Plane Wave

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Plane Wave

Sana Zabat and Lyazid Chetouani

Département de Physique, Faculté des Sciences Exactes, Université Mentouri, 25000 Constantine, Algeria

Reprint requests to L. C; E-mail: lyazidchetouani@gmail.com

Z. Naturforsch.65a,431 – 444 (2010); received March 25, 2009 / revised August 20, 2009

The Green functions for Klein-Gordon and Dirac particles in a weak gravitational field are deter- mined exactly by the path integral formalism. By using simple changes, it is shown that the classical trajectories play an important role in determining these Green functions.

Key words:Path Integral; Dirac Equation; Exact Solution.

PACS numbers:04.30.-w, 03.65.Ca, 03.65.Db, 03.65.Pm

1. Introduction

We know that the Dirac equation is one of the funda- mental equations of physics which allows not only to determine the value¹₂ of the electron spin, but predict the existence of an antiparticle. However, this equation is not free of difficulties related to states of negative energy, and in order to explain certain phenomena, the use of field theory becomes more than necessary. A good approximation for the treatment of certain problems, when the interaction is not strong enough, is the direct use of the Dirac equation, which is sufficient in this case. For example, for a particle of spin¹₂, the cor- rection due to the polarization of the vacuum can be obtained, following Pauli’s results, in a simple modification of the Dirac equation, where the movement of the particle is corrected by an additional term describing the anomaly of the magnetic moment of the particle.

In addition, we know that the operators of quantum mechanics play a central role in the quantization; later, another means of quantization has been proposed using the path integral. If, in non-relativistic case, the deep link with classical mechanics has been established with the path integral formulation, in the relativistic case there are various formulations. For the Dirac equation, for example, which is a matrix equation in which the spin is described by the matrices γ^µ (which do not commutate), the formalism of path integrals according to Fradkin and Gitman [1, 2] – object of this paper – uses two types of variables: bosonic for the external motion and fermionic (Grassmann) for the in-

0932–0784 / 10 / 0500–0431 $ 06.00 c2010 Verlag der Zeitschrift f¨ur Naturforschung, T ¨ubingen·http://znaturforsch.com

terior motion of the particle. Two types of representations of path integrals relative to the equation of Dirac exist at present: one, the so-called global one, is based on the “square” of the Dirac equation, i. e. the Klein- Gordon equation with an additional spin-field term; the other one, the so-called local one, which uses a matrix γ⁵[3] for reasons of homogenization, gives the physi- cal states directly and not artificially, in contrary to the global case.

The purpose of this paper is to examine the possi- bilities of solving simple problems of particles inter- acting with an external classical field, using these two representations, equivalent by construction to the Dirac equation. We consider in this paper the Klein-Gordon and the Dirac particles moving in a gravitational field which is relatively weak. Our object is to show by a simple modification, that we can easily obtain from the classical trajectories the Green functions solution of the Klein-Gordon and the Dirac equations.

In the absence of gravitation we have an Euclidian metricηµν. When the gravitation is weak, the metric is not flat but can be chosen such that the Green functions have analytical expressions. In first approximation the metric is [4]

g_µν(x) =ηµν+h_µν(kx) or

g^µν(x) =η^µν−h^µν(kx), (1) where the matrixhis considered as a perturbation and is chosen in a simple form. It only depends on the prod- uctkx, wherekandxare the 4-vectors related to the wave number and the position of the particle.

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It is further supposed that the form ofh[4] is h^µν(kx) =a^µνF(kx), (2) where the wave number 4-vectorskand the elements of the(4×4)matrixastatisfy the condition

k²=0, k^µa_µν=a_µνk^ν=0, Tra=0. (3) We begin with the simple case of the Klein-Gordon particles (without spin) to determine the Green function, and then we pass to the Green function for the Dirac equation following successively the two representations, without using theγ⁵matrix for the global approach and the local approach, where the matrix γ⁵[1] is necessary. Let us note that this problem has been the object of some other works [4, 5], calculating the Green functions for particles of spin 0 and spin ¹₂ by solving the Klein-Gordon and Dirac equations via the solutions of the respective Heisenberg equations.

The same problem has recently been considered with a stochastic approach [6] and with the formalism of path integrals [7], using only the global representation.

