# Green’s Function for Dirac Particle in a Non-AbelianSU(2) Field: A Path Integral Approach

Authors: Sami Boudieb; Lyazid Chetouani
Journal: ISRN High Energy Physics
 (2013)

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## Abstract

The Green function for a Dirac particle moving in a non-AbelianSU(2) 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.


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## Body

## 1. Introduction

In the present paper, we suggest to determine the Green function for Dirac particle of massM in a SU(2) non-Abelian field by the path integral approach. The SU(2) non-Abelian field in question has the form [1, 2]
(1)Aμa=[(k1x)bma+(k2x)cma+ha]kμ,
where(i)
k1,  k2, and k are, respectively, the polarizations of color wave and the isotropic wave such as
(2)kk1=kk2=k1k2=0,(ii)
b=b(kx),  c=c(kx) are functions which depend only of φ=kx,(iii)
ma are generators of SU(2) group which satisfy the condition
(3)ma(φ)·ma(φ)=1,(iv)
ha  is a gauge function [3] such as
(4)ha=1gεabcdmb(φ)dφmc(φ)=1gεabcm′b(φ)mc(φ),(v)
anda=1,2,3 is a color index.With this form of field, the Lorentz condition is satisfied:(5)∂μAμa=0.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 SU(2) by a bilinear expression
(6)ma=14ΓαtaβαΓβ,
where taβα are traceless Hermitian matrices and Γα are the operators which satisfy the following anticommutation relations:
(7){Γα,Γβ}=2δαβ.First, let us write the equation satisfied by Green’s functionS related to the Dirac particle submitted to the action of non-Abelian field [1]
(8)[(d(ΓαtβαΓβ)dφ(ΓαtβαΓβ))-(γp^)+g8(γk)(b(k1x)+c(k2x))(ΓαtβαΓβ)+++g32g(γk)(dΓαtβαΓβdφ(ΓαtβαΓβ))-M(d(ΓαtβαΓβ)dφ(ΓαtβαΓβ))]××S(xb,xa)=δ4(xb-xa),
where γμ are usual Dirac matrices obeying to anticommutation relations
(9){γμ,γν}=2gμν,μ,ν=0,1,2,3
with gμν is the metric tensor of Minkowski space (diaggμν=(1,-1,-1,-1)).In order to determine the Green function, we follow the usual construction method [11] of path integral form.First, we know that the kernel(10)S(xb,xa)=〈xb|S^|xa〉
is a matrix element of operator S~, the solution of the following symbolic equation:
(11)(×(dΓαtβαΓβdφ(ΓαtβαΓβ))-M-(γp^)+g8(γk)(b(k1x)+c(k2x))×××(ΓαtβαΓβ)+132g(γk)×××(dΓαtβαΓβdφ(ΓαtβαΓβ))-M)S^=I,
or
(12)S^=I×(((d(ΓαtβαΓβ)dφ(ΓαtβαΓβ))-(γ~·p^)+g8(γ~·k)&××××××(b(k1·x)+c(k2·x))(ΓαtβαΓβ)&×××××+(γ~·k)32g(dΓαtβαΓβdφ(ΓαtβαΓβ))××××××-Mγ5(d(ΓαtβαΓβ)dφ(ΓαtβαΓβ)))(d(ΓαtβαΓβ)dφ(ΓαtβαΓβ))((d(ΓαtβαΓβ)dφ(ΓαtβαΓβ))-(γ~·p^)+g8(γ~·k))-1γ5=S~γ5,
where γ5 is introduced in order to homogenize the variables
(13)γ5=γ0γ1γ2γ3,(γ5)2=-1.
