(p.209) Appendix A The Standard Two-Component Spinor Formalism
(p.209) Appendix A The Standard Two-Component Spinor Formalism
In chapter 6, we presented Weyl spinors in ; now, the connection with the ordinary notation in either field theory or supersymmetry books is briefly presented.
A spacetime vector can be expressed as where () denotes components of with respect to an orthonormal basis . Null vectors are isotropic vectors and satisfy . They present null directions in with respect to the origin of an arbitrary frame in .
The space of null directions that are future (past) pointed are denoted by , and represented by the intersections  of the future (past) light cones with the hyperplanes ().1 The space is a sphere with the equation where () are coordinates in Penrose and Rindler 1984.
Generally, the direction of any null vector , unless the vector is an element of the plane defined by the equation , can be represented by two points. This description results from the intersection of and the hyperplanes . The future-pointed is thus represented by (). The inner points of () represent the set of future-pointed (past-pointed) light-like directions.
By considering 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 .
By defining the complex number
we obtain and, consequently,
The correspondence between points of and the Argand–Gauss plane is injective if the point is added to the complex plane, making it correspond to the north pole with components . However, to avoid this point, it is convenient to associate a point of not to a complex number β but to a pair of complex numbers2 , where (p.210)
The pairs and where , represent the same point in . Such components are called projective coordinates.
The point corresponds to the point of coordinates . The equations in (A.2) can then be expressed as
The point 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 . In particular, if a point R is taken in the line by multiplying P by the factor , then R has coordinates
Unlike the point P, the point R is not invariant under , although it is independent of phases .
Consider now the following complex linear transformation
where satisfy , so that the transformation is invertible. It can be rewritten as
and called a Möbius transformation, from the set to . Moreover, if , and , then f is an injective function from the complex plane, compactified by the point at the infinity; this point is denoted by ().
Hence, the space of light-like vectors on Minkowski spacetime is naturally a Riemann sphere. The restricted Lorentz group is, on the other hand, the automorphism group of the Riemann sphere. The equations in (A.6), when taken with the condition
are called spinor transformations, where is related to the null vectors by the equations in (A.5), implying that
(p.211) The spinor matrix is defined as
The equations in (A.6), with respect to , read
The spinor matrices induce the same transformation of . The equations in (A.5) yield
Hence, up to a factor , it follows that
The transformation acting on the point is real and preserves the light-cone structure . Thus, this relation defines a restricted Lorentz transformation.
Hence, the group SL(2, ) is the twofold covering of the restricted Lorentz group .
A more general case than this is the spin space , which has three basic operations (Penrose and Rindler, 1984):
• multiplication by scalars: ()
• sum: ()
• scalar product: ()
The dual spin space is similarly defined: . Thus, 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 is defined by the application such that
Instead of by the notation , this transformation is denoted by , according to the Penrose notation, characterised by the composition between the -conjugation and the transposition. From now on, the notation
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 and :
where () stands for the dual pair of (). Moreover, the antisymmetry of (A.15) implies that . The pair (), 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)
The antisymmetric element
is responsible for lowering and raising indices, such that
We can write
since, for any spin basis, it follows that
Here, denotes the tensor product . The dual tensor satisfies
where is defined. Similarly, we can express
Now, given arbitrary spinors such that , we can write (p.213)
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 present a spinor description via spin vectors. Latin indices are employed here to label the elements of a real vector space, when the indices α and are grouped together. Moreover, the notation will be used. The null tetrad ()
and the metric can be defined by verifying that the vectors of the null tetrad are null vectors with respect to :
Similarly, In addition, the following expressions hold:
It is sometimes convenient to define another tetrad () as
We obtain from (A.28) that
Consequently, the metric components (A.28) have the form , thus identifying the tetrad () as the Minkowski tetrad. Hence, we can write
By considering the spin basis , we can express the vector with respect to the basis as
When the two last equations are compared, it follows that (p.214)
where the sign defines a future (+) (past (−)). The vector is real and null, since . If , and , it follows that
which causes eqn (A.33) to be led to
where ; ; ; and .
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 ), the 2-fold covering of the restricted Lorentz group . Linear transformations with a unit determinant, with respect to the spin space, determine the group SL. We have already shown that there are two non-equivalent representations of SL; these are denoted by and , respectively, and the elements of the carrier space associated with them are called Weyl spinors.
