## 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

Source:
An Introduction to Clifford Algebras and Spinors
Publisher:
Oxford University Press

In chapter 6, we presented Weyl spinors in $Cℓ3,0$; now, the connection with the ordinary notation in either field theory or supersymmetry books is briefly presented.

A spacetime vector $v∈R1,3$ 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 $v∈R1,3$, 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)
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we obtain $ββ¯=x2+y2(1−z)2$ and, consequently,

(A.2)
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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)
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The pairs $(ξ,η)$ and $(λξ,λη),$ where $λ∈C$, 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)
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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)
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Unlike the point P, the point R is not invariant under $(ξ,η)↦(rξ,rη),r∈R$, although it is independent of phases $(ξ,η)↦(eiθξ,eiθη),θ∈R$.

Consider now the following complex linear transformation

(A.6)
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where $α,μ,γ,δ∈C$ satisfy $αδ−μγ≠0$, so that the transformation is invertible. It can be rewritten as

(A.7)
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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

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

(A.8)
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(p.211) The spinor matrix is defined as

(A.9)
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The equations in (A.6), with respect to $A$, read

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

(A.10)
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Hence, up to a factor $1/2$, it follows that

(A.11)
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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: $C×Gα→Gα$ ($λ∈C,kα∈Gα↦λkα∈Gα$)

• sum: $Gα×Gα→Gα$ ($kα,ωα∈Gα↦kα+ωα∈Gα$)

• scalar product: $Gα×Gα→C$ ($kα,ωα∈Gα↦{kα,ωα}∈C$)

The dual spin space $Gα$ is similarly defined: $Gα∋πα:Gα→C$. 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 $ϱ:Gα→Gα˙$ such that

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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)
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(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)
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(A.14)
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(A.15)
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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)
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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)
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Hence, $kα=k0oα+k1ια$.

The antisymmetric element

(A.18)
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is responsible for lowering and raising indices, such that

(A.19)
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We can write

(A.20)
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since, for any spin basis, it follows that

(A.21)
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Here, $Gαβ$ denotes the tensor product $Gα⊗Gβ$. The dual tensor $Gαβ∋ϵαβ:Gβ→Gα$ satisfies

(A.22)
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where $Gα→Gβ→Gγ$ is defined. Similarly, we can express

(A.23)
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and

(A.24)
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Now, given arbitrary spinors $kα,ωα∈Gα$ such that $kαωα=1$, we can write (p.213)

(A.25)
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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)
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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)
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Similarly, $nμnμ=mμmμ=m¯μm¯μ=0.$ In addition, the following expressions hold:

(A.28)
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It is sometimes convenient to define another tetrad ($tμ,xμ,yμ,zμ$) as

(A.29)
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We obtain from (A.28) that

(A.30)
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Consequently, the metric components (A.28) have the form , thus identifying the tetrad ($tμ,xμ,yμ,zμ$) as the Minkowski tetrad. Hence, we can write

(A.31)
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By considering the spin basis ${oα,ια}$, we can express the vector $Kμ$ with respect to the basis as

(A.32)
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When the two last equations are compared, it follows that (p.214)

(A.33)
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Thus,

(A.34)
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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)
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which causes eqn (A.33) to be led to

(A.36)
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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)
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where

(A.38)
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The spinor ζ‎ is represented by the column vector

(A.39)
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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 $R∈SL(2,C)$ as

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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 $C♭2$, and are defined by

(A.40)
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which implies that $ζ♭=ζ†J.$ Hence, the spinor $ζ♭$ is represented by

(A.41)
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This spinor can be identified with its algebraic counterpart in eqn (6.155). For a spinor metric to be invariant under $R∈SL(2,C)$, it is necessary that

(A.42)
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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 $(C˙2)∋ζ˙$, $ζ∈C2$, endowed with the spinor metric

(A.43)
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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)
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which implies that $χ˙♭=J(χ˙)†$; a covariant dotted spinor is represented by

(A.45)
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Hence, we can identify it with its algebraic equivalent counterpart in eqn (6.160).

Clearly, the transformation rule for dotted spinors under the transformation $R∈SL(2,C)$ is provided by

(A.46)
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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)
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(A.48)
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(A.49)
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(A.50)
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These transformations are emulated in eqns (6.1626.165), in the algebraic spinor framework.