So, S~  can be written in the exponential form
(14)S~=∫0∞dλ∫dχ∫exp(-iλH^l(χ,p^,x^)),
where λ and χ are, respectively, bosonic and fermionic variables [16, 17]. The Grassmann variables have the following properties:
(15)χ2=0,∫dχ=0,∫χdχ=1,
and the Hamiltonian system which governs the movement of the Dirac particle in the non-Abelian field is given by
(16)Hl=p2-M2Hl=-116(d(ΓαtβαΓβ)dφ(p^·k^)mmmmmmm×(σ(dΓαtβαΓβdφ(ΓαtβαΓβ)))mmmmmmm+(p^k^)(b(k1x)+c(k2x))mmmmmmm×(σ(14ΓαtβαΓβ))d(ΓαtβαΓβ)dφ)-ig(k^σk^)Hl=×(d(ΓαtβαΓβ)dφ(132gdΓαtβαΓβdφ(ΓαtβαΓβ))mmmmm+{(b(k1x)+c(k2x))(18ΓαtβαΓβ)}′d(ΓαtβαΓβ)dφ)Hl=+χ(d(ΓαtβαΓβ)dφ-p^+g2k^(b(k1x)+c(k2x))mmmmm×(σ(14ΓαtβαΓβ))+k^2mmmmm×(σ(116gdΓαtβαΓβdφ(ΓαtβαΓβ)))-Mγ5d(ΓαtβαΓβ)dφd(ΓαtβαΓβ)dφ).In the configuration space, the Green function is [4]
(17)S~(xb,xa)=∫0∞dλ∫dχ∫〈xb|exp(-iλH^l(χ,p^,x^))|xa〉,
and the path integral formulation consists of(i)
subdividing the interval[xa,xb] into N equals intervals of length Δτ=λ/N,(ii)
inserting the projectors(18)∫‍d4x|x〉〈x|=1,d4p|p〉〈p|=1,∫〈x∣p〉=1(2π)4exp(ipx),by using the actions on kets
(19)x^μ|x〉=xμ|x〉,p^μ|p〉=pμ|p〉,
in order to eliminate the operators x^, p^.So the Green function can be written (with the midpoint prescription) as following(20)Gc(xb,xa)==iT1T2∫0∞dλ∫Dx∫Dp×expi×××∫0λdτ[(σ·(116gd(ΓαtβαΓβ)dφ(ΓαtβαΓβ)))-Mγ5)px˙+(p2-M2)-116×mmmm×[(σ(d(ΓαtβαΓβ)dφ(ΓαtβαΓβ)))(p^k^)(σ(dΓαtβαΓβdφ(ΓαtβαΓβ)))mmmmmmmmmm+(p^k^)(b(k1x)+c(k2x))mmmmmmmmmm×(σ(14ΓαtβαΓβ))(σ(d(ΓαtβαΓβ)dφ(ΓαtβαΓβ)))]×mmmm-ig(k^σk^)(×(18ΓαtβαΓβ)(132gd(ΓαtβαΓβ)dφ(ΓαtβαΓβ))}′(132gdΓαtβαΓβdφ(ΓαtβαΓβ))×mmmmmmmmmmm+{(132gd(ΓαtβαΓβ)dφ(ΓαtβαΓβ))(b(k1x)+c(k2x))×mmmmmmmmmmmmmm×(18ΓαtβαΓβ)(132gd(ΓαtβαΓβ)dφ(ΓαtβαΓβ))}′)×mmmm+χ((σ·(116gd(ΓαtβαΓβ)dφ(ΓαtβαΓβ)))-p^+g2k^(b(k1·x)+c(k2·x))×mmmmmmmm×(σ·(14ΓαtβαΓβ))+k^2×mmmmmmmm×(σ·(116gdΓαtβαΓβdφ(ΓαtβαΓβ)))×mmmmmmmm-Mγ5(σ·(116gd(ΓαtβαΓβ)dφ(ΓαtβαΓβ))))],
where the T1T2-products are introduced because of the presence of matrices γμ and operators Γ which do not commutate. To eliminate the T1T2-products, we used the following form path integral:
(21)T1exp{∫0λ(ρ1·γ)dτ}==exp(iγδLδθ)×××∫E1‍DΨexp{∫0λ((Ψ·Ψ˙)-2i(ρ1·Ψ))dτmmmmmmmmmm+(Ψ(λ)·Ψ(0))∫0λ((Ψ.Ψ˙)-2i(ρ1.Ψ))}|Ψ0+Ψ1=θ,(22)T2exp{∫0λ(ρ2·Γ)dτ}=exp(iΓδLδϑ)×∫E2D‍Υexp{∫0λ‍(ρ1·γ)∫0λ((Υ·Υ˙)-2i(ρ2·Υ))dτ&∫E1‍DΨexp+(Υ(λ)·Υ(0))∫0λ‍((Υ·Υ˙)-2i(ρ2·Υ))}|Υ0+Υ1=ϑ,E1={Ψ(τ) where Ψ(0)+Ψ(λ)=θ},  E2={Υ(τ) where Υ(0)+Υ(λ)=ϑ} which are the domains of integration. ρ1,  ρ2  are coupled currents to matrices γ and operators Γ,  and θ, ϑ are Grassmann variables.So, we obtain(23)S~(xb,xa)=exp(iγδLδθ)exp(iΓδLδϑ)×∫0∞dλ∫dχ∫Dx∫Dp∫E1DΨ∫E2DΥ×exp[d(ΓαtβαΓβ)dφi∫0λdτmmmmmm×[px˙+(d(ΓαtβαΓβ)dφp2-M2-i16(pΨ)(kΨ)mmmmmmmmmmmmm×(tβαΥα′Υβ)-ig4(pΨ)(kΨ)mmmmmmmmmmmmm×(b(k1x)+c(k2x))(tβαΥαΥβ)d(ΓαtβαΓβ)dφ)mmmmmmmmmm+χ(∫0λdτ-(pΨ)+i32(kΨ)(Ψ(tβαΥα′Υβ))mmmmmmmmmmmmmm+ig8(kΨ)(b(k1x)+c(k2x))mmmmmmmmmmmmmm×(tβαΥαΥβ)-MΨ5∫0λdτ)mmmmmmmmmm-iΨΨ˙-iΥΥ˙d(ΓαtβαΓβ)dφi16]+Ψ(λ)Ψ(0)mmmmmm+Υ(λ)Υ(0)d(ΓαtβαΓβ)dφ]|θ=ϑ=0,
which is the Green function to calculate in the local approach of path integral.Having obtained the path integral form for the Green functionS~, let us perform the integrations.We notice that the field depends on productskx and kix, with i=1,2. First, we put φ=kx and φi=kix, and then the variables φ and φi are rendered, respectively, independent of kx and of kix by introducing the following three identities [11, 12]:
(24)∫dφb∫dφaδ(φa-kxa)∫Dφ×××∫Dpφexp{i∫0λpφ(φ˙-kx˙)dτ}=1,∫dφib∫dφiaδ(φia-kixa)∫Dφi×××∫Dpφiexp{i∫0λpφi(φ˙i-kix˙)dτ}=1.
