Conformally Covariant Vector-Spinor Field in de Sitter Space

# Conformally Covariant Vector-Spinor Field in de Sitter Space

N. Fatahi Department of Physics, Science and Research branch, Islamic Azad University, Tehran, Iran    M.V. Takook Department of Physics, Science and Research branch, Islamic Azad University, Tehran, Iran    M.R. Tanhayi Department of Physics, Islamic Azad University, Central Tehran Branch, Tehran, Iran
July 19, 2019
###### Abstract

In this paper, we study conformally invariant field equations for vector-spinor (spin-) field in de Sitter space-time. The solutions are also obtained in terms of the de Sitter-Dirac plane waves. The related two-point functions are calculated in both de Sitter ambient space formalism and intrinsic coordinate. In order to study the conformal invariance, Dirac’s six-cone formalism is utilized in which the field equations are expressed in a manifestly conformal way in dimensional conformal space and then followed by the projection to de Sitter space.

## I Introduction

Recent astrophysical data indicate that our universe is undergoing an accelerated expansion Riess () and at the first approximation the background space-time can be well described by the de Sitter (dS) space. The dS universe is a maximally symmetric curved space with the same degrees of symmetry as the Minkowskian space. It also plays an important role in the inflation scenario where the cosmic dynamics is assumed to be dominated by a term that acting like a cosmological constant linde (). Thus, constructing a formulation of dS quantum field theory seems to be important which has been studied by many authors qft (); qft1 (); qft2 (); qft3 (); entropy (); ta1403 (). The high level of symmetry in this curved space-time can be a guideline for constructing the quantum fields and the underlining symmetry of the background may be well established by group theory. Actually revealing the group theoretical and geometrical structures underlying the theory is of great importance. We use this idea to study the higher-spin free field in de Sitter space. Basically studying the free higher-spin fields may shed some light to our better understanding of the interactions in higher-spin field theories. As one knows, the construction of a consistent interacting theory of higher-spin fields is one of the oldest long standing problems in theoretical physics of particles and fields.

The higher-spin field equations have been considered in wigner1 (). It was proved that a given field of rest mass and spin can be represented by a completely symmetric multi-spinor wigner3 (). Manifestly covariant massless fields for half-integer spin in Anti-de Sitter background have been studied in fronsdal12 (). Massive spin- field in dS space has been considered in baba () and the extension to the massless fields is done in azizi (). Here, within the language of the group theory in de Sitter space, we study the conformal invariance of vector-spinor field by finding the field equations, solutions and two-point function.

As it is known, the massless fields which live on the light-cone, are conformally invariant Wald (), generally, all massless gauge fields in four dimensional constant curvature spaces are conformally invariant at the level of degrees of freedom. This indicates that the respective representations of (A)dS or Poincaré group can be extended to the representations of the conformal group ta1403 (). The details for higher-spin gauge fields can be found in new (); new2 (). For our purpose, in dS space-time the conformal invariance may be understood by the concept of light-cone propagators. Noting that in dS space, mass is not a well-defined invariant parameter for a set of observers transforming under the dS group , the masslessness is used in reference to the conformal invariance (propagation on the dS light-cone). From the group theoretical point of view, the massive representations refer to those that coincide to the massive representation of the Poincaré group at the zero curvature limit (appendix A). Our motivation to study the spin-vector conformal invariance field equation in dS space is that in the supergravity model this field is the partner of the gravitational field and it must propagate on the dS light-cone.

In flat space, according to the group representation theory and Wigner’s interpretation of the elementary systems, the field operators of any kind of spin transform according to the unitary irreducible representation (UIR) of the Poincaré group (the kinematical group of Minkowski space), and their corresponding field equations are well established by the Casimir operators. Being interested in symmetry of space, the isometry group of dS space is which may be viewed as a deformation of the proper orthochronous Poincaré group. There are two Casimir operators in dS group and it has been shown that the massive scalar, vector and spin- fields can be associated with the UIRs of dS group qft2 (); re (); chli (); ta1 (); anilto (). The massless fields can be associated with an indecomposable representation of dS group gagarota (). The covariant quantum field theory for massive and massless conformally coupled scalar field in dS space have been studied in bgm (); qft2 () and also for the massless minimally coupled scalar field in grt (). In Ref.s new3 (); new4 (), one can find the conformally covariant quantization of the gauge field in dS space.

