ISRN High Energy Physics

Volume 2013 (2013), Article ID 751963, 15 pages

http://dx.doi.org/10.1155/2013/751963

Research Article

Green’s Function for Dirac Particle in a Non-Abelian Field: A Path Integral Approach

Sami Boudieb and Lyazid Chetouani

Département de Physique, Faculté des Sciences Exactes, Université Constantine 1, Constantine, Algeria

Received 16 April 2013; Accepted 8 May 2013

Academic Editors: G. Bonvicini, F.-H. Liu, L. Marek-Crnjac, and O. A. Sampayo

Copyright © 2013 Sami Boudieb and Lyazid Chetouani. 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.

The Green function for a Dirac particle moving in a non-Abelian field and having a particular form is exactly determined by the path integral approach. The wave functions were deduced from the residues of Green’s function. It is shown that the classical paths contributed mainly to the determination of the Green function.

1. Introduction

In the present paper, we suggest to determine the Green function for Dirac particle of mass in a non-Abelian field by the path integral approach. The non-Abelian field in question has the form [1, 2]

where

 (i),  , and are, respectively, the polarizations of color wave and the isotropic wave such as

 (ii),   are functions which depend only of , (iii) are generators of group which satisfy the condition

 (iv)  is a gauge function [3] such as

 (v) and is a color index.

With this form of field, the Lorentz condition is satisfied:

From the point of view of the literature, we can notice that there exists only one limiting number of cases of interactions where the equation of Dirac has analytical and exact solution. We can cite, for example, the simple cases such as the constant field [4], the plane wave [5, 6], the crossed electric and magnetic fields [7, 8], and the combination of constant homogeneous electromagnetic field with a plane wave field [9, 10]. With the path integral formalism, we can cite [11–14], where the interactions have classical form.

For the non-Abelian case, at our knowledge, there exist no explicit calculations of propagators by the path integral formalism, except the formalism given by Fradkin et al. [15] and which we use in the present paper.

Note that for this problem treated in [1] the wave functions for a particle of spin 1/2 are determined by solving the Yang-Mills equation.

Also, in this paper we propose to consider the non-Abelian field described previously in order to determine analytically Green’s function by using the approach of path integral, to extract then the wave functions.

2. Green’s Function for Dirac Particle: Local Approach

First, let us proceed as indicated in the book [15] by replacing the generators of by a bilinear expression

where are traceless Hermitian matrices and are the operators which satisfy the following anticommutation relations:

First, let us write the equation satisfied by Green’s function related to the Dirac particle submitted to the action of non-Abelian field [1]

where are usual Dirac matrices obeying to anticommutation relations

with is the metric tensor of Minkowski space .

In order to determine the Green function, we follow the usual construction method [11] of path integral form.

First, we know that the kernel

is a matrix element of operator , the solution of the following symbolic equation:

or

where is introduced in order to homogenize the variables

So,   can be written in the exponential form

where and are, respectively, bosonic and fermionic variables [16, 17]. The Grassmann variables have the following properties:

and the Hamiltonian system which governs the movement of the Dirac particle in the non-Abelian field is given by

In the configuration space, the Green function is [4]

and the path integral formulation consists of

 (i) subdividing the interval into equals intervals of length , (ii) inserting the projectors

by using the actions on kets

in order to eliminate the operators , .

So the Green function can be written (with the midpoint prescription) as following

where the -products are introduced because of the presence of matrices and operators which do not commutate. To eliminate the -products, we used the following form path integral:

where ,   where which are the domains of integration. ,    are coupled currents to matrices and operators ,  and , are Grassmann variables.

So, we obtain

which is the Green function to calculate in the local approach of path integral.

Having obtained the path integral form for the Green function , let us perform the integrations.

We notice that the field depends on products and , with . First, we put and , and then the variables and are rendered, respectively, independent of and of by introducing the following three identities [11, 12]:

Then, is rewritten as

Using a shift on defined by

we obtain

where ,  .

If we integrate by part the first term in the action and then integrate the variables , a Dirac function appeared, and the integrations on all the reduce to a simple integral ; that is, the momentum of the particle is conserved (constant) during the movement.

Thus, we have

We notice also, that there is dependence of product of in the action. So if we put  , these new variables are rendered independent [11, 12] by the introduction of the following relation:

with and being Grassmann variables.

Then , with these new variables, is rewritten as

Let us remark that the integrations over the Grassmann variables are subjected to the boundary conditions. We can render free these integrations, if we use new variables (velocities) defined by [15, 18]

or by integration we have the relation

between and .

Since

where the following notation is used:

we obtain for

By using a shift for

the Green function can be simplified and take the following form:

At this level, to integrate velocities , first, we insert the integral representation of the Dirac function

with   being Grassmann variable; then we have

We notice that the integration over can be eliminated thanks to the following result:

and for other variables , the integrations have the following forms:

where

After we have integrated over the ,   is reduced to

and the integrations on variables   give

Thus, becomes

The integrations over   give the following result:

and the Green function is reduced to

By integrations on ,  , and again, three Dirac functions appear also

that is, the equations

for solutions

contribute to the determination of the Green function.

Now, we introduce the integral representations of three :

After having changed into , the relations (22) are reused to reintroduce the generators of   group since there is no order problem; the -product is omitted, then the Green function symmetrized with respect to positions and takes the following form:

The variable [16, 19] can be eliminated by integration, and we obtain

where the calculations have been simplified thanks to proprieties of field.

The matrices are reintroduced by using the following relations:

where

Then

and after having eliminated by integration, we have

and an integration by part of the exponent of exponential has the following form:

and after some manipulations, the Green function for the Dirac particles moving in the non-Abelian field in the local approach is finally

which is obviously symmetrical with respect to and .

From the residues related to two poles

and after having introduced the projectors on states of positive and negative energy [20],

we obtain the following spectral decomposition:

where finally the wave functions related to particle and to antiparticle are, respectively,

and   are the spinors solutions of the free Dirac equation and are

3. Conclusion

In the present paper, the Green function related to Dirac particle moving in a non-abelian field has been determined in the local approach of path integral. The obtained wave functions from the spectral decomposition are in accordance with those obtained by direct resolution of Dirac equation [1].

Using only shifts related to properties of field, the integrations have been easily performed, and it is remarkable to notice that the selected equations by arguments of   functions which are appeared during calculation can be also obtained from the classical mechanics (cf. Appendix).

Appendix

In local approach, the action is

and the equations of movement are, respectively,

After multiplication by and , we obtain the following relations:

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