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Pseudoclassical description of the massive Dirac particles in odd dimensions
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Pseudoclassical description of the massive Dirac particles in odd
dimensions
D.M. Gitman
Instituto de F??sica, Universidade de Sa?o Paulo
P.O. Box 66318, 05389-970 Sa?o Paulo, SP, Brazil
A.E. Gonc?alves
Departamento de F??sica, Universidade Estadual de Londrina
P.O. Box 6001, 86051-970 Londrina, Pr, Brazil
(February 1, 2008)
Abstract
A pseudoclassical model is proposed to describe massive Dirac ( spin one-half)
particles in arbitrary odd dimensions. The quantization of the model repro-
duces the minimal quantum theory of spinning particles in such dimensions.
A dimensional duality between the model proposed and the pseudoclassical
description of Weyl particles in even dimensions is discussed.
11.10.Ef, 03.65.Pm
Typeset using REVTEX
1
I. INTRODUCTION
As it is known one can construct a pseudoclassical model to describe massive Dirac (spin
one-half) particles in 3 + 1 dimensions [1]. Its generalization to the case of even dimensions
D = 2n , n = 3, 4, . . . , can be done by means of the direct dimensional extension [2]. The
corresponding action has the form
S =
∫ 1
0
[
?
x?2
2e
? e
m2
2
+ ?
(
x?μψ
μ
e
?mψD
)
χ? ?ψaψ?
a
]
dτ , (1)
where x?2 = x?μx?
μ; the Greek (Lorentz) indices μ, ν, . . ., run over 0, 1, . . . , d?1, whereas the
Latin ones a, b, run over 0, 1, . . . , d; ημν = diag(1,?1, . . . ,?1︸ ︷︷ ︸
D
), ηab = diag(1,?1, . . . ,?1︸ ︷︷ ︸
D+1
).
The variables xμ and e are even and ψn, χ are odd. The quantization of the model leads to
the Dirac quantum theory of spin one-half particle (to the Dirac equation).
Attempts to extend the pseudoclassical description to the arbitrary odd-dimensional case
had met some problems, which are connected with the absence of an analog of γ5?matrix in
such dimensions. For instance, in 2n+1 dimensions the direct generalization of the standard
action [1] does not reproduce a minimal quantum theory of spinning particle, where particles
with spin 1/2 and ?1/2 have to be cons
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