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An Introduction to Clifford Algebras and Spinors$

Jayme Vaz, Jr. and Roldão da Rocha, Jr.

Print publication date: 2016

Print ISBN-13: 9780198782926

Published to Oxford Scholarship Online: August 2016

DOI: 10.1093/acprof:oso/9780198782926.001.0001

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(p.209) Appendix A The Standard Two-Component Spinor Formalism

(p.209) Appendix A The Standard Two-Component Spinor Formalism

Source:
An Introduction to Clifford Algebras and Spinors
Author(s):

Jayme Vaz

Roldão da Rocha

Publisher:
Oxford University Press

In chapter 6, we presented Weyl spinors in C3,0; now, the connection with the ordinary notation in either field theory or supersymmetry books is briefly presented.

A spacetime vector vR1,3 can be expressed as v=x0e0+x1e1+x2e2+x3e3, where (x0,x1,x2,x3) denotes components of v with respect to an orthonormal basis {e0,e1,e2,e3}. Null vectors are isotropic vectors and satisfy (x0)2(x1)2(x2)2(x3)2=0. They present null directions in R1,3 with respect to the origin O of an arbitrary frame in R1,3.

The space of null directions that are future (past) pointed are denoted by S+ [S], and represented by the intersections E+ [E] of the future (past) light cones with the hyperplanes x0=1 (x0=1).1 The space S± is a sphere with the equation x2+y2+z2=1, where (x,y,z) are coordinates in E± Penrose and Rindler 1984.

Generally, the direction of any null vector vR1,3, unless the vector is an element of the plane defined by the equation x0=0, can be represented by two points. This description results from the intersection of v and the hyperplanes x0=±1. The future-pointed v is thus represented by (x1/x0,x2/x0,x3/x0). The inner points of E+ (E) represent the set of future-pointed (past-pointed) light-like directions.

By considering E+ and by performing a stereographic projection on the Argand–Gauss plane, we obtain a representation of the union between the set of complex numbers and the point at infinity, the latter corresponding to the north pole of E+.

By defining the complex number

(A.1)
β=x+iy1z,

we obtain ββ¯=x2+y2(1z)2 and, consequently,

(A.2)
x=β+β¯ββ¯+1,y=ββ¯i(ββ¯+1),z=ββ¯1ββ¯+1.

The correspondence between points of E+ and the Argand–Gauss plane is injective if the point β is added to the complex plane, making it correspond to the north pole with components (1,0,0,1). However, to avoid this point, it is convenient to associate a point of E+ not to a complex number β‎ but to a pair of complex numbers2 (ξ,η), where (p.210)

(A.3)
β=ξ/η.

The pairs (ξ,η) and (λξ,λη), where λC, represent the same point in E+. Such components are called projective coordinates.

The point β=ξ/η corresponds to the point of coordinates ξη=10. The equations in (A.2) can then be expressed as

(A.4)
x=ξη¯+ηξ¯ξξ¯+ηη¯,y=ξη¯ηξ¯i(ξξ¯+ηη¯),z=ξξ¯ηη¯ξξ¯+ηη¯.

The point P=(1,x,y,z) is an arbitrary point of the light-cone transversal section with constant time and represents a null future-pointed direction, which can be represented by any point of the line OP. In particular, if a point R is taken in the line OP by multiplying P by the factor (ξξ¯+ηη¯)/2, then R has coordinates

(A.5)
x1=12(ξη¯+ηξ¯), x2=1i2(ξη¯ηξ¯),x3=12(ξξ¯ηη¯), x0=12(ξξ¯+ηη¯).

Unlike the point P, the point R is not invariant under (ξ,η)(rξ,rη),rR, although it is independent of phases (ξ,η)(eiθξ,eiθη),θR.

Consider now the following complex linear transformation

(A.6)
ξξ˜=αξ+μη,ηη˜=γξ+δη,

where α,μ,γ,δC satisfy αδμγ0, so that the transformation is invertible. It can be rewritten as

(A.7)
βf(β)=αβ+μγβ+δ,

and called a Möbius transformation, from the set C{δ/γ} to C{α/γ}. Moreover, if f(δ/γ), and f()α/γ, then f is an injective function from the complex plane, compactified by the point at the infinity; this point is denoted by (C{}).

Hence, the space of light-like vectors on Minkowski spacetime is naturally a Riemann sphere. The restricted Lorentz group L+ is, on the other hand, the automorphism group of the Riemann sphere. The equations in (A.6), when taken with the condition

αδμγ=1

are called spinor transformations, where β= ξ/η is related to the null vectors by the equations in (A.5), implying that

(A.8)
β=x1+ix2x0x3=x0x3x1ix2.