The main goal of this paper is to study conformal invariance of spin- field in de Sitter space, we use group theoretical approach and hence the de Sitter invariance becomes manifest. The paper is organized in seven sections: The conformally invariant spin- field equations are obtained in section II and III. The solutions are found in section IV, as it will be shown, they can be written in terms of the de Sitter-Dirac plane waves. In section V and VI, we find the two-point function in both embedding space namely ambient space of dS and also in 4 dimensional intrinsic space. We discuss the results in the conclusion part. Finally, some useful relations are presented in the appendixes.

## Ii massless spinor field equations in de Sitter space

The dS metric is the unique solution of Einstein’s equation in vacuum with positive cosmological constant , in which dS space may be visualized as the hyperboloid embedded in a five-dimensional Minkowski space

 XH={x∈IR5;x2=ηαβxαxβ=−H−2}, (II.1)

is the Hubble constant, hereafter for the sake of simplicity, we set . The de Sitter metric is

 ds2=ηαβdxαdxβ∣x2=−1=gdSμνdXμdXν, (II.2)

where , and . We use for ambient space formalism whereas stand for de Sitter intrinsic coordinates. For simplicity the dot product is shown as . We define the transverse derivative in de Sitter space as , where is the projection operator and note that . Working in embedding space has two advantages, first it is close to the group theoretical language and second the equations are obtained in an easer way than they might be found in de Sitter intrinsic space.

There are two Casimir operators for dS group, these operators commute with all the action of the group generators and thus they are constant on each representation. In this section we briefly recall the notations of the Casimir operator and more details can be found in bida (); dixmier (); sep (); re (). The second and forth order Casimir operators are:

 (II.3)

where and is the antisymmetric tensor in the ambient space notation with . The generator of de Sitter group is defined by

 Lαβ=Mαβ+Sαβ,

where the "orbital" part is

 Mαβ=−i(xα∂β−xβ∂α)=−i(xα∂Tβ−xβ∂Tα), (II.4)

and the "spinoral" part which acts on the spinor field () is bida ()

 Sαβ=−i4[γα,γβ]. (II.5)

In this case the five matrices of are the generators of the Clifford algebra which are constructed as

 γαγβ+γβγα=2ηαβ,γα†=γ0γαγ0. (II.6)

Based on spectrum of the possible values of Casimir eigenvalues, the UIRs of de Sitter group can be classified as the principal, complementary and discrete series (Appendix A). In the principal series which belongs to the massive representation of dS space and tends to the massive representation of Poincaré group at the zero curvature limit, the eigenvalues of the Casimir operators can be written as dixmier ()

 =(94+ν2−s(s+1)), =((14+ν2)s(s+1)),

where stands for the spin, is a real positive parameter. The second order field equations can be written as111Note that in writing the field equations we use only the second order Casimir operator, because the forth order one leads to higher derivative equations

 (Q(1)s−)ψ=0. (II.7)

For example for , one has

 (Q(1)12−32)ψ=ν2ψ, (II.8)

stands for a spinor field with arbitrary degree of homogeneity: . The second order Casimir operator for spin- is given by

 Q(1)12=−12MαβMαβ−12SαβSαβ−SαβMαβ, (II.9)

where one can show

 12SαβSαβ=52,SαβMαβ=−i2γαγβMαβ=−x/∂/T, (II.10)

note that . After making use of above relations, equation (II.8) can be written in terms of the scalar Casimir operator as follows

 (Q0+x/∂/T−4)ψ=ν2ψ, (II.11)

where is the spin-less Casimir operator. If one defines the de Sitter-Dirac operator as

 D≡−i2γαγβMαβ+2=−x/∂/T+2, (II.12)

then (II.11) can be written as follows

 (iD−ν)(iD+ν)ψ(x)=0. (II.13)

This relation is similar to the standard relation of the spinor field in flat space, . The first order field equation for a field of spin- in dS space becomes

 (II.14)

The massless case in de Sitter space belongs to the discrete series and the eigenvalue of the second order Casimir operator is given by

 =2(1−s2)I1. (II.15)

Plugging this value for in (II.7) and in terms of the operator, one obtains

 iDψ(x)=0,or equivalently(Q0−2)ψ=0. (II.16)

Similarly for spin- field equation in dS space, two types of UIRs of dS group are characterized, the principal and discrete series

• The unitary irreducible representations of the principal series,

note that and are equivalent this kind of representation belong to the massive case.

• The unitary irreducible representations of the discrete series,

 =−52, (II.17)

the sign stands for the helicity.