Then, S~ is rewritten as
(25)S~(xb,xa)=exp(iγδLδθ)exp(iΓδLδϑ)×∫0∞dλ∫dχ∫Dx∫Dp×∫dφb∫dφaδ(φa-kxa)∫Dφ∫Dpφ×∏i=12‍∫dφib∫dφiaδ(φia-kixa)∫Dφi×∫Dpφi∫E1DΨ∫E2DΥ×exp[∫0λ‍dτi∫0λdτ[(p2-M2-ig32(pΨ)(k·Ψ)(tβαΥα′Υβ)mmmmmmmmmmm+(p-kpφ-k1pφ1-k2pφ2)x˙mmmmmmmmmmm+pφφ˙+pφ1φ˙1+pφ2φ˙2mmmmmmmmmmm-ig(pΨ)(kΨ)(bφ1+cφ2)(tβαΥαΥβ)g32(pΨ)(k·Ψ)(tβαΥα′Υβ))mmmmmmmmmm-χ[(pΨ)-ig2(kΨ)(Ψ(tβαΥα′Υβ))mmmmmmmmmmmmmm-ig2(kΨ)(bφ1+cφ2)mmmmmmmmmmmmmm×(tβαΥαΥβ)+MΨ5ig2(Ψ(tβαΥα′Υβ))(tβαΥα′Υβ)(Ψ(tβαΥα′Υβ))]mmmmmmmmmm-iΨΨ˙-iΥΥ˙g32(pΨ)(k·Ψ)(tβαΥα′Υβ)]mmmmm+Ψ(λ)Ψ(0)+Υ(λ)Υ(0)∫0λ‍dτ]|θ=ϑ=0.
Using a shift on p defined by
(26)p-kpφ-k1pφ1-k2pφ2⟶p,
we obtain
(27)S~(xb,xa)=exp(iγδLδθ)exp(iΓδLδϑ)×∫0∞dλ∫dχ∫Dx∫Dp×∫dφb∫dφaδ(φa-kxa)∫Dφ∫Dpφ×∏i=12‍∫dφib∫dφiaδ(φia-kixa)×∫Dφi∫Dpφi∫E1DΨ∫E2DΥ×exp[i∫0λ‍dτi∫0λdτmmmmmm×[g32(g32px˙+p2-M2mmmmmmmmmm-ig32(pΨ)(kΨ)(tβαΥα′Υβ)mmmmmmmmmm-ig(pΨ)(kΨ)(bφ1+cφ2)(tβαΥαΥβ)g32)mmmmmmmm+pφ(φ˙+2kp)+pφ1(φ˙1+2k1p)mmmmmmmm+pφ2(φ˙2+2k2p)mmmmmmmm-χ[(pΨ)-ig2(kΨ)(Ψ(tβαΥα′Υβ))mmmmmmmmmmmn-ig2(kΨ)(bφ1+cφ2)(tβαΥαΥβ)mmmmmmmmmmmn+MΨ5ig2(kΨ)(Ψ(tβαΥα′Υβ))]-iΨΨ˙-iΥΥ˙g32]mmmmmm+Ψ(λ)Ψ(0)+Υ(λ)Υ(0)i∫0λ‍dτ]|θ=ϑ=0,
where b=b(φ),  c=c(φ).If we integrate by part the first term in the action and then integrate the variablesx(τ), a Dirac function δ(p˙) appeared, and the integrations on all the pn reduce ∫‍Dp to a simple integral ∫‍d4p/(2π)4; that is, the momentum of the particle is conserved (constant) during the movement.Thus, we have(28)S~(xb,xa)=exp(iγδLδθ)exp(iΓ·δLδϑ)×∫0∞‍dλ∫‍dχ∫‍d4p(2π)4×exp{ip(xb-xa)+iλ(p2-M2)}×∫‍dφb∫‍dφaδ(φa-kxa)∫‍Dφ∫‍Dpφ×∏i=12‍∫‍dφib∫‍dφiaδ(φia-kixa)×∫‍Dφi∫‍Dpφi∫E1‍DΨ∫E2‍DΥ×exp[i∫0λdτmmmmmm×[∫0λdτ[(-ig32(pΨ)(kΨ)(tβαΥα′Υβ)(-ig32(pΨ)(kΨ)(tβαΥα′Υβ)mmmmmmmmm-ig(pΨ)(kΨ)(bφ1+cφ2)(tβαΥαΥβ)g32)mmmmmmmm+pφ(φ˙+2kp)+pφ1(φ˙1+2k1p)mmmmmmmm+pφ2(φ˙2+2k2p)mmmmmmmm-χ[(pΨ)-ig2(kΨ)(Ψ(tβαΥα′Υβ))mmmmmmmmmmm-ig2(kΨ)(bφ1+cφ2)mmmmmmmmmmm×(tβαΥαΥβ)+MΨ5ig2]-iΨΨ˙-iΥΥ˙i∫0λdτ[(-ig32(pΨ)(kΨ)(tβαΥα′Υβ)]mmmmmm+Ψ(λ)Ψ(0)+Υ(λ)Υ(0)i∫0λdτ[(-ig32(pΨ)(kΨ)(tβαΥα′Υβ)]|θ=ϑ=0.