(p.211) The spinor matrix A SL(2,C) is defined as

(A.9)
A=αμγδ,det A=1.

The equations in (A.6), with respect to A, read

ξ˜η˜=A ξη.

The spinor matrices {±A} induce the same transformation of β=ξ/η. The equations in (A.5) yield

(A.10)
12x0+x3x1+ix2x1ix2x0x3=ξξ¯ ξη¯ηξ¯ηη¯=ξη(ξ¯η¯).

Hence, up to a factor 1/2, it follows that

(A.11)
(x0+x3x1+ix2x1ix2x0x3)  (x0˜+x3˜x1˜+ix2˜x1˜ix2˜x0˜ x3˜)=A (x0+x3x1+ix2x1ix2x0x3) A.

The transformation acting on the point v=(x0,x1,x2,x3) is real and preserves the light-cone structure (x0)2(x1)2(x2)2(x3)2=0. Thus, this relation defines a restricted Lorentz transformation.

Hence, the group SL(2, C) is the twofold covering of the restricted Lorentz group SO+(1,3)L+.

A more general case than this is the spin space Gα, which has three basic operations (Penrose and Rindler, 1984):

  • multiplication by scalars: C×GαGα (λC,kαGαλkαGα)

  • sum: Gα×GαGα (kα,ωαGαkα+ωαGα)

  • scalar product: Gα×GαC (kα,ωαGα{kα,ωα}C)

The dual spin space Gα is similarly defined: Gαπα:GαC. Thus, kαωα{kα,ωα}C. With respect to the null vectors of Minkowski spacetime, Penrose proposed that the algebra of null vectors must be contained in the algebra of spinors. The spin space Gα˙ is defined by the application ϱ:GαGα˙ such that

ϱ(kα+ωα)=ϱ(kα)+ϱ(ωα),ϱ(λkα)=λˉϱ(kα),kα,ωαGα,λC (λˉ is the C-conjugate of λ).

Instead of by the notation ϱ(kα), this transformation is denoted by kα¯, according to the Penrose notation, characterised by the composition between the C-conjugation and the transposition. From now on, the notation

(A.12)
kα¯kˉα˙Gα˙

(p.212) shall be used. From (A.10), the space R1,3 endowed with the coordinates (x0,x1,x2,x3) can be expressed from the components of the spin vector k (ξ=k0,η=k1). The basic operations in this space are defined by

(A.13)
λ(k0,k1)=(λk0,λk1),

(A.14)
(k0,k1)+(ω0,ω1)=(k0+ω0,k1+ω1),

(A.15)
{(k0,k1),(ω0,ω1)}=k0ω1k1ω0.

With the antisymmetric bilinear form in eqn (A.15), the representation of a spin vector is given by the choice of a pair of normalised spin vectors oα and ια:

(A.16)
{oα,ια}=oαια=1=ιαoα={ια,oα},

where (oα,ια) stands for the dual pair of (oα,ια). Moreover, the antisymmetry of (A.15) implies that oαoα=ιαια=0. The pair (oα,ια), with the condition in eqn (A.16), is called the spin basis, and the components of k with respect to the spin basis are provided by (Figueiredo, de Oliveira, and Rodrigues, 1990)

(A.17)
k0={kα,ια},k1={kα,oα}.

Hence, kα=k0oα+k1ια.

The antisymmetric element

(A.18)
Gαβ αβ:GαGβ       kαkβ=kααβ,

is responsible for lowering and raising indices, such that

(A.19)
{kα,ωα}=ϵαβkαωβ=k0ω1k1ω0={ωα,kα}.

We can write

(A.20)
ϵαβ=oαιβιαoβ

since, for any spin basis, it follows that

(A.21)
kα=k0oα+k1ια=(oαιβιαoβ)(k0oβ+k1ιβ)   =ϵαβkβ.

Here, Gαβ denotes the tensor product GαGβ. The dual tensor Gαβϵαβ:GβGα satisfies

(A.22)
ϵαβϵγβ=δα γ,

where GαGβGγ is defined. Similarly, we can express

(A.23)
ϵα β=oαιβιαoβ,

and

(A.24)
{k,ω}=kβωβ=k0ω0+k1ω1=k0ω1ω0k1k0=k1k1=k0.