The massless spin- field in dS space becomes baba (); azizi ()

 (Q(1)32+52)Ψα(x)=0, (II.18)

where

 Q(1)32Ψα(x)= (−12MαβMαβ+i2γαγβMαβ−112)Ψα(x) −2∂αx⋅Ψ(x)+2xα∂⋅Ψ(x)+γαγ⋅Ψ(x). (II.19)

Similar to the massless vector-spinor fields equations in Minkowski space, the solutions of this filed equation possess a singularity due to the divergencelessness condition . Then the gauge invariant field equation is azizi ()

 (Q(1)32+52)Ψα(x)−D(32)α∂T⋅Ψ(x)=0, (II.20)

where This equation is invariant under the following gauge transformation

 Ψα(x)→Ψ′α(x)=Ψα(x)+D(32)αζ, (II.21)

where is an arbitrary spinor field. In order to fix the gauge, is introduced and then one has

 (Q(1)32+52)Ψα(x)−cD(32)α∂T⋅Ψ(x)=0. (II.22)

Similar to the spin- case, one can write the first order field equation as

 x/∂/TΨα(x)−3Ψα(x)−xαx/Ψ/−∂Tαx/Ψ/=0. (II.23)

This equation is invariant under the gauge transformation . There exist another first order field equation ta1403 ():

 x/∂/TΨα(x)−Ψα(x)−xαx/Ψ/+D(32)αx/Ψ/=0, (II.24)

which is invariant under the gauge transformation . It is worth to mention that (II.24) appears in the conformal invariant field equation.

## Iii Conformal invariance and Dirac’s six-cone formalism

Massless field equations are expected to be conformally invariant (CI). A trivial example is the Maxwell’s equations where in Cunningham and Bateman showed that these equations are covariant under the larger 15-parameter conformal group as well as 10-parameter Poincaré group stwi (). Fields with spin are invariant under the gauge transformation as well. In Dirac used a manifestly conformally covariant formulation namely, the conformal space notation, to write down wave equations in Minkowski space Dirac (). The conformal group acts non-linearly on Minkowski coordinates, Dirac used coordinates which the conformal group acts linearly on them. This actually reassembles the conformal space and the theory is defined on a dimensional hypercone (hereafter named as Dirac’s six-cone) or equivalently, in a dimensional conformal space. Within this formalism, he obtained scalar, spinor and vector conformally invariant fields in flat space-time. This theory developed in some papers (Mack () and references therein). The generalization to dS space was done in gareta1 (); ta4 () to obtain CI field equations for scalar, vector and symmetric rank-2 tensor fields. Here, we use this approach to study the spinor fields () in de Sitter space. First let us recall this method briefly.

### iii.1 Dirac’s six-cone

Basically, the special conformal transformation acts non-linearly on 4-dimensional coordinates. In conformal space Dirac proposed the coordinates , where acts linearly on them. Dirac’s six-cone is then defined as a 5-dimensional hyper-surface in satisfying following constraint

 u2=ηabuaub=u20−→u2+u25=0,a,b=0,1,...,5, (III.1)

this is obviously invariant under the conformal transformation. A given operator say as is said to be intrinsic if it satisfies

 ^Au2ϕ=u2^´Aϕ,

where is a function in . One should write all the wave equations, subsidiary conditions and etc., in terms of operators that are defined intrinsically on the cone. The following CI system which is defined on the cone is well established this goal222In fact, this approach to conformal symmetry leads to the best path to exploit the physical symmetry and it provides a rather simple way to write the conformally invariant field equations. Moreover, it is important to mention that on the cone , the second-order Casimir operator of conformal group, , is not a suitable operator to obtain CI wave equations. Because it is proved on the cone , it reduces to a constant, consequently, this operator cannot lead to the wave equations on the cone. The well-defined operators exist only in exceptional cases. For tensor fields of degree , the intrinsic wave operators are respectively. This method previously was established for -de Sitter linear gravity in Ref. new5 (). gareta1 ()

 {(∂a∂a)nΦ=0,^N5Φ=(n−2)Φ, (III.2)

where the powers of d’Alembertian act intrinsically on fields of conformal degree , and is a tensor or spinor field of a definite rank and symmetry. The conformal-degree operator is given by

 ^N5≡ua∂a. (III.3)

One can add the following CI conditions to restrict the space of solution

• transversality:

• divergencelessness:

• tracelessness:

• for tensor-spinor field:

In conformal coordinate , the definition of is given by il ()

 ∇a≡ua∂b∂b−(2^N5+4)∂a.

