AMPAdvances in Mathematical Physics1687-91391687-9120Hindawi Publishing Corporation68036710.1155/2011/680367680367Research ArticleRelativistic Spinning Particle without Grassmann Variables and the Dirac EquationDeriglazovA. A.AncoStephen1Departamento de MatematicaInstituto de Ciências Exatas (ICE)Universidade Federal de Juiz de Fora, 36039-900 Juiz de ForaMGBrazilufjf.br20112410201120113103201130072011230820112011Copyright © 2011 A. A. Deriglazov.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 , a lot of efforts have been spent in attempts to understand behaviour of a particle with spin on the base of semiclassical mechanical models .

In the course of canonical quantization of a given classical theory, one associates Hermitian operators with classical variables. Let zα stands for the basic phase-space variables that describe the classical system, and {zα,zβ} 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 , the operators ẑα must be chosen to obey the quantization rule[ẑα,ẑβ]=i{zα,zβ}|zẑ. 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 Ŝi=(/2)σi (σi are the Pauli matrices (1.5)) form a simple algebra with respect to commutator[Ŝi,Ŝj]-=iϵijkŜk, as well as to anticommutator[Ŝi,Ŝj]+=22δij. 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)(ẋi)2+(i/2)ξiξ̇i, where the spin inner space is constructed from vector-like Grassmann variables ξi, ξiξj=-ξjξi. Since the Lagrangian is linear on ξ̇i, their conjugate momenta coincide with ξ, πi=L/ξ̇i=iξi. 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{ξi,ξj}DB=iδij. Dealing with the Dirac bracket, one can resolve the constraints, excluding the momenta from consideration. So, there are only three spin variables ξi 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 Ŝi proportional to the Pauli σ-matrices, Ŝi=(/2)σi, [σi,σj]+=2δij,σ1=(0110),σ2=(0-ii0),σ3=(100-1), acting on two-dimensional spinor space Ψα(t,x). Canonical quantization of the particle on an external electromagnetic background leads to the Pauli equationiΨt=(12m(p̂i-ecAi)2-eA0-e2mcBiσi)Ψ. It has been denoted that pi=-i(/xi), (A0,Ai) is the four-vector potential of electromagnetic field, and the magnetic field is Bi=ϵijkjAk, where ϵijk 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  and references therein), there appears to be no wholly satisfactory solution to date. It seems to be surprisingly difficult  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  starting from the LagrangianL=m2(ẋi)2+ecAiẋi+eA0+12g(ω̇i-emcϵijkωjBk)2+3g28a2+1ϕ((ωi)2-a2). The configuration-space variables are xi(t), ωi(t), g(t), ϕ(t). Here xi represents the spatial coordinates of the particle with the mass m and the charge e, ωi 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, Ai 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  that spin lives on two-dimensional surface of six-dimensional spin phase space ωi, πi. The surface can be parameterized by the angular-momentum coordinates Si=ϵijkωjπk, subject to the condition S2=32/4. They obey the classical brackets {Si,Sj}=ϵijkSk. Hence we quantize them according the rule SiŜi.

The model leads to reasonable picture both on classical and quantum levels. The classical dynamics is governed by the Lagrangian equationsmẍi=eEi+ecϵijkẋjBk-emcSkiBk,Ṡi=emcϵijkSjBk. It has been denoted that E=-(1/c)(A/t)+A0. Since S22, the S-term disappears from (1.8) in the classical limit 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 Ŝi (1.2) either into the Frenkel tensor Φμν, Φμνuν=0, or into the Bargmann-Michel-Telegdi four-vector (The conditions Φμνuν=0 and Sμuμ=0 guarantee that in the rest frame survive only three components of these quantities, which implies the right nonrelativistic limit). Sμ, Sμuμ=0, where uμ 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 Ŝi into the SO(3,2) angular-momentum tensor L̂AB of five-dimensional Minkowski space A=(μ,5)=(0,i,5)=(0,1,2,3,5), ηAB=(-+++-).

