The two-spinor calculus: $\epsilon$-spinors, abstract indices, and the spinor–tensor dictionary

Intuition Beginner

There is a number system, older than quaternions, in which the square root of a vector makes sense. Start with an arrow in space and time. The two-spinor calculus says that arrow can, in a precise way, be written as a product of two more basic objects, each living in a smaller two-dimensional world of complex numbers. Those smaller objects are the spinors. A vector is built from a spinor times its mirror image.

The smaller world is just a plane of pairs of complex numbers. What makes it a spinor world rather than an ordinary plane is the single extra rule for measuring: instead of an ordinary length, every pair gets paired with every other pair by an antisymmetric rule. Swap the two inputs and the answer flips sign. A direct consequence is that each spinor paired with itself gives zero. Every spinor is, in this measuring rule, like a direction along the light cone: it has no length of its own.

The pleasant surprise is that vectors, which do have length, are recovered as products. Two spinors with no length each can multiply to give a genuine vector that points somewhere and has a real size. Length emerges from the pairing of a thing with its mirror.

Visual Beginner

Picture two small complex planes side by side. The left plane holds an object we draw as a short flag on a pole; the right plane holds its mirror copy. A faint double-headed arrow connects each plane to itself: that arrow is the antisymmetric pairing rule, and its key feature is that it gives zero when a flag is paired with its own self.

Between the two planes sits a bridge labelled "soldering". The bridge takes one flag from the left and one mirror flag from the right and welds them into a single straight arrow drawn in a spacetime box below. The picture records the whole story of this unit in one move: two flags, each with no length of its own, combine across the bridge into one spacetime arrow that does have a length and a direction. Reading the picture backwards is the harder art, and it is the engine of the whole subject: take an arrow apart into the two flags that built it.

Worked example Beginner

Take the simplest spinor, the pair $(1,0)$, and its mirror, also $(1,0)$. The soldering bridge welds these into a spacetime arrow whose four components in a standard frame are computed by a fixed multiplication table. For this pair the result is the arrow with time component $1$ and one space component $1$, all others zero.

Check its size. In spacetime the squared size of an arrow is time-squared minus space-squared. Here that is $1-1=0$. The arrow has zero spacetime length: it points straight along the light cone. This is forced, not accidental: an arrow welded from a single flag and its own mirror always lands on the light cone, because the flag has no length of its own to give.

Now mix two flags. Weld the flag $(1,0)$ with the mirror flag $(0,1)$ and add the welding of $(0,1)$ with $(1,0)$. The two pieces combine into a real arrow pointing in a second space direction, again of zero spacetime length. To get an arrow inside the light cone, with positive size, you add together weldings of several flags. A timelike arrow is a sum of light-cone arrows, exactly as a clock's tick is the sum of two light-bounces. The light-cone pieces are the atoms; everything else is built by adding them.

Check your understanding Beginner

Formal definition Intermediate+

Fix a two-dimensional complex vector space $\mathbb{S}$, the spin space, with elements written using a capital abstract index, ${\xi }^A$ (Penrose's abstract-index convention: the index names the type, not a chosen basis). Its dual is ${\mathbb{S}}^{\ast }$, with elements ${\eta }_A$. The defining extra structure is a symplectic form ${\epsilon }_{AB}$ on $\mathbb{S}$: a nondegenerate antisymmetric bilinear form, ${\epsilon }_{AB}=-{\epsilon }_{BA}$, with $\det \epsilon \ne 0$. In a spin basis $(o^A,{\iota }^A)$ normalised by ${\epsilon }_{AB}{o}^A{\iota }^B=1$, the form has components ${\epsilon }_{01}=-{\epsilon }_{10}=1$, ${\epsilon }_{00}={\epsilon }_{11}=0$.

The special linear group $\mathrm{SL}(2,\mathbb{C})$ acts on $\mathbb{S}$ by ${\xi }^A\mapsto t^A{}_B{\xi }^B$, and the condition $\det t=1$ is exactly the condition that ${\epsilon }_{AB}$ is preserved: $t^A{}_C{t}^B{}_D{\epsilon }_{AB}={\epsilon }_{CD}$. Thus ${\epsilon }_{AB}$ is the $\mathrm{SL}(2,\mathbb{C})$-invariant pairing, the spinor analogue of the metric.