Let us begin with the simple case of the Klein- Gordon particles.

2. Green Function for the Klein-Gordon Particle In this section, we propose to determine the Green function for a particle in a weak gravitational field, which solves the Klein-Gordon equation

pˆ²_b−pˆ_bh_bpˆ_b−m²

G^c(x_b,x_a) =δ⁴(x_b−x_a). (4) Symbolically we have

Gˆ^c= I ˆ

p²−phˆ pˆ−m²,

and with the help of a parameterλ, we obtain for ˆG^c the exponential form

G^c(x_b,x_a) =i

_∞

0

dλx_b|exp

−iλ^H^ˆ(pˆ,xˆ)

|x_a, where

Hˆ(pˆ,xˆ) =m²−pˆ²+paˆ pFˆ (kx), following [4].

Let us pass now to the path integral formulation. To construct the Green function, first let us eliminate the operators. We use the usual procedure: subdivide the

interval[x_a,x_b]intoNequal intervals of lengthτ=

λ

N, then insert the closed relations

d⁴x|xx|=1, d⁴p|pp|=1, with the scalar product

x|p= 1 (2π)⁴e^ipx,

which allows to pass from one base to another one.

Next we eliminate the operators by using ˆ

x^µ|x=x^µ|x, pˆ^µ|p=p^µ|p.

Now we choose the Weyl order or the mid-point pre- scription, and the expression of the Green function then is

G^c(x_b,x_a) =i

_∞

0

dλ ^Dx ^Dp

·exp

i _λ

0

px˙+p²−m²−papF(kx) dτ

.

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Let us now proceed to its evaluation: As the action depends onkx, let us defineϕ=kxas a new variable. We introduce the identity [8 – 10]

dϕb

dϕaδ(ϕa−kx_a) Dϕ ^Dpϕ

·exp

i _λ

0

p_ϕ(ϕ˙−kx)dτ

=1.

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Then (5) becomes G^c(x_b,x_a) =i

_∞

0

dλ

Dx

Dp

dϕb

· dϕaδ(ϕa−kx_a) Dϕ ^Dpϕ

·exp

i

_λ

0

(p−p_ϕk)x˙+p_ϕϕ˙+p²

−paF(ϕ)p−m² dτ

.

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Introducing p=P+p_ϕk, the measure remains un- changed. We then have

G^c(x_b,xa) =i

_∞

0

dλ ^Dx ^Dp ^dϕb

· dϕaδ(ϕa−kx_a) Dϕ ^Dpϕ

·exp

i

_λ

0

px˙+p²+ (ϕ˙+2pk)p_ϕ

−F(ϕ)pap−m² dτ

.

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Let us integrate the first term of the action by part, _λ

0

pxd˙ τ= (p_bx_b−p_ax_a)− ^λ

0

˙ pxdτ,

and then integrate over the pathsx(τ), G^c(x_b,x_a) =i

_∞

0

dλ ^Dpδ(p˙) dϕb

·

dϕaδ(ϕa−kxa)

Dϕ ^Dpϕ

·exp

i

(x_bp_b−x_ap_a) + ^λ

0

p²+ (2pk+ϕ˙)p_ϕ

−F(ϕ)pap−m² dτ. The Dirac functionδ(p˙)expresses that the momentum of the particle remains constant during the motion,

p₁=p₂=...=p_n=p=const. (9) Integrating successively over all the p_n, the integral Dpis finally reduced to a simple integral ^dp₂_π.

G^c(x_b,xa) =i

_∞

0

dλ ^dp 2π

dϕb

dϕaδ(ϕa−kxa)

· Dϕ ^Dpϕexp{i(x_bp_b−x_ap_a)}δ(p˙)

·exp

i

_λ

0 (p²+ (2pk+ϕ˙)p_ϕ−F(ϕ)pap−m²)dτ

. (10)

Integrating over p_ϕ also leads to a Dirac function δ(2pk+ϕ˙). The integration over the variables ϕ shows that the essential contribution to the Green function comes from the trajectory

dϕ

dτ ⁼⁻²^pk^, ⁽¹¹⁾

i. e. from the path described by

ϕ(τ) =−2pkτ+c^te, (12) which is obviously that of a line.

