šœ”š“šœ‹š“, šœ‹š“šœ‹š“. We choose conventionally the surface determined by the equationsšœ”š“šœ”š“šœ”+š‘…=0,(2.13)š“šœ‹š“=0,šœ‹š“šœ‹š“=0,(2.14)š‘…=const>0. The quantities š½5šœ‡, š½0š‘– form a coordinate system of the spin-space surface. So, we can quantize them instead of the initial variables šœ”š“, šœ‹šµ.

According to (1.1), (2.6), (2.9), quantization is achieved replacing the classical variables š½5šœ‡, š½šœ‡šœˆ on Ī“-matricesš½5šœ‡āŸ¶ā„Ī“šœ‡,š½šœ‡šœˆāŸ¶ā„Ī“šœ‡šœˆ.(2.15) It implies, that the Dirac equation can be produced by the constraint (we restate that š½5šœ‡ā‰”2(šœ”5šœ‹šœ‡āˆ’šœ”šœ‡šœ‹5))š‘šœ‡š½5šœ‡+š‘šš‘ā„=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 š‘‘=4 Poincare-invariant Lagrangian1šæ=āˆ’2š‘’2ī‚ƒī€·Ģ‡š‘„šœ‡+š‘’3šœ”šœ‡ī€ø2āˆ’ī€·š‘’3šœ”5ī€ø2ī‚„āˆ’šœŽš‘šš‘ā„2šœ”5+1šœŽī€ŗī€·Ģ‡š‘„šœ‡+š‘’3šœ”šœ‡ī€øĢ‡šœ”šœ‡āˆ’š‘’3šœ”5Ģ‡šœ”5ī€»āˆ’š‘’4ī€·šœ”š“šœ”š“ī€ø,+š‘…(3.1) written on the configuration space š‘„šœ‡, šœ”šœ‡, šœ”5, š‘’š‘–, šœŽ, where š‘’š‘–, šœŽ are the auxiliary variables. The variables šœ”5, š‘’š‘–, šœŽ are scalars under the Poincare transformations. The remaining variables transform according to the ruleš‘„ā€²šœ‡=Ī›šœ‡šœˆš‘„šœˆ+š‘Žšœ‡,šœ”ā€²šœ‡=Ī›šœ‡šœˆšœ”šœˆ.(3.2) Local symmetries of the theory form the two-parameter group composed by the reparametrizationsš›æš‘„šœ‡=š›¼Ģ‡š‘„šœ‡,š›æšœ”š“=š›¼Ģ‡šœ”š“,š›æš‘’š‘–=ī€·š›¼š‘’š‘–ī€ø.,š›æšœŽ=(š›¼šœŽ).,(3.3) as well as by the local transformations with the parameter šœ–(šœ) (below we have denoted š›½ā‰”Ģ‡š‘’4šœ–+(1/2)š‘’4Ģ‡šœ–)š›æš‘„šœ‡=0,š›æšœ”š“=š›½šœ”š“,š›æšœŽ=š›½šœŽ,š›æš‘’2=0,š›æš‘’3=āˆ’š›½š‘’3+š‘’2šœŽĢ‡š›½,š›æš‘’4=āˆ’2š‘’4ī‚µš‘’š›½āˆ’2Ģ‡š›½2šœŽ2ī‚¶ā‹….(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 (š‘„šœ‡,šœŽ,š‘’3)ā†’(Ģƒš‘„šœ‡,Ģƒš‘„5,Ģƒš‘’3), where Ģƒš‘„šœ‡=š‘„šœ‡āˆ’(š‘’2/šœŽ)šœ”šœ‡, Ģƒš‘„5=āˆ’(š‘’2/šœŽ)šœ”5, Ģƒš‘’3=š‘’3+(š‘’2/šœŽ), it reads1šæ=āˆ’2š‘’2ī€·š·Ģƒš‘„š“ī€ø2+ī€·Ģƒš‘„5ī€ø22š‘’2ī€·šœ”5ī€ø2ī€·Ģ‡šœ”š“ī€ø2+š‘’2š‘šš‘ā„Ģƒš‘„5āˆ’š‘’4ī€·šœ”š“šœ”š“ī€ø+š‘…,(3.5) where the covariant derivative is š·Ģƒš‘„š“=Ģ‡Ģƒš‘„š“+Ģƒš‘’3šœ”š“ (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 š‘’š‘–, šœŽ can be omitted from consideration after the partial fixation of a gauge. After that, Hamiltonian of the model reads 1š»=2šœ‹š“šœ‹š“+12ī€·š‘šœ‡š½5šœ‡ī€ø.+š‘šš‘ā„(3.6) Since šœ”š“, šœ‹š“ are not šœ–-invariant variables, their equations of motion have no much sense. So, we write the equations of motion for šœ–-invariant quantities š‘„šœ‡, š‘šœ‡, š½5šœ‡, š½šœ‡šœˆĢ‡š‘„šœ‡=12š½5šœ‡,Ģ‡š½(3.7)5šœ‡=āˆ’š½šœ‡šœˆš‘šœˆ,Ģ‡š½šœ‡šœˆ=š‘šœ‡š½5šœˆāˆ’š‘šœˆš½5šœ‡,(3.8)Ģ‡š‘šœ‡=0.(3.9) They imply Ģˆš‘„šœ‡1=āˆ’2š½šœ‡šœˆš‘šœˆ.(3.10) In three-dimensional notations, the equation (3.8) read Ģ‡š½50Ģ‡š‰=āˆ’(š–š©),5=āˆ’š‘0Ģ‡š–+šƒĆ—š©,š–=š‘0š‰5āˆ’š½50š©.(3.11)