Now, given arbitrary spinors kα,ωαGα such that kαωα=1, we can write (p.213)

(A.25)
ϵαβ=kαωβωαkβ=(k0ια+k1oα)(ω0ιβ+ω1oβ) =(k1ω0k0ω1)oαιβ(k1ω0k0ω1)ιαoβ =oαιβιαoβ,

where the equivalence to eqn (A.20) is then accomplished.

Now the spacetime metric and the spacetime vectors as well can be constructed from spin vectors. Spacetime vectors of R1,3 present a spinor description via spin vectors. Latin indices are employed here to label the elements xμ of a real vector space, when the indices α‎ and α˙ are grouped together. Moreover, the notation {a,b,}={αα˙,ββ˙,} will be used. The null tetrad (lμ,nμ,mμ,m¯μ)

(A.26)
lμ=oαoα˙,nμ=ιαια˙,mμ=oαια˙,m¯μ=ιαoα˙

and the metric ημν=ϵαβϵα˙β˙ can be defined by verifying that the vectors of the null tetrad are null vectors with respect to ηab:

(A.27)
ημνlμlν=lμlμ=0.

Similarly, nμnμ=mμmμ=m¯μm¯μ=0. In addition, the following expressions hold:

(A.28)
lμnμ=1,  mμm¯μ=1,  lμmμ=lμm¯μ=nμmμ=nμm¯μ=0,ημνημρgρν=nμlν+lμnνm¯μmνmμm¯ν.

It is sometimes convenient to define another tetrad (tμ,xμ,yμ,zμ) as

(A.29)
tμ=12(lμ+nμ)=12(oαoα˙+ιαια˙),xμ=12(mμ+m¯μ)=12(oαια˙+ιαoα˙),yμ=i2(mμ+m¯μ)=i2(oαια˙ιαoα˙),zμ=12(lμnμ)=12(oαoα˙ιαια˙).

We obtain from (A.28) that

(A.30)
tμxμ=tμyμ=tμzμ=xμyμ=yμzμ=zμxμ=0,tμtμ=1=xμxμ=yμyμ=zμzμ.

Consequently, the metric components ημ ν (A.28) have the form ημ ν=tμtνxμxνyμyνzμzν, thus identifying the tetrad (tμ,xμ,yμ,zμ) as the Minkowski tetrad. Hence, we can write

(A.31)
Kμ=K0tμ+K1xμ+K2yμ+K3zμ.

By considering the spin basis {oα,ια}, we can express the vector Kμ with respect to the basis as

(A.32)
Kμ=K00˙lμ+K01˙mμ+K10˙m¯μ+K11˙nμ.

When the two last equations are compared, it follows that (p.214)

(A.33)
Kμ=12K0+K3K1+iK2K1iK2K0K3=K00˙K01˙K10˙K11˙.

Thus,

(A.34)
Kμ=±kαkˉα˙,

where the sign defines a future (+) (past (−)). The vector Kμ is real and null, since KμKμ=|kαkα|2=0. If ξ=k0, and η=k1, it follows that

(A.35)
K00˙=ξξ¯,K01˙=ξη¯,K10˙=ηξ¯,K11˙=ηη¯,

which causes eqn (A.33) to be led to

(A.36)
12x0+x3x1+ix2x1ix2x0x3=ξξ¯ ξη¯ηξ¯ηη¯=ξη(ξ¯η¯),

where x0=K0; x1=K1; x2=K2; and x3=K3.

A.1 Weyl Spinors

Given the formalism presented for spin vectors, which are known as 2-spinors, it is also necessary to study then from the point of view of representations of the Lorentz group SL (2,C), the 2-fold covering of the restricted Lorentz group L+SO+(1,3). Linear transformations with a unit determinant, with respect to the spin space, determine the group SL(2,C). We have already shown that there are two non-equivalent representations of SL(2,C); these are denoted by D(1/2,0) and D(0,1/2), respectively, and the elements of the carrier space associated with them are called Weyl spinors.

Both the left-handed D(1/2,0) and the right-handed D(0,1/2) representations of the Lorentz group determine the rules of transformation obeyed by fermions of spin-1/2. It is well known that the Hermitian conjugation can be used to interchange these two representations. Dirac spinors take into account reducible representations of the form D(1/2,0)D(0,1/2). From here on, D(1/2,0) spinors will carry undotted indices α,β,=1,2, and D(0,1/2) spinors will carry dotted indices α˙,β˙,=1,2.

A.2 Contravariant Undotted Spinors

Contravariant undotted spinors are elements of a complex two-dimensional space endowed with the spinor metric

(A.37)
G:2×2,   (ζ,χ)G(ζ,χ)=ζJχ,

where

(A.38)
J=0110.