ThusG^cbecomes G^c(x_b,xa) =i

_∞

0

dλ ^d^p 2π

dϕb

dϕaδ(ϕa−kxa)

·δ(ϕb−kx_a−(2pk)λ)exp

i

p(x_b−x_a) + ^λ

0 (p²−m²)dτ ^Dϕ^exp

ipap 2pk

_ϕ_b kxa

F(ϕ)dϕ

. (13)

This Green functionG^c(x_b,x_a)has a form which is not symmetrical with respect to the positionsxa and x_b. In order to symmetrize this expression, let us introduce

the integral representation of theδ ^function δ(ϕb−kx_a−(2pk)λ)

= 1 2π

dp_ϕ_bexp

ip_ϕ_b[ϕb−kx_a−(2pk)λ] .

Let us change thenpintop−kp_ϕ_b, so we have G^c(x_b,x_a) =i

_∞

0

dλ ^d^p 2π^exp

i

p(x_b−x_a) + ^λ

0 (p²−m²)dτ ^exp ipap

2pk _kx_b

kxa

F(ϕ)dϕ

. (14)

After transformingpinto−p, we integrate overλ^{. The} result is

G^c(x_b,x_a) = 1 (2π)⁴

d⁴p

p²−m²

·exp

i

p(x_b−xa) +(pap) 2pk

_kx

b

kxa

F(ϕ)dϕ . (15)

A simple integration over p⁰ leads to the following form:

G^c(x_b,x_a) =−i

2 _ε=±

∑

^θ^[^ε⁽^t^b⁻^t^a^)] ₍^d₂³_π^p₎³

·exp

−iεω(t_b−t_a) +i^→p(x^→_b−x^→_a)

·exp

i

εω^k⁰−2pk(pap) ^kx^b

kxa

F(ϕ)dϕ

, (16)

which allows us to extract the wave functions related to the Klein-Gordon particle in a weak gravitational field:

Ψp^±(x) = 1

(2π)²³ 1 (2p⁰)¹² exp

∓i

px

+(pap) 2pk

_kx

F(ϕ)dϕ

p⁰=_→

p²+m²1/2. (17)

3. Green Function for a Dirac Particle:

Global and Local Approaches

Let us pass now to the Dirac particle: We consider the motion of a Dirac particle in a weak gravitational field using the path integral approach. The Green func- tionSwhich we propose is the solution of the following equation:

−γ^p^ˆb+1

2F(kx)γ^a^p^ˆb−m S(x_b,x_a)=δ⁴(x_b,x_a), (18)

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whereγ^µ are the usual Dirac matrices obeying the relations

γ^µγ^ν+γ^νγ^µ=2g^µν, µ,ν=0,1,2,3. We know that in the global approach of the path integral formulation of the Green function it is not necessary to introduce the matrixγ⁵[3] to the global approach contrary to the local approach.

Thus, in the global approach we can obtain the Green function according to [2, 3] by

S(x_b,x_a) =

−γ^p^ˆb+1

2F(kx_b)γ^a^p^ˆb+m G(x_b,x_a), (19) where

G(x_b,x_a) =i exp

iγ^µ δL

δθ^µ _∞

0

dλ ^Dx ^Dp

E

DΨ^exp

i

_λ

0

dτ

px˙−iΨµΨ˙^µ+p²−m²

−papF−1

2(paΨ)ξ^F +Ψµ(λ)Ψ^µ(0)

|_θ₌0

. (20)

In the local approach we find S˜(x_b,x_a) =exp

iγⁿ δL

δθⁿ _∞

0

dλ ^dχ ^Dx

·

Dp

E

DΨ^exp

i

_λ

0

dτ^p^x^˙−iΨnΨ˙ⁿ+p²

−m²−papF−1

2(paΨ)(kΨ)F+

−(pΨ) +1

2F(apΨ)−mΨ⁵χ+Ψn(λ)Ψⁿ(0) θ=0.