Raising and lowering. The form ${\epsilon }_{AB}$ and its inverse ${\epsilon }^{AB}$ (fixed by ${\epsilon }^{AB}{\epsilon }_{CB}={\delta }^A{}_C$) convert between $\mathbb{S}$ and ${\mathbb{S}}^{\ast }$. Because $\epsilon$ is antisymmetric, the order of contraction matters, and one fixes the conventions $$ \xi_A = \xi^B \varepsilon_{BA}, \qquad \xi^A = \varepsilon^{AB}\xi_B $$ (the "see-saw" rule: lower on the second slot, raise on the first). A direct consequence of antisymmetry is the central identity $$ \xi^A \xi_A = \xi^A \xi^B \varepsilon_{BA} = 0, $$ since ${\xi }^A{\xi }^B$ is symmetric and ${\epsilon }_{BA}$ antisymmetric. Every spinor is null with respect to its own induced pairing.

Primed spinors and conjugation. Complex conjugation carries $\mathbb{S}$ to a different space, the conjugate spin space $\bar{\mathbb{S}}$, whose elements carry primed indices ${\bar{\xi }}^{A^{\prime }}$. The operation is antilinear: $\overline{\lambda {\xi }^A}=\bar{\lambda }{\bar{\xi }}^{A^{\prime }}$ for $\lambda \in \mathbb{C}$. The conjugate space carries its own ${\epsilon }_{A^{\prime }{B}^{\prime }}$, the conjugate of ${\epsilon }_{AB}$. Under $\mathrm{SL}(2,\mathbb{C})$, $\bar{\mathbb{S}}$ carries the complex-conjugate representation; the two representations $\mathbb{S}=(\frac{1}{2},0)$ and $\bar{\mathbb{S}}=(0,\frac{1}{2})$ are the two fundamental Weyl spinors of 07.07.09.

The soldering form. A real tangent space $T_{p}M$ of Minkowski signature is identified with the Hermitian part of $\mathbb{S}\otimes \bar{\mathbb{S}}$ by the Infeld–van der Waerden symbols (the soldering form) ${\sigma }^a{}_{A{A}^{\prime }}$, a fixed collection of four Hermitian $2\times 2$ matrices satisfying $$ \sigma^a{}{AA'},\sigma^b{}{BB'},\varepsilon^{AB}\varepsilon^{A'B'} = g^{ab},\qquad V^a ;\longleftrightarrow; V^{AA'} = V^a,\sigma_a{}^{AA'}, $$ giving the real isomorphism $T_p{M}_{\mathbb{C}}\cong \mathbb{S}\otimes \bar{\mathbb{S}}$. In the standard frame ${\sigma }^a{}_{A{A}^{\prime }}=\frac{1}{\sqrt{2}}(1,{\sigma }^1,{\sigma }^2,{\sigma }^3)$ with ${\sigma }^i$ the Pauli matrices — the same identification $X=x^{\mu }{\sigma }_{\mu }$ of 07.07.09, now read as a frame field.

Key theorem with proof Intermediate+

Theorem (the spinor–tensor dictionary). Under the soldering isomorphism $V^a\leftrightarrow V^{A{A}^{\prime }}$:

1. A vector $V^a$ is real if and only if its spinor form is Hermitian, $V^{A{A}^{\prime }}=\overline{V^{A^{\prime }A}}$ (equivalently $\overline{V^{A{A}^{\prime }}}=V^{A^{\prime }A}$, reading the bar as exchanging primed and unprimed).
2. A nonzero real vector is null and future-pointing if and only if $V^{A{A}^{\prime }}=\pm {\xi }^A{\bar{\xi }}^{A^{\prime }}$ for some spinor ${\xi }^A$, the sign fixing the time orientation.
3. An antisymmetric 2-form $F_{ab}=-F_{ba}$ decomposes uniquely as$$F_{ab}={\varphi }_{AB}{\epsilon }_{A^{\prime }{B}^{\prime }}+{\bar{\varphi }}_{A^{\prime }{B}^{\prime }}{\epsilon }_{AB},{\varphi }_{AB}={\varphi }_{(AB)}=\frac{1}{2}{F}_{ab}{\epsilon }^{A^{\prime }{B}^{\prime }},$$ with ${\varphi }_{AB}$ a symmetric spinor (the Maxwell/anti-self-dual spinor) and ${\bar{\varphi }}_{A^{\prime }{B}^{\prime }}$ its conjugate.