In the passage from nonrelativistic to relativistic spin, we replace the Pauli equation by the Dirac one(p̂μΓμ+mc)Ψ(xμ)=0, where p̂μ=-iμ. We use the representation with Hermitian Γ0 and anti-Hermitian ΓiΓ0=(100-1),Γi=(0σi-σi0), then [Γμ,Γν]+=-2ημν, ημν=(-+++), and Γ0Γi, Γ0 are the Dirac matrices  αi, βαi=(0σiσi0),β=(100-1). We take the classical counterparts of the operators x̂μ and p̂μ=-iμ in the standard way, which are xμ, pν, with the Poisson brackets {xμ,pν}PB=ημν.

Let us discuss the classical variables that could produce the Γ-matrices. To this aim, we first study their commutators. The commutators of Γμ do not form closed Lie algebra, but produce SO(1,3)-Lorentz generators[Γμ,Γν]-=-2iΓμν, where it has been denoted Γμν(i/2)(ΓμΓν-ΓνΓμ). The set Γμ, Γμν form closed algebra. Besides the commutator (2.4), one has[Γμν,Γα]-=2i(ημαΓν-ηναΓμ),[Γμν,Γαβ]-=2i(ημαΓνβ-ημβΓνα-ηναΓμβ+ηνβΓμα). The algebra (2.4), (2.5) can be identified with the five-dimensional Lorentz algebra SO(2,3) with generators L̂AB[L̂AB,L̂CD]-=2i(ηACL̂BD-ηADL̂BC-ηBCL̂AD+ηBDL̂AC), assuming ΓμL̂5μ, ΓμνL̂μν.

To reach the algebra starting from a classical-mechanics model, we introduce ten-dimensional “phase” space of the spin degrees of freedom, ωA, πB, equipped with the Poisson bracket{ωA,πB}PB=ηAB. Then Poisson brackets of the quantitiesJAB2(ωAπB-ωBπA) read{JAB,JCD}PB=2(ηACJBD-ηADJBC-ηBCJAD+ηBDJAC). Below we use the decompositionsJAB=(J5μ,Jμν)=(J50,  J5iJ5,  J0iW,  Jij=ϵijkDk).

The Jacobian of the transformation (ωA,πB)JAB has rank equal seven. So, only seven among ten functions JAB(ω,π), A<B, are independent quantities. They can be separated as follows. By construction, the quantities (2.8) obey the identityϵμναβJ5νJαβ=0,Jij=1J50(J5iJ0j-J5jJ0i), that is, the three-vector D can be presented through J5, W asD=1J50J5×W. Further, (ωA,πB)-space can be parameterized by the coordinates J5μ, J0i, ω0, ω5, π5. We can not yet quantize the variables since it would lead to the appearance of some operators ω̂0, ω̂5, π̂5, 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 ωA, πB. The only SO(2,3) quadratic invariants that can be constructed from ωA,πB are ωAωA, ωAπA, πAπA. We choose conventionally the surface determined by the equationsωAωA+R=0,ωAπA=0,πAπA=0,R=const>0. The quantities J5μ, J0i form a coordinate system of the spin-space surface. So, we can quantize them instead of the initial variables ωA, πB.

According to (1.1), (2.6), (2.9), quantization is achieved replacing the classical variables J5μ, Jμν on Γ-matricesJ5μΓμ,JμνΓμν. It implies, that the Dirac equation can be produced by the constraint (we restate that J5μ2(ω5πμ-ωμπ5))pμJ5μ+mc=0.

