Spinors in Relativity

Jörg Frauendiener

December 13, 2018

Department of Mathematics and Statistics University of Otago Since this script is available online I would appreciate any feedback from whoever reads it. Please send comments, suggestions, corrections to

[email protected]. Contents

1 Introduction and Literature 4

2 The geometry of Minkowski space-time 8 2.1 Afﬁne space ...... 8 2.2 Metric space ...... 10 2.3 Basic notions ...... 11

3 Spin transformations 15 3.1 Representation of the sphere ...... 15 3.2 Null directions and the celestial sphere ...... 17 3.3 Lorentz and spin transformations ...... 18 3.4 Lorentz transformations: a closer look ...... 21

4 Spinorial objects and spin vectors 23 4.1 Null ﬂags ...... 23 4.2 Spin vectors ...... 27 4.3 Some global considerations ...... 28

5 Spinor algebra 30 5.1 Foundations of tensors ...... 30 5.2 Spinors ...... 35 5.2.1 ε spinors ...... 36 5.2.2 Complex conjugate spinors ...... 37 5.2.3 Spin frames ...... 39

6 4-vectors as spinors 40 6.1 Spin frames and their tetrads ...... 40 6.2 Van-der-Waerden symbols ...... 42 6.3 Clifford algebra, Dirac spinors, Weyl spinors ...... 46 6.4 Null vectors, null ﬂags, etc...... 50

3 1 Introduction and Literature

Einstein’s theory of special relativity is of fundamental importance in today’s mod- ern Physics. It is not a theory of interaction, i.e., describing a fundamental force such as for instance electrodynamics. Rather it is a kinematical theory which describes the framework within which Physics takes place. It takes the place of classical Newtonian Physics which was shaped by Galilei’s principle of relativity. This principle postulates that all reference frames which are related by Galilei transformations are equivalent for the description of physical phenomena. This stipulation distinguishes a class of preferred reference frames, called inertial frames. It was expected that all laws of Nature “look the same” with respect to each of these inertial frames, i.e., that they are invariant under Galilei transformations. However, after the realisation that Maxwell’s equations of electrodynamics are not Galilei invariant, it was Einstein to dissolved the ensuing dilemma (parts of Physics being Galilei invariant, others not) by postulating a different principle of relativity. According to his principle of relativity not the Galilei transformations map inertial frames to inertial frames but it is Poincaré transformations, also known as inhomogeneous Lorentz transformations.

The space-time continuum of relativistic (quantum-) mechanics is the Minkowski space- time, an affine space which is equipped with a bilinear form of signature −2. The invariance group of this space is the Poincaré group, the group of inhomogeneous Lorentz transformations. The Lorentz group itself is the subgroup of the Poincaré group which consists of transformations which keep one selected point fixed. It can be regarded as the invariance group of the Minkowski vector space of position vectors based on the selected point. The importance of the Poincaré group is highlighted by the fact that elementary particles are classified in relativistic quantum mechanics according to its irreducible representations.

The fact that the Poincaré group is intimately connected to the global structure of Minkowski space (which can even be defined as a homogeneous space on which the Poincaré group acts) already suggests that there might be difficulties in the transition to Einstein’s theory of gravity. Now the space-time is no longer an affine space but a more general (curved) manifold whose curvature is determined by the matter contents in the space-time. The only commonality with Minkowski space is the fact that at every point there is attached a Minkowski vector space which also carries a bilinear form of signature −2. Accordingly, the importance of the Poincaré group evaporates because only in the rarest cases will there exists a global invariance group for the space-time which in addition will not agree with the Poincaré group. Furthermore, in this general case the notion of an elementary particle gets blurred. Together with other difficulties

4 this is a manifestation of the current incompatibility of quantum theory and general relativity. However, the Lorentz group does survive in the transition from ﬂat to curved space. It remains the invariance group of the Minkowski vector space and, hence, only acts at each point individually. It is the truly fundamental group of relativistic physics and its representations should be of physical relevance in all aspects of Physics. The representations of the Lorentz group are treated at length in [4].

The best known representation of the Lorentz group is its defining representation in the 4-dimensional Minkowski vector space. It is the basis for the usual tensorial representations and the mathematical process leads naturally to the development of tensor algebra. The Lorentz group is a semi-simple Lie group whose finite dimensional representations are fully reducible. Hence, it is enough to find and classify the irreducible representations. Among those there are two distinguished ones: the two 2-dimensional representations are fundamental representations, i.e., all representations of the Lorentz group can be obtained as appropriate tensor products of only those two. The vectors of these representation spaces are the primed and unprimed spin-vectors and the tensor algebra built upon the fundamental representations is called spinor algebra. This means that spin-vectors can be used to construct all other tensors.

The 2-dimensionality of the spin-vectors makes calculating with them quite easy. This is a peculiarity of the four dimensions of the world we live in. In higher dimensions one can ﬁnd analogous spinor representations related to the corresponding invariance groups. However, their dimension grows exponentially with the dimension of the space-time so that already for space-time dimension 8 the corresponding spinor representation is also 8-dimensional. Only for space-time dimensions 3 and 4 can we ex- pect to gain simpliﬁcations in the study of space-time geometry using the calculus of spinors.

The theory of spinors was already developed and applied at the beginning of the last century. The general notion of a spinor was introduced by Cartan [1, 2] while van der Waerden [11] developed the calculus of two-component spinors. Dirac discovered the fundamental importance of spinors for relativistic physics when he found the equation the electron. With the work of Penrose [7] the calculus of spinors was ﬁrmly introduced into General Relativity. This formalism as well as the related Newman-Penrose (NP) formalism was very successfully applied in many different areas of gravitational theory, in particular in the treatment of asymptotically ﬂat space-times, the analysis of gravitational waves and last but not least in the proof of the positive-energy-theorem of General Relativity, one of the most fundamental results in that area.

The two volumes by Penrose and Rindler [8] contain an impressive introduction into the theory of spinors and many application thereof. They also explain the relationship between spin-vectors and space-time geometry. In the book by Stewart [10] one can ﬁnd the treatment of asymptotically ﬂat space-times using the calculus of two-component spinors. Spinors are also discussed in the book by Naber [6].

5 Another good source for the mutual relations between spinors, relativity and quantum theory is the book by Sexl and Urbantke [9]. It discusses in particular the group theoret- ical aspects. Finally, it is worthwhile to point out a classic which does not deal explicitly with spinors but presents a very good treatment of the foundations of General Relativ- ity: the book by Hermann Weyl [12] is warmly recommended. It appeared first in 1918 and has been published now in its 7th edition. The following course will provide an introduction into the calculus of two-component spinor-vectors. The aim of the course is to reduce the timidity towards and overcome the hesitation to apply the formalism which may seem daunting at the first encounter. The first chapters deal with the fundamental properties of Minkowski geometry. How- ever, the point of view is a bit different from other approaches. We regard the geometry of light rays as fundamental. The “celestial sphere” of past directed light rays is introduced and with its aid the null flags are defined. These are the geometric imprints of spin-vectors. Then we adopt a more abstract point of view and introduce the spinor algebra. After a chapter on the main tools of spinor calculus (principal null directions, dualisation, symmetry operations) the results are transformed to manifolds. The last three chapters are devoted to applications, the classification of the Weyl tensor, the NP formalism, and finally a discussion of some spinor equations. Except for some parts of the last chapter the whole material is standard and can be found in the cited literature.

6 Bibliography

[1] É. CARTAN, Les groupes projectifs qui ne laissent invariante aucune multiplicité plane, Bull. Soc. Math. France, 41 (1913), pp. 53–96. [2] , The Theory of Spinors, Hermann, Paris, 1966.

[3] T. FRIEDRICH, Dirac-Operatoren in der Riemannschen Geometrie, Verlag Vieweg, Braunschweig, 1997.

[4] I. M. GELFAND,R.A.MINLOS, UND Z. Y. SHAPIRO, Representations of the Rotation and Lorentz Group and Their Applications, Pergamon Press, Oxford, 1963. [5] H. B. Lawson und M.-L. Michelsohn, Spin Geometry, Princeton University Press, 1989.

[6] G. L. NABER, The Geometry of Minkowski Spacetime, Springer-Verlag, Berlin, 1992.

[7] R. PENROSE, A spinor approach to general relativity, Ann. Phys., 10 (1960), pp. 171– 201.

[8] R. PENROSEUND W. RINDLER, Spinors and Spacetime, vol. 1,2, Cambridge Univer- sity Press, 1984, 1986.

[9] R. SEXLUND H.URBANTKE, Relativität, Gruppen, Teilchen, Springer-Verlag, Wien, 3. Auﬂage, 1992.

[10] J. STEWART, Advanced general relativity, Cambridge University Press, Cambridge, 1990.

[11] B. VAN DER WAERDEN, Spinoranalyse, Nachr. Akad. Wiss. Götting. Math.-Physik. Kl., (1929), pp. 100–109.

[12] H. WEYL, Raum, Zeit, Materie, Springer-Verlag, Berlin, 7. Auﬂage., 1988.

7 2 The geometry of Minkowski space-time

In this ﬁrst chapter we will present a short representation of the geometrical foundations of special relativity. This theory lives in the simplest model of a space-time contin- Minkowski space-time uum, the so called Minkowski space-time, named after HERMANN MINKOWSKI who was the ﬁrst to describe its geometric properties.1

The structure of the world of special relativity follows from Einstein’s empirical principle of relativity which, mathematically formulated, is formulated as follows

The world is a 4-dimensional affine space equipped with an indefinite quadratic form with signature −2 which defines a metric.

This formulation is taken from the book Space - Time - Matter by HERMANN WEYL, which despite of its age (ﬁrst published in 1918) even today provides a clear and trans- parent presentation of the mathematical and physical notions underlying both theories of relativity.

2.1 Afﬁne space

What is the contents of the principle mentioned above? We first remind ourselves about the notion of an affine space. It is well known that there are two categories of “things” point in affine geometry — points and vectors (translations) — and relations between them. vector An affine space M can be characterised by the following three requirements

1. Vectors: The set of vectors is a real vector space V of dimension n < ∞.

2. Points: −→ a) For any two points P, Q there exists exactly one vector, denoted by PQ.

b) If u is any vector and P is any point then there exists a unique point Q, so −→ that u = PQ holds. −→ −→ −→ c) If u = PQ and v = QR, then PR = u + v.

1H. Minkowski, Raum und Zeit, Talk at 80. Versammlung Deutscher Naturforscher und Ärzte zu Cöln on 21. September 1908

8 Strictly speaking we need to distinguish between the space of points and the space of vectors. This is often suppressed. The relationship between these two spaces is charac- −→ terised by the property (2). Every vector PQ may be regarded as a translation which sends the point P to the point Q. The main property of afﬁne space is its homogeneity: homogeneity no point is preferred over any other. This is contrary to a vector space where the zero- vector is different from all others since it is the only one with vanishing components in every basis. At every point O the vector space V can be represented as the set of −→ position vectors OP. In this sense, we can regard V to be attached to every point of the position vector afﬁne space.

An afﬁne coordinate system C consists of a point O together with a basis (e1,..., en) of coordinate system V. Each vector v can be expressed in exactly one way as a linear combination

i v = v ei. (2.1.1)

(We use the Einstein summation convention throughout). The numbers vi are the coordinates of V with respect to C. The coordinates of a point P with respect to cC are coordinates −→ deﬁned as the coordinates of the vector OP.

