Cent. Eur. J. Phys. • 7(1) • 2009 • 184-192 DOI: 10.2478/s11534-008-0131-0

Central European Journal of Physics

Green’s function for an electron model with a plane wave Research Article

Hassen K. Ould Lahoucine1 , Lyazid Chetouani2∗ 1 Département de Physique, Université de Sétif, Sétif, Algeria 2 Département de Physique, Faculté des sciences exactes, Université Mentouri, Constantine, Algeria

Received 8 May 2008; accepted 19 September 2008

Abstract:

The Green function for a Dirac particle subject to a plane wave field is constructed according to the path integral approach and the Barut’s electron model. Then it is exactly determined after having fixed a matrix U chosen so that the equations of motion are those of a free particle, and by using the properties of the plane wave and also with some shifts.

PACS (2008): 03.65.Ca; 03.65.Db; 03.65.Pm Keywords:

electron model • path integral • Volkov wave • Green’s function © Versita Warsaw and Springer-Verlag Berlin Heidelberg.

1.

Introduction

The path integral formulation of quantum mechanics is known to be based on the classical action [1].Two types of formulations for spinorial particles in path integral are known, the bosonic and the fermionic classes, respectively. In the fermionic model for 1/2 spin particle proposed by Berezin and Marinov [2], the grassmannian anticommuting variables were used. Fradkin and Gitman [3] succeeded to derive the Dirac equation by the path integral method and also to calculate the Feynman propagator for a number of different interactions. The bosonic formulation proposed by Barut and Zanghi [4] uses the commuting variables and the action is formulated in a non-standard quadratic form. This model has been adopted by Barut and Duru [5] ∗

E-mail: [email protected]

184

to calculate the Dirac propagator for an electron in the free case. In the present paper, the modelization of the interaction radiation-matter via the Barut’s electron model is proposed and the case of the Dirac’s particle subject to a plane wave field is examined. The plane wave is characterized by two important properties: • the first one concerns the field Aµ (φ) which depends only on the quantity φ = kx, being the product of two variables k (with k 2 = 0) and x, the 4-vectors propagation and position, respectively. • the second property is that it satisfies also the Lorentz gauge ∂µ Aµ = 0 ⇐⇒ kA = 0.

The Green’s function expansion of QED has been derived by the use of the perturbative method [6]. However, the

Hassen K. Ould Lahoucine, Lyazid Chetouani

aim of this paper is to propose a new way to derive the exact Green function with contrast with the former which is sometimes complicated. In the present paper, we adopt a different approach which is an exact calculation to obtain the same result. It concerns the use of a FoldyWouthuysen [7] transformation type in order to obtain the propagator for the free case which is exactly calculable in path integral. In fact, the following relation µ ν

ν µ

with π/ = γ µ (pµ − eAµ ). Thus, the relation (2) can be written as: G (xb , xa ) = hxb p G p xa i = hxb p fr It is easy to show that: dτ exp (imτ) 
 hxb | fr . exp −iτfl p/ − eA/ fr .fl | xa i, 0

is still satisfied under the present new representation ∼µ

γ µ → γ = U −1 γ µ U where the quantity U can played the role of a transformation which verify the following conditions: • The transformation is uniquely a function of the ˆ which commute in order to quantities kˆ x and k p obtain a well-ordered Green function in the path integral formulation. • The transformation must reproduce the classical equations of motion of the free particle in order to use the Green function expression of the free case which is exactly solvable. The first part of this paper is devoted to the construction of the Green function for an electron subject to a plane wave field where we will choose a particular form of the transformation U which is adapted to the studied interaction. The Green function is obtained in the path integral form after an extension to the complex space C4 and the insertion of projectors. Besides the pair of conjugate variables (x, p) ∈ M4 , the internal degrees of freedom of the particle are expressed by a pair of complex variables (z, −iz) ∈ C4 which represent the conjugate moment and the internal position, respectively. The second part of the paper concerns the determination of the exact Green function by the use of the properties of the plane wave.

