# Relativistic Spinning Particle without Grassmann Variables and the Dirac Equation

Authors: A. A. Deriglazov
Journal: Advances in Mathematical Physics
 (2011)

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## 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.


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## Body

## 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. Letzα 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 [3], the operators ẑα must be chosen to obey the quantization rule(1.1)[ẑα,ẑβ]∓=iℏ{zα,zβ}|z→ẑ.
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=(ℏ/2)σi (σi are the Pauli matrices (1.5)) form a simple algebra with respect to commutator(1.2)[Ŝi,Ŝj]-=iℏϵijkŜk,
as well as to anticommutator(1.3)[Ŝi,Ŝj]+=ℏ22δij.
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(1.4){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=(ℏ/2)σi, [σi,σj]+=2δij,(1.5)σ1=(0110),σ2=(0-ii0),σ3=(100-1),
acting on two-dimensional spinor space Ψα(t,x⃗). Canonical quantization of the particle on an external electromagnetic background leads to the Pauli equation(1.6)iℏ∂Ψ∂t=(12m(p̂i-ecAi)2-eA0-eℏ2mcBiσi)Ψ.
It has been denoted that pi=-iℏ(∂/∂xi), (A0,Ai) is the four-vector potential of electromagnetic field, and the magnetic field is Bi=ϵijk∂jAk, 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 [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 Lagrangian(1.7)L=m2(ẋi)2+ecAiẋi+eA0+12g(ω̇i-emcϵijkωjBk)2+3gℏ28a2+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 [18] 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=3ℏ2/4. They obey the classical brackets {Si,Sj}=ϵijkSk. Hence we quantize them according the rule Si→Ŝi.The model leads to reasonable picture both on classical and quantum levels. The classical dynamics is governed by the Lagrangian equations(1.8)mẍi=eEi+ecϵijkẋjBk-emcSk∂iBk,(1.9)Ṡi=emcϵijkSjBk.
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 vectorŜi (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(2.1)(p̂μΓμ+mc)Ψ(xμ)=0,
where p̂μ=-iℏ∂μ. We use the representation with Hermitian Γ0 and anti-Hermitian Γi(2.2)Γ0=(100-1),Γi=(0σi-σi0),
then [Γμ,Γν]+=-2ημν, ημν=(-+++), and Γ0Γi, Γ0 are the Dirac matrices [3] αi, β(2.3)αi=(0σiσi0),β=(100-1).
We take the classical counterparts of the operators x̂μ and p̂μ=-iℏ∂μ 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 theΓ-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(2.4)[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(2.5)[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(2.6)[L̂AB,L̂CD]-=2i(ηACL̂BD-ηADL̂BC-ηBCL̂AD+ηBDL̂AC),
assuming Γμ≡L̂5μ, Γμν≡L̂μν.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(2.7){ωA,πB}PB=ηAB.
Then Poisson brackets of the quantities(2.8)JAB≡2(ωAπB-ωBπA)
read(2.9){JAB,JCD}PB=2(ηACJBD-ηADJBC-ηBCJAD+ηBDJAC).
Below we use the decompositions(2.10)JAB=(J5μ,Jμν)=(J50,J5i≡J5,J0i≡W,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 identity(2.11)ϵμναβJ5νJαβ=0,⟺Jij=1J50(J5iJ0j-J5jJ0i),
that is, the three-vector D can be presented through J5, W as(2.12)D=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 equations(2.13)ωAωA+R=0,(2.14)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-matrices(2.15)J5μ⟶ℏΓμ,Jμν⟶ℏΓμν.
It implies, that the Dirac equation can be produced by the constraint (we restate that J5μ≡2(ω5πμ-ωμπ5))(2.16)pμJ5μ+mcℏ=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 followingd=4 Poincare-invariant Lagrangian(3.1)L=-12e2[(ẋμ+e3ωμ)2-(e3ω5)2]-σmcℏ2ω5+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 rule(3.2)x′μ=Λμνxν+aμ,ω′μ=Λμνων.
Local symmetries of the theory form the two-parameter group composed by the reparametrizations(3.3)δxμ=αẋμ,δωA=αω̇A,δei=(αei).,δσ=(ασ).,
as well as by the local transformations with the parameter ϵ(τ) (below we have denoted β≡ė4ϵ+(1/2)e4ϵ̇)(3.4)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 reads(3.5)L=-12e2(Dx̃A)2+(x̃5)22e2(ω5)2(ω̇A)2+e2mcℏx̃5-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 [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 variablesei, σ can be omitted from consideration after the partial fixation of a gauge. After that, Hamiltonian of the model reads
(3.6)H=12πAπA+12(pμJ5μ+mcℏ).
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, Jμν(3.7)ẋμ=12J5μ,(3.8)J̇5μ=-Jμνpν,J̇μν=pμJ5ν-pνJ5μ,(3.9)ṗμ=0.
They imply
(3.10)ẍμ=-12Jμνpν.
In three-dimensional notations, the equation (3.8) read
(3.11)J̇50=-(Wp),J̇5=-p0W+D×p,Ẇ=p0J5-J50p.Relativistic Invariance
While the canonical momentum ofxμ 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 variablesxμ with position of the charge, (3.7) implies that the rest frame is characterized by the conditions
(3.12)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 [7]) x̃μ=xμ+(1/2p2)Jμνpν obeys x̃̇μ=-(mcℏ/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 takeSμ=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 [16] is widely used [19–24] 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 J5μ↔vμ, Jμν↔Sμν. Besides, our model implies the equations (J5μ/π5)2=-4R,pμJ5μ+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.)



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