The spinor ζ‎ is represented by the column vector

(A.39)
ζ=ζ1ζ2.

This spinor can be identified with its algebraic counterpart in eqn (6.154).

(p.215) It also carries the D(1/2,0) representation of SL(2,C) and is transformed under RSL(2,C) as

ζRζ.

Moreover, this transformation corresponds to the rule established in chapter 6, eqn (6.162).

A.3 Covariant Undotted Spinors

Covariant undotted spinors are elements of a complex two-dimensional dual space C2, and are defined by

(A.40)
2ζ:2,    χζ(χ)=ζχ=G(ζ,χ)=ζJχ,

which implies that ζ=ζJ. Hence, the spinor ζ is represented by

(A.41)
ζ=(ζ1,ζ2)=(ζ2,ζ1).

This spinor can be identified with its algebraic counterpart in eqn (6.155). For a spinor metric to be invariant under RSL(2,C), it is necessary that

(A.42)
ζζR1,RSL(2,C),

which corresponds to the transformation in eqn (6.163).

Contravariant undotted spinors and covariant undotted spinors represent respectively elements of Gα and Gα.

A.4 Contravariant Dotted Spinors

Covariant dotted spinors are elements of a complex two-dimensional space (C˙2)ζ˙, ζC2, endowed with the spinor metric

(A.43)
G˙:2.×2.,  (ζ˙,χ˙)G˙(ζ˙,χ˙)=ζ˙Jχ˙.

A covariant dotted spinor is represented by ζ˙=(ζ¯1˙,ζ¯2˙) and can be identified with its algebraic counterpart in eqn (6.158).

(p.216) A.5 Covariant Dotted Spinors

Covariant dotted spinors are elements of a complex two-dimensional dual space C˙2, which is defined by

(A.44)
2.χ˙:2.,     ζ˙ζ˙(χ˙)=ζ˙χ˙=G˙(ζ˙,χ˙)=ζ˙Jχ˙,

which implies that χ˙=J(χ˙); a covariant dotted spinor is represented by

(A.45)
χ˙=χ¯1˙χ¯2˙=χ¯2˙χ¯1˙.

Hence, we can identify it with its algebraic equivalent counterpart in eqn (6.160).

Clearly, the transformation rule for dotted spinors under the transformation RSL(2,C) is provided by

(A.46)
ζ˙ζ˙R,ζ˙(R)1ζ˙,

carrying the D(0,1/2) representation of SL(2,C). In fact, it corresponds to the transformation in eqn (6.164).

Dotted spinors are elements of Gα˙ and Gα˙. The action of the Lorentz group on Weyl spinors can be depicted as follows:

(A.47)
ζRζ,

(A.48)
ζζR1,

(A.49)
ζ˙ζ˙R,

(A.50)
ζ˙(R)1ζ˙.

These transformations are emulated in eqns (6.1626.165), in the algebraic spinor framework.

From 2-spinors, Dirac spinors can defined as elements of GαGα˙. They are classically realised as elements of C4, equipped with the spinor metric

(A.51)
G:4×4 (ψ1,ψ2)G(ψ1,ψ2)=ψ1Jdψ2,

where ψ‎ is defined by

(A.52)
C2(C˙2)ψ=ζ+χ˙=ζ1ζ2χˉ1˙χˉ2˙.

With respect to the standard basis of C4, the matrix Jd is the representation Jd=diag(J,J), where J denotes the symplectic matrix defined by eqn (A.38).

(p.217) Dirac spinors carry the D(1/2,0)D(0,1/2) representation of SL(2,C). Under the condition G(ψ1,ψ2)=G(ρ(R)ψ1,ρ(R)ψ2), requiring that the spinor metric G be invariant under R, the following important representation is obtained:

(A.53)
ρ(R)=R00(R)1,RSL(2,C).

A.6 Null Flags and Flagpoles

This section describes the classical framework corresponding to the algebraic formulation provided by section 6.9.

We have already associated, via eqn (A.34), a future-pointed null vector, which contains components Kμ, with the spin-vector kα, which has coordinates (ξ,η). From eqn (A.35), the 2-uple (ξ,η) can be further identified as the coordinates associated with the components Kμ, which are invariant under transformations (ξ,η)(eiθξ,eiθη). Moreover, this ambiguity can be reduced up to a sign by introducing a structure that is composed of a null vector K=Kμeμ, called a pole, and a null half-plane – tangent to the light cone and having K as the intersection – called a flagpole.