Proof. For (1): the soldering symbols are Hermitian, so reality of the four real components $V^a$ is exactly Hermiticity of the matrix $V^{A{A}^{\prime }}=V^a{\sigma }_a{}^{A{A}^{\prime }}$. For (2): a Hermitian $2\times 2$ matrix has $\det V^{A{A}^{\prime }}=\frac{1}{2}{V}^a{V}_a$ up to a fixed positive factor (from the determinant identity for the Pauli basis, 07.07.09); the matrix has rank one if and only if its determinant vanishes, that is if and only if $V^a$ is null. A rank-one Hermitian matrix factorises as an outer product $V^{A{A}^{\prime }}=\pm {\xi }^A{\bar{\xi }}^{A^{\prime }}$, the sign being the sign of its single nonzero eigenvalue, which records future versus past. For (3): the index pair $ab$ corresponds to the four primed-unprimed slots $A{A}^{\prime }B{B}^{\prime }$. Any spinor with these indices splits by the decomposition lemma below into parts symmetric and antisymmetric in $AB$ and in $A^{\prime }{B}^{\prime }$ separately. Antisymmetry of $F_{ab}$ in $ab$ removes the part symmetric in both unprimed and primed pairs and the part antisymmetric in both; what survives is the part symmetric in $AB$, antisymmetric in $A^{\prime }{B}^{\prime }$, plus its mirror. An antisymmetric pair in two-dimensional spin space is a multiple of $\epsilon$: $F_{[A^{\prime }{B}^{\prime }]}=\frac{1}{2}{\epsilon }_{A^{\prime }{B}^{\prime }}(F\epsilon )$. Carrying this out on both index pairs yields $F_{A{A}^{\prime }B{B}^{\prime }}={\varphi }_{AB}{\epsilon }_{A^{\prime }{B}^{\prime }}+{\psi }_{A^{\prime }{B}^{\prime }}{\epsilon }_{AB}$ with ${\varphi }_{AB}={\varphi }_{(AB)}$; reality of $F_{ab}$ forces ${\psi }_{A^{\prime }{B}^{\prime }}={\bar{\varphi }}_{A^{\prime }{B}^{\prime }}$. $\square$

Bridge. This dictionary builds toward the whole apparatus of relativistic field theory in spinor form, and the symmetric spinor ${\varphi }_{AB}$ extracted here appears again in the spin-$s$ field equations and in the Petrov classification of the Weyl tensor 03.02.18, where the analogous fully symmetric ${\Psi }_{ABCD}$ does the work. The foundational reason the bookkeeping closes is that spin space is two-dimensional: every antisymmetric pair of spinor indices collapses to a multiple of the single object $\epsilon$, so there is nowhere for index structure to hide. This is exactly the algebraic miracle that the corpus elsewhere meets as the $\mathfrak{so}(1,3{)}_{\mathbb{C}}\cong \mathfrak{sl}(2)\oplus \mathfrak{sl}(2)$ splitting of 07.07.09; here it generalises to a clean statement about all tensors at once, not just the Lie algebra. The central insight is that a tensor is a reducible object and a symmetric spinor is its irreducible core; putting these together, the bridge is that contracting tensor indices into primed-and-unprimed spinor pairs and symmetrising performs the $\mathrm{SL}(2,\mathbb{C})$ irreducible decomposition mechanically, which is the engine that the next units in this wave 03.02.42 03.02.43 turn into field equations and a computational formalism.

Exercises Intermediate+

Advanced results Master

The structural heart of the calculus is the canonical (symmetric) decomposition of an arbitrary spinor. Because $\mathbb{S}$ is two-dimensional, the only $\mathrm{SL}(2,\mathbb{C})$-invariant tensor available for contracting indices is ${\epsilon }_{AB}$; consequently every reduction of a spinor proceeds by symmetrising index groups and extracting $\epsilon$'s from the antisymmetric remainder. The precise statement: any spinor ${\chi }_{A_1\cdots A_p}{}^{}{}_{A_1^{\prime }\cdots A_q^{\prime }}$ equals a sum of terms, each a totally symmetric spinor (symmetric separately in its unprimed and in its primed indices) multiplied by a product of $\epsilon$'s carrying the leftover antisymmetric index pairs. The totally symmetric pieces are the $\mathrm{SL}(2,\mathbb{C})$ irreducibles: a symmetric spinor with $2j_1$ unprimed and $2j_2$ primed indices is precisely the $(j_1,j_2)$ representation of 07.07.09. The two-spinor calculus thus performs Clebsch–Gordan decomposition by inspection — no character theory, just symmetrisation and $\epsilon$-extraction.