Construction

(1)

(4)

τ being the total time transition, fl and fr are operators of matricial type which, for the specific case of the plane wave and to avoid some complications, are chosen function of two operators (ik.∂) and (k.b x ) which commute [kµ x µ , k ν (i∂ν )] = 0

(5)

because of the property k 2 = 0 of the plane wave: (

fl = fl (ik∂, kx, γ) fr = fr (ik∂, kx, γ) ,

(6)

where γ represents Dirac’s matrices. Moreover, the propagator being exactly calculable in the free case, we must have fl = fr = I when A = 0. The relation (4) represents the Green function written in the M4 space. In order to obtain its expression in the M4 ⊗ C4 configuration space we write: Z ∞ Z (x ) (−i) (imτ) G b , xa = dτ exp dz b dza 0  +
 hzb p hxb p fr . exp iτa fl p/ − eA/ fr a .fl p xa i p za i,

(7)

where a+ and a two operators acting in the complex space C4 and generating the complex variables z and z respectively. Let us divide the time interval [0, τ] in (N + 1) subintervals of an elementary length ε and let us introduce the following projection relations:   

R dp p pihp p= 1 R dx p xihx p= 1   R i
dzdz p zihz p exp (−zz) = 1. 2π

The Green function we want to determine is solution of the equation: (π/ b − m) G (xb , xa ) = δ (xb − xa ) ,

∞

Z G (xb , xa ) = i

µν

γ γ + γ γ = 2g

2.

1 fl p xa i. fl (π/ − m) fr (3)

Notation:

(8)

or, in another way, it is formally the matrix element b solution of the G (xb , xa ) = hxb |G| xa i of the operator G equation:

  xN+1 = xb    x =x 0 a  zN+1 = zb    z =z . 0 a

(π/ − m) G = I,

Then, we obtain the Green function in a discrete form as:

(2)

185

Green’s function for an electron model with a plane wave

Z G (xN+1 , x0 ) = (−i) lim

ε→0 N→∞

∞

Z dτ

dz N+1 dz0

Z N+1 N Y d4 pj Z Y j=1

0

(2π)4

d 4 xj

j=1

   Z Y N  i 1 (z N+1 zN+1 + z 0 z0 ) dz j dzj exp (imτ) . exp 2π 2 j=1  N+1 X 
fr (kxN+1 , kpN+1 , γ) exp i −pj xj − xj−1  j=1


 +εz j fl kxj , kpj , γ p/ j − eA/ kxj fr kxj , kpj , γ zj−1 

 1  z j − z j−1 zj−1 − z j zj − zj−1 + fl (kx0 , kp1 , γ) . 2i

(9)

or in a compact form such as: ∞

 Z Z 1 (z b zb + z a za ) D 4 pD 4 x dτ exp (imτ) DzDz exp 2 0  Z τ  . fr (kxb , kpb , γ) exp i ds −px 0 
1 . . zz − z z fl (kxa , kpa , γ) . +zfl (kx, kp, γ) p/ − eA/ (kx) fr (kx, kp, γ) z + 2i Z

G (xb , xa ) = (−i)

By construction, we have obtained a modified action (compared to the Barut’s one of course) and adapted to the plane wave case

Z S =

τ


. ds −px + zfl (kx, kp, γ) p/ − eA/ fr (kx, kp, γ) z 0    . 1 . zz − z z . (11) + 2i

3. Green’s function for an electron subject to a plane wave field Now, before making integrations, let us construct the form of the functions fl and fr , taking into account that: 1. the end points a and b play the same role in the Green function expression and this is the reason justifying that (

As specified before, since fl = fr = I when A = 0, the action is reduced to the Barut’s one in the free case where the equations of motion are  .µ  p = 0 → p = const.   .µ  x = zγ µ .z .  z = −izγ µ pµ   .  z = iγ µ pµ z,

..

186

U=

X

cA γ A ,

(14)

A

h. pµ pµ i x µ (0) H + x µ (0) − 2 H cos 2mτ + sin 2mτ, 2 m m 2m (13) which exhibits the famous phenomenon of zitterbewegung. .

fl = U fr = U −1 .

2. the transformation U being a matrix it can be expanded in terms of the 16 complete 4 × 4 matrices

(12)

and by integration we obtain following O. Barut the equation of the velocity xµ =

(10)

where cA = cA (kp, kx) the 16 coefficients which are chosen such as ( =0 if γ A 6= I cA = (15) =1 if γ A = I, for Aµ = 0.