We notice also, that there is dependence of product of kΨa in the action. So if we put  ξa=kΨa, these new variables are rendered independent [11, 12] by the introduction of the following relation:
(29)∫dξadξbδ(ξa-(k·Ψa))××∫Dpξ∫Dξexp{i∫0λpξ(ξ˙-(k·Ψ˙))dτ}=1,
with ξ and pξ being Grassmann variables.ThenS~, with these new variables, is rewritten as
(30)S~(xb,xa)=exp(iγδLδθ)exp(iΓδLδϑ)×∫0∞dλ∫dχ∫d4p(2π)4×exp{ip(xb-xa)+iλ(p2-M2)}×∫‍dφb∫‍dφaδ(φa-kxa)∫‍Dφ∫‍Dpφ×∏i=12‍∫‍dφib∫‍dφiaδ(φia-kixa)∫‍Dφi×∫‍Dpφi∫‍dξadξbδ(ξa-(k·Ψa))×∫‍Dξ∫‍Dpξ∫E1‍DΨ∫E2‍DΥ×exp[∫0λ‍dτi∫0λdτmmmmmm×[(-ig32(pΨ)(kΨ)(tβαΥα′Υβ)mmmmmmmlm-ig(pΨ)(kΨ)(bφ1+cφ2)(tβαΥαΥβ)g32)mmmmmmllm+pφ(φ˙+2kp)+pφ1(φ˙1+2k1p)mmmmmmllm+pφ2(φ˙2+2k2p)+pξ(ξ˙-(kΨ˙))mmmmmmllm-χ[(pΨ)-ig2(kΨ)(Ψ(tβαΥα′Υβ))mmmmmmmllmmm-ig2(kΨ)(bφ1+cφ2)(tβαΥαΥβ)mmmmmmmllmmm+MΨ5g32(Ψ(tβαΥα′Υβ))]-iΨΨ˙-iΥΥ˙-ig32]mmmmmm+Ψ(λ)Ψ(0)+Υ(λ)Υ(0)∫0λ‍dτ]|θ=ϑ=0.
Let us remark that the integrations over the Grassmann variables Ψ are subjected to the boundary conditions. We can render free these integrations, if we use new variables ω (velocities) defined by [15, 18]
(31)ω(τ)=dΨ(τ)dτ,
or by integration we have the relation
(32)Ψ(τ)=12∫0λε(τ-τ′)ω(τ′)dτ′+θ2
between Ψ and ω.Since(33)∫0λdτΨΨ˙+Ψ(λ)Ψ(0)==∫0λ∫0λ′dτdτ′ω(τ)ε(τ-τ′)ω(τ′)==-12ω*ε*ω,∫EDΨ=∫EDω,
where the following notation is used:
(34)a*b*c=∫∫a(τ)b(τ,τ′)c(τ′)dτdτ′,
we obtain for S~(35)S~(xb,xa)=exp(iγδLδθ)exp(iΓδLδϑ)××∫0∞‍dλ∫‍dχ∫‍d4p(2π)4××exp{ip(xb-xa)+iλ(p2-M2)}××∫‍dφb∫‍dφaδ(φa-kxa)∫‍Dφ∫‍Dpφ××∏i=12‍∫‍dφib∫‍dφiaδ(φia-kixa)∫‍Dφi∫‍Dpφi××∫‍dξa∫‍dξb∫‍Dpξ∫‍Dξ××∫E1‍Dωδ(ξa+12(k(ω-θ)))∫E2‍DΥ××exp[i∫0λdτmmmmm×(-i2gp(ε*ω)ξmmmmmmn-i2gp(ε*ω)ξ(bφ1+cφ2)mmmmmmn-i2g(pθ)(bφ1+cφ2)ξ+pφ(φ˙+2kp)mmmmmmn+pφ1(φ˙1+2k1p)+pφ2(φ˙2+2k2p)mmmmmmn+pξξ˙-iΥΥ˙mmmmmmn+χ[-12p(ε*ω)mmmmmmmmmn-12(pθ)+i4g(ε*ω)(tβαΥα′Υβ)mmmmmmmmmn+i4g(ε*ω)ξ(bφ1+cφ2)mmmmmmmmmn+i4g(θξ)(bφ1+cφ2)mmmmmmmmmn×(tβαΥαΥβ)+M(ε*ω5+θ5)i2]i2)mmmmm-12(ω*ε*ω)+(Υ(λ)·Υ(0))i∫0λdτ]|θ=ϑ=0.