This is what makes the tensor dictionary computational rather than merely existential. A symmetric spinor ${\varphi }_{A_1\cdots A_n}$ of valence $n$ further factorises, by the fundamental theorem of algebra applied to the associated homogeneous polynomial, into a symmetrised product of $n$ one-index spinors, $$ \phi_{A_1\cdots A_n} = \alpha_{(A_1}\beta_{A_2}\cdots\nu_{A_n)}, $$ the principal spinors. Their associated null directions ${\alpha }^A{\bar{\alpha }}^{A^{\prime }},\dots$ are the principal null directions of the field. For the Maxwell spinor ${\varphi }_{AB}$ this yields two principal null directions (the two roots of a quadratic), distinguishing null fields (a repeated root) from generic ones; for the Weyl spinor ${\Psi }_{ABCD}$ it yields four, and their coincidence pattern is the Petrov classification of 03.02.18. The algebraic classification of fields by principal-spinor multiplicity is one uniform theorem, instantiated at each valence.

The soldering form is not merely a frame-by-frame device: promoted to a covariantly constant field ${\nabla }_a{\sigma }^b{}_{B{B}^{\prime }}=0$, it links the spinor connection to the Levi-Civita connection of 13.02.03. The spinor covariant derivative ${\nabla }_{A{A}^{\prime }}$ then acts on spinor fields with ${\nabla }_{A{A}^{\prime }}{\epsilon }_{BC}=0$, so raising and lowering commute with differentiation. The curvature of this connection, applied to the $\epsilon$-preservation condition, splits the Riemann tensor into its irreducible spinor parts: the totally symmetric Weyl spinor ${\Psi }_{ABCD}$, the trace-free Ricci spinor ${\Phi }_{AB{A}^{\prime }{B}^{\prime }}$, and the scalar $\Lambda$ — the spinor face of the Ricci decomposition that the Newman–Penrose formalism 03.02.43 turns into a system of scalar equations.

A subtlety of global existence: a spin structure (a consistent choice of $\mathbb{S}$ over the whole manifold, 03.09.05) exists if and only if the second Stiefel–Whitney class $w_2$ vanishes; on a non-spin manifold the $\epsilon$-spinor calculus is only local. Where it does globalise, the sign ambiguity ${\xi }^A\mapsto -{\xi }^A$ — invisible to every tensor, since tensors are even in spinors — is the topological residue of the two-to-one cover $\mathrm{SL}(2,\mathbb{C})\to {\mathrm{SO}}^{+}(3,1)$ of 07.07.09.

Synthesis. The two-spinor calculus is the foundational reason the representation theory of the Lorentz group becomes a calculus of indices rather than a table of Clebsch–Gordan coefficients: the single invariant ${\epsilon }_{AB}$, the two-dimensionality of spin space, and the soldering form together reduce every tensor operation to symmetrisation and $\epsilon$-extraction. This is exactly the mechanism by which the abstract $(j_1,j_2)$ reps of 07.07.09 are realised as symmetric spinors, and it generalises the single-vector identification $V^a\leftrightarrow {\xi }^A{\bar{\xi }}^{A^{\prime }}$ into a complete dictionary for tensors of every rank. The central insight is that a symmetric spinor is the irreducible atom and a tensor is a molecule, with $\epsilon$'s the bonds; putting these together, the principal-spinor factorisation makes the algebraic classification of the electromagnetic and gravitational fields a single theorem about polynomial roots, which is dual to the geometric statement, in 03.02.18, that the Weyl tensor has four principal null directions. The bridge is that the same apparatus, differentiated, gives the spinor connection and the irreducible split of curvature, so that the field equations of 03.02.42 and the spin-coefficient equations of 03.02.43 are not new formalisms but this dictionary set in motion.

Full proof set Master

The dictionary theorem and its proof are given in the Key theorem section. The two structural pillars of the calculus are recorded here.

Proposition (canonical decomposition into symmetric parts). Let ${\chi }_{AB}$ be a spinor with two unprimed indices. Then ${\chi }_{AB}={\chi }_{(AB)}+\frac{1}{2}{\epsilon }_{AB}{\chi }_C{}^C$, and more generally any spinor decomposes uniquely as a sum of totally symmetric spinors times products of $\epsilon$'s, one term for each way of contracting index pairs.