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[(ẋμ+e3ωμ)2-(e3ω5)2]-σmc2ω5+1σ[(ẋμ+e3ωμ)ω̇μ-e3ω5ω̇5]-e4(ωAωA+R), written on the configuration space xμ, ωμ, ω5, ei, σ, where ei, σ are the auxiliary variables. The variables ω5, ei, σ are scalars under the Poincare transformations. The remaining variables transform according to the rulexμ=Λμνxν+aμ,ωμ=Λμνων. Local symmetries of the theory form the two-parameter group composed by the reparametrizationsδxμ=αẋμ,δωA=αω̇A,δei=(αei).,δσ=(ασ)., as well as by the local transformations with the parameter ϵ(τ) (below we have denoted βė4ϵ+(1/2)e4ϵ̇)δxμ=0,δωA=βωA,δσ=βσ,δe2=0,δe3=-βe3+e2σβ̇,δe4=-2e4β-(e2β̇2σ2). 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 (xμ,σ,e3)(x̃μ,x̃5,ẽ3), where x̃μ=xμ-(e2/σ)ωμ, x̃5=-(e2/σ)ω5, ẽ3=e3+(e2/σ), it readsL=-12e2(Dx̃A)2+(x̃5)22e2(ω5)2(ω̇A)2+e2mcx̃5-e4(ωAωA+R), where the covariant derivative is Dx̃A=x̃̇A+ẽ3ωA (The change is an example of conversion of the second-class constraints in the Lagrangian formulation ).

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 ei, σ can be omitted from consideration after the partial fixation of a gauge. After that, Hamiltonian of the model reads H=12πAπA+12(pμJ5μ+mc). Since ωA, πA are not ϵ-invariant variables, their equations of motion have no much sense. So, we write the equations of motion for ϵ-invariant quantities xμ, pμ, J5μ, Jμνẋμ=12J5μ,J̇5μ=-Jμνpν,J̇μν=pμJ5ν-pνJ5μ,ṗμ=0. They imply ẍμ=-12Jμνpν. In three-dimensional notations, the equation (3.8) read J̇50=-(Wp),J̇5=-p0W+D×p,Ẇ=p0J5-J50p.

Relativistic Invariance

While the canonical momentum of xμ is given by pμ, the mechanical momentum, according to (3.7), coincides with the variables that turn into the Γ-matrices in quantum theory, (1/2)J5μ. Due to the constraints (2.13), (2.14), J5μ obeys (J5μ/π5)2=-4R, which is analogy of p2=-m2c2 of the spinless particle. As a consequence, xi(t) cannot exceed the speed of light, (dxi/dt)2=c2(ẋi/ẋ0)2=c2(1-((π5)24R/(J50)2))<c2. Equations (3.10), (3.11) mean that both xμ-particle and the variables W, J5 experience the Zitterbewegung in noninteracting theory.

Center-of-Charge Rest Frame

Identifying the variables xμ with position of the charge, (3.7) implies that the rest frame is characterized by the conditions J50=const,J5=0. According to (3.12), (2.12), only W survives in the nonrelativistic limit.

The Variables Free of Zitterbewegung

The quantity (center-of-mass coordinate ) x̃μ=xμ+(1/2p2)Jμνpν obeys x̃̇μ=-(mc/2p2)pμ, so, it has the free dynamics x̃̈μ=0. Note also that pμ represents the mechanical momentum of x̃μ-particle.

As the classical four-dimensional spin vector, let us take Sμ=ϵμναβpνJαβ. It has no precession in the free theory, Ṡμ=0. In the rest frame, it reduces to S0=0, S=p×W.

Comparison with the Barut-Zanghi (BZ) Model

The BZ spinning particle  is widely used  for semiclassical analysis of spin effects. Starting from the even variable zα, where α=1,2,3,4 is SO(1,3)-spinor index, Barut and Zanghi have constructed the spin-tensor according to Sμν=(1/4)iz¯γμνz. We point out that (3.7)–(3.9) of our model coincide with those of BZ-model, identifying J5μvμ, JμνSμν. Besides, our model implies the equations (J5μ/π5)2=-4R,pμJ5μ+mc=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 , the mass of the spinning particle is not fixed from the model.)

Acknowledgment

This work has been supported by the Brazilian foundation FAPEMIG.

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