The quantities which are evaluated on the cone should be projected to de Sitter space, first, one needs a relation between the coordinates

 {xα=(u5)−1uα,x5=u5, (III.4)

therefore, the intrinsic operators turn to gareta1 (); il ():

 ^N5 = x5∂∂x5, (∂a∂a)n = −x−2n5n∏j=1[Q0+(j+1)(j−2)], (III.5)
 ∇α=−x−15[xα[Q0−^N5(^N5−1)]+2∂Tα(^N5+1)],

where is the Casimir operator in de Sitter space. First, one should write the equations in coordinates where the conformal invariance is manifest (III.2). Then by the help of the above mentioned relations, the obtained equations are related to the embedding space ones. Finally, the desired de Sitter relations can be obtained via the projection. This approach provides a simple way to study the conformal invariance in de Sitter space.

### iii.2 conformally invariant spinor field equations

Spinor field : The simplest conformally invariant system is obtained by setting and in (III.2) which after making use of (III.1) and in language of the Casimir operator of dS group, it turns to

 (Q0−2)ψ(x)=0. (III.6)

This equation is obviously conformally invariant, and stands for a massless conformally spinor field in dS space (see Eq.II.16). After making use of (II.11), the first order field equation in this case, becomes as follows

 (x/∂/T−2)ψ(x)=0. (III.7)

Indeed, the field , associates with the UIR of dS group and propagates on the dS light cone.

Vector-spinor : In this case one should classify the degrees of freedom of vector-spinor field on the cone in terms of the dS fields, this can be done as below

 Ψα=x5(Φα+xαx⋅Φ),K1=x5Φ5,K2=x5x⋅Φ, (III.8)

where and are two spinor fields and is a vector-spinor field. Note that indicates that lives on dS hyperboloid. To obtain CI field equations by choosing in (III.2), one obtains

 (Q0−2)Φα=0, (III.9)

where is a spin- field on the cone. After doing some tedious but straightforward calculation which partially is given in appendix C, following CI system of field equations is obtained

 ⎧⎪⎨⎪⎩(Q0−2)K1=0,(Q0−2)K2=0,(Q0−2)Ψα+2xα∂T⋅Ψ+∂Tα∂T⋅Ψ=0 (III.10)

that indicates: and are both CI massless spinor fields gareta1 (). By using the transversality condition on the cone and (III.9) we obtain (see appendix C)

 (Q0−2)∂T.Ψ=0. (III.11)

However, one can use (II) to write the third line of (III.10) as

 (Q(1)32−x/⧸∂T+72)Ψα+∂Tα∂T⋅Ψ−γαΨ/=0. (III.12)

Using the equations (II.18,II.24) the above field equation can be rewritten as

 (Q(1)32+52)Ψα−(x/⧸∂TΨα−Ψα−xαx/Ψ/−∂Tα∂T⋅Ψ+γTαΨ/)=0. (III.13)

As previously mentioned, the fields should be projected to the de Sitter space, the transverse projection implies the transversality of fields, , so that from the homogeneity condition, one obtains . These two conditions impose the following constrains on the projected fields: , and consequently At the appendix C, it is shown that the CI divergencelessness condition on the cone, namely , results in , which indicates the divergenceless fields are only mapped from the cone on dS hyperboloid.

For simplicity and irreducibility of vector-spinor field representation, the CI condition on the cone is imposed, this leads to , which is the conformally invariant condition on the de Sitter hyperboloid. Imposing this condition and irreducibility (see ( II.18)), one receives the following first and second order CI field equations

 (x/∂/T−1)Ψα(x)=0,and(Q(1)32+52)Ψα=0. (III.14)

In this case associates with the UIR of dS group, namely , and note that it propagates on the dS light cone. In the following sections, we find the solution and also obtain the two-point function of this vector-spinor field.