Given a contravariant undotted spinor kα, a geometric object can be constructed, namely, the flagpole. As in eqn (A.36), we shall change the notation by establishing Kμ=xμ. The pole is defined by eqn (A.34), namely,

xμ=12xαα˙=kαkˉα˙=12x0+x3x1+ix2x1ix2x0x3=k1kˉ1˙ k1kˉ2˙k2kˉ1˙k2kˉ2˙.

The vector xμ is dilated by ρ2, when kα is multiplied by λ=ρeiθ. Notwithstanding, the vector xμ does not change its direction and is independent of the choice of θ‎. Hence, the null vector xμ is uniquely determined by the spinor kα. However, the spinor kα is not uniquely determined by xμ, which corresponds to a family of spinors. They form a projective space and differ from each other by a phase eiθ.

The momentum is defined as

(A.54)
Fμν=Fαβα˙β˙=kαkβϵα˙β˙+ϵαβkˉα˙kˉβ˙.

The antisymmetric tensor Fab is real and determines a half-plane that is tangent to the light cone along the vector xμ=kαkˉα˙.

By taking a spin basis {kα,ωα}, where kαωα=1, we find that ϵαβ=kαωβωβkα. Thus, the quantity Fμν can be characterised as the angular momentum, since

(A.55)
Fμν=Fαβα˙β˙=kαkβ(k¯α˙ω¯β˙ω¯α˙k¯β˙)+(kαωβωαkβ)k¯α˙k¯β˙ =kαk¯α˙(kβω¯β˙ωβk¯β˙)+(kαω¯α˙+ωαk¯α˙)kβk¯β˙=Xαα˙Yββ˙Yαα˙Xββ˙ =xμyνyμxν.

The tensor Fab hence represents a bivector constituted of two vectors with components xμ and yμ in R1,3. The pole xμ is the null flagpole vector, uniquely determined by the spinor oα. The second vector, given by (p.218)

(A.56)
yμ=Yαα˙=(kαωˉα˙+ωαkˉα˙),

is also determined by kα, although not uniquely, since the pair (kα,ωα) is not the only way to construct ϵαβ. Indeed, any spinor of type

(A.57)
ω0α=ωα+λkα,λC

satisfies kαω0α=1. With this freedom, the vector yμ transforms as y0μ=yμ+(λ+λˉ)xμ. Each scalar (λ+λˉ) can be thus associated to a family of coplanar vectors y0μ. This is the flagpole, as proposed by Penrose. Some prominent properties can be now derived.

The vector yμ is orthogonal to the null vector xμ. Indeed,

(A.58)
xμyμ=xμyμ=12Xαα˙Yαα˙=12kαk¯α˙(kαω¯α˙+ωαk¯α˙)=0.

Moreover, yμ is a space-like unit vector. In fact,

(A.59)
yμyμ=yμyμ=12(kαω¯α˙+ωαk¯α˙)(kαω¯α˙+ωαk¯α˙) =12(kαωα)(ω¯α˙k¯α˙)+12(ωαkα)(k¯α˙ω¯α˙)=1.

By multiplying the spinor kα by eiθ, the vector yμ spins around the pole by the angle 2θ. Actually, we have

(A.60)
yrotμ=YrotAα˙=e2iθkαω¯α˙+e2iθωαk¯α˙ =cos2θ(kαω¯α˙+ωαk¯α˙)+sin2θ(ikαω¯α˙iωαk¯α˙) =yμcos2θ+zμsin2θ.

In addition, zμ=i(kαωˉα˙ωαkˉα˙) is a space-like unit vector, orthogonal to the vectors xμ and yμ. Together with yμ, it constitutes the flagpole.

In order to fix the notation, we know that two-dimensional spinor representations of the Lorentz group can be derived from the property that, under a Lorentz transformation, a contravariant 4-vector xμ transforms as xμxμ=Rμνxν, where R SO(1,3) satisfies RμνημρRρλ=ηνλ. The corresponding covariant 4-vector xμημνxν satisfies xν=xμRμν. The most general proper orthochronous Lorentz transformation, corresponding to a rotation by an angle of θ‎ about an axis nˆ, where θ=θnˆ, and a boost vector ζvˆtanh1β, where vˆ=v/v and where β=v, is a 4×4 matrix given by

(A.61)
R=expi2θρσSρσ=expiθSiζK,

where θi=12ϵijkθjk; ζi=θi0=θ0i; Si=12ϵijkSjk; Ki=S0i=Si0; and

(A.62)
(Sρσ)μ ν=i(ηρ μ ησνησ μ ηρν).