0 0 0 0 If C = (O ; e1,..., en) is another afﬁne coordinate system then equations of the form

i 0 ek = A kei, (2.1.2)

i hold, where the number A k form a matrix with non-vanishing determinant. For the coordinates of the vector v with respect to C0 we obtain the expressions

0i i k v = A kv . (2.1.3)

When xi are the coordinates of a point P with respect to C then the coordinates of the −→ −−→ −→ same point with respect to C0 are obtained from the relation O0P = O0O + OP as

0i i k i x = A kx + α . (2.1.4)

Here, the αi are the coordinates of O with respect to C0. We see again that there is indeed a difference between points and vectors, they behave differently under the change of coordinate system: the coordinates of points transform inhomogeneously, while those of vectors are subject to homogeneous transformations.

The formulae (2.1.4) indicate to transition from one coordinate system to another. This is the passive point of view of a transformation in which the description of an object is changed. But there is another interpretation of the same formulae: according to the ac- tive point of view they define an affine transformation. This is a bijective map of affine affine transformation space M onto itself which induces a bijective linear transformation on the vector space V. For, if xi are the coordinates of P with respect to C then (2.1.4) defines the coordinates of another point P0 with respect to the same coordinate system C. The assignment

9 P 7→ P0 deﬁned in this way gives a map f : M → M and a related map fˆ : V → V. It is easy to see that this map has the property that −→ −−−−−−→ v = PQ ⇐⇒ fˆ(v) = f (P) f (Q).

Indeed, every map with this property can be expressed by (2.1.4) with respect to an affine coordinate system C. The set of affine transformations form a group, the affine affine group group of dimension n. It is the invariance group of affine space M. invariance group In a sense, the affine geometry abstracts the measurement process. This is most easily seen in the 1-dimensional case. Measuring the distance between two points consists of comparing the segment between the points to a given standard ruler which is freely moving in space. In a sense this ruler corresponds to the vector between the endpoints of the standard segment. By extending and subdividing the standard ruler one can obtain the law of addition of vectors and multiplication of vectors with rational numbers. The extension to real numbers is then a mathematical sophistication. In higher dimensions one can also orient the standard ruler in different directions. With the affine framework there is no criterion to select a particular class of standard rulers. Therefore, there is no “absolute” notion of length or distance in this geometry, only a “relative” one: for any two segments it is not possible to determine their length individually, one can only say by which factor they differ. To make such a statement it is not relevant which standard ruler is used to measure the two segments as long as it is the same for both. These considerations extends in an analogous way to the higher dimensional notions such as area and volume. In order to go from the affine to the metric geometry one needs an additional structure which allows one to assign to each segment between two points a unique length and in higher dimensions to any two vectors an angle.

2.2 Metric space

A metric space in an afﬁne space M on whose vector space V there is deﬁned a real scalar product. To any pair of vectors U, V is assigned a real number (U · V) in such a way that this map is linear in both arguments and symmetric, i.e., (U · V) = (V · U) regular for all U, V ∈ V. Finally, we require that this bilinear form is regular in the sense that U · V = 0 for all V ∈ V implies U = 0. After choosing a coordinate system C the bilinearity implies

i k i k (U · U) = (ei · ek)U U = gikU U . (2.2.1)

The number gik = (ei · ek) form a symmetric matrix with non-vanishing determinant. From (2.2.1) we see that the scalar product deﬁne a quadratic form of the coordinates of a vector.

10 Since the choice of a coordinate system is entirely arbitrary one can ask whether there are any particularly useful choices. Indeed, it is well known that one can always ﬁnd a linear transformation of the coordinates Ui (i.e., a basis of V) of U so that

2 2 2 2 2 (U · U) = U1 + U2 + ··· + Ur − Ur+1 − · · · − Un (2.2.2) for some 1 ≤ r ≤ n holds. This means that one can always find a coordinate system so that the matrix of the gik has the diagonal form diag(1, . . . , 1, −1, . . . , −1). The signature signature | {z } | {z } p q of a quadratic form is the difference p − q between the number of positive and negative entries in this diagonal form. Together with the dimension n = p + q a regular bilinear form is uniquely fixed by its signature. All bilinear forms on a metric space with the same signature are equivalent and cannot be distinguished. The bilinear form on V allows us to define a distance between any two points of M. This is achieved by assigning to any two points P and Q their squared distance squared distance −→ −→ Φ(P, Q) = PQ · PQ. In space with a definite bilinear form (signature ±n) this expression has always the same sign and one can take the square root without any problem. This yields the geometric distance between the points. In a similar way one can also define angles.

The existence of such a bilinear form, also called metric, restricts the invrariance group metric of the space. Now only those afﬁne transformations are allowed which in addition leave the metric invariant. If the corresponding linear transformation is given by (2.1.3) then we need to require in addition that

l m glm A i A k = gik (2.2.3) holds. The afﬁne group is reduced to the subgroup of isometries. isometries

A special example for a metric space is Euclidean space for which the signature is Euclidean space equal to the dimension of the space, and Lorentzian space with signature ±(n − 1). Lorentzian space In Euclidean spaces we have Euclidean geometry and the invariance group is called the Euclidean group. In Lorentz spaces we have Lorentzian geometry with the Poincaré group or inhomogeneous Lorentz group as invariance group. The Lorentz space with Poincaré group dimension 4 and metric signature ±2 is called Minkowski space-time. We will choose Lorentz group signature −2, i.e., the metric can be put into diagonal form diag(1, −1, −1, −1). The reason for this will become clear later. Minkowski space-time

2.3 Basic notions

First, we want to have a closer look at the Minkowski vector space V. It appears at several places in Relativity, not only the space of position vectors in Minkowski space- time but also as tangent space to a space-time manifold or as the space of 4-momenta.

11 As we have already pointed out the 4-dimensionality of space-time means nothing else than the existence of four linearly independent vectors (e0,..., e3). Occasionally, we will denote these also by (t, x, y, z) in accord with the meaning of the four dimensions. tetrad A basis in Minkowski vector space is also often called a tetrad. Each vector U can be written as a linear combination with respect to a basis

0 1 2 3 i U = U e0 + U e1 + U e2 + U e3 = U ei. (2.3.1)

0 1 2 3 contravariant The numbers U , U , U , U are the contravariant coordinates of U with respect to the tetrad (e0,..., e3).

Any two tetrads (ei), (fk) are related by an invertible linear map

i fk = A kei. (2.3.2)

i The determinant of the matrix A k is different from zero, so it is either strictly positive, or strictly negative. Depending on the sign the two bases are called similarly or oppositely oriented. The set of all bases of V splits into exactly two disjoint classes. Within one class the bases are similarly oriented. The restriction to bases from only one orientation of the classes is called an orientation of V. In the following we will assume that such a choice has been made and we call tetrads from the selected class to be oriented. The scalar product on V allows us to also select another class of tetrads. As we have already described one can always ﬁnd a basis (t, x, y, z) of V so that

t · t = −x · x = −y · y = −z · z = 1 (2.3.3) orthonormal and all other combinations vanish. Such a basis is called a (pseudo)-orthonormal frame

Minkowski tetrad or Minkowski tetrad. The matrix for the metric with respect to such a tetrad has the diagonal form

 1 0 0 0   0 −1 0 0  η =   , (2.3.4) ik  0 0 −1 0  0 0 0 −1 and the scalar product of two vectors U and V can be written as

U · V = U0V0 − U1V1 − U2V2 − U3V3. (2.3.5)

A non-vanishing vector U is called   time-like,   > 0 light-like, iff U · U = 0 (2.3.6) space-like,   < 0

12 A non space-like vector is also called causal. Obviously, for a causal vector the inequal- causal ity (U0)2 ≥ (U1)2 + (U2)2 + (U3)2 (2.3.7) holds. For any two causal vectors we have the chain of inequalities (U0V0)2 ≥ (U1)2 + (U2)2 + (U3)2 (V1)2 + (V2)2 + (V3)2 2 ≥ U1V1 + U2V2 + U3V3 , where we have used the Cauchy-Schwarz inequality. With this we have

U0V0 ≥ U1V1 + U2V2 + U3V3. (2.3.8)

Equality holds if and only if both U and V are light-like and proportional to each other. This is also the only case where two causal vectors can be orthogonal to each other, i.e., where U · V = 0. Analogously to above, we can now split the set of all causal vectors into exactly two classes within which the scalar product of any two vectors is positive. The selection of one such class is called a time orientation and the causal vectors from this selected time orientation class are called future pointing. The vectors from the other class are then called past future pointing pointing. past pointing A Minkowski tetrad is called orthochronous if its time-like “leg” t is future pointing. orthochronous With respect to an orthochronous Minkowski tetrad the causal vectors are exactly those for which U0 = U · t > 0 holds. Note, that this orientation applies only to causal vectors. There are no future or past pointing space-like vectors. However, using the orientation of the full vector space V and the time orientation we can define also a space orientation: a set of three space-like linearly independent vectors (x, y, z), a triad, is called right-handed if there is a future triad pointing time-like vector t such that the tetrad (t, x, y, z) is oriented. Any two of the right-handed orientations defined in this way imply the third. We have now introduced even more structures beyond the scalar product and we have to ask ourselves what the invariance group looks like. As always, this is the group of transformations which leaves all structures invariant. In the present case, these are the affine structure, the scalar product and the orientations. An automorphism of V which leaves the scalar product invariant is called Lorentz transformation and if it also Lorentz transformation preserves the orientations it is called orthochronous Lorentz transformation. A map of the Minkowski space-time M to itself which preserves the affine structure and the squared distance Φ(P, Q) between any two points P and Q is called a Poincaré transformation. Since every such transformation induces a Lorentz transformation of Poincaré transformation V so that it can therefore also be called orthochronous provided that the Lorentz transformation is. The invariance group of Minkowski space-time is the group of Poincaré

13 transformations, the invariance group of the oriented Minkowski space-time is the orthochronous Poincaré group. Finally, we want to have a look at how to describe orthochronous Lorentz transformations with respect to a Minkowski tetrad. The invariance of the scalar product together with the diagonal form of the metric (2.3.4) implies with (2.2.3)

l m ηlm L i L k = ηik, (2.3.9)

i where L k is the matrix L of a Lorentz transformation with respect to a Minkowski tetrad. If, in addition, this transformation leaves the orientation of V invariant then

i det(L k) > 0, (2.3.10) and if, furthermore, also the time orientation is invariant then

0 L 0 > 0. (2.3.11)

This follows from the fact, that the image vector L t must be future pointing so that 0 L 0 = Lt · t > 0 must hold.

14 3 Spin transformations

The Minkowski tetrads that we introduced in the last chapter correspond to coordinates which are specially adapted to the metric-afﬁne structure of Minkowski space-time. In this chapter we focus on another geometric structure of Minkowski space, the geometry of light rays. In order to describe this adequately it is useful to have several different representations of the sphere available.

3.1 Representation of the sphere

The first representation is also the definition: in 3-dimensional space with Cartesian coordinates (x, y, z) we define the unit-sphere as S = {(x, y, z)|x2 + y2 + z2 = 1} = S2. Further coordinate representations are (among others)

• Polar coordinates: They are deﬁned as usual by x = sin θ cos φ, y = sin θ sin φ, z = cos θ, for θ ∈]0, π[, φ ∈]0, 2π[. Obviously, these coordinates do not cover the entire sphere but only points outside the half-plane {x ≥ 0} and one needs to choose other coordinate ranges (charts) in order to achieve a complete covering of the sphere.