Hassen K. Ould Lahoucine, Lyazid Chetouani

3. the 16 coefficients can be reduced if we consider
that the 4-components of k µ = k 0, k 1 , k 2 , k 3 and
of Aµ = A0 , A1 , A2 , A3 should be represented in a symmetric way in the expression of U. Consequently, relatively to the properties of the plane wave field and the γ matrices, we can suppose that

Using the last equation we can write


dpρ ek ρ d 2Ap − eA2 / = z kz dτ 2pk d (kx)
ek ρ d 2Ap − eA2 , = 2pk dτ

/ + c3 k/ A/ U = I + c1 A/ + c2 G / A/ + c5 k/ A/ G / + ... +c4 G

(25)

(16) from which we can put

where G = G(kp) and A = A(kx) are functions of the quantities kp and kx. In addition, the condition lim U = I can also reduce the form of the

d dτ



pρ −

Aµ →0


ek ρ 2Ap − eA2 2pk

 = 0.

(26)

transformation to Thus, the following quantity is a constant of motion / U = I + c3 k/ A.

(17)

4. The last form is also justified by the fact that we must obtain the classical equations of the free particle. Let us suppose that all coefficients vanish, except c3 which we will determine to obtain the classical equations of motion of the free particle. Then, the field Aµ can be eliminated for c3 = −e/2pk and we get


/ p/ − eA/ (1 − c3 k/ A) / (1 + c3 k/ A)   e2 k/ epA − A2 , = p/ − kp 2

(27)

For the 4. position, since P ρ a constant of motion, we also have dx ρ dkx = zγ ρ z − dτ dτ


e ek ρ 2 (2Aρ ) − 2Ap − eA 2pk 2 (pk)2 Z kx e d dφ = zγ ρ z − 2Pk dτ  
kρ 2Aρ − 2AP − eA2 . (28) Pk 

(18)

where we used the property a/ b/ + b/ a/ = 2ab with 2ab a quantity without Dirac’s matrices. The classical equations of motion present an interesting form such as
dpρ ek ρ d 2Ap − eA2 / = z kz dτ 2pk d (kx) " e dx ρ / (2Aρ ) = zγ ρ z − z kz dτ 2pk #
ek ρ 2 − 2Ap − eA 2 (pk)2  
e dz − = −iz p/ − k/ 2Ap − eA2 dτ 2pk  
dz e = i p/ − k/ 2Ap − eA2 z. dτ 2pk


ek ρ 2Ap − eA2 = C te. 2pk

P ρ = pρ −

(19)

(20)

(21)

Then, if we put e X =x + Pk ρ

ρ

Z

kx

  kρ  e 2 ρ dφ A − AP − A , (29) Pk 2

we finally obtain by the use of the new variables P and X the same classical equations of motion for the free particle

dP ρ = 0 dτ dX ρ = zγ ρ z dτ

(30) (31)

−

Then, if we multiply by kρ , since k = 0 and kA = 0, we obtain

dz − / = −iz {P} dτ dz / z, = i {P} dτ

dpk = 0 dτ dkx / = z kz. dτ

where we used the properties pk = Pk and pA = PA and one can verify that for A = 0 the last equations remain those of a free particle.

(22)

2

(23) (24)

(32) (33)

187

Green’s function for an electron model with a plane wave


ek ρ ek ρ kµ 2 − − Aµ + 2AP − eA Pk 2 (Pk)2  
ek µ kρ ek µ 2 Aρ + 2AP − eA − Pk 2 (Pk)2 # 

We also note that the transformation (x, p) → (X , P) is canonical since the Jacobian ∂ (X , P) = 1. J = ∂ (x, p)

(34)

+ ... = 1,

In fact, it is easy to see that we have since k 2 = kA = 0, and we verify effectively that J = 1. Now, by the use of the equations

∂ (X , P) J = ∂ (x, p) ∂ (X , P, ) ∂ (x, P) = ∂ (x, P) P=const ∂ (x, p) x=cont ρ ρ ∂X ∂P = Det µ Det µ ∂x ∂p = Det (I + B) Det (I + C ) .

