Hindawi Publishing Corporation
Advances in Mathematical Physics
Volume 2011, Article ID 680367, 10 pages
doi:10.1155/2011/680367
Research Article
Relativistic Spinning Particle without Grassmann
Variables and the Dirac Equation
A. A. Deriglazov
Departamento de Matematica, Instituto de Ciências Exatas (ICE), Universidade Federal de Juiz de Fora,
36039-900 Juiz de Fora, MG, Brazil
Correspondence should be addressed to A. A. Deriglazov, alexei.deriglazov@ufjf.edu.br
Received 31 March 2011; Revised 30 July 2011; Accepted 23 August 2011
Academic Editor: Stephen Anco
Copyright q 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 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 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 rule
zα , zβ ∓ i zα , 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.
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Advances in Mathematical Physics
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 classicalmechanics system whose classical bracket obeys 1.1 for the known left-hand side.
Components of the nonrelativistic spin operator Si /2σ i σ i are the Pauli matrices 1.5
form a simple algebra with respect to commutator
Si , Sj iijk Sk ,
1.2
−
as well as to anticommutator
2
Si , Sj δij .
2
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/2ẋi 2 i/2ξi ξ̇i , where the spin inner space is constructed from vector-like Grassmann
variables ξi , ξi ξj −ξj ξi . 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 .
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 /2σ i ,
σ i , σ j 2δij ,
σ 1
0 1
1 0
,
σ 2
0 −i
i 0
,
σ 3
1 0
0 −1
,
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 equation
∂Ψ
i
∂t
2
1
e
e
i
pi − Ai − eA0 −
Bi σ Ψ.
2m
c
2mc
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 ijk ∂j Ak , where ijk represents the totally
antisymmetric tensor with 123 1. Relativistic spin is described in a similar way 13, 14.
Advances in Mathematical Physics
3
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
L
2
3g2 1 e
1
e
m
2
2
.
ω̇i −
ijk ωj Bk −
a
ẋi 2 Ai ẋi eA0 ω
i
2
c
2g
mc
φ
8a2
1.7
The configuration-space variables are xi t, ωi t, gt, φt. 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 twodimensional surface of six-dimensional spin phase space ωi , πi . The surface can be
parameterized by the angular-momentum coordinates Si ijk ωj πk , subject to the condition
S2 32 /4. They obey the classical brackets {Si , Sj } ijk Sk . Hence we quantize them
according the rule Si → Si .
The model leads to reasonable picture both on classical and quantum levels. The
classical dynamics is governed by the Lagrangian equations
e
e
Sk ∂i Bk ,
mẍi eEi ijk ẋj Bk −
c
mc
e
Ṡi ijk Sj Bk .
mc
1.8
1.9
It has been denoted that E −1/c∂A/∂t ∇A0 . Since S2 ≈ 2 , 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.
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Advances in Mathematical Physics
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 , Φμν uν 0, or into the Bargmann-Michel-Telegdi
four-vector The conditions Φμν uν 0 and Sμ uμ 0 guarantee that in the rest frame
survive only three components of these quantities, which implies the right nonrelativistic
limit. Sμ , Sμ uμ 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 SO3, 2
AB of five-dimensional Minkowski space A μ, 5 0, i, 5 angular-momentum tensor L
AB
0, 1, 2, 3, 5, η − −.
In the passage from nonrelativistic to relativistic spin, we replace the Pauli equation
by the Dirac one
pμ Γμ mc Ψxμ 0,
2.1
where pμ −i∂μ . We use the representation with Hermitian Γ0 and anti-Hermitian Γi
Γ 0
1 0
0 −1
Γ 0
i
,
σi
,
−σ i 0
2.2
then Γμ , Γν −2ημν , ημν − , and Γ0 Γi , Γ0 are the Dirac matrices 3 αi , β
α i
0 σi
σi 0
,
β
1 0
0 −1
.
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 SO1, 3-Lorentz generators
Γμ , Γν − −2iΓμν ,
2.4
where it has been denoted Γμν ≡ i/2Γμ Γν − Γν Γμ . The set Γμ , Γμν form closed algebra.
Besides the commutator 2.4, one has
Γμν , Γα − 2i ημα Γν − ηνα Γμ ,
Γμν , Γαβ 2i ημα Γνβ − ημβ Γνα − ηνα Γμβ ηνβ Γμα .
−
2.5
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5
The algebra 2.4, 2.5 can be identified with the five-dimensional Lorentz algebra SO2, 3
AB
with generators L
CD 2i ηAC L
BD − ηAD L
BC − ηBC L
AD ηBD L
AC ,
AB , L
L
−
2.6
5μ , Γμν ≡ L
μν .
assuming Γμ ≡ L
To reach the algebra starting from a classical-mechanics model, we introduce tendimensional “phase” space of the spin degrees of freedom, ωA , π B , equipped with the Poisson
bracket
ωA , π B
PB
ηAB .