• Stereographic projection from the north pole: let N = (0, 0, 1) be the north pole of the sphere. We assign to each point P of S a point Q in the plane Σ := {(x, y, z)|z = 0} in such a way that N, P and Q are collinear. This means that Q is the intersec-

Figure 3.1: Stereographic projection of the sphere onto the plane

15 tion point of the line through N and P with Σ. When Q = (X, Y, 0), then obviously

X 1 Y = = . (3.1.1) x 1 − z y

Interpreting Σ as the complex plane and writing ζ = X + iY yields

x + iy ζ = , (3.1.2) 1 − z

for all points of S except for the north pole. This gives a covering of the sphere except for one single point. The inverse of the transformation (3.1.2) is

ζ + ζ ζ − ζ ζζ − 1 x = , y = −i , z = . (3.1.3) ζζ + 1 ζζ + 1 ζζ + 1

From this we see that all points except for N with z = 1 are reached by (3.1.3). The north pole can only be reached formally for “ζ = ∞”. Thus, by adding this “in- ﬁnitely distant” point one can “compactify” the complex plane to the Riemann Riemann sphere sphere of complex numbers. So we can always imagine the sphere to be represented as the complex plane plus the point at inﬁnity. The relationship with the polar coordinates is given by

x + iy ζ = = cot(θ/2)eiφ. (3.1.4) 1 − z

• Homogeneous coordinates: Sometimes it is annoying having to treat “ζ = ∞” as a special case and it is desirable to have representations of the sphere which treat all points in the same way. Such a possibility is the one given as the deﬁnition of sphere. Here we mention another one which has far reaching consequences.

In C2 − {(0, 0)} we deﬁne an equivalence relation according to which two pairs of complex numbers (ξ1, η1) and (ξ2, η2) are equivalent if there exists a number λ 6= 0 so that (ξ2, η2) = (λξ1, λη1). The set of equivalence classes {[ξ, η]} is the 1 1 projective space projective space CP of one (complex) dimension. One can see easily that CP has a spherical topology: when η 6= 0 then [ξ, η] = [ξ/η, 1]. When ξ and η range over all possible values with η 6= 0 then ζ := ξ/η ranges over the entire complex plane. The case η = 0 yields only one further equivalence class since [ξ, 0] = [1, 0] and one can see that it corresponds formally to the “point at inﬁnity” ζ = ∞. The representation of the sphere as a projective complex space is very useful and leads to important consequences in particular in connection with the Minkowski geometry.

16 3.2 Null directions and the celestial sphere

We choose an event (point) O in Minkowski space-time and we consider the light-like directions at that point O. By this null direction we mean a null vector (light-like vec- null direction tor) with all its positive multiples. Thus, there are future and past null directions. With null vector respect to a Minkowski tetrad (t, x, y, z) every null vector L has components (T, X, Y, Z) satisfying T2 − X2 − Y2 − Z2 = 0. The null direction which is deﬁned by L can be rep- −→ resented in particular by the vector multiple OQ of L which has its endpoint either on the plane {T = 1} or on {T = −1} depending on the time-orientation of L. For future pointing directions the point Q has the coordinates (1, X/T, Y/T, Y/T) which therefore satisfy the equation x2 + y2 + z2 = 1. Consequently, all these endpoints corresponding to different future pointing null directions lie on a sphere. The corresponding statement holds for the past pointing null directions. These concrete spheres S± are the corresponding representations of the (abstract) spheres S± of future and past null directions.

The objects just defined can be given a concrete physical meaning. Consider an observer located at O and moving in the direction of t. Everything he sees at the event O (i.e., at the location in space and instant of time defined by O) arrives along light rays from the past. Therefore, his entire field of vision consists of the past directed null directions, i.e., of S−. The sphere S− at the time T = −1 gives an exact geometric image of what he actually sees. After all, the observers considers himself to be in the centre of a sphere onto which he projects his field of vision. For this reason the two spheres S− and S− are called celestial sphere. celestial sphere

With every null direction represented by the null vector L one obtains a null direction with the opposite time orientation deﬁned by the null vector −L. The assignment L 7→ −L gives maps S± → S∓ and S± → S∓. For the concrete spheres S± these maps are given by the antipodal map (x, y, z) 7→ (−x, −y, −z). With respect to the ζ coordinate antipodal map the antipodal map is given by

1 θ ζ 7→ − = − tan eiφ. (3.2.1) ζ¯ 2

What does S+ look like in homogeneous coordinates? The points of S+ which represent different null directions have coordinates

ξη¯ + ηξ¯ 1 ξη¯ − ηξ¯ ξξ¯ − ηη¯ (1, x, y, z) = 1, , , , (3.2.2) ξξ¯ + ηη¯ i ξξ¯ + ηη¯ ξξ¯ + ηη¯ where we used (3.1.3) and ζ = ξ/η. This representation holds for all values of ξ and η since they cannot vanish simultaneously and therefore it covers the entire sphere. In addition, this representation is invariant under the scaling (ξ, η) 7→ (λξ, λη) with an arbitrary non-vanishing number λ ∈ C.

17 For representing null directions we need not restrict ourselves to points on S+. Instead, we can use any multiples of the corresponding null vectors. In the present case it is −→ useful to take null vectors K = OR which have the coordinates 1 1 T = √ ξξ¯ + ηη¯ , X = √ ξη¯ + ξη¯ , 2 2 (3.2.3) i 1 Y = −√ ξη¯ − ηξ¯ , Z = √ ξξ¯ − ηη¯ . 2 2 √ with respect to the chosen Minkowski tetrad. The factor 1/ 2 is conventional. It is easy to see that K · K = 0 and that the null direction deﬁned in this way does not depend on the scaling of (ξ, η), i.e., it depends only on the equivalence class [ξ, η]. However, the null vector instead depends on the scaling. This means that when we replace (ξ, η) with (λξ, λη) with λ 6= 0 then we obtain a null vector which is extended by a factor λλ¯ compared to K. Therefore, when ξ and η range over all admissible values then (3.2.3) yields the entire future light-cone of O. Exercise 3.1: Show that the equivalence relation

(ξ, η) ≡ (ξeiφ, ηeiφ), φ ∈ R describes exactly the future light-cone in Minkowski vector space.

3.3 Lorentz and spin transformations

Orthochronous Lorentz transformations map null vectors to null vectors. Since they are linear they also map null directions to null directions. Therefore, they induce maps from S± onto itself. What do these maps look like? We start with the projective representation of S+ as the set of all null vectors K with components given by (3.2.3). When we apply a complex linear, non-singular transformation to (ξ, η) which has the form ξ 7→ αξ + βη, η 7→ γξ + δη (3.3.1) where αδ − βγ 6= 0, then the action of this map on the null directions expressed in terms of the coordinate ζ is given by αζ + β ζ 7→ . (3.3.2) γζ + δ This map also makes sense for ζ = ∞ if one deﬁnes ∞ 7→ α/γ. Furthermore, ∞ can be reached as an image point when ζ = −δ/γ. Obviously, this map is not sensitive to the scaling of α, β, γ und δ: λα, λβ, λγ and λδ yield the exact same image for any λ 6= 0. Therefore, we may assume that αδ − βγ = 1. We call the matrix α β A = (3.3.3) γ δ

18 a spin matrix when it is normalised by the condition that det A = 1. The transformation spin matrix (3.3.1) deﬁned by the matrix A is called a spin transformation. spin transformation

Exercise 3.2: Show that the spin transformations form a group which is isomorphic to the group SL(2, C). The normalisation achieves that the assignment between a spin transformation and the transformations of null directions, i.e., the points on S+ given by ζ, is as unique as possible. However, the kernel of this group homomorphism is still not trivial: we ask for the spin transformation which gives the identity map on S+ and ﬁnd from

αζ + β ζ = (3.3.4) γζ + δ by comparing coefficients of ζ that β = γ = 0 and α = δ. The normalisation condition implies α = δ = ±1. This means that a spin transformation is fixed by its action on null directions up to a sign. The transformation in (3.3.2) are known from complex analysis. There they are called Möbius transformations. They are the biholomorphic (analytic) Möbius transformation maps from the Riemann sphere to itself, i.e., its automorphisms preserving its complex structure. We see here for the first time a natural relationship between the complex structure of the Riemann sphere and the geometry of Minkowski space. This interplay is even stronger than what it looks like at the moment.

The equations (3.3.1) deﬁne via (3.2.3) a mapping of the light cone, i.e., the set of null vectors, onto itself. This map is linear and invertible. Furthermore, it leaves the expression T2 − X2 − Y2 − Z2 invariant. This is easiest to see by reformulating the formulae (3.2.3) into

1 1 ξξ¯ = √ (T + Z) , ξη¯ = √ (X + iY) , 2 2 1 1 ηξ¯ = √ (X − iY) , ηη¯ = √ (T − Z) . (3.3.5) 2 2

We can write this in a more compact form as

1 T + ZX + iY ξ √ = ξ¯, η¯ . (3.3.6) 2 X − iY T − Z η

However, note that this can be done only for null vectors. Exercise 3.3: Why? Hint: Consider the determinant of the matrices on the right and the left. Now let a spin transformation be given by

ξ ξ 7→ A (3.3.7) η η

19 then the corresponding transformation of the null vector is

T + ZX + iY T + ZX + iY 7→ A A?. (3.3.8) X − iY T − Z X − iY T − Z

We have obtained this formula by considering the case of a null vector i.e., where (3.3.6) holds. However, the formula is obviously linear in the components (T, X, Y, Z) of the vector. Furthermore, every vector of V can be written as a linear combination of null vectors. Exercise 3.4: Show that this is true. Hint: When U ∈ V is time-like we can always ﬁnd a tetrad so that U = at and similarly for space-like vectors.

These facts allow us to extend the transformation formula (3.3.8) to arbitrary vectors U ∈ V. Since T + ZX + iY det = T2 − X2 − Y2 − Z2 (3.3.9) X − iY T − Z it follows from det A = 1 and (3.3.8) that the scalar product in V is invariant under these maps. Therefore, a spin transformation induces a Lorentz transformation of Minkowski space V. One can now determine the matric L of this Lorentz transformation to ﬁnd its explicit dependence on the spin matrix A, i.e., so that

 T   T   X   X    7→ L   . (3.3.10)  Y   Y  Z Z

Exercise 3.5: Do that. The result is a complicated 4 × 4 matrix with entires which are all quadratic expressions in the matrix elements of A and their complex conjugates (see [8] vol. 1, eq.(1.2.26)). This form also reveals that the Lorentz transformation coming from a spin transformation is orthochronous.

This fact also follows from a topological argument using the fact that the continuous image of a connected set is connected. The spin group SL(2, C) is a connected group and the map A 7→ L is continuous (the matrix elements of L are all polynomials in those in A). Therefore, the image of the map is also connected. Therefore, it must be contained in the connected component of the Lorentz group which contains the identity. But this is exactly the subgroup of orthochronous Lorentz transformations. Conversely, one can show that every orthochronous Lorentz transformation can be obtained from a spin matrix A (and −A). Therefore, there is a 2 − 1 relationship between spin and orthochronous Lorentz transformations.

