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Advances in Mathematical Physics
Volume 2011 (2011), Article ID 680367, 10 pages
http://dx.doi.org/10.1155/2011/680367
Research Article

Relativistic Spinning Particle without Grassmann Variables and the Dirac Equation

Departamento de Matematica, Instituto de Ciências Exatas (ICE), Universidade Federal de Juiz de Fora, 36039-900 Juiz de Fora, MG, Brazil

Received 31 March 2011; Revised 30 July 2011; Accepted 23 August 2011

Academic Editor: Stephen Anco

Copyright © 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.

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 [16], a lot of efforts have been spent in attempts to understand behaviour of a particle with spin on the base of semiclassical mechanical models [724].

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 [3], the operators ̂𝑧𝛼 must be chosen to obey the quantization rulê𝑧𝛼,̂𝑧𝛽𝑧=𝑖𝛼,𝑧𝛽||𝑧.̂𝑧(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 𝑆𝑖=(/2)𝜎𝑖 (𝜎𝑖 are the Pauli matrices (1.5)) form a simple algebra with respect to commutator𝑆𝑖,𝑆𝑗=𝑖𝜖𝑖𝑗𝑘𝑆𝑘,(1.2) as well as to anticommutator𝑆𝑖,𝑆𝑗+=22𝛿𝑖𝑗.(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)(̇𝑥𝑖)2+(𝑖/2)𝜉𝑖̇𝜉𝑖, where the spin inner space is constructed from vector-like Grassmann variables 𝜉𝑖, 𝜉𝑖𝜉𝑗=𝜉𝑗𝜉𝑖. Since the Lagrangian is linear on ̇𝜉𝑖, their conjugate momenta coincide with 𝜉, 𝜋𝑖̇𝜉=𝜕𝐿/𝜕𝑖=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 reads𝜉𝑖,𝜉𝑗DB=𝑖𝛿𝑖𝑗.(1.4) 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 𝑆𝑖 proportional to the Pauli 𝜎-matrices, 𝑆𝑖=(/2)𝜎𝑖, [𝜎𝑖,𝜎𝑗]+=2𝛿𝑖𝑗,𝜎1=0110,𝜎2=0𝑖𝑖0,𝜎3=1001,(1.5) acting on two-dimensional spinor space Ψ𝛼(𝑡,𝑥). Canonical quantization of the particle on an external electromagnetic background leads to the Pauli equation𝑖𝜕Ψ=1𝜕𝑡2𝑚̂𝑝𝑖𝑒𝑐𝐴𝑖2𝑒𝐴0𝑒𝐵2𝑚𝑐𝑖𝜎𝑖Ψ.(1.6) It has been denoted that 𝑝𝑖=𝑖(𝜕/𝜕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 [715] 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 Lagrangian𝑚𝐿=2̇𝑥𝑖2+𝑒𝑐𝐴𝑖̇𝑥𝑖+𝑒𝐴0+12𝑔̇𝜔𝑖𝑒𝜖𝑚𝑐𝑖𝑗𝑘𝜔𝑗𝐵𝑘2+3𝑔28𝑎2+1𝜙𝜔𝑖2𝑎2.(1.7) 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=32/4. They obey the classical brackets {𝑆𝑖,𝑆𝑗}=𝜖𝑖𝑗𝑘𝑆𝑘. Hence we quantize them according the rule 𝑆𝑖𝑆𝑖.