Here, the indices i,j,k=1,2,3, and ϵ123=+1 (Dreiner, Haber, and Martin, 2010).

It follows from (A.61, A.62) that an infinitesimal orthochronous Lorentz transformation is given by Rμνδνμ+θμν. Moreover, the infinitesimal boost parameter reads (p.219) βvˆ, since β1 for an infinitesimal boost. Hence, the actions of the infinitesimal boosts and rotations on the spacetime coordinates are respectively given by

(A.63)
xxx+(θ×x), (ttt),xxx+βt, (ttt+βx).

For contravariant 4-vectors, the reasoning is similar.

With respect to the Lorentz transformation R, a general n-component field Φ‎ transforms according to a representation R of the Lorentz group as Φ(xμ)Φ(xμ)=[R]Φ(xμ), where [R] is the corresponding (finite) d-dimensional matrix representation. Equivalently, the functional form of the transformed field Φ‎ obeys

(A.64)
Φ(xμ)=[R]Φ([R1]μνxν).

For proper orthochronous Lorentz transformations,

(A.65)
R=expi2θμνJμνId×diθJiζK,

where Id×d is the d×d identity matrix, and θμν parameterises the Lorentz transformation R by (A.61). The six independent components of the matrix-valued antisymmetric tensor Jμν are the d-dimensional generators of the Lorentz group and satisfy the commutation relations

(A.66)
[Jμν,Jλκ]=i(gμκJνλ+gνλJμκgμλJνκgνκJμλ).

The vectors J and K are defined as the generators of rotations parameterised by θ and the boosts parameterised by ζ, respectively, where Ji=12ϵijkJjk, and Ki=J0i.

Here, we focus on the inequivalent non-trivial irreducible representations of the Lorentz algebra D(1/2,0) and D(0,1/2). In the D(1/2,0) representation, J=σ/2, and K=iσ/2, in eq. (A.65), so

(A.67)
R(12,0)RI2×2iθσ/2ζσ/2,

where σ=(σ1,σ2,σ3) represents the Pauli matrices. The transformation R carries undotted spinor indices, as indicated by Rαβ. A two-component spinor in the D(1/2,0) representation is already denoted by ψα, which transforms as ψαRαβψβ.

On the other hand, in the D(0,1/2) representation, J=σ/2, and K=iσ/2, in eqn (A.65). Hence, its representation matrix is R, the complex conjugate of eqn (A.67). By definition, the indices carried by R are dotted, as indicated by (R)α˙β˙. It is already known that a two-component D(0,1/2) spinor ψα˙ transforms as ψα˙(R)α˙ β˙ψβ˙.

It follows that the D(1/2,0) and D(0,1/2) representations are related by Hermitian conjugation. In fact, if ψα denotes a D(1/2,0) spinor, then (ψα) transforms as a D(0,1/2) spinor. In combining spinors to make Lorentz tensors, it is useful to regard ψα˙ as a row vector, and ψα as a column vector, with

(A.68)
ψα˙(ψα).

The Lorentz transformation property of ψα˙ then follows from (ψα)(ψβ)(R)β˙α˙, where (R)β˙α˙=(R)α˙β˙.

(p.220) In the dotted-index notation, the dagger is used to denote Hermitian conjugation, as in (A.68). In fact, the dagger is used to denote the Hermitian conjugation of spinors in most textbooks (Srednicki, 2007). However, it is worth emphasising that many references in supersymmetry e.g. (Sohnius, 1985; Srivastava, 1986; West, 1990; Wess and Bagger, 1992; Bailin and Love, 1994; Mohapatra, 2003) employ the Wess and Bagger (1992) notation, where ψ¯α˙ψα˙=(ψα).

There are two additional spin-1/2 irreducible representations of the Lorentz group, namely, (R1) and (R1), However, they are equivalent to the D(1/2,0) and the D(0,1/2) representations, respectively. The spinors that transform under these representations have the raised spinor indices ψα and ψα˙, respectively, with the transformation laws ψα[(R1)]αβψβ, and ψα˙[(R1)]α˙β˙ψβ˙, respectively. Lorentz tensors can be derived from spinors by regarding ψα as a row vector, and ψα˙ as a column vector, with

(A.69)
ψα˙(ψα).

The Lorentz transformation property ψα˙ then follows from

(ψα)[(R1)]α˙β˙(ψβ).

The spinor indices are raised and lowered with the two-index antisymmetric epsilon symbol with non-zero components ϵ12=ϵ21=ϵ21=ϵ12=1, and similar sign conventions for the dotted spinor indices. In particular, ϵα˙β˙=(ϵαβ), and ϵα˙β˙=(ϵαβ), as well.