20 3.4 Lorentz transformations: a closer look

To end this chapter we want to look more closely at some properties of Lorentz transformations. We do this using the Möbius transformations of the celestial sphere. It is a well known theorem of topology that each homeomorphism of the sphere has at least one fixed point. How is this reflected in the transformation (3.3.2)? These formulae imply γζ2 + (δ − α)ζ − β = 0. (3.4.1) For spin transformations which are not proportional to the identity this equation has at most two different solutions. Thus, we obtain the result that every orthochronous Lorentz transformation leaves at least one and at most two null directions fixed. Thus, there are two kinds of such Lorentz transformations, those with one and those with two fixed null directions.

Let us first consider the case of two invariant directions. These span a 2-dimensional subspace of V and we can choose a Minkowski tetrad in such a way that t and x lie in that subspace. Exercise 3.6: Why does this subspace carry a Lorentz metric? Then t ± z each point into one of the invariant null directions and we can assume that the fixed points on the celestial are antipodes. We choose the coordinate ζ so that the fixed points lie at the poles of the sphere. Then we are faced with Möbius transformations which keep ζ = 0 and ζ = ∞ fixed. Exercise 3.7: Show that the most general spin transformation which keeps ζ = 0 and ζ = ∞ fixed is defined by the matrix α 0 A = , 0 α−1 so that ζ 7→ α2ζ (3.4.2) . From (3.3.8) we obtain the corresponding Lorentz transformation

T + ZX + iY αα¯ (T + Z) α/α¯ (X + iY) 7→ X − iY T − Z α¯ /α(X − iY)(αα¯ )−1(T − Z)

We see that T and Z transform with the squared modulus of α while X and Y transform with the double argument of α. Therefore, we put α = weiφ and we obtain the transformation

1 1 1 T 7→ w2 + T + w2 − Z , X 7→ cos 2φ X − sin 2φ Y 2 w2 w2 (3.4.3) 1 1 1 Z 7→ w2 − T + w2 + Z , Y 7→ sin 2φ X + cos 2φ Y. 2 w2 w2

21 Obviously, the class of orthochronous Lorentz transformations with two fixed null directions consists of two classes: rotations ζ 7→ e2iφζ and boosts, ζ 7→ w2ζ, i.e., pure velocity transformations. The fact that these are indeed velocity transformations can be verified easily by putting 1 + v w2 = (3.4.4) 1 − v We find that these transformation keep the null directions fixed but not the null vectors pointing into these directions. This stands in contrast to the rotations. The general Lorentz transformation with two fixed points which consists of a combination of boosts four-screw and rotations and corresponds to (3.4.2) is called a four-screw. From the formula ζ˜ = wζ for a pure boost one can very quickly obtain the aberration aberration formula formula for light. We consider two observers which move relative to each other with a velocity V in the direction given by θ = 0, in the direction of their z-axes. Both observe light approaching from a direction θ resp. θ˜. Since the incoming light is described by the past celestial sphere S− we need to compose with the antipodal map first so that we obtain for the map between the two angles the expression

θ˜ θ tan = w tan , (3.4.5) 2 2

q 1−V where w = 1+V . For an observer that is moving fast and also accelerates, all objects seem to move radially towards a common point. Finally, we discuss the case of transformations with one fixed null directions. We can arrange in a similar way to above that the fixed null direction points through the north pole of the celestial sphere. Then we ask for the most general Möbius transformations which fix the north pole i.e., ζ = ∞ and only the north pole. These are the translations of the complex plane ζ 7→ ζ + β. We write β = a + ib and obtain the corresponding Lorentz transformation X 7→ X + a(T − Z), Y 7→ Y + b(T − Z), 1 Z 7→ Z + aX + bY + (a2 + b2)(T − Z), (3.4.6) 2 1 T 7→ T + aX + bY + (a2 + b2)(T − Z). 2 This transformation does not only fix the direction given by t + z but the vector itself is null rotations fixed. Such Lorentz transformations are called null rotations. They form a subgroup of the Lorentz group which is isomorphic to the additive group C of complex numbers.

22 4 Spinorial objects and spin vectors

In the previous chapter we have seen that one can describe the spheres S+ and S+ of (future-pointing) null directions by equivalence classes of pairs [ξ, η]. We have also seen that one can describe every null vector attached to a point O by giving a pair (ξ, η), as long as one identifies (ξ, η) and (eiφξ, eiφη) for real φ, see (3.2.3). In both representation we lose information: the pair (ξ, η) contains four real numbers, while a point on the sphere needs only two and a null vector is characterised by three real numbers. This raises the question as to whether a pair (ξ, η) can be given a geometric interpretation without loss of information. We are looking for a geometric object whose geometric description in terms of coordinates is given directly by (ξ, η). This means that the object does not change when we change (ξ, η) by a spin transformation, because this leads to a Lorentz transformation leaving the Minkowski geometry invariant. Obviously, when considering a null vector we lose a phase. Thus, we need to find a way to represent the information contained in that phase. This leads to the notion of null flags.

4.1 Null ﬂags

We ﬁrst discuss the situation on S+, the sphere of future null directions. Each point L ∈ S+, is given by an equivalence class of pairs [ξ, η] or (except for one point) by the quotient ζ = ξ/η. This exhausts one half of the the available information in (ξ, η), namely two real numbers. The other two numbers are hidden in a natural way in the tangent vectors to the sphere. A tangent vector to S+ at a point with the coordinate ζ can be written as

M = f (ξ, η)∂ζ + f (ξ, η)∂ζ. (4.1.1) Here, f is an arbitrary (smooth) complex valued function of (ξ, η). The information contained in f (ξ, η) amounts to two real numbers. We want to interpret L together with M as the geometric object that we look for. For this to be feasible it is necessary that it does not change when we apply a spin transformation (3.3.1) to (ξ, η). Then (ξ˜, η˜) = (αξ + βη, γξ + δη) is another coordinate rep- ˜ αζ+β resentation for L, M, and hence ζ = γζ+δ is another complex stereographic coordinate for L.

23 Under the change from ζ to ζ˜ the derivative ∂ζ transforms according to

1 η2 ∂ = ∂ = ∂ . (4.1.2) ζ (γζ + δ)2 ζ˜ η˜2 ζ˜

For M to be invariant requires

f (ξ, η)∂ + f (ξ, η)∂ = f (ξ˜, η˜)∂ + f (ξ˜, η˜)∂ , (4.1.3) ζ ζ ζ˜ ζ˜ so that η2 f (ξ, η) = η˜2 f (ξ˜, η˜). Thus, we can put f (ξ, η) = g(ξ, η)/η2 where g(ξ, η) = g(ξ˜, η˜). Exercise 4.1: Show that the invariance of g implies, that g must be constant. It is conventional to choose g(ξ, η) = − √1 and this results in 2

1 1 1 M = −√ ∂ζ + ∂ . (4.1.4) 2 η2 η¯2 ζ

We have found that a pair (ξ, η) determines a null direction L and a tangent vector M to S+ at the point L. Conversely, when L and M are given, then we first find the quotient ζ = ξ/η and then, by comparing coefficients, the components of M, i.e., η2. With this we also know ηξ = η2ζ and ξ2 = η2ζ2. This allows us to determine ξ and η up to a common sign: (ξ, η) and (−ξ, −η) determine the same L and M.

Figure 4.1: A point on the celestial sphere with a tangent vectors

So much about the celestial sphere S+ see Fig. 4.1.

24 How is this situation represented in Minkowski space? The coordinate descriptions of S+ and S+ are the same so that we can simply take over the expressions for L and M. The null direction L corresponds to a null vector with endpoint on S+ and M corresponds to a tangent vector to S+. However, here one needs to be careful: (4.1.4) deﬁnes a tangent vector to the sphere but not a vector in Minkowski space because those have four components. When regarded as differential operators these vectors are given by expressions of the form

∂ U = Ui = U0∂ + U1∂ + U2∂ + U3∂ . (4.1.5) ∂Xi T X Y Z The Minkowski coordinates Xi = (T, X, Y, Z) of points on S+ are given in terms of ζ and ζ by (see (3.1.3) and (3.2.2))

ζ + ζ ζ − ζ ζζ − 1 T = 1, X = , Y = −i , Z = , (4.1.6) ζζ + 1 ζζ + 1 ζζ + 1 and this implies

2 ∂T ∂X 1 − ζ2 ∂Y 1 + ζ ∂Z 2ζ = 0, = , = , = . (4.1.7) ∂ζ ∂ζ (1 + ζζ)2 ∂ζ i(1 + ζζ)2 ∂ζ (1 + ζζ)2

This gives us the expressions

∂ 1 1 ∂Xi 1 ∂Xi ∂ M = Mi = −√ + (4.1.8) ∂Xi 2 η2 ∂ζ η¯2 ∂ζ ∂Xi for the vector M in terms of the Minkowski basis and this results in the components

ξ2 + ξ¯2 − η2 − η¯2 M0 = 0, M1 = √ , 2(ξξ¯ + ηη¯)2 (4.1.9) ξ2 − ξ¯2 + η2 − η¯2 2(ξη + ξ¯η¯) M2 = √ , M3 = −√ . 2i(ξξ¯ + ηη¯)2 2(ξξ¯ + ηη¯)2

Obviously, this vector is space-like with a squared length 2 M · M = − . (4.1.10) (ξξ¯ + ηη¯)2

Notice that these formula are valid for arbitrary values of ξ and η, including for η = 0. Using them we can now detach ourselves from the concrete sphere S+ and represent L by the null vector with components√ (3.2.3) and by deﬁning M by (4.1.9). Now, M is a unit-vector exactly when ξξ¯ + ηη¯ = 2, i.e., exactly when L has its endpoint on S+. We ﬁnd that the length of M varies inversely proportionally to the extent of L. extent Now we have assigned to each pair (ξ, η) a pair of vectors L and M in Minkowski space. If this pair would be an invariant object then it should not change under a change of

25 Minkowski tetrad, i.e., a passive (orthochronous) Lorentz transformation. But this is not the case. In order to see this it is sufﬁcient to consider a special case. We choose the pair (ξ, 0) corresponding to the north pole of the sphere S+ of null directions. The corresponding pair of vectors has the components     1 √0 1 0  2 L = √   M =   2 0  0  1 0

Now we consider spin transformations which map (1, 0) to itself. The corresponding Lorentz transformations are just the null rotations introduced in the previous chapter which leave the null vector L invariant, see (3.4.6). Now it is straightforward to verify that under these transformations the vector M is mapped to √   b√ 2  2  M˜ =   = M + 2bL  √0  b 2 This shows us that the pair of vectors does not remain invariant under Lorentz transformations. However, what does remain unchanged is the 2-dimensional subspace which is spanned by L and M. In order to account for the orientation of M we only admit half space positive multiples of M so that we obtain a half space, bounded by multiples of L

Π = {aL + bM|a, b ∈ R, b > 0}. (4.1.11) null flag This half space is the null flag that we were looking for. It is this object, that can be defined in terms of a pair of complex numbers (ξ, η). The half space touches the null cone of O along the null direction given by L. Every other vector in Π not parallel to L is space-like and orthogonal to L. The half space Π determines (ξ, η) up to a common sign. Since L is the boundary of Π we know with L the pair (ξ, η) up to a common phase. Then Π gives us the orientation of any other vector M in Π and therefore the phase due to (4.1.4). Again, we find that the geometric representation of (ξ, η) can be fixed only up to a sign. This arbitrariness is essential — it can not be removed within the Minkowski geometry. This can be clarified with the following argument. iφ iφ 1 2 Consider the pairs (e ξ, e η) with 0 ≤ φ ≤ 2 π. These form a closed loop in C along which the starting pair (ξ, η) is “rotated” around by an angle of 2π ending up at the same pair. The corresponding null flags rotate around the common null vector L. The importance of the rotation of the flags is that it happens “twice as fast” than the corresponding rotation of the complex pairs: when φ runs from 0 to π then (ξ, η) moves to (−ξ, −η) while the null flag rotates through a full angle of 2π returning to their starting position. When φ continues on from π to 2π the pairs finish their rotation back to the starting point while the null flags do another full rotation through 2π.