− pµ =

− Xµ =

(35)

The quantities ∂X ρ = Iµρ + Bµρ ∂x µ   kρ  e  e k µ Aρ − AP − A2 , (36) = δµρ + Pk Pk 2 ∂P ρ = Iµρ + Cµρ ∂pµ
ek ρ kµ ek ρ Aµ + 2AP − eA2 , (37) = δµρ − 2 Pk (Pk) 2 are the determinant elements which are calculable by the use of the well-known formula Det (I + B) = exp Tr (Ln (I + B)) .

∂F ekµ = −Pµ − ∂x µ kp

(38)

 ePA −

 e2 2 A , 2

∂F , ∂P µ

(41)

(42)

we can derive the function which generates the transformation. It can be written as

F (x, P, τ) = −Px −

e 2Pk

Z

kx


du 2AP − eA2 + g (τ) ,

(43) where g an arbitrary function of τ. After justifying our choice for U, we will proceed to the determination of the Green function. The plane wave being characterized by the quantity φ = kx, it seems complicated to perform the Dx integration. However, if we introduce the following identity [8]:

Z dφb dφa δ (φa − kxa ) δ (φb − φa − k (xb − xa )) = 1,

After an expansion we have respectively 
 expTrLn (I + B) = exp Tr B − B2 + ... "   e kρ  e  = exp kρ Aρ − AP − A2 Pk Pk 2   e 2  ρ  k e  − k µ Aρ − AP − A2 Pk Pk 2 #    µ  k e 2 µ kρ A − AP − A + ... Pk 2

(39)

(44) the variable φ can be considered independent of x and the four-dimensional motion is then reduced to the unidimensional one. On the other hand, the identity written above is a constraint which is reflected on the particle dynamics. Then, according to the adopted discretization we can write δ (φb − φa − k (xb − xa )) Z Y N N+1 Y

= dφj δ φj − φj−1 − k xj − xj−1 ,

= exp 0 = 1,

j=1

= expTrLn (I + C ) "
ek ρ kρ ek ρ = exp − Aρ + 2AP − eA2 − 2 Pk 2 (Pk) 188

j=1

Z Y N

and

(40)

Z N+1 Y dpφj 
 exp ipφj ∆φj − k∆xj . dφj 2π j=1 j=1

(45)

Hassen K. Ould Lahoucine, Lyazid Chetouani

The Green function becomes after arrangement ∞

 Z Z Z Z 1 (z b zb + z a za ) D 4 pD 4 x DφDpφ dτ exp (imτ) dφb dφa δ (φa − kxa ) DzDz exp 2 0  τ   Z h  .
.
e 1 . . Ub−1 exp i ds − p + kpφ x + pφ φ +z p/ − zz − z z Ua k/ 2Ap − eA2 z +  0 2pk 2i Z

G (xb , xa ) = (−i)

(46)

By transforming p + kpφ into p and performing the Dx integration we get  Z Z ∞ Z 1 (z b zb + z a za ) dφb dφa δ (φa − kxa ) G (xb , xa ) = (−i) dτ exp (imτ) DzDz exp 2 0  Z τ  Z Z  . 1 . .
. DφDpφ D 4 p exp [−i (pb xb − pa xa )] .δ p Ub−1 exp i ds pφ φ + zz − z z 2i 0   
e / φ− + z p/ − kp k/ 2Ap − eA2 z Ua . 2pk

(47)

The presence of the Dirac distributions implies the conservation of the 4-moment (p = cte) during the motion of the particle (see also Appendix). The integration with respect to the variable p leads to ∞

  Z Z d4 p k/ A/ (φb ) (x )] (φ ) exp [−ip − x dφ dφ δ − kx DφDp 1 + e b a b a a a φ 2kp (2π)4 0    Z τ  Z   . 1 1 . . (z b zb + z a za ) exp i DzDz exp ds pφ φ + zz − z z 2 2i 0       k/ k/ A/ (φa ) e2 / φ− epA − A2 z 1−e . (48) + z p/ − kp kp 2 2kp Z

G (xb , xa ) = (−i)

Let us transform pφ −

Z

dτ exp (imτ)

 1 kp

epA −

e2 2 A 2

Z



into pφ . It follows that

∞

Z

d4 p exp [−ip (xb − xa )] (2π)4 0  Z Z Z 1 (z b zb + z a za ) dφb dφa δ (φa − kxa ) DφDpφ DzDz exp 2      Z φb e2 2 i k/ A/ (φb ) exp − dφ epA − A 1+e 2kp kp φa 2  Z τ       . .
1 k/ A/ (φa ) . / φ z exp i ds pφ φ + zz − z z + z p/ − kp 1−e , 2i 2kp 0

G (xb , xa ) = (−i)

dτ exp (imτ)

(49)

  · 2 1 where the quantity kp epA − e2 A2 φ is a phase of which the integral depends only of the end points since the plane wave considered in the problem is the classical one. The integration now with respect to the variables φ and pφ (which is conserved) leads to

Z

 Z Dpφ Dφ exp i

τ

.