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We propose to evaluate the Green function with the method already used in [9] which mainly consists of introducing two identities: The first one depends on the variable which characterizes the gravitation, the second one describing the spin in a form similar to the first. In order to obtain the expressions for the Green functions let us return to the definition of S(x_b,x_a) which is the matrix element

S(x_b,x_a) =x_b|Sˆ|x_a

of an operatorS. Symbolically, we have

−γ^p^ˆ+1

2F(kx)γ^a^p^ˆ−m

Sˆ=I, (22) or

Sˆ= I

−γ^p^ˆ+¹₂F(kx)γ^a^p^ˆ−m

= I

−γ˜^p^ˆ+¹₂Fγ˜^a^p^ˆ−mγ⁵γ⁵=S˜γ⁵. (23) Let us remind that the matrixγ⁵is defined by

γ⁵=γ⁰γ¹γ²γ³,

γ⁵²=−1.

Then, we have for the global case the following equal- ities:

I

−γ^p^ˆ+¹₂F(kx)γ^a^p^ˆ−m

=

−γ^p^ˆ+1

2F(kx)(γ^a^p^ˆ] +m

G,

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with the relation between ˆSand ˆGas S(x_b,xa) =

−γ^p^ˆb+1

2F(kx_b)γ^a^p^ˆb+m G(x_b,xa), (25) where

G=i _∞

0

dλ^exp

iλ

−γ^p^ˆ +1

2F(kx)γ^a^p^ˆ−m

−γ^p^ˆ+1

2F(kx)γ^a^p^ˆ+m . (26)

For the local case the Green function to be evaluated is the following:

S˜= ^∞

0

dλ ^dχ^exp

i

ˆ

p²−m²−paˆ pFˆ +i

2(paˆ σ^k)F+χ

−(γ˜^p^ˆ)+1

2F(γ˜^a^p^ˆ)−mγ⁵ , (27)

where the parameters λ ^and χ are the bosonic and fermionic variables, respectively.

The kernels are in the global cases G=i

_∞

0

dλ^exp−iλ^H^ˆg(pˆ,xˆ)

, (28)

with

H_g=p²−m²−paˆ pFˆ +i

2paˆ σ^kF, (29) and in the local case

S˜= ^∞

0

dλ ^exp−iλ^H^ˆl(χ,pˆ,xˆ)

dχ, (30)

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with

Hˆ_l=p²−m²−paˆ pFˆ +i 2paˆ σ^kF +χ

−(γ^p^ˆ) +1

2F(γ^a^p^ˆ)−mγ⁵

. (31)

Let us then pass to the path integral formulation. We first have, for the global case,

G(x_b,x_a) =i

_∞

0

dλx_b|exp

−iλ^Hg(pˆ,xˆ)

|x_a, (32) and for the local case

S˜(x_b,x_a) = ^∞

0

dλx_b|exp

−iλ^H^ˆl(χ,pˆ,xˆ)

|x_adχ. (33) Following the usual procedure (Trotter formula, inser- tion of projectors, etc.. . . ) we obtain the Green functions in the two cases,

i) global G(x_b,xa) =iT

_∞

0

dλ ^Dx ^Dp^exp

i _λ

0

dτ

·

px˙+p²−m²−papF+i

2paσ^kF , (34)

ii) local S˜(x_b,x_a) =T

_∞

0

dλ ^dχ ^Dx ^Dp

·exp

i _λ

0

dτ

px˙+p²−m²−papF+i

2(paσ^k)F +

−(γ^p) +1

2F(γ^ap)−mγ⁵

χ . (35)

Here the time-ordered productT is introduced because of the presence of the matricesγ^.

In order to eliminate theT-product, in a first step we shift theT-symbol by using the following identity:

Texp{F(γⁿ(τ))}= exp

F

δL

δρn

Texp

_λ

0 ρnγⁿ^dτ

ρ=0. Then, in a second step, theT-product is replaced by a path integral

Texp _λ

0 ρnγⁿ^dτ

=exp

iγⁿ δL

δθⁿ

DΨ|_Ψ

0+Ψ₁=θ

·exp _λ

0 (ΨnΨ˙ⁿ−2iρnΨⁿ)dτ+Ψn(λ)Ψⁿ(0)

. (36)