26 The deeper reason for the behaviour is the 2-1 relationship between the spin transformations and the orthochronous Lorentz transformations: the identity in orthochronous Lorentz group corresponds to ±I in SL(2, C). Therefore, a pair (ξ, η) can not be represented bijectively by an object of Minkowski geometry, since these map to themselves when (ξ, η) is replaced by (−ξ, −η). If we want to remove the ambiguity of the sign then we are forced to go beyond the Minkowski geometry by assigning those objects a geometrical meaning which are car- ried to themselves not necessarily after a rotation of 2π but only under rotation by 4π. Such objects are called spinorial objects. A special example are the spin vectors. spinorial object

4.2 Spin vectors

The simplest spinorial objects are spin vectors. They are the geometric objects which are directly described by the coordinate pair (ξ, η) of complex numbers. Up to a sign they are represented by null flags in Minkowski space. We call the set S• of all spin vectors spin space. The elements of spin space are the spin vectors κ. spin space Which operations are defined on spin vectors? Since we can assign a null-flag to each pair of complex numbers each such pair is an admissible representation of a spin vector. Therefore, each operation on pairs of complex numbers is transferable to null flags as long as they are compatible with spin transformations. Thus we need to find properties of the spin space which exhibit the spin transformations as its natural invariance group. The most obvious property is that spin transformations are linear maps so that sum and complex multiples are compatible with them.

Exercise 4.2: Show that for any pair (ξ1, η1) and (ξ2, η2) the expression

ξ1η2 − ξ2η1 is invariant under spin transformations. This combination of two pairs is obviously bilinear, anti-symmetric and regular. The so described operations on pairs of complex numbers give rise to operations on spin vectors. These are • addition of two spin vectors (κ, ω) 7→ κ + ω, • multiplication with complex numbers κ 7→ λκ, for λ ∈ C, • the "‘symplectic product"’ (κ, ω) 7→ {κ, ω} ∈ C. The following algebraic rules hold: 1. S• is a complex vector space with respect to addition and scalar multiplication. 2. {·, ·} is a bilinear and anti-symmetric form {κ, ω} = −{ω, κ}.

27 3. The identity {κ, ω}τ + {ω, τ}κ + {τ, κ}ω = 0 holds for all spin vectors κ and ω.

Exercise 4.3: Prove this identity. The space of spin vectors is 2-dimensional since for any two spin vectors κ and ω with {κ, ω} 6= 0 it follows that every other spin vector is a linear combination of these two. • spin frame A basis for (o, ι) for S is called a spin frame, or spin basis or spin dyad if {o, ι} = 1. Every spin vector can be written as a linear combination with respect to such a spin frame κ = κ0o + κ1ι and this yields the expression for the symplectic product between two spin vectors with respect to to the spin frame {κ, ω} = κ0ω1 − κ1ω0. (4.2.1)

The spin frame√ vector o has the components√ (1, 0), i.e., it is assigned to the null vector L = (t + z)√/ 2 and the vector M = 2x.√ Similarly, the spin frame vector ι is mapped to (t − z)/ 2 and the space-like vector − 2y.

4.3 Some global considerations

In chapter 2 we had seen that Minkowski space M is an affine space. Each event O −→ has attached a Minkowski vector space of position vectors OP. We have defined null directions light cones, null flags etc with respect to these vectors spaces. Therefore, these objects exist at each separate event. Similarly, we have to imagine now that at each event P of M there is attached a separate spin space whose elements can be put into relation with the objects in the Minkowski vector space at the same event. This gives us an entire collection of spin spaces, each indexed (so to speak) by the event that it is spin bundle attached to. The existence of such a spin bundle is of fundamental importance for all considerations in relativity, and quantum field theory, that refer to spinors. For Minkowski space M the existence of this spin bundle is not a problem. Restriction arise when we ask for spin bundles on arbitrarily curved manifolds M with Lorentz metric. Here also, one can attach to each point of M a vector space, the tangent space of M, which is eqipped with a bilinear form of signature −2. And also in this case there are null flags which can be characterised up to a sign by a pair of complex numbers. However, the question arises now as to whether it is possible to consistently and continuously select from the pair of numbers exactly one. Two necessary criteria that the manifold M must satisfy are obvious: the spin transformation generate only orthochronous Lorentz transformations therefore the tangent spaces are automatically time and space-time oriented. Therefore M must be orientable and time-orientable. But this is not all. There is another restriction which is of a topological nature and which we can not discuss here in any further detail. Essentially the following condition must hold

28 On every closed 2-dimensional surface S in M there exist three continuous vector ﬁelds which are linearly independent at each point on S. For non-compact space-times the theorem by Geroch1 holds

Theorem 4.1. For M a non-compact space-time the necessary and sufﬁcient condition for the existence of a spin bundle is the existence of four continuous, linearly independent vector ﬁelds, which form a Minkowski tetrad at every point of M.

In the following, when discussing curved manifolds, we will always assume that these conditions are satisﬁed. However, most of the time our discussions will be limited to a local context anyway where there is no restriction on the existence of a spin bundle.

1R. Geroch (1968), Spin-structure of space-times in general relativity, J. Math. Phys. 9, 1739–1744

29 5 Spinor algebra

In the previous chapter we introduced the space of spin vectors which is attached to every event in Minkowski space-time, and more generally, at every point of a general curved space-time. A spin vector is essentially a null flag but with the understanding that a rotation around any axis through the angle of 2π does not lead back to the original spin vector but to its negative, which however, is represented by the same null flag. Only a further rotation around 2π will bring the spin vector back to its original state. The spin vectors form a 2-dimensional complex vector space which carries as an additional structure a symplectic product. All operations which are possible for the spin vectors (addition, scalar multiplication and symplectic product) can be interpreted in a geometric way (see [8], vol. 1, chap. 1.6). So far the approach towards spin vectors has been entirely geometric. For later applications however it is useful to also have algebraic tools for computations available. Therefore, in this chapter we present the algebra of spinors. This is essentially tensor algebra constructed from the basic vector space of spin vectors. Thus, spinors are tensors over spin space S•. Most statements will be valid not only for spinors at a space-time point (event) but more generally for spinor fields over a general manifold M (where we of course assume that M must admit a spin structure). With the algebraic approach we will be able to derive the geometric properties and constructions in Minkowski space that we had used in order to get to the spin vectors. The geometry of space-time can be completely reconstructed from the algebra of spinors. In this sense the spinor algebra is more fundamental than the tensor algebra of vectors over Minkowski space.

5.1 Foundations of tensors abstract index From now on we will use the abstract index notation for tensors and spinors. This is a convention (or formalism) which guarantees not only the basis independence of the formalism but also the easy manipulation of tensor expressions. A symbol Vα does not represent an n-tuple of numbers (V1,..., Vn) but it stands for a speciﬁc element V in an n-dimensional vector space. The index α does not assume the concrete sequential values 1, . . . , n, but it is a label for the type of object, in this case a vector. The same is true for the symbol Vβ, which by convention we agree to stand for the same element V as Vα. Obviously, the symbols Vα and Vβ are different, but they

30 refer to the same element. The reason for this convention becomes clear when considering tensor expressions such as UαVβ − VαUβ. If the respective symbols were the same then this expression would vanish identically. Instead, this expression symbolises an anti-symmetrisation, U ⊗ V − V ⊗ U. Certainly, this notation needs some getting used to but it is logically consistent and can be handled quite intuitively after some time. Abstract indices can be introduced formally in a rigorous way (again, see [8]).

Tensors are usually introduced for vector spaces over a number field. However, if one wants to treat tensor fields on the same footing as tensors then one needs to generalise the notions “vector space” over a “field” to “module” over a “ring”. For example, when dealing with C∞ tensor fields the field of scalars is replaced by the ring of C∞-functions over a manifold. In that case, the functions serve as the scalars with which we multiply the tensor fields. In all what follows it is however advantageous to think about a vector space over R or C.

We denote the set of scalars by S. They form an Abelian ring with identity 1, characterised by the following properties for any a, b, c ∈ S

(i) a + b = b + a.

(ii) a + (b + c) = (a + b) + c.

(iii) ab = ba.

(iv) a(bc) = (ab)c.

(v) a(b + c) = ab + ac.

(vi) There exists 0 ∈ S mit 0 + a = a for all a ∈ S.

(vii) There exists 1 ∈ S mit 1a = a for all a ∈ S.

(viii) For every a ∈ S there exists (−a), so that a + (−a) = 0.

The space for which we want to deﬁne tensors will be denoted by Sα (in the sense of the abstract index notation). It is a module over S. As such it is characterised by

(i) Uα + Vα = Vα + Uα.

(ii) Uα + (Vα + Wα) = (Uα + Vα) + Wα.

(iii) a (Uα + Vα) = aUα + aVα.

(iv) (a + b)Uα = aUα + bUα.

(v) (ab)Uα = a (bUα) .

(vi) 1Uα = Uα.

(vii) 0Uα = 0α(= 0).

31 In this connection the elements of S are referred to as scalars or tensors of rank 0, those of Sα are called contravariant vectors or contravariant tensors of rank 1. In order to α also reach covariant tensors we consider the space Sα which is dual to S . This is the α α α space of all S-linear maps S → S. When Qα ∈ Sα, then Qα(aU ) = aQα(U ) and α α α α Qα(U + V ) = Qα(U ) + Qα(V ) hold. Addition and multiplication in Sα are deﬁned "pointwise", i.e., for all Uα ∈ Sα

α α α (Qα + Rα)(U ) = Qα(U ) + Rα(U ), (5.1.1) α α (aQα)(U ) = aQα(U ). (5.1.2)

This makes Sα into a S-module. Often the parentheses are left off and it is written α β γ simply QαU = QβU = QγU = ....