Z

dspφ φ (...) =

  Z φ Dpφ Dφ exp i pφ φ φba −

τ

.

dspφ φ

 (...)

0

0

Z

o .
φ = Dpφ δ pφ exp ipφ φ φba (...) Z dpφ = exp {ipφ (φb − φa )} (...) pφ =cst. . 2π n

(50)

189

Green’s function for an electron model with a plane wave

Thus ∞

  Z 1 d4 p ) (z (x )] exp z + z z exp [−ip − x DzDz b b a a b a 2 (2π)4 0    Z Z φb  2 i e dφb dφa δ (φa − kxa ) exp − epA − A2 dφ kp φa 2    Z τ       .
k/ A/ (φb ) k/ A/ (φa ) 1 . / φ z 1+e exp i zz − z z + z p/ − kp . ds 1−e 2kp 2i 2kp 0 Z

G (xb , xa ) = (−i)

Z

dτ exp (imτ)

(51)

By changing p − kpφ into p and integrating with respect to the variable pφ it follows that ∞

 1 (z b zb + z a za ) 2 0    Z Z φb  e2 i dφb dφa δ (φa − kxa ) δ (φb − φa − k (xb − xa )) exp − epA − A2 dφ kp φa 2   Z τ       
k/ A/ (φb ) 1 . k/ A/ (φa ) . exp i ds , 1+e zz − z z + z p/ z 1−e 2kp 2i 2kp 0 Z

G (xb , xa ) = (−i)

Z

dτ exp (imτ)

d4 p exp [−ip (xb − xa )] (2π)4



Z

DzDz exp

(52)

or ∞

    Z kxb  d4 p i e2 2 k/ A/ (kxb ) (x ) exp −ip − x − epA − A dφ 1 + e b a kp kxa 2 2kp (2π)4 0     Z τ     Z 
k/ A/ (kxa ) 1 1 . . (z b zb + z a za ) exp i . (53) DzDz exp ds zz − z z + z p/ z 1−e 2 2i 2kp 0 Z

G (xb , xa ) = (−i)

Z

dτ exp (imτ)

It remains now to perform integration over the complex variables z and z. For this purpose, let us write explicitly Z

Z DzDz(...) = lim

ε→0 N→∞

dz N+1 dz0

 Z Y N   
 
 i dz j dzj exp −iz 1 i I + iεp/ z0 − iz1 − iz 2 i I + iεp/ z1 − iz2 + ... 2π j=1 
 
 −iz N i I + iεp/ zN−1 − iz2 − iz N+1 i I + iεp/ zN

(54)

The integration on the variables z N , z N−1,..., z 1 , on the variables zN , zN−1,..., z1 and then with respect to the variables z N+1 and z0 leads to Z

Z DzDz(...) = lim

dz N+1 dz0

N→∞ ε→0

Z Y N j=1

Z = lim

ε→0 N→∞

dz N+1 dz0 exp

= lim

ε→0 N→∞

190

  

N+1 Y

Consequently



 
 dzj δ zj − I + iεp/ zj−1 exp −iz N+1 i I + iεp/ zN 

N+1 Y

−iz N+1 i

j=1


1
= exp −iτ p/ , I + iεp/

  I + iεp/ z0  


(55)

Hassen K. Ould Lahoucine, Lyazid Chetouani

   Z kxb  i e2 2 d4 p (x )] exp [−ip − x exp − A dφ epA − b a kp kxa 2 (2π)4 0     
 k/ A/ (kxb ) k/ A/ (kxa ) 1+e exp −iτ p/ − m 1−e . 2kp 2kp Z

∞

G (xb , xa ) = (−i)