By using a shift for ω(36)ω-(τ)⟶ω(τ)+ik(ε-1*pξ)+++ik1(ε-1*pξ1)+ik2(ε-1*pξ2),
the Green function can be simplified and take the following form:
(37)S~(xb,xa)=exp(iγδLδθ)exp(iΓδLδϑ)××∫0∞‍dλ∫‍dχ∫‍d4p(2π)4××exp{ip(xb-xa)+iλ(p2-M2)}××∫‍dφb∫‍dφaδ(φa-kxa)∫‍Dφ∫‍Dpφ××∏i=12‍∫‍dφib∫‍dφiaδ(φia-kixa)××∫‍Dφi∫‍Dpφi∫‍dξa∫‍dξb∫‍Dpξ∫‍Dξ××∫E1‍Dωδ(ξa+12(k·(ω-θ)))∫E2‍DΥ××exp[i∫0λdτmmmmm×(-i2g(p(ε*ω-i(k·pξ)-i(k1·pξ1)mmmmmmmmnmmm-i(k2·pξ2))ξ)mmmmmmnm-i2gp(ε*ω-i(k·pξ)-i(k1·pξ1)mmmmmmnmmmmmn-i(k2·pξ2))mmmmmmnm×ξ(b·φ1+c·φ2)-i2g(p·θ)mmmmmmnm×(bφ1+cφ2)ξ+pφ(φ˙+2(k·p))mmmmmmnm+pφ1(φ˙1+2k1p)+pφ2(φ˙2+2k2p)mmmmmmnm+pξξ˙-iΥΥ˙mmmmmmnm+χ[-12pmmmmmmnmmmm×(ε*ω-i(kpξ)-i(k1pξ1)-i(k2pξ2))mmmmmmnmmmm-12(pθ)+i4gmmmmmmnmmmm×(i(k2pξ2)ε*ω-i(k·pξ)mmmmmmnmmmmmm-i(k1pξ1)-i(k2pξ2))mmmmmmnmmmm×(tβαΥα′Υβ)+i4gmmmmmmnmmmm×(ε*ω-i(kpξ)-i(k1pξ1)-i(k2pξ2))mmmmmmnmmmm×ξ(bφ1+cφ2)mmmmmmnmmmm+i4g(θξ)(bφ1+cφ2)(tβαΥαΥβ)mmmmmmnmmmm+M(ε*ω5+θ5)12](p(ε*ω-i(k·pξ)-i(k1·pξ1)-i(k2·pξ2))ξ)i2)mmmmmmnm-12(ω*ε*ω-2i(kω)pξmmmmmmnmmmm-2i(k1ω)pξ1-2i(k1ω)pξ2)mmmmmmnm+Υ(λ)Υ(0)∫0λdτ]|θ=ϑ=0.
At this level, to integrate velocities ω, first, we insert the integral representation of the Dirac function
(38)δ(ξa+12(k(ω-θ)))==∫‍dpξaexp{ipξa(ξa+12(k(ω-θ)))},
with pξa  being Grassmann variable; then we have
(39)S~(xb,xa)=exp(iγδLδθ)exp(iΓδLδϑ)××∫0∞‍dλ∫‍dχ∫‍d4p(2π)4××exp{ip(xb-xa)+iλ(p2-M2)}××∫‍dφb∫‍dφaδ(φa-kxa)∫‍Dφ∫‍Dpφ××∏i=12‍∫‍dφib∫‍dφiaδ(φia-kixa)∫‍Dφi××∫‍Dpφi∫‍dξa∫‍dpξa∫‍dξb∫‍Dpξ∫‍Dξ××∫E1‍Dωδ(ξa+12(k(ω-θ)))∫E2‍DΥ××exp[∫0λdτi∫0λdτmmmmm×(-i2gmmmmmmm×(i2p(i2ε*ω-i(kpξ)-i(k1pξ1)mmmmmmmmmmm-i(k2pξ2)i2)ξi2)mmmmmmm-i2gp(ε*ω-i(kpξ)mmmmmmmmmmml-i(k1pξ1)-i(k2pξ2))mmmmmmm×ξ(bφ1+cφ2)-i2g(pθ)(bφ1+cφ2)ξmmmmmmm+pφ(φ˙+2kp)+pφ1(φ˙1+2k1p)mmmmmmm+pφ2(φ˙2+2k2p)+pξξ˙+pξammmmmmm×(ξa+12(k(ω-θ)))-iΥΥ˙mmmmmmm+χ[-12p(ε*ω-i(kpξ)mmmmmmmmmmmmml-i(k1pξ1)-i(k2pξ2))mmmmmmmmmm-12(pθ)+i4gmmmmmmmmmm×(ε*ω-i(kpξ)mmmmmmmmmmmm-i(k1pξ1)-i(k2pξ2))mmmmmmmmmm×(tβαΥα′Υβ)+i4gmmmmmmmmmm×(ε*ω-i(kpξ)mmmmmmmmmmmm-i(k1pξ1)-i(k2pξ2))mmmmmmmmmm×ξ(bφ1+cφ2)+i4g(θξ)(bφ1+cφ2)mmmmmmmmmm×(tβαΥαΥβ)+M(ε*ω5+θ5)12])mmmmm-12(ω*ε*ω-2i(kω)pξmmmmmmmm-2i(k1ω)pξ1-2i(k1ω)pξ2)mmmmm+Υ(λ)Υ(0)∫0λdτ]|θ=ϑ=0.