Since Sα is a module we can construct its own dual space. In general, this is a space containing Sα. However, in all situations of interest here, this is not the case and con- α α struction of the double dual of S , i.e., the dual of Sα returns S . This is called the property of reﬂexivity. Tensors of arbitrary rank are deﬁned most easily by multilinear maps. Every map

α...β γ δ Tγ...δ : Sα × ... × Sβ × S × ... × S → S, (5.1.3) | {z } | {z } p q

p tensor which is S-linear in each argument is called a tensor of type q . There are several other definitions for tensors. However, in our context these are all equivalent. We write α...β p α Sγ...δ for the set of tensors of type q over S . tensor operations We now discuss the tensor operations. First, there is addition: the sum of two tensors of the same type is the map which maps to the sum of the images of the two tensors (“pointwise definition”). In the same way we can define the outer product or tensor tensor product p r product of two tensors of arbitrary type q and s which results in a tensor of type p+r α q+s by multiplying their images. As an example consider the two tensors A and Bβγ. Their outer product is defined by the equation

α β γ α β γ A Bβγ U V Qα = (A Qα) BβγU V . (5.1.4)

In general we have the rules

α... γ... γ... α... Aβ...Bδ... = Bδ... Aβ..., (5.1.5) α... γ... γ... α... γ... α... γ... Aβ... Bδ... + Cδ... = Aβ...Bδ... + Aβ...Cδ... . (5.1.6)

A special case of the outer product is the scalar multiplication when one of the factors 0 α...β is of type 0 ), i.e., an element of S. With this operation each space Sγ...δ of tensors of the same type acquires the structure of an S-module.

32 Index substitution is a formal operation in the abstract index formalism. It consists only in replacing an index in a general tensor expression by another symbol. It origi- nates from the definition of a tensor as a multilinear map and corresponds to an operation on the arguments. For example, consider the tensor Aαβ ∈ Sαβ. We can define another tensor Bαβ ∈ Sαβ by putting Bαβ := Aβα. In terms of the definition as a map this relationship corresponds to the equation B(U, V) = A(V, U). This defines a new map which is obtained from the old one by switching the arguments. In the index notation this is denoted by interchanging the indices.

Finally, there are the contractions. To define these we observe that under the current conditions (reflexivity) every tensor can be written as a linear combination of simple tensors: i i i i i i α...βζ = α β ζ Aγ...δη ∑ U ... V W Qγ ... RδSη . (5.1.7) i With this representation we can obtain a contraction as a map defined by

α...βζ α...β Sγ...δη → Sγ...δ ! i i i i i i α...βζ 7→ α...βη = η α β Aγ...δη Aγ...δη ∑ W Sη U ... V Qγ ... Rδ . i

Now one needs to and, in fact, can show that this definition does not depend on the representation (5.1.7) so that the contraction is well defined. Using index permutations αβ βα one can define contractions over arbitrary indices. For instance, Aγα = Bγα, where αβ βα Bγδ := Aγδ. It is easy to see that contractions are compatible with addition and multiplication. Sometimes, the combination of outer product with following contraction is βγ called a transvection as illustrated in the following example: Aα is transvected with δ α βγ V , by forming the expression V Aα .

An important tensor is the Kronecker delta (here we really mean a map and not the Kronecker delta usual symbol, see below). One of the many equivalent deﬁnitions is as follows: consider the map

β S × Sα → S β α U , Vα 7→ U Vα

α α 1 It is bilinear, hence a tensor δβ ∈ Sβ of type 1 . Thus,

α α β U Vα = δβ U Vα.

This holds for all Vα. Therefore, we can also write

α α β U = δβ U .

33 We can interpret this equation as the action of the identity in Sα acting on an element. α Therefore, the tensor δβ also represents the identity map. We can also read it as a manifestation of the index substitution. Even though there have been (too?) many indices in the presentation so far we have never mentioned the notion of a basis. Everything that was said until now is entirely coordinate and basis independent. When we now introduce a basis then for the sole purpose to show how one can decompose tensor expressions into components. basis 1 α α α α A ﬁnite basis of Sα is a set of n elements δ1 , δ2 ,..., δn, such that every element V of Sα can be written as an unique linear combination

α 1 α 2 α n α V = V δ1 + V δ2 + ... + V δn (5.1.8) α dimension Then the number n is characteristic for S and is called the dimension of the S-module. We now assume the existence of such a basis (this is true in all cases discussed here) and we denote the concrete indices 1, 2, . . . , n collectively by underlined letters α, β,.... α α Then a basis of S is written as (δα )α=1,...,n. We introduce the Einstein convention for concrete indices and then (5.1.9) can be written simply as

α α α V = V δα. (5.1.9) Thus, a basis simply converts contravariant, concrete indices to abstract indices (this also explains the use of the symbol δ for a basis). α dual basis To each basis S we can introduce its dual basis. This is a basis in the dual space Sα α α α which is deﬁned a maps which assign to each V = V δα a scalar as follows

α α α α δα : S → S, V 7→ V . (5.1.10)

α α Therefore, δα is the map which assigns to each V its component in the direction of the α basis vector δα α α α δαV = V . (5.1.11)

α Exercise 5.1: Show that (δα )α=1,...,n is indeed a basis of Sα. The dual basis is seen to convert covariant abstract indices into concrete ones. Clearly, we have α α α δα δβ = δβ. (5.1.12) α Here, (δβ)α,β=1:n with concrete indices is the n × n identity matrix, the usual well known Kronecker symbol Kronecker symbol. One can obtain the components of vα by transvecting with the dual basis see (5.1.11). β Similarly, for Qβ we have Qβ = Qβ δβ and also for tensors of higher rank. For example

1This is neither the Kronecker delta nor the usual Kronecker symbol. The kind of indices is important.

34 we can regard (5.1.12) as the determination of the components of the Kronecker delta with respect to a basis α α α β δβ = δβδα δβ. (5.1.13) Or, more general, α...β α...β α β γ δ Aγ...δ = Aγ...δ δα δβ δγ δδ. (5.1.14)

When the components of a tensor with respect to a basis are known then one can write it as a linear combination α...β α...β α β γ δ Aγ...δ = Aγ...δ δα δβ δγ δδ. (5.1.15) To write tensor expressions with respect to a basis we replace each tensor by a linear combination (5.1.15) to obtain expressions which only contain components. For example for the scalar product we obtain

α β α β 1 2 n QαU = (Qβδα ) U = QβU = Q1U + Q2U + ... + QnU . (5.1.16)

α α α α αˆ When δα and δαˆ are two bases for S then there are matrices s αˆ and t α, inverse to each other so that

α α γˆ α α β β β αˆ γˆ γˆ α δα = δγˆ t α, δαˆ = δβ s αˆ , δα = s αˆ δα, δα = t α δα. (5.1.17)

For the components of a tensor there result very characteristic transformation laws un- α der a change of basis. For example we obtain for the components of a tensor Tβγ the transformation law ˆ Tα = Tαˆ sα tβ tγˆ (5.1.18) βγ βˆ γˆ αˆ β γ α αˆ We observe, that the appearance of the matrices s αˆ and t α is governed entirely by the number and positions of the indices i.e., the type of the tensor. Hence, tensors can be deﬁned by reference to these transformation laws. This has been the approach in many older physics textbooks.

5.2 Spinors

The general constructions of the last section can now be used to introduce spinors and spinor fields. For the first case we take the complex numbers C as the scalar range S and the spin vector space S• as S module (in this case as a C vector space). In the second case, we use for S the complex-valued C∞ functions on a manifold M and we take the spin vector fields on M as the S-module, i.e. the sections in the spin bundle over M. In the previous chapter we introduced three operations that can be performed with spin vectors, namely addition, scalar multiplication and the symplectic product. These operations are initially only defined for a vector space. However, by performing them in a

35 continuous and sufficiently smooth manner at any point of a manifold, such operations can also be performed on spin vector fields. Spinors are denoted by Latin capital letters as indices. So we have SA as a module and generally call its elements κA, ωB, or similar. Analogous to the discussions in the previous section, we can now define tensors over SA.

5.2.1 ε spinors

The symplectic product has an important consequence. It implies the existence of a special ε-spinor. Obviously, the symplectic product of two spin vectors is a bilinear • • map S × S → S. Consequently, there is an element ε AB ∈ SAB, so that for all ω, κ ∈ S• A B {κ, ω} = ε ABκ ω = −{ω, κ}. (5.2.1)

As a consequence ε AB anti-symmetric, εAB = −εBA. The symplectic product can be used to assign a spin covector to each spin vector κ, i.e. the linear map

ω 7→ {κ, ω}. (5.2.2)

B A B In index notation this is ω 7→ κ ε ABω . We denote the element from the dual space SB A A of S deﬁned in this way by κB, so that we obtain the equation κB = κ ε AB. This shows that the ε spinor serves the same purpose as the metric in Riemannian geometry, it is used to “lower the index”, i.e., to identify vector space and dual space. Applying (the B linear map) κB to ω yields, when we introduce a spin frame, see (5.1.16) and (4.2.1)

B B 0 1 κBω = κBω = κ0ω + κ1ω , (5.2.3) and with it the coordinate representation of this isomorphism between SA and its dual 1 0 AB AB as κ0 = −κ , κ1 = κ . This is clearly invertible so that there must be a spinor ε ∈ S which achieves the inverse map. A AB κ = ε κB. (5.2.4)

This gives the following relations which express the fact that the two ε-spinors ε AB and εAB are inverse to each other:

AB B CB C ε ε AC = δC, ε ABε = δA. (5.2.5)

However, we may also interpret these in the following way: in the ﬁrst equation ε AC pulls the index A down, while in the second equation εCB raises the index B. This yields equations of the form B B C εC = δC = −ε B. (5.2.6) B B Often, people write εC instead of δC. From (5.2.5) ensues the relation

AC BD AB ε ε εCD = ε , (5.2.7)

36 AB which implies that one can regard ε as ε AB with its indices raised. This implies the anti-symmetry of εAB. Raising and lowering of spinor indices needs some care because since the anti-symmetry of the ε-spinors has to be accounted for. Here are the conventions again B BA C T = ε TA, TA = T εCA, (5.2.8) i.e., when the indices are located directly next to each other the contraction operates from the upper left to the lower right. We can write the symplectic product also in the form

A B B {κ, ω} = ε ABκ ω = κBω , (5.2.9) and we can reformulate the triple identity of section 4.2

A B C A B C A B C ε ABκ ω τ + ε ABω τ κ + ε ABτ κ ω = 0. (5.2.10)

Renaming (substituting) the indices appropriately and introducing a second εCD and lowering all indices we can write

ε ABεCD + εBCε AD + εCAεBD = 0. (5.2.11)

Here, we have used the fact the identity holds for all spin vectors κA, ωB, τC and therefore cane be reduced to an identity involving the ε’s alone. A very useful property of the spinor algebra follows from (5.2.11) when we raise two indices:

B D D B BD ε A εC − ε A εC = ε ACε . (5.2.12)

Transvecting this identity with an arbitrary spinor TBD yields

D TAC − TCA = ε ACTD . (5.2.13) this means that anti-symmetrisation is equivalent to taking a trace. In particular, when D TAB = −TBA, then TAB = 1/2ε ABTD , i.e., all anti-symmetric elements in SAB are proportional to each other.