Moreover, the Kronecker delta symbol reads δα˙β˙=(δαβ). The epsilon symbols with undotted and with dotted indices, respectively, satisfy

(A.70)
ϵαβϵγδ=δαγδβδ+δαδδβγ,ϵα˙β˙ϵγ˙δ˙=δα˙γ˙δβ˙δ˙+δα˙δ˙δβ˙γ˙,

yielding the so-called Schouten identities

(A.71)
ϵαβϵβγ=ϵγβϵβα=δαγ,

(A.72)
ϵαβϵγδ+ϵαγϵδβ+ϵαδϵβγ=0.

The same equations hold for dotted indices. To construct Lorentz invariant Lagrangians and observables, in particular, Lorentz vectors are obtained by introducing the sigma matrices σαβ˙μ and σˉμα˙β defined by

(A.73)
σ0=σ¯0=(1001),  σ1=σ¯1=(0110),σ2=σ¯2=(0ii0),  σ3=σ¯3=(1001).

Hence, eqn (A.73) is equivalent to σμ=(I2×2,σ), and σˉμ=(I2×2,σ), which can be related by (p.221)

(A.74)
σαα˙μ=ϵαβϵα˙β˙σ¯μ β˙β, σ¯μ α˙α=ϵαβϵα˙β˙σββ˙μ,ϵαβσβα˙μ=ϵα˙β˙σ¯μβ˙α, ϵα˙β˙σαβ˙μ=ϵαβσ¯μα˙β.

There is a one-to-one correspondence between each 2-spinor construction Vαβ˙ and the associated Lorentz 4-vector Vμ, provided by the Infeld–van der Waerden symbols

(A.75)
Vμ=12σˉμβ˙αVαβ˙,Vαβ˙=Vμσμαβ˙.

In particular, if Vμ is a real 4-vector, then Vαβ˙ is Hermitian. Moreover, it is often useful to further simplify the notation by defining Vα˙β=(Vαβ˙). In this notation, an Hermitian 2-spinor satisfies Vαβ˙=Vα˙β. Then,

(V)αβ˙=Vβα˙,(V)α˙β=(Vαβ˙),(V)αβ˙=(Vβα˙)=(V)β˙α.

A Hermitian 2-spinor satisfies V=V or, equivalently, Vαβ˙=(V)β˙α.

In addition, 2-spinors can be interpreted as 2×2 matrices. It is indeed convenient to define the following:

(V)α β=Vβ α, (V)α˙ β˙=(Vα β), (V)β˙ α˙=(Vα β)=(V)α˙ β˙.

Note that the matrix transposition of Vαβ interchanges the rows and columns of W without modifying the relative heights of the α‎ and β‎ indices. Similar results hold for Vαβ and Vαβ by either lowering or raising the spinor indices.

For an anti-commuting two-component spinor ψ‎, the product ψαψβ is antisymmetric with respect to the interchange of the spinor indices α‎ and β‎. Hence, it must be proportional to ϵαβ. Similar conclusions hold for the corresponding spinor products with raised undotted indices and with lowered and raised dotted indices, respectively. Thus,

(A.76)
ψαψβ=12ϵαβψψ,  ψαψβ=12ϵαβψψ,ψα˙ψβ˙=12ϵα˙β˙ψψ,  ψα˙ψβ˙=12ϵα˙β˙ψψ,

where ψψ=ψαψα and ψψ=ψα˙ψα˙.

The van der Waerden symbols in eqns (A.75) provide antisymmetrised products, from the sigma matrices (Dreiner, Haber, and Martin, 2010):

(A.77)
(σμν)α β=i4(σαρ˙μσ¯νρ˙βσαρ˙νσ¯μρ˙β),

(A.78)
(σ¯μν)α˙ β˙=i4(σ¯μα˙ρ σρβ˙νσ¯να˙ρ σρβ˙μ).

Now we can introduce the infinitesimal forms for the 4×4 Lorentz transformation matrix; the corresponding matrices R and (R1) which transform the D(12,0) and D(0,12) spinors, respectively, are given by (p.222)

(A.79)
Rμ νδνμ+12(θανgαμθνβgβμ),

(A.80)
RI2×212iθμνσμν,

(A.81)
(R1)I2×212iθμνσ¯μν.