We notice that the integration over ω5 can be eliminated thanks to the following result:
(40)∫‍Dω5exp{-12ω5*ε*ω5-iM2ε*ω5χ}=1,
and for other variables ωμ(μ=0,1,2,3), the integrations have the following forms:
(41)∫Dωexp{∫0λ[-12ωμεωμ+Lμωμ]dτ},
where
(42)Lμ(τ)=-ig2∫0λpμξ(s)(1+(bφ1+cφ2))ε(τ-s)ds+χ∫0λ(-12pμ+i4gkμmmmmn×((tβαΥα′Υβ)+ξ(s)(bφ1+cφ2))12)mmn×ε(τ-s)ds+12kμpξa.
After we have integrated over the ω,  S~ is reduced to
(43)S~(xb,xa)=exp(iγδLδθ)exp(iΓδLδϑ)××∫0∞‍dλ∫‍dχ××∫‍d4p(2π)4exp{ip(xb-xa)+iλ(p2-M2)}××∫‍dφb∫‍dφaδ(φa-kxa)××∫‍Dφ∫‍Dpφ∏i=12‍∫‍dφib××∫‍dφiaδ(φia-kixa)∫‍Dφi∫‍Dpφi××∫‍dξa∫‍dpξa∫‍dξb∫‍Dpξ∫‍Dξ∫E2‍DΥ××exp[∫0λ‍dτi∫0λdτmmmmm×(×(tβαΥαΥβ)+Mθ512]pφ(φ˙+2kp)+pφ1(φ˙1+2k1p)mmmmmmm+pφ2(φ˙2+2k2p)+pξξ˙+pξammmmmmm×(ξa-12kθ)-iΥΥ˙mmmmmmm+χ[-12(pθ)+i4g(tβαΥαΥβ)mmmmmmmllmm+i4gξ(bφ1+cφ2)+i4g(θξ)(bφ1+cφ2)mmmmmmllmmm×(tβαΥαΥβ)+Mθ512])mmmmm+Υ(λ)Υ(0)∫0λ‍dτ]|θ=ϑ=0,
and the integrations on variables pξ  give
(44)ξ˙=0,thatis,ξa=ξb=cst.
Thus, S~ becomes
(45)S~(xb,xa)=exp(iγδLδθ)exp(iΓδLδϑ)××∫0∞‍dλ∫‍dχ××∫‍d4p(2π)4exp{ip(xb-xa)+iλ(p2-M2)}××∫‍dφb∫‍dφaδ(φa-kxa)∫‍Dφ∫‍Dpφ××∏i=12‍∫‍dφib∫‍dφiaδ(φia-kixa)∫‍Dφi××∫‍Dpφi∫‍dξa∫‍dpξa∫‍Dξ∫E2‍DΥ××exp[i∫0λdτmmmmmmn×(12pφ(φ˙+2(kp))mmmmnmmmn+pφ1(φ˙1+2k1p)+pφ2(φ˙2+2k2p)mmmmnmmmn+pξa(ξa-12kθ)-iΥΥ˙mmmmnmmmn+χ[-12(pθ)+i4g(tβαΥα′Υβ)mmmmnmmmnmmn+i4gξ(bφ1+cφ2)+i4g(θξ)mmmmnmmmnmmn×(bφ1+cφ2)(tβαΥαΥβ)+Mθ512])mmmmmmn+Υ(λ)Υ(0)∫0λdτ]|θ=ϑ=0.
The integrations over pξa  give the following result:
(46)ξa-12(kθ)=0,thatis,ξa=ξb=12(kθ),
and the Green function is reduced to
(47)S~(xb,xa)=exp(iγδLδθ)exp(iΓδLδϑ)××∫0∞‍dλ∫‍dχ∫‍d4p(2π)4××exp{ip(xb-xa)+iλ(p2-M2)}××∫‍dφb∫‍dφaδ(φa-kxa)∫‍Dφ∫‍Dpφ××∏i=12‍∫‍dφib∫‍dφiaδ(φia-kixa)××∫‍Dφi∫‍Dpφi∫‍dξa∫‍Dξ∫E2‍DΥ××exp[i∫0λdτmmmmmn×(12pφ(φ˙+2kp)mmmmmmlln+pφ1(φ˙1+2k1p)mmmmmmlln+pφ2(φ˙2+2k2p)-iΥΥ˙mmmmmmlln+χ[-12(pθ)+i4g(tβαΥα′Υβ)mmmmmmllnmmm+i4gξ(bφ1+cφ2)+i4g(θξ)mmmmmmllnmmm×(bφ1+cφ2)(tβαΥαΥβ)+Mθ512])mmmmmn+Υ(λ)Υ(0)∫0λdτ]|θ=ϑ=0.