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

(A.51)
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where ψ‎ is defined by

(A.52)
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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)
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# 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,

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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)
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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 $ϵαβ=kαωβ−ωβkα$. Thus, the quantity $Fμν$ can be characterised as the angular momentum, since

(A.55)
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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)
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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)
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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)
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Moreover, $yμ$ is a space-like unit vector. In fact,

(A.59)
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By multiplying the spinor $kα$ by $eiθ$, the vector $yμ$ spins around the pole by the angle $2θ$. Actually, we have

(A.60)
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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 $xμ↦x′μ=Rμνxν$, where $R∈$ SO(1,3) satisfies $RμνημρRρλ=ηνλ$. The corresponding covariant 4-vector $xμ↦ημνxν$ satisfies $xν=xμ′Rμν.$ The most general proper orthochronous Lorentz transformation, corresponding to a rotation by an angle of θ‎ about an axis $nˆ$, where $θ⃗=θnˆ$, and a boost vector $ζ↦vˆtanh−1β$, where $vˆ=v/∥v∥$ and where $β=∥v∥$, is a $4×4$ matrix given by

(A.61)
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where $θi=12ϵijkθjk$; $ζi=θi0=−θ0i$; $Si=12ϵijkSjk$; $Ki=S0i=−Si0$; and

(A.62)
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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 $Rμν≈δνμ+θμν$. Moreover, the infinitesimal boost parameter reads (p.219) $βvˆ$, since $β≪1$ for an infinitesimal boost. Hence, the actions of the infinitesimal boosts and rotations on the spacetime coordinates are respectively given by

(A.63)
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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 $Φ(xμ)↦Φ′(x′μ)=[R]Φ(xμ)$, where [R] is the corresponding (finite) d-dimensional matrix representation. Equivalently, the functional form of the transformed field Φ‎ obeys

(A.64)
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For proper orthochronous Lorentz transformations,

(A.65)
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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)
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The vectors $J⃗$ and $K⃗$ are defined as the generators of rotations parameterised by $θ⃗$ and the boosts parameterised by $ζ⃗$, 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, $J⃗=σ⃗/2$, and $K⃗=−iσ⃗/2$, in eq. (A.65), so

(A.67)
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where $σ⃗=(σ1,σ2,σ3)$ 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 $ψα↦Rαβψβ$.

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 .

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)
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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, $(R−1)⊺$ and $(R−1)†$, 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 $ψ†α˙$, respectively, with the transformation laws $ψα↦[(R−1)⊺]αβψβ$, and $ψ†α˙↦[(R−1)†]α˙β˙ψ†β˙$, respectively. Lorentz tensors can be derived from spinors by regarding $ψα$ as a row vector, and $ψ†α˙$ as a column vector, with

(A.69)
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The Lorentz transformation property $ψ†α˙$ then follows from

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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)
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yielding the so-called Schouten identities

(A.71)
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(A.72)
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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)
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Hence, eqn (A.73) is equivalent to $σμ=(I2×2,σ⃗)$, and $σˉμ=(I2×2,−σ⃗),$ which can be related by (p.221)

(A.74)
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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)
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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,

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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:

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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)
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where $ψψ=ψαψα$ and $ψ†ψ†=ψα˙†ψ†α˙$.

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

(A.77)
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(A.78)
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Now we can introduce the infinitesimal forms for the $4×4$ Lorentz transformation matrix; the corresponding matrices R and $(R−1)†$ which transform the $D(12,0)$ and $D(0,12)$ spinors, respectively, are given by (p.222)

(A.79)
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(A.80)
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(A.81)
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The inverses of these quantities are obtained up to the first order in θ‎ by replacing $θ↦−θ$ in the formulæ. Equations (A.80) and (A.81) yield

(A.82)
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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)
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Using the Lorentz transformation properties of the undotted and dotted two-component spinor fields, eqn (A.83) yields the proof that the spinor products $ξ†σˉμη$ 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: $p0=Ep=(∥p∥2+m2)1/2$. The matrices $Rαβ$ and $[(R−1)†]α˙β˙$, 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)
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where

(A.85)
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According to (A.84), the spinor index structure of $p⋅σ$ and $p⋅σˉ$ corresponds to that of $Rαβ$ and $[(R−1)†]α˙β˙$, respectively. Hence, the equations in (A.85) yield

(A.86)
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(A.87)
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since $σ0=σˉ0=I2×2$.

# (p.223) A.7 The Supersymmetry Algebra

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

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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)
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(A.89)
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(A.90)
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(A.91)
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(A.92)
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The first three equations describe the usual Poincaré algebra.

A spinor $Qα$ transforms under an infinitesimal Lorentz transformation as

(A.93)
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From the operator point of view, it transforms, by denoting $U=e−i2ωμνMμν$, as

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

(A.95)
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which implies that

(A.96)
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The commutator for the right-handed representation reads

(A.97)
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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.