Z

dτ

(56)

Finally, the integration on the total time transition leads to    Z kxb  i e2 2 d4 p (x )] exp [−ip − x exp − A dφ epA − b a kp kxa 2 (2π)4     k/ A/ (kxb ) 1 k/ A/ (kxa ) 1+e , 1−e 2kp 2kp p/ − m Z

G (xb , xa ) =

(57)

i.e. the Green function for an electron subject to a plane wave field as presented in the literature [9]. Herein, we can make an important remark. Let us introduce the time parameter τ in the free part of the propagator such as 1 = p/ − m




1 p2 − m2 Z
∞ 
 = i p/ + m dτ exp −iτ p2 − m2 p/ + m

0

Then, noting p by P, the Green function takes the form

    Z kxb  e2 2 1 d4 P 2 G (xb , xa ) = i dτ exp iτm ePA − A dφ + τP exp −i P (xb − xa ) + kP kxa 2 (2π)4 0     / / k/ A (kxb ) k/ A (kxa ) (P/ + m) 1 − e . 1+e 2kP 2kP Z

∞

2




Z

By identifying the quantity τP 2 to g (τ) in the exponential, one can find the expression of the function F (x, P, τ) F which generates the transformation (x, p) → (X , P). Following our choice g = P 2 τ, (59) we have ∂F = P2 = ∂τ

 p−


2 ek 2Ap − eA2 2pk

= (p − eA)2 ,

(60)

which is also written as 2  ∂F ∂F − − − eA + =0 ∂x ∂τ

4.

(58)

Conclusion

In this paper, we studied by the path integral approach the movement of relativistic electron of Dirac by using for the description the Lagragian of Barut. Knowing that only the Dirac propagator has been calculated exactly in this approach, we introduced and chosen an operator U so that the new Hamiltonian/Lagrangian is more suitable to calculations. Then, with a auxiliary variable φ we reduced the 4-dimensional movement to that of one dimension and by successive integrations, we have determined exactly the Greens’ function of our problem in question.

(61)

i.e. F is solution of the Hamilton-Jacobi equation relatively to the Klein-Gordon particle.

Its expression is the same that obtained by other approaches [10]. 191

Green’s function for an electron model with a plane wave

Appendix

and after multiplying equation relatively to the position kµ and since kpµH = cste,the latter equation becomes

The p = cst may still be justified as follows. At first, let us note by

  2 epA − e2 A2 dpµH µ d   =k dτ dτ kp

ˆ (τ) = eiHτ a ˆ e−iHτ aH = a ˆ

ˆ

H

ˆ is the Hamiltonian. where H Starting from the Hamiltonian

or

ˆ/ ˆ = −(π), H

(A1)

by construction and following the path integral approach corresponds the expression (with fl = fr = I ) of the action (11) and after choosing U, to the operator ˆ
+ k/ ˆ0 = − p H / ˆ kp

 ˆ− eˆ pA

 e2 ˆ 2 A , 2

(A2)

which describes the particle subjected to the 4-potential µ

k ˆ kp

 ˆ− eˆ pA

e ˆ2 A 2 2

Z

S = 0

τ

"

   k/ e2 . ds − px + z p/ − epA − A2 z kp 2 #    . 1 . zz − z z . + 2i

In the Heisenberg picture, we have with the expression of ˆ the following equations of motion for the position and H momentum operators, respectively h i dxHµ ˆ xµ = γµ , = i H, H H dτ h i dpµH ˆ pµ = ek µ = i H, H dτ

ˆ/ d(A) d (kx)

(A3)

! 6= 0,

(A4)

H

ˆ the quantity pµ is not where we can see that, by using H, H conserved. ˆ 0 , since the equaBy contrast, if we use the expression of H tion relatively to the position is unchanged , the other equation becomes h i dpµH ˆ 0 , pµ = =i H H dτ    e2 2   d epA − A 2 k/   . kµ kp H d (kx) H

192

 
d kµ pµH − 2epH AH − e2 A2H = 0. dτ 2kpH Thus, the quantity PHµ = pµH −


kµ 2epH AH − e2 A2H 2kpH

(A7)

is conserved in the Heisenberg picture.

References



it corresponds the expression of another action given by 0

(A6)

(A5)

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