By integrations on pφ,  pφ1, and pφ2 again, three Dirac functions appear also
(48)δ(φ˙+2(kp)),δ(φ˙i+2(kip)),(i=1,2);
that is, the equations
(49)dφdτ=-2(kp),dφidτ=-2(kip),
for solutions
(50)φ(τ)=-2(kp)τ+cte,φi(τ)=-2(kip)τ+cte
contribute to the determination of the Green function.Now, we introduce the integral representations of threeδ:
(51)δ(φa-kxa-(kp)λ)===12π∫‍dpφ××××exp{ipφ(φa-kxa-(kp)λ)},δ(φia-kixa-(kip)λ)===12π∫‍dpφi××××exp{ipφi(φia-kixa-(kip)λ)}.
After having changed p into p-kpφb-k1pφ1b-k2pφ2b, the relations (22) are reused to reintroduce the generators Γα of SU(2)  group since there is no order problem; the T2-product is omitted, then the Green function S~ symmetrized with respect to positions xa and xb takes the following form:
(52)S~(xb,xa)=exp(iγδLδθ)∫0∞‍dλ∫‍dχ××∫‍d4p(2π)4exp{ip(xb-xa)+iλ(p2-M2)}××exp[∫0λ‍dτi∫φaφbdφ(kθ)××××××(d(ΓαtβαΓβ)dφ-ig8(kp)(kp)×××××××××(dΓαtβαΓβdφ(ΓαtβαΓβ))-14(kp)2×××××××××p2(1-(1/2)χ)χ××××××××-ig16(dΓαtβαΓβdφ(ΓαtβαΓβ))χd(ΓαtβαΓβ)dφ)×××××-i2((pθ)+Mθ5)χ×××××+(kθ)(kp)((d(ΓαtβαΓβ)dφ(ΓαtβαΓβ))2-132∫φaφbdφ(d(ΓαtβαΓβ)dφ(ΓαtβαΓβ))××××××××××××(kϑ)χ+k2128∫φaφbdφp2(kp)××××××××××××(d(ΓαtβαΓβ)dφ(ΓαtβαΓβ))2+ig16××××××××××××∫φaφbdφp2(kp)××××××××××××(d(ΓαtβαΓβ)dφ(ΓαtβαΓβ))χ×××××××××××+g64×∫φaφbdφ1(kp)××××××××××××(d(ΓαtβαΓβ)dφ(ΓαtβαΓβ))2χ)×××××+g264∫φ1aφ1bdφ1××××××∫φ2aφ2bdφ21(kp)(bφ1+cφ2)(kθ)(ΓαtβαΓβ)∫0λ‍dτ]|θ=ϑ=0.
The variable χ [16, 19] can be eliminated by integration, and we obtain
(53)S~(xb,xa)=exp(iγδLδθ)∫0∞‍dλ××∫‍d4p(2π)4exp{ip(xb-xa)+iλ(p2-M2)}××(12((pθ)+Mθ5))××exp[((kθ)+i(kθ)(kp)2p2+g8(kθ))-g16((kθ)+i(kθ)(kp)2p2+g8(kθ))××××××∫φaφbdφ(d(ΓαtβαΓβ)dφ(ΓαtβαΓβ))×××××+132(kθ)(kp)∫φaφbdφ(d(ΓαtβαΓβ)dφ(ΓαtβαΓβ))××××××(k·ϑ)+(kθ)(kp)(-g64(kp)+(kp)128)××××××∫φaφbdφ(d(ΓαtβαΓβ)dφ(ΓαtβαΓβ))2+g264(kθ)(kp)××××××∫φ1aφ1bdφ1××××××∫φ2aφ2bdφ2(bφ1+cφ2)(ΓαtβαΓβ)((kθ)+i(kθ)(kp)2p2+g8(kθ))]|θ=ϑ=0,
where the calculations have been simplified thanks to proprieties of field.The matricesγμ are reintroduced by using the following relations:
(54)exp(i(γδLδθ))f(θ)|θ=0===f(∂L∂θ)exp(i(γθ))|θ=0,∂L2∂θα∂θβθμθνγμγν=-2iσαβ,exp(i(γθ))=1+i(γθ)exp(i(γθ))=-12θμθνγμγνexp(i(γθ))=+i6θμθνθσγμγνγσexp(i(γθ))=+θ0θ1θ2θ3γ5,
where
(55)f(θ)=12((pθ)+Mθ5)××exp{((kθ)+i(kθ)(kp)2p2+g8(kθ))-g16((kθ)+i(kθ)(kp)2p2+g8(kθ))×××××××∫φaφbdφ(d(ΓαtβαΓβ)dφ(ΓαtβαΓβ))××××××+132(kθ)(kp)×××××××∫φaφbdφ(d(ΓαtβαΓβ)dφ(ΓαtβαΓβ))×××××××(k·ϑ)+(kθ)(kp)(-g64(kp)+(kp)128)×××××××∫φaφbdφ(d(ΓαtβαΓβ)dφ(ΓαtβαΓβ))2××××××+g264(kθ)(kp)∫φ1aφ1bdφ1×××××××∫φ2aφ2bdφ2(bφ1+cφ2)(ΓαtβαΓβ)g16((kθ)+i(kθ)(kp)2p2+g8(kθ))}.