Both the left-handed and the right-handed 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 . From here on, spinors will carry undotted indices , and spinors will carry dotted indices .
A.2 Contravariant Undotted Spinors
Contravariant undotted spinors are elements of a complex two-dimensional space endowed with the spinor metric
The spinor ζ is represented by the column vector
This spinor can be identified with its algebraic counterpart in eqn (6.154).
(p.215) It also carries the representation of SL and is transformed under as
A.3 Covariant Undotted Spinors
Covariant undotted spinors are elements of a complex two-dimensional dual space , and are defined by
which implies that Hence, the spinor is represented by
This spinor can be identified with its algebraic counterpart in eqn (6.155). For a spinor metric to be invariant under , it is necessary that
which corresponds to the transformation in eqn (6.163).
Contravariant undotted spinors and covariant undotted spinors represent respectively elements of and .
A.4 Contravariant Dotted Spinors
Covariant dotted spinors are elements of a complex two-dimensional space , , endowed with the spinor metric
A covariant dotted spinor is represented by 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 , which is defined by
which implies that ; a covariant dotted spinor is represented by
Hence, we can identify it with its algebraic equivalent counterpart in eqn (6.160).
Clearly, the transformation rule for dotted spinors under the transformation is provided by
carrying the representation of SL. In fact, it corresponds to the transformation in eqn (6.164).
Dotted spinors are elements of and . The action of the Lorentz group on Weyl spinors can be depicted as follows:
From 2-spinors, Dirac spinors can defined as elements of . They are classically realised as elements of , equipped with the spinor metric
where ψ is defined by
With respect to the standard basis of , the matrix is the representation , where denotes the symplectic matrix defined by eqn (A.38).
(p.217) Dirac spinors carry the representation of SL. Under the condition requiring that the spinor metric be invariant under R, the following important representation is obtained:
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 , with the spin-vector , which has coordinates (). From eqn (A.35), the 2-uple ) can be further identified as the coordinates associated with the components , which are invariant under transformations (). Moreover, this ambiguity can be reduced up to a sign by introducing a structure that is composed of a null vector , called a pole, and a null half-plane – tangent to the light cone and having as the intersection – called a flagpole.
Given a contravariant undotted spinor , a geometric object can be constructed, namely, the flagpole. As in eqn (A.36), we shall change the notation by establishing . The pole is defined by eqn (A.34), namely,
The vector is dilated by , when is multiplied by . Notwithstanding, the vector does not change its direction and is independent of the choice of θ. Hence, the null vector is uniquely determined by the spinor . However, the spinor is not uniquely determined by , which corresponds to a family of spinors. They form a projective space and differ from each other by a phase .
The momentum is defined as
The antisymmetric tensor is real and determines a half-plane that is tangent to the light cone along the vector .
By taking a spin basis , where , we find that . Thus, the quantity can be characterised as the angular momentum, since
The tensor hence represents a bivector constituted of two vectors with components and in . The pole is the null flagpole vector, uniquely determined by the spinor . The second vector, given by (p.218)
is also determined by , although not uniquely, since the pair is not the only way to construct . Indeed, any spinor of type
satisfies . With this freedom, the vector transforms as . Each scalar () can be thus associated to a family of coplanar vectors . This is the flagpole, as proposed by Penrose. Some prominent properties can be now derived.
The vector is orthogonal to the null vector . Indeed,
Moreover, is a space-like unit vector. In fact,
By multiplying the spinor by , the vector spins around the pole by the angle . Actually, we have
In addition, is a space-like unit vector, orthogonal to the vectors and . Together with , 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 transforms as , where SO(1,3) satisfies . The corresponding covariant 4-vector satisfies The most general proper orthochronous Lorentz transformation, corresponding to a rotation by an angle of θ about an axis , where , and a boost vector , where and where , is a matrix given by
where ; ; ; ; and
Here, the indices , and (Dreiner, Haber, and Martin, 2010).