The model leads to reasonable picture both on classical and quantum levels. The classical dynamics is governed by the Lagrangian equations𝑚̈𝑥𝑖=𝑒𝐸𝑖+𝑒𝑐𝜖𝑖𝑗𝑘̇𝑥𝑗𝐵𝑘𝑒𝑆𝑚𝑐𝑘𝜕𝑖𝐵𝑘,̇𝑆(1.8)𝑖=𝑒𝜖𝑚𝑐𝑖𝑗𝑘𝑆𝑗𝐵𝑘.(1.9) 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 𝑆𝑖 (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 𝑆𝑖 into the 𝑆𝑂(3,2) angular-momentum tensor 𝐿𝐴𝐵 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 onê𝑝𝜇Γ𝜇Ψ+𝑚𝑐(𝑥𝜇)=0,(2.1) where ̂𝑝𝜇=𝑖𝜕𝜇. We use the representation with Hermitian Γ0 and anti-Hermitian Γ𝑖Γ0=1001,Γ𝑖=0𝜎𝑖𝜎𝑖0,(2.2) then [Γ𝜇,Γ𝜈]+=2𝜂𝜇𝜈, 𝜂𝜇𝜈=(+++), and Γ0Γ𝑖, Γ0 are the Dirac matrices [3] 𝛼𝑖, 𝛽𝛼𝑖=0𝜎𝑖𝜎𝑖0,𝛽=1001.(2.3) We take the classical counterparts of the operators ̂𝑥𝜇 and ̂𝑝𝜇=𝑖𝜕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 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 generatorsΓ𝜇,Γ𝜈=2𝑖Γ𝜇𝜈,(2.4) where it has been denoted Γ𝜇𝜈(𝑖/2)(Γ𝜇Γ𝜈Γ𝜈Γ𝜇). The set Γ𝜇, Γ𝜇𝜈 form closed algebra. Besides the commutator (2.4), one hasΓ𝜇𝜈,Γ𝛼=2𝑖(𝜂𝜇𝛼Γ𝜈𝜂𝜈𝛼Γ𝜇Γ),𝜇𝜈,Γ𝛼𝛽𝜂=2𝑖𝜇𝛼Γ𝜈𝛽𝜂𝜇𝛽Γ𝜈𝛼𝜂𝜈𝛼Γ𝜇𝛽+𝜂𝜈𝛽Γ𝜇𝛼.(2.5) The algebra (2.4), (2.5) can be identified with the five-dimensional Lorentz algebra SO(2,3) with generators 𝐿𝐴𝐵𝐿𝐴𝐵,𝐿𝐶𝐷𝜂=2𝑖𝐴𝐶𝐿𝐵𝐷𝜂𝐴𝐷𝐿𝐵𝐶𝜂𝐵𝐶𝐿𝐴𝐷+𝜂𝐵𝐷𝐿𝐴𝐶,(2.6) assuming Γ𝜇𝐿5𝜇, Γ𝜇𝜈𝐿𝜇𝜈.

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 bracket𝜔𝐴,𝜋𝐵PB=𝜂𝐴𝐵.(2.7) Then Poisson brackets of the quantities𝐽𝐴𝐵𝜔2𝐴𝜋𝐵𝜔𝐵𝜋𝐴(2.8) read𝐽𝐴𝐵,𝐽𝐶𝐷PB𝜂=2𝐴𝐶𝐽𝐵𝐷𝜂𝐴𝐷𝐽𝐵𝐶𝜂𝐵𝐶𝐽𝐴𝐷+𝜂𝐵𝐷𝐽𝐴𝐶.(2.9) Below we use the decompositions𝐽𝐴𝐵=𝐽5𝜇,𝐽𝜇𝜈=𝐽50,𝐽5𝑖𝐉5,𝐽0𝑖𝐖,𝐽𝑖𝑗=𝜖𝑖𝑗𝑘𝐷𝑘.(2.10)