Let us note in passing that the termsσ^µν will be replaced by

σ^µν→iΨ^µΨ^ν, (37) and that the term

i

2(paσ^k)F→ −1

2(paΨ)(kΨ)F

is a product of factors. Thus, we have in the global case G(x_b,x_a) =i exp

iγⁿ δL

δθⁿ _∞

0

dλ ^Dx ^Dp

·

E

DΨ^exp

i _λ

0

dτ

px˙+p²−m²−(pap)F

−1

2(paΨ)(kΨ)F−iΨnΨ˙ⁿ +Ψn(λ)Ψⁿ(0)|_θ=0

, (38)

and in the local case S˜(x_b,x_a) =exp

iγⁿ δL

δθⁿ _∞

0

dλ ^dχ ^Dx ^Dp

·

E

DΨ^exp

i

_λ

0

dτ

px˙+p²−m²−(pap)F

−1

2(paΨ)(kΨ)F+

−(pΨ) +1

2(Ψ^ap)F−mΨ⁵

χ

−iΨnΨ˙ⁿ +Ψn(λ)Ψⁿ(0)|_θ=0

. (39)

Having formulated the Green function in the path integral approach, let us proceed now to their calculation.

For that, let us use the properties of their dependence on the gravitation [8]. Therefore, we insert the identity

dϕa

dϕbδ(ϕa−kx_a) Dp_ϕ

Dϕ

·exp

i _λ

0

p_ϕ(ϕ˙−kx˙)dτ

=1.

(40)

This leads for the global case to G(x_b,x_a) =i exp

iγⁿ δL

δθⁿ _∞

0

dλ ^Dx ^Dp

· dϕa

Dϕ

E

DΨ

·exp

i

_λ

0

dτ

p²−m²−papF−1

2(paΨ)ξ^F + (p−kp_ϕ)x˙+p_ϕϕ˙−iΨnΨ˙ⁿ +Ψn(λ)Ψⁿ(0)|_θ=0

, (41)

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and for the local case to S˜(x_b,x_a) =exp

iγⁿ δL

δθⁿ _∞

0

dλ ^dχ ^Dx

· Dp

dϕa

Dϕ

·

E

DΨ^exp

i _λ

0

dτ

2(paΨ)ξ^F +

−(pΨ) +1

2F(apΨ)−mΨ⁵

χ+ (p−kp_ϕ)x˙ +p_ϕϕ˙−iΨnΨ˙ⁿ +Ψn(λ)Ψⁿ(0)|_θ=0

. (42)

Let us make the following transformation:

p=P+kp_ϕ. (43)

Then, we obtain G(x_b,x_a) =i exp

iγⁿ δL

δθⁿ _∞

0

dλ ^Dx ^Dp

· dϕa

Dϕ

E

DΨ

·exp

i _λ

0

dτ

px˙+

p²−m²−papF

−1

2(paΨ)(kΨ)F

+p_ϕ(ϕ˙+2pk)−iΨnΨ˙ⁿ +Ψn(λ)Ψⁿ(0)|_θ=0

(44)

and

S˜(x_b,x_a) =exp

iγⁿ δL

δθⁿ _∞

0

dλ ^dχ ^Dx ^Dp

· dϕa

dϕbδ(ϕa−kxa) Dp_ϕ

Dϕ

E

DΨ

·exp

i _λ

0

dτ

px˙+

2(paΨ)ξ^F +

−pΨ−kp_ϕΨ+1

2F(Ψ^ap)−mΨ⁵

χ +p_ϕ(ϕ˙+2kp)−iΨnΨ˙ⁿ +Ψn(λ)Ψⁿ(0)|_θ=0

. (45) Let us integrate the first term of the action by parts then integrate over the paths x(τ). This leads in the two cases to

i) global G(x_b,x_a) =i exp

iγⁿ δL

δθⁿ _∞

0

dλ ^Dp ^dϕa

dϕb

·δ(ϕa−kx_a) Dp_ϕ

Dϕ

E

DΨ^exp{i(x_bp_b−x_ap_a)}

·δ(p˙)exp

i _λ

0

dτ

p²−m²−(pap)F

−1

2(paΨ)(kΨ)F

+p_ϕ(ϕ˙+2pk)−iΨnΨ˙ⁿ +Ψn(λ)Ψⁿ(0)|_θ=0

, (46)

ii) local S˜(x_b,x_a) =exp

iγⁿ δL

δθⁿ _∞

0

dλ ^dχ ^Dp

·

dϕa

dϕbδ(ϕa−kx_a)

Dp_ϕ

Dϕ

E

DΨ

·exp{i(x_bp_b−xapa)}δ(p˙)exp

i _λ

0

dτ

p²−m²

−papF−1

2(paΨ)(kΨ)F

−

pΨ−1

2F(Ψ^ap) +ξ^pϕ+mΨ⁵

χ+p_ϕ(ϕ˙+2kp)−iΨnΨ˙ⁿ +Ψn(λ)Ψⁿ(0)|_θ=0

. (47)

The Dirac functionδ(p˙)expresses that the momentum is constant during the motion, i. e.,

p₁=p₂=...=p_n=p.