Proof. For two indices: decompose ${\chi }_{AB}={\chi }_{(AB)}+{\chi }_{[AB]}$ into symmetric and antisymmetric parts. The antisymmetric part lives in ${\Lambda }^2{\mathbb{S}}^{\ast }$, which is one-dimensional (since $\dim \mathbb{S}=2$) and spanned by ${\epsilon }_{AB}$; thus ${\chi }_{[AB]}=c{\epsilon }_{AB}$. Contracting with ${\epsilon }^{AB}$ and using ${\epsilon }^{AB}{\epsilon }_{AB}=2$ gives $c=\frac{1}{2}{\epsilon }^{AB}{\chi }_{AB}=\frac{1}{2}{\chi }_C{}^C$. For higher valence, apply this two-index reduction repeatedly to each antisymmetric pair: any pair of indices that is not yet symmetrised is split into a symmetric part and a multiple of $\epsilon$, the $\epsilon$ removed from the sum, and the procedure iterated on the now-shorter spinor. Because each antisymmetric two-dimensional pair reduces to exactly one $\epsilon$ and one trace, the procedure terminates with a unique sum of totally symmetric spinors times $\epsilon$-products; uniqueness follows from the uniqueness of the symmetric/antisymmetric split at each step and the one-dimensionality of each ${\Lambda }^2$. $\square$

Proposition (factorisation of a symmetric spinor into principal spinors). A totally symmetric spinor ${\varphi }_{A_1\cdots A_n}$ ($\varphi ={\varphi }_{(A_1\cdots A_n)}$, valence $n$) factorises as ${\varphi }_{A_1\cdots A_n}={\alpha }_{(A_1}\cdots {\nu }_{A_n)}$ for one-index spinors ${\alpha }_A,\dots ,{\nu }_A$, unique up to scaling and order.

Proof. Fix a spin basis and contract ${\varphi }_{A_1\cdots A_n}$ with $n$ copies of a variable spinor ${\zeta }^A=x{o}^A+y{\iota }^A$. The contraction ${\varphi }_{A_1\cdots A_n}{\zeta }^{A_1}\cdots {\zeta }^{A_n}$ is a homogeneous polynomial of degree $n$ in $(x,y)$, whose coefficients are the components of $\varphi$ (total symmetry guarantees no information is lost). Over $\mathbb{C}$, by the fundamental theorem of algebra this binary form factors into $n$ linear factors ${\prod }_k(a_{k}x+b_{k}y)$. Each linear factor $a_{k}x+b_{k}y$ equals ${\beta }_A^{(k)}{\zeta }^A$ for a one-index spinor ${\beta }_A^{(k)}=(a_k,b_k)$ in the dual basis. Therefore ${\varphi }_{A_1\cdots A_n}{\zeta }^{A_1}\cdots {\zeta }^{A_n}={\prod }_k({\beta }_A^{(k)}{\zeta }^A)$ for all $\zeta$, and since both sides are symmetric multilinear in the $\zeta$'s, polarisation gives ${\varphi }_{A_1\cdots A_n}={\beta }_{(A_1}^{(1)}\cdots {\beta }_{A_n)}^{(n)}$. The principal spinors ${\beta }^{(k)}$ are the roots of the binary form, unique up to scale and the order of listing. $\square$

Connections Master

Representations of the Lorentz group, $\mathrm{SL}(2,\mathbb{C})$, and the $(j_1,j_2)$ reps 07.07.09 are the algebraic foundation this unit makes geometric. There the double cover $\mathrm{SL}(2,\mathbb{C})\to {\mathrm{SO}}^{+}(3,1)$ acts on Hermitian matrices by $X\mapsto AX{A}^{†}$ and the finite-dimensional irreducibles are labelled $(j_1,j_2)$; here those same irreducibles are realised concretely as totally symmetric spinors with $2j_1$ unprimed and $2j_2$ primed indices, and the Pauli identification $X=x^{\mu }{\sigma }_{\mu }$ becomes the soldering form ${\sigma }^a{}_{A{A}^{\prime }}$ read as a frame field. This unit supplies the index calculus that turns that representation theory into computation.

The spinor bundle 03.09.05 is what lets the pointwise spin space $\mathbb{S}$ be assembled into a global field. The existence of a consistent $\epsilon$-spinor over a whole spacetime is exactly the existence of a spin structure, obstructed by the second Stiefel–Whitney class; this unit's local algebra globalises precisely when that obstruction vanishes, and the sign ambiguity ${\xi }^A\mapsto -{\xi }^A$ is the topological shadow of the two-to-one cover the bundle trivialises.