2.7
Then Poisson brackets of the quantities
J AB ≡ 2 ωA π B − ωB π A
2.8
read
J AB , J CD
PB
2 ηAC J BD − ηAD J BC − ηBC J AD ηBD J AC .
2.9
Below we use the decompositions
J AB J 5μ , J μν J 50 , J 5i ≡ J5 , J 0i ≡ W, J ij ijk Dk .
2.10
The Jacobian of the transformation ωA , 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 identity
μναβ J 5 ν Jαβ 0,
⇐⇒ J ij 1 5i 0j
J J − J 5j J 0i ,
50
J
2.11
that is, the three-vector D can be presented through J5 , W as
D
1 5
J × W.
J 50
2.12
Further, ωA , π B -space can be parameterized by the coordinates J 5μ , J 0i , ω0 , ω5 , π 5 . We can
not yet quantize the variables since it would lead to the appearance of some operators ω
0,
5
5
ω
, π , 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 , π 5 , restricting our model to
live on seven-dimensional surface of ten-dimensional phase space ωA , π B . The only SO2, 3
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Advances in Mathematical Physics
quadratic invariants that can be constructed from ωA , π B are ωA ωA , ωA πA , π A πA . We choose
conventionally the surface determined by the equations
2.13
ωA ωA R 0,
ωA πA 0,
2.14
π A πA 0,
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 ωA , π B .
According to 1.1, 2.6, 2.9, quantization is achieved replacing the classical
variables J 5μ , J μν on Γ-matrices
J 5μ −→ Γμ ,
2.15
J μν −→ Γμν .
It implies, that the Dirac equation can be produced by the constraint we restate that J 5μ ≡
2ω5 π μ − ωμ π 5 2.16
pμ J 5μ 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 Lagrangian
L−
2 σmc 1 1
μ
μ
5 5
A
−
e
ω
e
ω
−
e
ω
ω̇
ω
R
,
−
ω̇
ẋ
ẋμ e3 ωμ 2 − e3 ω5
3
μ
3
4
A
2e2
σ
2ω5
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 rule
x Λμ ν xν aμ ,
μ
ω Λμ ν ων .
μ
3.2
Local symmetries of the theory form the two-parameter group composed by the reparametrizations
δxμ αẋμ ,
δωA αω̇A ,
δei αei . ,
δσ ασ. ,
3.3
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as well as by the local transformations with the parameter τ below we have denoted
˙
β ≡ ė4 1/2e4 δxμ 0,
δωA βωA ,
δe3 −βe3 δσ βσ, δe2 0,
·
e2 β̇
δe4 −2e4 β −
.
2σ 2
e2
β̇,
σ
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 /σωμ , x5 −e2 /σω5 ,
e3 e3 e2 /σ, it reads
5 2 2
x
1 A 2
e2 mc
A
A
L−
Dx
ω̇
ω
−
e
ω
R
,
4
A
2e2
x5
2e2 ω5 2
3.5
A
where the covariant derivative is DxA x˙ e3 ωA 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
H
1 A
1
π πA pμ J 5μ mc .
2
2
3.6
Since ωA , π A are not -invariant variables, their equations of motion have no much sense. So,
we write the equations of motion for -invariant quantities xμ , pμ , J 5μ , J μν
ẋμ 1 5μ
J ,
2
3.7
J̇ 5μ −J μν pν ,
J˙ μν pμ J 5ν − pν J 5μ ,
3.8
ṗμ 0.
3.9
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Advances in Mathematical Physics
They imply
1
ẍμ − J μν pν .
2
3.10
In three-dimensional notations, the equation 3.8 read
J˙ 50 −Wp,
J̇5 −p0 W D × p,
Ẇ p0 J5 − J 50 p.
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, 1/2J 5μ .
Due to the constraints 2.13, 2.14, J 5μ obeys J 5μ /π 5 2 −4R, which is analogy of
p2 −m2 c2 of the spinless particle. As a consequence, xi t cannot exceed the speed of light,
dxi /dt2 c2 ẋi /ẋ0 2 c2 1−π 5 2 4R/J 50 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
J 50 const,
J5 0.
3.12
According to 3.12, 2.12, only W survives in the nonrelativistic limit.
The Variables Free of Zitterbewegung
μ
The quantity center-of-mass coordinate 7 xμ xμ 1/2p2 J μν pν obeys x˙
μ
2 μ
μ
−mc/2p p , so, it has the free dynamics x¨ 0. Note also that p represents the mechanical
momentum of xμ -particle.
As the classical four-dimensional spin vector, let us take Sμ μναβ pν Jαβ . 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 SO1, 3-spinor index, Barut and
Zanghi have constructed the spin-tensor according to Sμν 1/4izγμν z. We point out that
3.7–3.9 of our model coincide with those of BZ-model, identifying J 5μ ↔ vμ , J μν ↔ Sμν .
Besides, our model implies the equations J 5μ /π 5 2 −4R, pμ J 5μ 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.
Advances in Mathematical Physics
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Acknowledgment
This work has been supported by the Brazilian foundation FAPEMIG.