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=1𝐽50𝐽5𝑖𝐽0𝑗𝐽5𝑗𝐽0𝑖,(2.11) that is, the three-vector 𝐃 can be presented through 𝐉𝟓, 𝐖 as1𝐃=𝐽50𝐉5×𝐖.(2.12) 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 𝜔0, 𝜔5, 𝜋5, 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 equations𝜔𝐴𝜔𝐴𝜔+𝑅=0,(2.13)𝐴𝜋𝐴=0,𝜋𝐴𝜋𝐴=0,(2.14)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.(2.15) It implies, that the Dirac equation can be produced by the constraint (we restate that 𝐽5𝜇2(𝜔5𝜋𝜇𝜔𝜇𝜋5))𝑝𝜇𝐽5𝜇+𝑚𝑐=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 Lagrangian1𝐿=2𝑒2̇𝑥𝜇+𝑒3𝜔𝜇2𝑒3𝜔52𝜎𝑚𝑐2𝜔5+1𝜎̇𝑥𝜇+𝑒3𝜔𝜇̇𝜔𝜇𝑒3𝜔5̇𝜔5𝑒4𝜔𝐴𝜔𝐴,+𝑅(3.1) 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.(3.2) Local symmetries of the theory form the two-parameter group composed by the reparametrizations𝛿𝑥𝜇=𝛼̇𝑥𝜇,𝛿𝜔𝐴=𝛼̇𝜔𝐴,𝛿𝑒𝑖=𝛼𝑒𝑖.,𝛿𝜎=(𝛼𝜎).,(3.3) as well as by the local transformations with the parameter 𝜖(𝜏) (below we have denoted 𝛽̇𝑒4𝜖+(1/2)𝑒4̇𝜖)𝛿𝑥𝜇=0,𝛿𝜔𝐴=𝛽𝜔𝐴,𝛿𝜎=𝛽𝜎,𝛿𝑒2=0,𝛿𝑒3=𝛽𝑒3+𝑒2𝜎̇𝛽,𝛿𝑒4=2𝑒4𝑒𝛽2̇𝛽2𝜎2.(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 (𝑥𝜇,𝜎,𝑒3)(̃𝑥𝜇,̃𝑥5,̃𝑒3), where ̃𝑥𝜇=𝑥𝜇(𝑒2/𝜎)𝜔𝜇, ̃𝑥5=(𝑒2/𝜎)𝜔5, ̃𝑒3=𝑒3+(𝑒2/𝜎), it reads1𝐿=2𝑒2𝐷̃𝑥𝐴2+̃𝑥522𝑒2𝜔52̇𝜔𝐴2+𝑒2𝑚𝑐̃𝑥5𝑒4𝜔𝐴𝜔𝐴+𝑅,(3.5) where the covariant derivative is 𝐷̃𝑥𝐴=̇̃𝑥𝐴+̃𝑒3𝜔𝐴 (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 𝑒𝑖, 𝜎 can be omitted from consideration after the partial fixation of a gauge. After that, Hamiltonian of the model reads 1𝐻=2𝜋𝐴𝜋𝐴+12𝑝𝜇𝐽5𝜇.+𝑚𝑐(3.6) 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, 𝐽𝜇𝜈̇𝑥𝜇=12𝐽5𝜇,̇𝐽(3.7)5𝜇=𝐽𝜇𝜈𝑝𝜈,̇𝐽𝜇𝜈=𝑝𝜇𝐽5𝜈𝑝𝜈𝐽5𝜇,(3.8)̇𝑝𝜇=0.(3.9) They imply ̈𝑥𝜇1=2𝐽𝜇𝜈𝑝𝜈.(3.10) In three-dimensional notations, the equation (3.8) read ̇𝐽50̇𝐉=(𝐖𝐩),5=𝑝0̇𝐖+𝐃×𝐩,𝐖=𝑝0𝐉5𝐽50𝐩.(3.11)

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(̇𝑥𝑖/̇𝑥0)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.(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]) ̃𝑥𝜇=𝑥𝜇+(1/2𝑝2)𝐽𝜇𝜈𝑝𝜈 obeys ̇̃𝑥𝜇=(𝑚𝑐/2𝑝2)𝑝𝜇, so, it has the free dynamics ̈̃𝑥𝜇=0. Note also that 𝑝𝜇 represents the mechanical momentum of ̃𝑥𝜇-particle.
As the classical four-dimensional spin vector, let us take 𝑆𝜇=𝜖𝜇𝜈𝛼𝛽𝑝𝜈𝐽𝛼𝛽. It has no precession in the free theory, ̇𝑆𝜇=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 [1924] 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)𝑖𝑧𝛾𝜇𝜈𝑧. 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 [16], the mass of the spinning particle is not fixed from the model.)

Acknowledgment

This work has been supported by the Brazilian foundation FAPEMIG.

References

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