Then
(56)S~(xb,xa)=∫0∞‍dλ××∫‍d4p(2π)4exp{ip(xb-xa)+iλ(p2-M2)}××exp{(g64(pσk)(kp)∫kxakxb‍dφi(kp)×××××××((g64(pσk)(kp)∫kxakxb‍dφg64(pσk)(kp)××××××××××∫kxakxbdφ(bk1k+ck2k)φdm(φ)dφ(g64(pσk)(kp)∫kxakxb‍dφ)},
and after having eliminated λ by integration, we have
(57)S~(xb,xa)=1(2π)4××∫‍d4p(-(γp)+M)p2-M2+iOexp{ip(xb-xa)}××exp{i(kp)××××××(g64(pσk)(kp)×××××××××∫kxakxbdφ(bk1k+ck2k)φdm(φ)dφ)},
and an integration by part of the exponent of exponential S~(xb,xa) has the following form:
(58)S~(xb,xa)=1(2π)4∫‍d4p(-(γp)+M)p2-M2+i0+exp{ip(xb-xa)}××exp{(k1kb(φb)+k2kc(φb))ig64(kp)2(pσk)mmmmm×[ig64(kp)2((k1kb(φb)+k2kc(φb))φbm(φb)mmmmmmmmm-(k1kb(φa)+k2kc(φa))φam(φa))mmmmmmm-∫kxakxbdφmmmmmmm×(k1k(b(φ)+φb′(φ))mmmmmmmmm+k2k(c(φ)+φc′(φ)))m(φ)ig64(kp)2]},
and after some manipulations, the Green function for the Dirac particles moving in the non-Abelian field in the local approach is finally
(59)S~(xb,xa)=∫‍d4p(2π)41p2-M2+i0+exp{ip(xb-xa)}××g641(kp)2××[(k1kb(φa)+k2kc(φa))(1+(pσk)(k1kb(φb)+k2kc(φb))φbm(φb))××××(-(γp)+M)××××(1-(pσk)(k1kb(φa)+k2kc(φa))φam(φa))]××exp{+k2k(c(φ)+φc′(φ))]m(φ)∫kxakxb‍dφ)]+k2k(c(φ)+φc′(φ))]m(φ))]i[+k2k(c(φ)+φc′(φ))]m(φ)∫kxakxb‍dφ)p(xb-xa)-g(pσk)64(kp)2mmmmmm×(∫kxakxb‍dφ∫kxakxbdφmmmmmmmm×[k1k(b(φ)+φb′(φ))+k2kmmmmmmmmmm×(c(φ)+φc′(φ))k1k]m(φ)∫kxakxb‍dφ)]},
which is obviously symmetrical with respect to xa and xb.From the residues related to two poles(60)p±0=±ω-i0+=±(p2+M2)-1/2-i0+,
and after having introduced the projectors on states of positive and negative energy [20],
(61)Λ+=∑±s‍u(p,s)u-(p,s)=γp+M2M,Λ-=∑±s‍v(p,s)v-(p,s)=-γp+M2M,
we obtain the following spectral decomposition:
(62)S(xb,xa)=-iθ(tb-ta)×∫‍d3p∑±s‍Ψs,p+(xb)Ψ-s,p+(xa)+iθ(tb-ta)∫‍d3p×∑±s‍Ψs,p-(xb)Ψ-s,p-(xa),
where finally the wave functions related to particle and to antiparticle are, respectively,
(63)Ψs,p+(x)=1(2π)3/2(Mp0)1/2××((k1kb(kx)+k2kc(kx))1+(pσk)××××(k1kb(kx)+k2kc(kx))φbm(kx))u(p,s)××exp{i[g(pσk)64(k·p)2px-g(pσk)64(kp)2mmmmmm×[+k2k(c(φ)+φc′(φ)))g(pσk)64(k·p)2∫kxdφmmmmmmmmm×(+k2k(c(φ)+φc′(φ)))m(φ)]k1k(b(φ)+φb′(φ))mmmmmmmmmmm+k2k(c(φ)+φc′(φ)))mmmmmmmmm×m(φ)g(pσk)64(k·p)2]]},Ψs,p-(x)=1(2π)3/2(Mp0)1/2××(k1kb(kx)1-(pσk)mmm×(k1kb(kx)+k2kc(kx))mmm×φam(kx)k1kb(kx))v(p,s)××exp{+k2k(c(φ)+φc′(φ)))m(φ)]]-i[g(pσk)64(k·p)2+k2k(c(φ)+φc′(φ)))m(φ)]px+g(pσk)64(k·p)2mmmmmmmm×[g(pσk)64(k·p)2+k2k(c(φ)+φc′(φ)))m(φ)∫kxdφmmmmmmmmmm×(k1k(b(φ)+φb′(φ))mmmmmmmmmmmm+k2k(c(φ)+φc′(φ)))m(φ)g(pσk)64(k·p)2]]}.u(p,s) and v(p,s)  are the spinors solutions of the free Dirac equation and are
(64)u-(p,s)u(p,s)=1,v-(p,s)v(p,s)=-1.


## 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 ofδ  functions which are appeared during calculation can be also obtained from the classical mechanics (cf. Appendix).



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*Source: 751963-2013-07-14.xml*