5.2.2 Complex conjugate spinors

The spinor algebra as it was built up so far is not yet complete. As can be seen from (3.2.3), we must also consider complex conjugate spinors. So we need an operation that assigns to each spin vector κA its complex conjugate counterpart κA. Of course, the complex conjugated coordinates κ0, κ1 should be able to serve as coordinates for the conjugated spin vector, and it should it is also be true that κA = κA as well as the anti-linearity λκA = λ¯ κA. The conjugated spin vectors form a vector space, the space 0 SA which is conjugate to SA. The complex conjugation forms an anti-isomorphism

37 between the two spaces. In the index formalism, we use primed spinor indices for 0 0 conjugated complex spin vectors. For example, we write µA ∈ SA and

0 κA = κ¯ A , (5.2.14)

0 in order to express the fact that the conjugated spin vector κ¯ A is the image of κA under complex conjugation. We then obtain the property of anti-linearity in the form of

0 0 ακA + βωA = α¯ κ¯ A + β¯ω¯ A . (5.2.15)

A0 Since S is a complex vector space there exist a space SA0 which is dual to it, whose 0 elements are C-linear maps on SA. The complex conjugation map SA → SA induces a complex conjugation map SA → SA0 , which assigns to each τA the element τ¯A0 via the equation 0 A A τ¯A0 κ¯ = τAκ (5.2.16) for arbitrary κA. The line on the right hand side is the usual complex conjugation of complex numbers. A A0 Now we can use the four spaces S , SA, S , SA0 to build up the general spinor algebra. We deﬁne a general spinor as a multilinear map of any Cartesian product of these p r four basic spaces into the scalars. In this way we obtain general spinors of type q s

A...BA0...B0 TC...DC0...D0 . (5.2.17)

As before, there are the operations of addition, outer multiplication and contraction, which are allowed for such general spinors (contractions, of course, only between primed or unprimed indices). However, there is now the additional complex conjugation, p r which can also be extended to general spinors. This turns a spinor of type q s into r p one of type s q . We illustrate the definition by way of an example. The general defi- AB0 nition is clear but complicated to write down explicitly. So let TCDE0 be a spinor of type 1 1 0 0 0 C D E ¯ BA 2 1 , i.e., a multi-linear map SA × SB × S × S × S → C. Then we define TEC0 D0 E C0 D0 as the map SB × SA0 × S × S × S → C, by the equation

BA0 E C0 D0 0 0 ¯ 0 AB 0 ¯ E C D TEC0 D0 κBω¯ A λ µ¯ τ¯ = TCDE0 κ¯B ωAλ µ τ . (5.2.18)

It is worthwhile to note that the sequence within a set of one kind is meaningful, but not within sets of different kinds. Thus, TABC0 = TAC0 B = TC0 AB, but in general it is not true that TABC0 = TBAC0 . Furthermore, is is irrelevant when an index appears with its primed version, i.e., TAA0 is exactly the same element as is TAB0 . 0 The complex conjugation transplants all the structures of SA to SA , in particular the symplectic product. Thus, there exist also primed ε-spinors

0 0 A B AB ε A0 B0 = ε AB, ε = ε . (5.2.19)

38 These can serve as their unprimed counterparts to raise and lower primed indices (i.e., A0 they deﬁne the isomorphisms between S and SA0 ) and the conventions that hold for the former also hold for the latter. Finally, we want to mention that everything that has been said about the complex conjugation of spinors can be transferred to the complex conjugation of spinor ﬁelds.

5.2.3 Spin frames

In section 4.2 we have introduced the notion of a spin basis or a spin frame (o, ι), i.e., a pair of spin vectors which satisfy the normalisation condition {o, ι} = 1. Now we can also write this as A A oAι = 1 = −ιAo , (5.2.20) and, due to the anti-symmetry of the symplectic product

A A oAo = ιAι = 0. (5.2.21) A A A A A One often writes ε A, A = 0, 1 for such a basis so that ε0 = o und ε1 = ι . For the dual A basis ε A A A A ε A εB = δB (5.2.22) 0 1 must hold and one ﬁnds very quickly that ε A = −ιA, ε A = oA. The components of a spinor are obtained as in section 5.1 by transvection with the basis spinors. For example, the components of the ε-spinors one ﬁnds immediately 0 1 ε = ε εA εB = . (5.2.23) AB AB A B −1 0 The components of the ε-spinors can be found in an analogous way. A 0 A 1 A The components of any spin vector κ are obtained from κ = −κ ιA und κ = κ oA so that A B A B A κ = − κ ιB o + κ oB ι . (5.2.24) This is another example for the triple identity of section 4.2. 0 0 A spin frame (oA, ιA) induces via complex conjugation a spin frame (oA , ιA ) in the complex conjugate spin space. Everything said about the unprimed spin frames is valid analogously for the primed spin frame. Finally, we quote a useful result: A A A Theorem 5.1. If αAβ = 0 (at a point), then α and β are proportional (at this point).

To prove this we may assume without loss of generality that αA 6= 0. We complement A A A A A A A α to a spin frame (α , γ ), so that αAγ = 1. Then we can write β = aα + bγ and A A the assumption implies via transvection with αA that b = 0, so β = aα .

39 6 4-vectors as spinors

In this chapter we want to show how the entire Minkowski geometry can be reconstructed from the spinor algebra. The equation (3.2.3) suggests that 4-vectors are ob- 1 1 tained by considering spinors of type 0 0 . In order to make the connection between spinors and 4-vectors we can proceed by combining one unprimed and one primed index into one unit and identify it with a vector index. Thus, we put a = AA0, b = BB0, 0 ABA0 B0 ab c = CC , etc. Then we can write the spinor ψCC0 also in the form ψc . Expressed in a 0 0 more conventional way: we consider the vector space SAA = SA ⊗ SA and the tensor algebra over it. Since the vector space itself is a tensor product of spin vector spaces we obtain a natural embedding of its tensor algebra into the spinor algebra. 0 The vector space SAA is a complex vector space. We call its elements the complex AA0 AA0 complex 4-vectors 4-vectorss. However, complex conjugation maps S into itself, i.e., with v also 0 0 0 0 vAA0 = v¯AA lies in SAA . Those elements of vAA of SAA which are invariant under 0 AA0 real 4-vector complex conjugation, i.e., for which vAA = v holds are called real 4-vectors. They form a 4-dimensional real vector space Ta. In order to show that Ta is indeed a Minkowski space one needs to ﬁnd candidates for the metric. So we look for gab ∈ Tab = SABA0 B0 . The only available option is

ab AB A0 B0 gab = ε ABε A0 B0 , g = ε ε . (6.0.1)

ab From the properties of the ε-spinors it follows immediately that gab and g are inverse a a AA0 and that ga = 4. Furthermore, we have as an example T gab = T ε ABε A0 B0 = TBB0 = ab Tb. This means that gab lowers the vector indices (and g raises them).

6.1 Spin frames and their tetrads

A A In order to ﬁnd out more about the properties of gab we select a spin frame (o , ι ). This spin frame and its complex conjugate can be used to construct four 4-vectors

0 0 la = oAoA , na = ιAιA , 0 0 ma = oAιA , ma = ιAoA .

We compute their scalar products using gab and obtain

a a a a lal = nan = mam = mam = 0. (6.1.1)

40 Thus, these four vectors are all null vectors. Here, la and na are real, while ma and m¯ a are conjugate to each other (as indicated by the notation). In addition,

a a lan = −mam = 1 (6.1.2) holds and all other combinations vanish. Four vectors with these properties are called AA0 null tetrad. They are a basis of C over C. For instance, we have null tetrad

b B B0 B B B0 B0 δa = ε A ε A0 = oAι − o ιA oA0 ι − o ιA0 (6.1.3) b b b b = lan + nal − mam − mam , (6.1.4)

And therefore, for every vector Va

b b a a b a b a b a b V = δa V = (V la)n + (V na)l − (V ma)m − (V ma)m . so that every vector can be written as a linear combination of these four null vectors. In order to get a real basis for Ta we form linear combinations 1 1 ta = √ (la + na) , za = √ (la − na) , 2 2 1 i xa = √ (ma + ma) , ya = √ (ma − ma) . 2 2 These four vectors are real, generate Ta and have the scalar products

a a a a tat = −xax = −yay = −zaz = 1, (6.1.5) with all other combinations vanishing. But this means that they form a Minkowski tetrad. The metric gab expressed with respect to this basis has the form diag(+, −, −, −), i.e., it has signature −2. Any vector ka in Ta can be expanded with respect to this Minkowski tetrad

Ka = K0 ta + K1 xa + K2 ya + K3 za, (6.1.6)

0 a 1 a 2 a 3 a where K = K ta, K = −K xa, K = −K ya und K = −K za. On the other hand, a AA0 AA0 A A A K = K ∈ S is a spinor which can be referred to the spin frame ε A = (o , ι ). Comparing these two possibilities, we obtain

0 0 0 0 0 0 0 0 0 Ka = KAA = K00 oAoA + K01 oAιA + K10 ιAoA + K11 ιAιA 0 0 0 0 = K00 la + K01 ma + K10 m + K11 na.

This yields the relationship between the spin components and the Minkowski components in the form

0 0 1 K0 + K3 K1 + iK2 K00 K01 √ 1 2 0 3 = 0 0 . (6.1.7) 2 K − iK K − iK K10 K11

41 When Ka is a (future pointing) null vector then

a 0 2 1 2 2 2 3 2 0 = KaK = (K ) − (K ) − (K ) − (K )

and hence 0 0 K0 + K3 K1 + iK2 K00 K01 0 = = 2 0 0 . K1 − iK2 K0 − K3 K10 K11 Therefore, the column vectors of the matrices in (6.1.7) are linearly dependent, so that we can write 0 ! 0 ! K00 ξ K01 ξ = α und = β , (6.1.8) K100 η K110 η

ξ for some column vector η . Since the Minkowski components are real numbers the hermiticity of the matrix of the spin components follows. This yields the reality conditions αξ = αξ, βη = βη, αη = βξ, (6.1.9) and it follows α = rξ¯ and β = rη¯ for some real number r, which can be absorbed into the deﬁnition of ξ and η. Now one can write (6.1.7) in the form

1 T + ZX + iY ξ √ = ξ¯, η¯ . (6.1.10) 2 X − iY T − Z η

where, in addition, we have put K0 = T, K1 = X, K2 = Y, K3 = Z. This is the same equation as (3.3.6). This shows the following. If we start with a Minkowski space M (or a manifold M which satisﬁes the global conditions given in 4 and introduce spinors using the Minkowski constructions as detailed in that chapter, then the spinor algabra which emerges from that yields back the 4-vectors that we started with. In particular, the metric (6.0.1) agrees with the Minkowski metric.