## Iv Solutions of the conformally invariant wave equations

A general solution of the field equation (III.14) can be written in terms of three spinor fields and as follows

 Ψα(x)=ZTαψ1+D(32)αψ2+γTαψ3, (IV.1)

where is an arbitrary five-component constant vector field and Now we should identify the introduced spinor fields and . If one demands that satisfies the second order field equation (III.14), then the spinor fields and obey the following equations

 (Q0+x/∂/T−3)ψ1=0,2(x⋅Z)ψ1+(Q0+x/∂/T)ψ2=0,[x/(Z.x)+Z/]ψ1+(Q0+x/∂/T)ψ3=0. (IV.2)

On the other hand, must satisfy the first order field equation (III.14), therefore one obtains:

 (x/∂/T−1)ψ1=0,(x/−2)ψ2−2x/ψ3=0,x/∂/Tψ3−(4x/+1)ψ2−x/x.Zψ1=0. (IV.3)

The second line of equation (IV.3) result to:

 ψ2=25(1−2x/)ψ3,orψ3=12(1+2x/)ψ2, (IV.4)

From the equations (IV.2) and (IV.3), satisfies the following first and second order field equations

 (Q0−2)ψ1=0,(x/∂/T−1)ψ1=0, (IV.5)

that indicates it can be regarded as a massless conformally coupled spinor field with homogeneity degree of and sep (). On the other hand, the homogeneity consideration reveals that the degrees of homogeneity of and are equal, note that all the sentences in (IV.1) have the same degrees of homogeneity and also the degrees of homogeneity and are zero.

Now let us multiply from the left on equations (IV.3), after doing some calculations, this yields

 Q0ψ2=25(1−2x/)ψ2+2[3x.Z−x/Z/T]ψ1, (IV.6) Q0ψ3=−25(2x/+7)ψ3+[3x.Z−x/Z/T]ψ1. (IV.7)

Inserting these results in equations (IV.2), and after making use of (IV.4), one can write and in term of as follows

 ψ2 = 12(1+3x/)[Z/+(1+3x/)x⋅Z]ψ1, (IV.8) ψ3 = −54(1−x/)[Z/+(1+3x/)x⋅Z]ψ1. (IV.9)

Using the divergencelessness condition and the above relations, one obtains

 (Z⋅∂T+3x⋅Z)ψ1+12(x/−3)[Z/+(1+3x/)x⋅Z]ψ1=0, (IV.10)

and making use of , leads one to write

 Z/Tψ1−(4x/+∂/T)ψ2+4ψ3=0,

and in terms of , this equation becomes

 Z/Tψ1+2(x/+5)[Z/+(1+3x/)x⋅Z]ψ1−25(1+2x/)x⋅Zψ1=0. (IV.11)

Actually what we have obtained is that by dealing with , the other two fields will be established as well. Consequently by gathering all the results, can be written as follows

 Ψα(x)=Dα(x,∂T,Z)ψ1, (IV.12)

where we have defined

 Dα(x,∂T,Z)≡ZTα+[12D(32)α(1+3x/)−54γTα(1−x/)][Z/+(1+3x/)x⋅Z].

In this formalism a given spin- field could be constructed from the multiplication of the polarization vector or ( degrees of freedom) with the spinor field ( degrees of freedom), which appears naturally polarization states. After making use of (IV.10) and (IV.11), the degrees of freedom are indeed reduced to the usual polarization states and , where two of them are the physical states ta1403 ().

Now, should be identified. Making use of the relation between the and spin-zero Casimir operators, equation (IV.5) can be written as

 (Q(1)12−12)ψ1(x)=0. (IV.13)

This means that and its related two-point functions can in fact be extracted from a massive spinor field in the principal series representation given by (II.8) by setting . Massive spinor field and its two-point functions has already been studied in sep (). Therefore, the solution of (IV.5) are found to be

 ψ1(x) = (x⋅ξ)−2V(x,ξ), ψ′1(x) = (x⋅ξ)−1U(ξ), (IV.14)

, these solutions are the de Sitter-Dirac plane waves. is a vector that lives in the positive sheet of the light cone, i.e.,

 ξ∈lC+={ξ;ηαβξαξβ=(ξ0)2−→ξ⋅→ξ−(ξ4)2=0,ξ0>0}.

plays the role of the energy momentum in the null curvature limit. Two spinors and are given by sep ()

 Ui(ξ)=ξ0−→ξ⋅→γγ0+1√2(ξ0+1)Ui(\lx@stackreloξ+),Vi(ξ)=1√2(ξ0+1)Ui(\lx@stackreloξ−),i=1,2, (IV.15)

where

 U1(\lx@stackreloξ+) = 1√2(χχ),U2(\lx@stackreloξ+)=1√2(χ′χ′), U1(\lx@stackreloξ−) = 1√2(χ−χ),U2(\lx@stackreloξ−)=1√2(χ′−χ′), (IV.16)

in which , and .