Let us now integrate onp, implying in the i) global case

G(x_b,x_a) =i exp

iγⁿ δL

δθⁿ _∞

0

dλ d⁴p

(2π)⁴

·exp

ip(x_b−xa) +iλ(p²−m²) dϕa

· dϕbδ(ϕa−kx_a) Dp_ϕ

Dϕ

E

DΨ

·exp

i

_λ

0

dτ

−papF−1

2(paΨ)(kΨ)F +p_ϕ(ϕ˙+2pk)−iΨnΨ˙ⁿ +Ψn(λ)Ψⁿ(0)|_θ=0

,

(48)

ii) local case S˜(x_b,x_a) =exp

iγⁿ δL

δθⁿ _∞

0

dλ ^dχ ₍^d⁴^p 2π)⁴

·exp

ip(x_b−x_a) +iλ(p²−m²) dϕa

· dϕbδ(ϕa−kx_a) Dp_ϕ

Dϕ

E

DΨ

·exp

i

_λ

0

dτ

−papF−1

2(paΨ)(kΨ)F

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−

pΨ−1

χ +p_ϕ(ϕ˙+2kp)−iΨnΨ˙ⁿ +Ψn(λ)Ψⁿ(0)|_θ=0

. (49) Now we use the factorization property of the spin-field interaction [8] by introducing the identity

dξadξbδ(ξa−kΨa) Dp_ξ

Dξ

·exp

i

_λ

0

p_ξ(ξ˙−kΨ˙)dτ

=1,

(50)

whereξ ^and^p_ξ are odd Grassmann variables.

ThenGandSbecome in the i) global case

G(x_b,x_a) =i exp

iγⁿ δL

δθⁿ _∞

0

dλ ₍^d₂⁴π^p)⁴

dϕa

· dϕbδ(ϕa−kx_a) dξadξbδ(ξa−kΨa) Dp_ϕ

·

Dϕ ^Dp_ξ ^Dξ

E

DΨ^exp^ip(x_b−xa) +iλ^p²−m²

) exp

i

_λ

0

dτ

−(pap)F(ϕ)

−1

2(paΨ)ξ^F+p_ϕ(ϕ˙+2kp)

+p_ξ(ξ˙−kΨ˙)−iΨµΨ˙^µ +Ψµ(λ)Ψ^µ(0)

,

(51)

ii) local case S˜(x_b,xa) =exp

iγⁿ δL

δθⁿ _∞

0

dλ ^dχ ₍^d⁴^p 2π)⁴

·exp

ip(x_b−x_a) +iλ^p²−m² dϕa

· dϕbδ(ϕa−kx_a) dξadξbδ(ξa−kΨa) Dp_ϕ

· Dϕ ^Dp_ξ ^Dξ

E

DΨ^exp

i

_λ

0

dτ

−papF

−1

2(paΨ)ξ^F

−

pΨ−1

χ

+p_ξ(ξ˙−kΨ˙) +p_ϕ(ϕ˙+2kp)−iΨnΨ˙ⁿ +Ψn(λ)Ψⁿ(0)|_θ=0

. (52)

At this state, it is preferable to use the velocitiesω by making change of variablesΨⁿ → ωⁿ^{, defined}

by

Ψⁿ(τ) =1 2

_λ

0 ε(τ−τ)ωⁿ(τ)dτ+θⁿ 2 . (53) We also adopt the condensed notation of [2] and obtain G(x_b,x_a) =i exp

iγⁿ δL

δθⁿ _∞

0

dλ ₍₂^d⁴π^p)⁴

·exp

·

dϕbδ(ϕa−kxa)

dξa

dξb

Dϕ ^Dpϕ

· Dξ ^Dp_ξ

E

Dωδ(ξa+k

2(ω−θ)