It follows from (A.61, A.62) that an infinitesimal orthochronous Lorentz transformation is given by . Moreover, the infinitesimal boost parameter reads (p.219) , since for an infinitesimal boost. Hence, the actions of the infinitesimal boosts and rotations on the spacetime coordinates are respectively given by
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 , where [R] is the corresponding (finite) d-dimensional matrix representation. Equivalently, the functional form of the transformed field Φ obeys
For proper orthochronous Lorentz transformations,
where is the identity matrix, and parameterises the Lorentz transformation R by (A.61). The six independent components of the matrix-valued antisymmetric tensor are the d-dimensional generators of the Lorentz group and satisfy the commutation relations
The vectors and are defined as the generators of rotations parameterised by and the boosts parameterised by , respectively, where , and
Here, we focus on the inequivalent non-trivial irreducible representations of the Lorentz algebra and . In the representation, , and , in eq. (A.65), so
where represents the Pauli matrices. The transformation R carries undotted spinor indices, as indicated by . A two-component spinor in the representation is already denoted by , which transforms as .
On the other hand, in the representation, , and , in eqn (A.65). Hence, its representation matrix is , the complex conjugate of eqn (A.67). By definition, the indices carried by are dotted, as indicated by . It is already known that a two-component spinor transforms as .
It follows that the and representations are related by Hermitian conjugation. In fact, if denotes a spinor, then transforms as a spinor. In combining spinors to make Lorentz tensors, it is useful to regard as a row vector, and as a column vector, with
The Lorentz transformation property of then follows from , where .
(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 .
There are two additional spin-1/2 irreducible representations of the Lorentz group, namely, and , However, they are equivalent to the and the representations, respectively. The spinors that transform under these representations have the raised spinor indices and , respectively, with the transformation laws , and , respectively. Lorentz tensors can be derived from spinors by regarding as a row vector, and as a column vector, with
The Lorentz transformation property then follows from
The spinor indices are raised and lowered with the two-index antisymmetric epsilon symbol with non-zero components and similar sign conventions for the dotted spinor indices. In particular, , and , as well.
Moreover, the Kronecker delta symbol reads . The epsilon symbols with undotted and with dotted indices, respectively, satisfy
yielding the so-called Schouten identities
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 and defined by
There is a one-to-one correspondence between each 2-spinor construction and the associated Lorentz 4-vector , provided by the Infeld–van der Waerden symbols
In particular, if is a real 4-vector, then is Hermitian. Moreover, it is often useful to further simplify the notation by defining . In this notation, an Hermitian 2-spinor satisfies . Then,
A Hermitian 2-spinor satisfies or, equivalently, .
In addition, 2-spinors can be interpreted as matrices. It is indeed convenient to define the following:
Note that the matrix transposition of interchanges the rows and columns of W without modifying the relative heights of the α and β indices. Similar results hold for and by either lowering or raising the spinor indices.
For an anti-commuting two-component spinor ψ, the product is antisymmetric with respect to the interchange of the spinor indices α and β. Hence, it must be proportional to . Similar conclusions hold for the corresponding spinor products with raised undotted indices and with lowered and raised dotted indices, respectively. Thus,
where and .
The van der Waerden symbols in eqns (A.75) provide antisymmetrised products, from the sigma matrices (Dreiner, Haber, and Martin, 2010):
Now we can introduce the infinitesimal forms for the Lorentz transformation matrix; the corresponding matrices R and which transform the and spinors, respectively, are given by (p.222)
Using the Lorentz transformation properties of the undotted and dotted two-component spinor fields, eqn (A.83) yields the proof that the spinor products and transform as Lorentz 4-vectors.
The usual framework use in field theory regards a pure boost from the rest frame to a frame where , which corresponds to , and . The so-called mass-shell condition is satisfied: . The matrices and , which describe Lorentz transformations of spinor fields, are given, respectively, for the and representations by
(p.223) A.7 The Supersymmetry Algebra
The two operators and in a graded Lie algebra satisfy
where either for even (bosonic) or for odd (fermionic) operators and the denote the algebra structure constants. The Poincaré generators in eqn (A.66), together with the , are bosonic generators. Nevertheless, in supersymmetry, fermionic generators and are introduced, respectively denoting elements of the and representations of the Lorentz group, and label the number of supercharges (West, 1990; Wess and Bagger, 1992).
For , the supersymmetry algebra reads
The first three equations describe the usual Poincaré algebra.
A spinor transforms under an infinitesimal Lorentz transformation as
From the operator point of view, it transforms, by denoting , as
which implies that
The commutator for the right-handed representation reads
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).