Therefore we have two solutions for which are as follows

 Ψ1α(x)=Dα(x,∂T,Z)V(x,ξ)(x⋅ξ)−2≡Vα(x,ξ,Z)(x⋅ξ)−2, (IV.17)

and

 Ψ2α(x)=Dα(x,∂T,Z)U(ξ)(x⋅ξ)−1≡Uα(x,ξ,Z)(x⋅ξ)−1, (IV.18)

After doing some calculation and taking the derivative, the explicit form of and are obtained in terms of and .

## V Two-point function in ambient space formalism

In this section, we deal with conformally invariant two-point functions of a massless spin- field. They are found in terms of the spinor (spin-1/2) two-point function. The two-point function of massless spin- field is given by

 Si¯jαα′(x,x′)=<Ω∣Ψiα(x)¯¯¯¯Ψ¯jα′(x′)∣Ω>, (V.1)

where , and stands for the vacuum state. Let us use the similar procedure of the previous section and write the desired two-point function in terms of three spinor two-point functions and impose some conditions to write down the two of them as a function of the third one. By using the equations (IV.1) and (V.1), the following form of two-point function is proposed

 Sαα′(x,x′)=θα⋅θ′α′S1(x,x′)−D(32)αS2(x,x′)γ4←D′(32)α′γ4−γTαS3(x,x′)γ4γ′Tα′γ4, (V.2)

where the prime operators act only on . This two-point function must satisfy the CI system of the field equations (III.14). If one demands that satisfies the second order field equation and using the identities of appendix B, the following relations are obtained

 (Q0+x/∂/T−3)S1(x,x′)=0, (V.3) 2(x⋅θ′α′)S1(x,x′)+(Q0+x/∂/T)S2(x,x′)γ4←D′(32)α′γ4=0, (V.4) [−3x/(x⋅θ′α′)+(θ′α′⋅γTα)]S1(x,x′)−(Q0+x/∂/T)S3(x,x′)γ4γ′Tα′γ4=0. (V.5)

On the other hand, must satisfy the first order equation, then one obtains

 (x/∂/T−1)S1(x,x′)=0,(x/−2)S2(x,x′)γ4←D′32α′γ4−2x/S3(x,x′)γ4γ′Tα′γ4=0,x/∂/TS3(x,x′)γ4γ′Tα′γ4−(x/+4)S2(x,x′)γ4←D′32α′γ4+xα′x/S1(x,x′)=0. (V.6)

We can write in term of and vice-versa by using the second equation (V.6) as fallows

 S2(x,x′)γ4←D′32α′γ4=25(1−2x/)S3(x,x′)γ4γ′Tα′γ4,
 S3(x,x′)γ4γ′Tα′γ4=12(1+2x/)S2(x,x′)γ4←D′32α′γ4. (V.7)

The first order equation of two-point function can be written as

 (x/∂/T−1)S1(x,x′)=0,
 (x/+2)∂/TS2(x,x′)γ4←D′32α′γ4+2(x/−8)S2(x,x′)γ4←D′32α′γ4+2xα′x/S1(x,x′)=0,
 x/∂/TS3(x,x′)γ4γ′Tα′γ4−25(6−7x/)S3(x,x′)γ4γ′Tα′γ4+xα′x/S1(x,x′)=0. (V.8)

The equation governing can be deduced from the equation (V.3) and the first equation of (V.6) as

 (Q0−2)S1(x,x′)=0,(x/∂/T−1)S1(x,x′)=0. (V.9)

In Ref. sep (), the solutions have been obtained, here, we only quote the result

 S1(x,x′)=i2(4π)2[P(7)−3(x⋅x′)x/−3P(7)−1(x⋅x′)x/′]γ4, (V.10)

are the generalized Legendre functions (see appendix D). Similar to the previous section, by acting on the first order equation (V.8), one finds the second order equation for and as follows

 Q0S2(x,x′)γ4←D′32α′γ4=−2(3x/+20)S2(x,x′)γ4←D′32α′γ4+25[2xα′(1+7x/)+(1−2x/)γTα′]S1(x,x′), (V.11)
 Q0S3(x,x′)γ4γ′Tα′γ4=25(7x/−16)S3(x,x′)γ4γ′Tα′γ4+[25xα′(6x/−7)+γTα′]S1(x,x′). (V.12)

After making use of the above relations in (V.5), one can write and in terms of as follows

 S2(x,x′)γ4←D′32α′γ4=−110(1−2x/)[−3x/(x.θ′α′)+(θ′α′.γTα)−15xα′(7x/−14)−γTα′]S1(x,x′),