·exp

i _λ

0

dτ

λ

−(pap)F(ϕ) +i

4(paθ)ξ^F +i

4pa(ε∗ω)ξ^F

+p_ξ(ξ˙−kω) +p_ϕ(ϕ˙+2kp) −¹₂ω∗ε∗ω

θ=0

(54)

and

S˜(x_b,x_a) =exp

iγⁿ δL

δθⁿ _∞

0

dλ ^dχ ₍₂^d⁴π^p)⁴

·exp

·

dϕbδ(ϕa−kxa)

dξadξb

Dp_ϕ

Dϕ ^Dp_ξ

· Dξ

E

Dωδ(ξa+k

2(ω−θ)

·exp

i _λ

0

dτ

λ

−papF(ϕ) +i 4(pa

ε∗ω +θ)ξ^F

−1 2

(ε∗ω+θ)p−1

2F(ε∗ω+θ)ap +2ξ^pϕ+m(ε∗ω⁵+θ⁵)

χ+p_ξ(ξ˙−kω) +p_ϕ(ϕ˙+2kp) −1

2ω∗ε∗ω

.

(55)

Let us now introduce the transformation ω¯^µ(τ)→ω^µ(τ) +ik^µ(ε⁻¹∗p_ξ), leading to

kω¯(τ) =kω(τ).

(8)

Since

ω¯∗ε∗ω¯ =ω∗ε∗ω+2ikω^p_ξ, ε∗ω¯ =ε∗ω+ikp_ξ,

we obtain for the two cases i) global

G(x_b,x_a) =i exp

iγⁿ δL

δθⁿ

d⁴p (2π)⁴

_∞

0

dλ ^dϕa

· dϕbδ(ϕa−kx_a) Dϕ ^Dpϕ

dξa

dξb

Dξ

·

Dp_ξ

E

Dωδ

ξa+k

2(ω−θ)

exp

ip(x_b−x_a) +iλ(p²−m²)

exp

i

_λ

0

dτ

−(pap)F(ϕ) +i

4(pa(ε∗ω+θ))ξ^F

+p_ξ(ξ˙−kω) +p_ϕ(ϕ˙+2kp) −1

2(ω∗ε∗ω−2ikω^p_ξ)

, (56)

iγⁿ δL

δθⁿ

d⁴p

(2π)⁴exp{ip(x_b−x_a) +iλ(p²−m²)} ^∞

0

dλ ^dχ ^dϕa

dϕbδ(ϕa−kx_a)

· Dϕ ^Dpϕ

dξa

dξb

Dξ ^Dp_ξ

E

Dωδ(ξa

+k

2(ω−θ))exp

i

_λ

0

dτ(p_ξ(ξ˙−kω) +p_ϕ(ϕ˙+2kp))

exp

i

_λ 0

dτ

−papF(ϕ) +i

4(pa(ε∗ω−ikp_ξ+θ))ξ^F

−1 2

(ε∗ω

−ikp_ξ+θ)p_µ−1

2F(ε∗ω−ikp_ξ+θ)ap+2ξ^pϕ

+m(ε∗ω⁵+θ⁵)

χ −1

2(ω∗ε∗ω−2ikω^p_ξ)

. (57)

One can replace the Dirac function by its integral representation

δ

ξa+k 2(ω−θ)

=

dp_ζ_aexp

ip_ζ_a

ξa+k

2(ω−θ)

,

(58)

wherep_ζ_ais a Grassmann variable.

We have for the two cases i) global

G(x_b,x_a) =i exp

iγⁿ δL

δθⁿ

d⁴p (2π)⁴

_∞

0

dλ ^dϕa

dξa

dξb

Dξ

· Dp_ξ

E

Dω^expⁱ^p(x_b−x_a) +iλ^p²−m²

·exp

i _λ

0

dτ

−papF(ϕ) +i 4

pa(ε∗ω

−ikp_ξ+θ) ξ^F

+p_ϕ(ϕ˙+2kp) +p_ξ(ξ˙−kω)

−1

2(ω∗ε∗ω−2ikω^p_ξ) +p_ζ_a

ξa+k

2(ω−θ) θ=0,

(59)

iγⁿ δL

δθⁿ

d⁴p

(2π)⁴exp{ip(x_b−x_a) +iλ(p²−m²)} ^∞

0

dλ ^dχ ^dϕa

dϕbδ(ϕa−kx_a)