The inverses of these quantities are obtained up to the first order in θ‎ by replacing θθ in the formulæ. Equations (A.80) and (A.81) yield

(A.82)
(R1)ρ σ=ϵσαRα βϵβρ,    (R1)ρ˙ σ˙=ϵσ˙α˙(R)α˙ β˙ ϵβ˙ρ˙.

These results prove the covariance of the spinor index raising and lowering properties of the epsilon symbols. The infinitesimal forms given by (A.79) and (A.81) imply that

(A.83)
Rσ¯μR=Rμ νσ¯ν,    R1σμ(R1)=Rμ ν σν.

Using the Lorentz transformation properties of the undotted and dotted two-component spinor fields, eqn (A.83) yields the proof that the spinor products ξσˉμη and ξσμη transform as Lorentz 4-vectors.

The usual framework use in field theory regards a pure boost from the rest frame to a frame where pμ=(Ep,p), which corresponds to θij=0, and ζi=θi0=θ0i. The so-called mass-shell condition is satisfied: p0=Ep=(p2+m2)1/2. The matrices Rαβ and [(R1)]α˙β˙, which describe Lorentz transformations of spinor fields, are given, respectively, for the D(1/2,0) and D(0,1/2) representations by

(A.84)
exp(i2θμνJμν)={R=exp(12ζσ)=pσm,(R1)=exp(12ζσ)=pσ¯m,

where

(A.85)
pσ=(Ep+m) I2×2σp2(Ep+m),pσ¯=(Ep+m) I2×2+σp2(Ep+m).

According to (A.84), the spinor index structure of pσ and pσˉ corresponds to that of Rαβ and [(R1)]α˙β˙, respectively. Hence, the equations in (A.85) yield

(A.86)
[pσ]α β=[pσ σ¯0]α β=(pσαα˙)σ¯0 α˙β+mδαβ2(Ep+m),

(A.87)
[pσ¯]α˙ β˙=[pσ¯ σ0]α˙ β˙=(pσ¯α˙α)σαβ˙0+mδβ˙α˙2(Ep+m),

since σ0=σˉ0=I2×2.

(p.223) A.7 The Supersymmetry Algebra

The two operators ϕa and ϕb in a graded Lie algebra satisfy

ϕaϕb(1)|a||b|ϕbϕa=Cabdϕd,

where either |a|=0 for even (bosonic) ϕa or |a|=1 for odd (fermionic) operators and the Cabd denote the algebra structure constants. The Poincaré generators Jμν in eqn (A.66), together with the Pμ, are bosonic generators. Nevertheless, in supersymmetry, fermionic generators QαA and Qˉα˙β are introduced, respectively denoting elements of the D(12,0) and D(0,12) representations of the Lorentz group, and A,B=1,,N label the number of supercharges (West, 1990; Wess and Bagger, 1992).

For N=1, the supersymmetry algebra reads

(A.88)
[Mμν,Mσρ]=i(Mμνηνρ+MνρημσMμρηνσMνσημρ),

(A.89)
[Pμ,Pν]=0=[Qα,Pμ]={Qα,Qβ},

(A.90)
[Mμν,Pσ]=i(PμηνσPνημσ),

(A.91)
[Qα,Mμν]=(σμν)αβQβ,

(A.92)
{Qα,Q¯β˙}=2(σμ)αβ˙Pμ.

The first three equations describe the usual Poincaré algebra.

A spinor Qα transforms under an infinitesimal Lorentz transformation as

(A.93)
QαQα=(ei2ωμνσμν)αβQβ(Ii2ωμνσμν)αβQβ.

From the operator point of view, it transforms, by denoting U=ei2ωμνMμν, as

(A.94)
QαQα=UQαUI+i2ωμνMμνQαIi2ωμνMμν.

When eqn (A.94) is compared to eqn (A.93), eqn (A.91) can be derived. Indeed,

(A.95)
Qαi2ωμν(σμν)α βQβ=Qαi2ωμν(QαMμνMμνQα)+O(ω2),

which implies that

(A.96)
[Qα,Mμν]=(σμν)α βQβ.

The commutator for the right-handed representation reads

(A.97)
[Qˉα˙,Mμν]=(σˉμν) β˙α˙Qˉβ˙.

For more detail, the reader can see, for example, the references by West (1990), Wess and Bagger (1992), Bailin and Love (1994), and Dreiner, Haber, and Martin (2010). Moreover, the Clifford–Hopf algebra associated with the super-Poincaré algebra was formulated in the article by da Rocha, Bernardini, and Vaz Jr (2010).

Notes:

(1) This space is a Riemann sphere.

(2) With the condition that both numbers are not simultaneously equal to 0.