References
1 A. H. Compton, “The magnetic electron,” Journal of The Franklin Institute, vol. 192, no. 2, pp. 145–155,
1921.
2 G. E. Uhlenbeck and S. Goudsmit, “Spinning electrons and the structure of spectra,” Nature, vol. 117,
pp. 264–265, 1926.
3 P. A. M. Dirac, The Principles of Quantum Mechanics, Oxford Clarendon Press, Oxford, UK, 1958.
4 J. Frenkel, “Die elektrodynamik des rotierenden elektrons,” Zeitschrift fur Physik, vol. 37, no. 4-5, pp.
243–262, 1926.
5 P. A. M. Dirac, “The quantum theory of the electron,” Proceedings of the Royal Society A, vol. 117, pp.
610–624, 1928.
6 P. A. M. Dirac, “The quantum theory of the electron,” Proceedings of the Royal Society A, vol. 118, pp.
351–361, 1928.
7 M. H. L. Pryce, “The mass-centre in the restricted theory of relativity and its connexion with the
quantum theory of elementary particles,” Proceedings of the Royal Society A, vol. 195, pp. 62–81, 1948.
8 T. D. Newton and E. P. Wigner, “Localized states for elementary systems,” Reviews of Modern Physics,
vol. 21, no. 3, pp. 400–406, 1949.
9 L. L. Foldy and S. A. Wouthuysen, “On the dirac theory of spin 1/2 particles and its non-relativistic
limit,” Physical Review, vol. 78, no. 1, pp. 29–36, 1950.
10 V. Bargmann, L. Michel, and V. L. Telegdi, “Precession of the polarization of particles moving in a
homogeneous electromagnetic field,” Physical Review Letters, vol. 2, no. 10, pp. 435–436, 1959.
11 T. F. Jordan and N. Mukunda, “Lorentz-covariant position operators for spinning particles,” Physical
Review, vol. 132, pp. 1842–1848, 1963.
12 A. O. Barut, Electrodynamics and Classical Theory of Fields and Particles, Macmillan, New York, NY, USA,
1964.
13 F. A. Berezin and Marinov M. S., “Classical spin and Grassmann algebra,” JETP Letters, vol. 21, pp.
320–321, 1975.
14 F. A. Berezin and M. S. Marinov, “Particle spin dynamics as the grassmann variant of classical
mechanics,” Annals of Physics, vol. 104, no. 2, pp. 336–362, 1977.
15 A. J. Hanson and T. Regge, “The relativistic spherical top,” Annals of Physics, vol. 87, pp. 498–566,
1974.
16 A. O. Barut and N. Zanghi, “Classical model of the dirac electron,” Physical Review Letters, vol. 52, no.
23, pp. 2009–2012, 1984.
17 D. M. Gitman and I. V. Tyutin, Quantization of Fields with Constraints, Springer, Berlin, Germany, 1990.
18 A. A. Deriglazov, “Nonrelativistic spin: a la Berezin-Marinov quantization on a sphere,” Modern
Physics Letters A, vol. 25, no. 32, pp. 2769–2777, 2010.
19 G. Salesi and E. Recami, “Effects of spin on the cyclotron frequency for a dirac electron,” Physics Letters
A, vol. 267, no. 4, pp. 219–224, 2000.
20 M. Pavsic, E. Recami, W. A. Rodrigues Jr., G. D. Maccarrone, F. Raciti, and G. Salesi, “Spin and electron
structure,” Physics Letters B, vol. 318, no. 3, pp. 481–488, 1993.
21 D. Hestenes, “Zitterbewegung in quantum mechanics,” Foundations of Physics, vol. 40, no. 1, pp. 1–54,
2010.
22 D. Singh and N. Mobed, “Effects of spacetime curvature on spin−1/2 particle zitterbewegung,”
Classical and Quantum Gravity, vol. 26, no. 18, Article ID 185007, 2009.
23 N. Unal, “Electromagnetic and gravitational interactions of the spinning particle,” Journal of
Mathematical Physics, vol. 47, no. 9, Article ID 092501, 2006.
24 J. R. Van Meter, A. K. Kerman, P. Chen, and F. V. Hartemann, “Radiative corrections in symmetrized
classical electrodynamics,” Physical Review E, vol. 62, no. 6, pp. 8640–8650, 2000.
25 A. A. Deriglazov and Z. Kuznetsova, “Conversion of second class constraints by deformation of
Lagrangian local symmetries,” Physics Letters B, vol. 646, no. 1, pp. 47–53, 2007.
10
Advances in Mathematical Physics
26 A. A. Deriglazov, Classical Mechanics Hamiltonian and Lagrangian Formalism, Springer, Heidelberg,
Germany, 2010.
27 J. D. Bjorken and S. D. Drell, Relativistic Quantum Fields, McGraw-Hill, New York, NY, USA, 1965.
28 J. J. Sakurai, Modern Quantum Mechanics, Addison-Wesley, 1994.
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