· Dϕ ^Dpϕ

dξa

dξb

Dξ ^Dp_ξ

E

Dω

·exp

i _λ

0

dτ^p_ξ(ξ˙−kω) +p_ϕ(ϕ˙+2kp)

·exp

i _λ

0

dτ

−papF(ϕ) +i

4(pa(ε∗ω

−ikp_ξ+θ))ξ^F

−1 2

(ε∗ω−ikp_ξ+θ)p

−1

2F(ε∗ω−ikp_ξ+θ)ap+2ξ^pϕ

+m(ε∗ω⁵+θ⁵)

χ −1

2(ω∗ε∗ω−2ikω^p_ξ) +p_ζ_a

ξa+k

2(ω−θ) θ=0.

(60)

The integration on the velocityω⁵^{is simple:}

Dω⁵^exp

−1 2

ω5∗ε∗ω⁵+imε∗ω⁵χ=1.(61)

The integrals overω^µ(µ=0,1,2,3)are

Dω^µ^exp _λ

0

−1

2ωµεω^µ+J_µω^µ

dτ

, where the sourcesJ_µ(τ)are in the cases

(9)

i) global

J_µ(τ) =−1 4

_λ

0

ds(pa)_µξ(s)F(s)ε(τ−s)+i 2k_µp_ξ_a.

ii) local J_µ(τ) =−1

4 _λ

0

ds(pa)_µξ(s)F(s)ε(τ−s)

−1 2χ ^λ

0 (p+ (ap))_µF(s)ε(τ−s)ds+i 2p_ξ_ak_µ, By using the propertieska=0 andk²=0, which allow to simplify the Green functions,G and ˜S have the following forms:

i) global case G(x_b,x_a) =i exp

iγⁿ δL

δθⁿ

d⁴p (2π)⁴

_∞

0

dλ ^dϕa

dϕbδ(ϕa−kx_a)

· Dϕ ^Dpϕ

dξa

dξb

Dξ ^Dp_ξ^exp^ip(x_b−x_a) +iλ(p²−m²)

·exp









 i

_λ

0

dτ

−(pap)F(ϕ) +p_ϕ(ϕ˙+2kp) +p_ξξ˙+p_ζ_a

ξa−k 2θ

+i

4(paθ)ξ^F +1

32

pa²p ^λ

0 _λ

0

dsdsε(s−s)ξ(s)ξ(s)F(s)F(s)











θ=0, (62)

ii) local case S˜(x_b,xa) =i exp

iγⁿ δL

δθⁿ

d⁴p (2π)⁴

_∞

0

dλ ^dϕa

dϕbδ(ϕa−kxa)

· Dϕ ^Dpϕ

dξa

dξb

Dξ ^Dp_ξ^exp^ip(x_b−x_a) +iλ(p²−m²)

·exp









 i

_λ

0

dτ







−papF(ϕ) +p_ϕ(ϕ˙+2kp) +p_ξ

ξ˙+1 2pkχ

+p_ζ_a

ξa−k

2θ

+i

4(paθ)ξ^F−1 2

θ^p+2ξ^pϕ+mθ−1 2F(θ^ap)

χ





 +1

32(pa²p) ^λ

0

_λ

0

dsdsε(s−s)ξ(s)ξ(s)F(s)F(s) +1

8((pap) +pa²p) _λ

0 _λ

0

dsdsε(s−s) F(s)F(s)ξ(s)χ+i

4(pk)p_ξ_aχ ^λ

0

dsF(s)











θ=0. (63)

Let us now integrate on thep_ξ. The two expressions (62) and (63) are then reduced to

i) global case G(x_b,x_a) =i exp

iγⁿ δL

δθⁿ

d⁴p (2π)⁴

_∞

0

dλ ^dϕa

dξaexp{ip(x_b−x_a)

+iλ(p²−m²)}exp

i

_λ

0

dτ

−(pap)F(ϕ)

+p_ϕ(ϕ˙+2kp) +p_ζ_a

ξa−k 2θ

+i

4(paθ)ξaF

θ=0

, (64)