The Cartan tetrad and spin connection 13.02.03 provide the differential side. The soldering form lives on the orthonormal frame defined there, and the spin connection of that unit is what differentiates spinor fields with ${\nabla }_{A{A}^{\prime }}{\epsilon }_{BC}=0$, so that the raising/lowering of this unit commutes with covariant differentiation. The covariantly-constant soldering form is the bridge welding the spinor connection to the Levi-Civita connection.

The Petrov classification of the Weyl tensor 03.02.18 is the most striking downstream payoff of the principal-spinor factorisation proved here. The Weyl spinor ${\Psi }_{ABCD}$ is a valence-four symmetric spinor; its factorisation into four principal spinors ${\alpha }_{(A}{\beta }_B{\gamma }_C{\delta }_{D)}$, and the coincidence pattern of the corresponding principal null directions, is the entire Petrov type list — a direct instance of this unit's factorisation theorem at valence four.

Zero-rest-mass field equations 03.02.42, co-produced in this wave, take the symmetric spinor ${\varphi }_{AB\cdots L}$ of the dictionary and impose ${\nabla }^{A{A}^{\prime }}{\varphi }_{AB\cdots L}=0$, recovering Maxwell, Weyl, and Dirac fields in one stroke; and the Newman–Penrose spin-coefficient formalism 03.02.43, also co-produced, is this calculus written out in a fixed spin frame, its twelve spin coefficients the components of the spinor connection introduced here.

Historical & philosophical context Master

The index calculus of dotted and undotted spinors was created by Bartel van der Waerden in 1929 (Spinoranalyse, Nachrichten der Gesellschaft der Wissenschaften zu Göttingen) ^{[van der Waerden 1929]}, at the prompting of Ehrenfest, who wanted a systematic notation for the two-component objects Pauli and Dirac had introduced into quantum mechanics. The remarkable historical inversion is that the spinor — discovered as a tool for the electron's quantum spin — turned out to be more fundamental than the spacetime vector it sits beneath: van der Waerden's calculus showed that every tensor of relativity is a composite of spinors. Roger Penrose, in the two-volume Spinors and Space-Time with Wolfgang Rindler (1984) ^{[Penrose-Rindler 1984]}, elevated this from notation to a worldview, developing the abstract-index formulation in which ${\xi }^A$ names a geometric object rather than a column of components, and making the symmetric-spinor decomposition the organising principle of relativistic field theory. Wald's General Relativity (1984) ^{[Wald 1984]} gives the compact modern textbook treatment that ties the soldering form to the metric.

The philosophical claim Penrose pressed is that spin space, not spacetime, may be the more primitive arena: spacetime points and vectors are derived — a real null vector is literally ${\xi }^A{\bar{\xi }}^{A^{\prime }}$, a spinor times its conjugate — which suggested to him that a discrete or combinatorial spinor structure (the spin networks of his later twistor and quantum-geometry programmes) might underlie the continuum. Whether or not one accepts that metaphysics, the calculus delivers a concrete methodological lesson: choosing the right primitive object — here the two-dimensional spin space with its single antisymmetric $\epsilon$ — collapses an apparently intricate theory (the Lorentz-tensor representation theory, the algebraic classification of fields) into mechanical symmetrisation. The two-dimensionality is doing the philosophical work: it is because spin space is two-dimensional that there is exactly one invariant form and that every antisymmetric pair collapses, and this small accident of dimension is what makes four-dimensional Lorentzian geometry uniquely amenable to spinor methods.

Bibliography Master

@book{penroserindler1984,
  author    = {Penrose, Roger and Rindler, Wolfgang},
  title     = {Spinors and Space-Time, Volume 1: Two-Spinor Calculus and Relativistic Fields},
  series    = {Cambridge Monographs on Mathematical Physics},
  publisher = {Cambridge University Press},
  year      = {1984}
}

@book{wald1984,
  author    = {Wald, Robert M.},
  title     = {General Relativity},
  publisher = {University of Chicago Press},
  year      = {1984}
}

@article{vanderwaerden1929,
  author  = {van der Waerden, Bartel L.},
  title   = {Spinoranalyse},
  journal = {Nachrichten der Gesellschaft der Wissenschaften zu G\"ottingen, Mathematisch-Physikalische Klasse},
  pages   = {100--109},
  year    = {1929}
}

@book{penroserindler1986,
  author    = {Penrose, Roger and Rindler, Wolfgang},
  title     = {Spinors and Space-Time, Volume 2: Spinor and Twistor Methods in Space-Time Geometry},
  series    = {Cambridge Monographs on Mathematical Physics},
  publisher = {Cambridge University Press},
  year      = {1986}
}