Relativistic Invariance
While the canonical momentum of š‘„šœ‡ is given by š‘šœ‡, the mechanical momentum, according to (3.7), coincides with the variables that turn into the Ī“-matrices in quantum theory, (1/2)š½5šœ‡. Due to the constraints (2.13), (2.14), š½5šœ‡ obeys (š½5šœ‡/šœ‹5)2=āˆ’4š‘…, which is analogy of š‘2=āˆ’š‘š2š‘2 of the spinless particle. As a consequence, š‘„š‘–(š‘”) cannot exceed the speed of light, (š‘‘š‘„š‘–/š‘‘š‘”)2=š‘2(Ģ‡š‘„š‘–/Ģ‡š‘„0)2=š‘2(1āˆ’((šœ‹5)24š‘…/(š½50)2))<š‘2. Equations (3.10), (3.11) mean that both š‘„šœ‡-particle and the variables š–, š‰šŸ“ experience the Zitterbewegung in noninteracting theory.

Center-of-Charge Rest Frame
Identifying the variables š‘„šœ‡ with position of the charge, (3.7) implies that the rest frame is characterized by the conditions š½50=const,š‰5=0.(3.12) According to (3.12), (2.12), only š– survives in the nonrelativistic limit.

The Variables Free of Zitterbewegung
The quantity (center-of-mass coordinate [7]) Ģƒš‘„šœ‡=š‘„šœ‡+(1/2š‘2)š½šœ‡šœˆš‘šœˆ obeys Ģ‡Ģƒš‘„šœ‡=āˆ’(š‘šš‘ā„/2š‘2)š‘šœ‡, so, it has the free dynamics ĢˆĢƒš‘„šœ‡=0. Note also that š‘šœ‡ represents the mechanical momentum of Ģƒš‘„šœ‡-particle.
As the classical four-dimensional spin vector, let us take š‘†šœ‡=šœ–šœ‡šœˆš›¼š›½š‘šœˆš½š›¼š›½. It has no precession in the free theory, Ģ‡š‘†šœ‡=0. In the rest frame, it reduces to š’0=0, š’=š©Ć—š–.

Comparison with the Barut-Zanghi (BZ) Model
The BZ spinning particle [16] is widely used [19ā€“24] for semiclassical analysis of spin effects. Starting from the even variable š‘§š›¼, where š›¼=1,2,3,4 is SO(1,3)-spinor index, Barut and Zanghi have constructed the spin-tensor according to š‘†šœ‡šœˆ=(1/4)š‘–š‘§š›¾šœ‡šœˆš‘§. We point out that (3.7)ā€“(3.9) of our model coincide with those of BZ-model, identifying š½5šœ‡ā†”š‘£šœ‡, š½šœ‡šœˆā†”š‘†šœ‡šœˆ. Besides, our model implies the equations (š½5šœ‡/šœ‹5)2=āˆ’4š‘…,š‘šœ‡š½5šœ‡+š‘šš‘ā„=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.)


This work has been supported by the Brazilian foundation FAPEMIG.


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