Alternative pseudoclassical action and spin factor
Stoian I. Zlatev*
Departamento de Matema ´tica, Universidade Federal de Sergipe, 49100-000 Sa ˜ o Cristo ´va ˜ o-SE, Brazil
Received 12 April 2000; published 23 October 2000
An alternative pseudoclassical action for a Dirac particle and a path integral for the propagator in a space-
time of arbitrary even dimensions are proposed. A new expression for the spin factor is obtained. It is shown
that it agrees with the known results in four-dimensional space-time.
PACS numbers: 11.10.Ef
I. INTRODUCTION
The first-quantized theory of relativistic particles is inter-
esting from conceptual and technical points of view. In par-
ticular, it exhibits a close similarity to string theory and, on
the other hand, can provide advantages in calculations see,
e.g., 1. In the case of a spinning particle the starting point
for the construction of the first-quantized theory is, usually, a
pseudoclassical action. Such an action for the Dirac electron
was first put forward by Berezin and Marinov 2 and has
been extensively studied by many authors 3. This action
can be readily generalized for any number of space-time di-
mensions. However, in odd-dimensional space-times it does
not produce a ‘‘minimal’’ model upon quantization. The rea-
son is that, as is well known, there is no matrix analogous to
5
in odd dimensions. Speaking less technically, there are
two inequivalent unitary irreducible representations of the
Poincare
´
group in odd dimensions, corresponding to particles
of opposite helicity 4,5. The quantization of the Berezin-
Marinov model yields an equation containing
5
or, respec-
tively, its analogue. Such an equation describes particles of
both helicities. Being completely equivalent to the Dirac
equation in even dimensions, it corresponds to a pair of
Dirac equations in odd-dimensional space-times.
In a recent paper by Gitman 6 path-integral representa-
tions for the propagator are derived and the corresponding
pseudoclassical description is given for spinning particles in
arbitrary space-time dimensions. While the even-
dimensional case is treated along the traditional lines leading
to a generalization of the Berezin-Marinov model, a new
approach is used in the odd-dimensional case. It is based on
the existing duality relations between the matrices which
allows one to represent the Dirac operator as an even func-
tion of the matrices. A conclusion has been made 6 that
the problem of the pseudoclassical description has different
solutions in even and odd dimensions.
However, the possibility to represent the Dirac operator as
an even function of matrices is not a specific feature of the
odd-dimensional case and also exists in even dimensions. In
the latter case one has to treat all the matrices including
the analogue of
5
) on equal footing.
In the present paper we obtain a path-integral representa-
tion for the propagator in space-time of arbitrary even di-
mension using the ‘‘even’’ form of the Dirac operator. The
corresponding pseudoclassical action is different from the
Berezin-Marinov one and is similar to the action proposed by
Gitman 6 for the odd-dimensional case. Using the path in-
tegral for the propagator we calculate the spin factor. It is
shown that in four dimensions the expression obtained
agrees with the known results.
II. DIRAC EQUATION
Dirac operator can be presented as an even function of
matrices in even-dimensional space-time as well. To this end
one has to treat all the matrices on equal footing, including
the matrix analogous to
5
. Let
0
,...,
2 d -1
be a set of
matrices for 2 d -dimensional space-time, i.e., a set of 2
d
2
d
matrices obeying the anticommutation relations
1
Let us define
2 d
=
i
d
2 d !
1
•••
2d
1
•••
2d
, 2
where
1
•••
2d
is the Levi-Civita
`
symbol normalized by
0, . . . ,2d -1
=1. Introducing the ‘‘extended’’ metric
and the corresponding Levi-Civita
`
symbol
n
1
••• n
2d+1
(
0, . . . ,2d
=1) one finds
n
=
i
d
2 d !
nm
1
••• m
2d
m
1
•••
m
2d
.
Using this identity one can put the Dirac equation
P
-m =0, 3
where P
=i
-gA
( A
being an external Abelian field
and g the particle charge, into the ‘‘even’’ form
i
d
2 d !
n
1
••• n
2d
n
1
•••
n
2d
P
-m
S
c
x , y =0. 4
*Email address: zlatev@ufs.br
PHYSICAL REVIEW D, VOLUME 62, 105020
0556-2821/2000/6210/1050206/$15.00 ©2000 The American Physical Society 62 105020-1