6.2 Van-der-Waerden symbols

0 The complex null tetrad and its related Minkowski tetrad are bases of Sa = SAA and Ta, induced from spin frames. It is not always possible to use these bases, however. In particular, when special coordinates have been introduced then other bases are better suited. The relationship between spin frames and general bases of Ta is mediated by van-der-Waerden symbols van-der-Waerden symbols.

a a a A A Let δa and δa be basis and dual basis in T , and, similarly, let ε A and ε A be bases for A a... a S . A tensor K b... can be decomposed with respect to the T bases into components a... AA0... K b... and, with respect to the spin frames into the components K BB0.... But it is

42 a... also possible to deﬁne mixed components such as for example K BB0.... In particular, we can decompose the Kronecker-Delta in this way and we obtain

0 0 b BB0 BB a B B0 b BB B B0 δa = δa = σa δa εBεB0 ⇐⇒ δa = σa εBεB0 (6.2.1) 0 0 b b b b A A b b A A = 0 = ⇐⇒ = δa δAA σAA0 δb ε Aε A0 δa σAA0 ε Aε A0 . (6.2.2)

We may write these equations explicitly and we obtain for the ﬁrst set of equations the explicit representation

a 000 A A0 010 A A0 100 A A0 110 A A0 000 a 010 a 100 a 110 a δ0 = σ0 o o + σ0 o ι + σ0 ι o + σ0 ι ι = σ0 l + σ0 m + σ0 m + σ0 n , a 000 A A0 010 A A0 100 A A0 110 A A0 000 a 010 a 100 a 110 a δ1 = σ1 o o + σ1 o ι + σ1 ι o + σ1 ι ι = σ1 l + σ1 m + σ1 m + σ1 n , a 000 A A0 010 A A0 100 A A0 110 A A0 000 a 010 a 100 a 110 a δ2 = σ2 o o + σ2 o ι + σ2 ι o + σ2 ι ι = σ2 l + σ2 m + σ2 m + σ2 n , a 000 A A0 010 A A0 100 A A0 110 A A0 000 a 010 a 100 a 110 a δ3 = σ3 o o + σ3 o ι + σ3 ι o + σ3 ι ι = σ3 l + σ3 m + σ3 m + σ3 n

0 a a a a AA for the four basis vectors (δ0, δ1, δ2, δ3). Obviously, the expansion coefﬁcients σa yield a four 2 × 2 matrices. The basis δa is real, and this means

0 0 a BC A A0 BC A A0 = 0 = δa σa εB εC σa εB εC0 . (6.2.3)

0 0 BC CB 010 100 Therefore, σa = σa , (e.g., σa = σa etc.), so that these four matrices are Hermitian. a Ta This also applies to the “dual” matrices σBB0 . Because of the duality of the bases in and in SA the equations

0 0 AA b b AA a B B0 = = 0 σa σAA0 δa, σa σBB0 ε A ε A (6.2.4) hold. The fundamental relationship (6.0.1) between metric and ε-spinors can also be written in components with the result

AA0 BB0 0 0 gab = ε ABε A B σa σb . (6.2.5)

This is the fundamental identity which is satisﬁed by the van-der-Waerden symbols van-der-Waerden symbols AA0 σa . We rewrite this relation a bit. The left hand side in (6.2.5) is symmetric in a and b. If we write the right hand side explicitly symmetric we obtain

A BA0 A BA0 0 0 ε AB σa A σb + ε AB σb A σa = 2gab.

Considering the left side of the equation without the ε we ﬁnd that

A BA0 A BA0 0 0 σa A σb + σb A σa

43 is anti-symmetric in (A, B). Therefore, we may write it also in the form

0 0 A BA A BA AB 0 0 σa A σb + σb A σa = gabε . Finally, we lower the index B to get

0 0 A A A A A 0 0 σa A σbB + σb A σaB = −gabεB .

A Regarding this as a matrix equation between the 2 × 2 matrices sa = σa A0 , we get

∗ ∗ sasb + sbsa = −gab · 1.

Here, the star indicates Hermitian conjugation and 1 the identity matrix in C2. Finally, we deﬁne the 4 × 4 matrices √ 0 sa γa = 2 ∗ . sa 0 Then, these γ matrices satisfy the anti-commutator relation

γaγb + γbγa = −2gab1. (6.2.6)

Dirac matrices Therefore, the γ matrices deﬁned in this way are automatically Dirac matrices; they

Clifford algebra satisfy the Dirac algebra also referred to as Clifford algebra. The 1 on the right hand side is the identity in C4. This already implies that Dirac spinors are “more compli- 2-component spinors cated” objects than the 2-component spinors we have discussed here. The fact, that Dirac spinors appear here as composition of 2-component spinors suggests that they can be replaced in many calculations by those. In this way one can avoid the explicit calculation with γ matrices in many cases. At the end of this section we display some explicit van-der-Waerden symbols. For the Minkowski tetrad of the previous section we obtain

0 0 AB 1 1 0 0 AB 1 0 1 1 σ = √ = σ 0 , σ = √ = σ 0 , 0 2 0 1 AB 1 2 1 0 AB 0 0 AB 1 0 −i 2 AB 1 1 0 3 σ = √ = −σ 0 , σ = √ = σ 0 . 2 2 i 0 AB 3 2 0 −1 AB

Pauli matrices These are the well-known Pauli matrices. We also determine the van-der-Waerden symbols for the following tetrad

Ta = ta, Ra = sin θ cos φxa + sin θ sin φya + cos θza, Θa = cos θ cos φxa + cos θ sin φya − sin θza, Φa = − sin φxa + cos φya.

44 This tetrad consists of the unit-vectors in radial, longitudinal and azimuthal direction with respect to the usual polar coordinates. With respect to the null tetrad above it can be written 1 Ta = √ (la + na) 2 1 Ra = √ cos θla + sin θe−iφma + sin θeiφma − cos θna , 2 1 Θa = √ − sin θla + cos θe−iφma + cos θeiφma + sin θna , 2 1 Φa = √ −ie−iφma + ieiφma . 2 Now we can read off the van-der-Waerden symbols for the polar tetrad

−iφ AB0 1 1 0 AB0 1 cos θ sin θe σ = √ σ = √ iφ 0 2 0 1 1 2 sin θe − cos θ −iφ −iφ AB0 1 − sin θ cos θe AB0 1 0 −ie σ = √ iφ σ = √ iφ . 2 2 cos θe sin θ 3 2 ie 0 We introduce another null tetrad by 1 1 La = √ (Ta + Ra) , Na = √ (Ta − Ra) , 2 2 1 1 Ma = √ (Θa + iΦa) , M¯ a = √ (Θa − iΦa) . 2 2 Obviously, this null tetrad belongs to a different spin frame (OA, I A). We want to determine the transformation between this and the original spin frame. To this end we ﬁnd the basis transformation between the two null tetrads. A short calculation yields 1 La = (1 + cos θ)la + (1 − cos θ)na + sin θe−iφma + sin θeiφma , 2 1 Na = (1 − cos θ)la + (1 + cos θ)na − sin θe−iφma − sin θeiφma , 2 1 Ma = − sin θla + sin θna + (cos θ + 1)e−iφma + (cos θ − 1)eiφma , 2 1 M¯ a = − sin θla + sin θna + (cos θ − 1)e−iφma + (cos θ + 1)eiφma . 2 We now observe, that 2 θ 2 θ θ θ (1 + cos θ) = 2 cos 2 , (1 − cos θ) = 2 sin 2 , sin θ = 2 sin 2 cos 2 and we quickly ﬁnd that

φ φ A θ −i 2 A θ i 2 A O = cos 2 e o + sin 2 e ι , φ φ A θ −i 2 A θ i 2 A I = sin 2 e o − cos 2 e ι .

45 Show that the van-der-Waerden symbols of the polar tetrad with respect to this new spin basis are also given by the Pauli matrices.

6.3 Clifford algebra, Dirac spinors, Weyl spinors

The Clifford relation (6.2.6) is the starting point for the investigation of Clifford alge- Clifford algebra bras. We want obtain a brief overview of their properties in this section. Thus, we consider an n-dimensional complex vector space V and a bilinear form g, a metric, on V1. The Clifford algebra C(g) is an associative algebra over C with a unit, which contains the vector space V, i.e., there exists a linear map

γ : V → C(g), v 7→ γ(v)

and γ(v)2 = −g(v, v)1 holds. Thus, using the map γ we may regard the vectors of V as elements of the algebra and we can multiply them. Then the Clifford relation holds which means that the product of each vector with itself yields a multiple of 1. Inserting v = v1 + v2 into this relation then we obtain on the left hand side

2 2 2 2 γ(v1 + v2) = (γ(v1) + γ(v2)) = γ(v1) + γ(v1)γ(v2) + γ(v2)γ(v1) + γ(v2)

and on the right

g(v1 + v2, v1 + v2) = g(v1, v1) + 2g(v1, v2) + g(v2, v2)

so that in total we obtain

γ(v1)γ(v2) + γ(v2)γ(v1) = −2g(v1, v2)1

Inserting here in particular the vectors (e1,..., en) of an orthonormal basis and writing γi = γ(ei) then we obtain the well-know Clifford relation

γiγk + γkγi = −2ηik1.

Thus, the elements γi anti-commute and their square is ±1. We can always think of the elements of the Clifford algebra as N × N-matrices (i.e., we consider a representation of the algebra). Then the Clifford elements are linear maps on an N-dimensional vector space where we do not kow anything yet about the dimension N. We will now assume that N is the smallest dimension in which the Clifford relation can be realised for a given algebra, i.e., we assume that the representation is irreducible. Dirac spinor This vector space is the space of Dirac spinors.

1Considering also the reality conditions obscures the discussion of the fundamental properties. Inter- ested readers should consult the books by Friedrich [3] and Lawson und Michelsohn [5].

46 One can show that for each bilinear form there exists a Clifford algebra and that this algebra is unique. In the discussions of the structure of a Clifford algebra the element

η := γ1γ2 ... γn plays a special role2. We ﬁnd the following properties for η 1. commutation property n ηγi + (−1) γiη = 0.

This means that η commutes with all γi when n is odd, while it anti-commutes with them when n is even. 2. Its square is proportional to 1

2 1 s(s+1) η = (−1) 2 1.

Here, s is the signature of g.

Let us ﬁrst discuss the case of odd n. Then η commutes with the γi and hence with all elements of the algebra. Because of our assumption of irreducibility of the representation it follows that η = a1 and due to the property (ii) it follows that

η = ±1 or η = ±i1.

The factor i appears according to the signature of the metric, when s ≡ 3 mod 4. The real case happens when s ≡ 1 mod 4. The sign is convention. For even dimension η2 = ±1 implies as above that η has eigen-values ±1 resp. ±i. In contrast to the odd case here both signs appear. For, if X is an eigen-vector with eigen-value λ then η(γiX) = −γiηX = −λ(γiX) for each γi. Therefore, γiX is an eigen-vector with eigen-value −λ. η is represented in its eigen-vector basis then it has the form

1 0 1 0 η = ± oder η = ±i . 0 −1 0 −1

Here, the 1 in the matrices is the identity matrix of dimension N/2. The full vecor space S splits into the direct sum of the eigen-spaces E± of η resp. of iη. The imaginary case occurs when s ≡ 2 mod 4 (i.e., in the case of a Lorentz metric) while the real case occurs when s is divisible by 4. The γi map the eigen-spaces onto each other. Due to the anti-commutation with η they must possess the form 0 σk γk = . σˆk 0

2 In the case n = 4 this element is often denoted by γ5.

47 The σk are N/2 × N/2 matrices, satisfying the relations

σiσˆk + σˆkσi = −2ηik1 und σˆiσk + σkσˆi = −2ηik1.

The eigen-spaces E± span the entire spin space S and each Dirac spinor can be split into two uniquely determined pieces from E+ and E−. Exercise 6.1: Show that 1 P± = (1 ± is/2η) 2 is a projector. What is its kernel, its image? ± The eigen-spaces E are called reduced spin spaces. In Physics η = γ5 is also referred ± chirality to as chirality. Elements from E are called reduced, chiral or Weyl spinors. Each element of the Clifford algebra can be written as a polynomial in the “basis elements” γi, i.e., is generated multiplicatively by the γi. Now we would like to know the dimension of the algebra considered as a vector space. A general element α has the form i ij i...j α = α0 + α γi + α γiγj + ··· + α γi ··· γj + ···

The product of the γi’s can not become arbitrarily long. It follows from the anti-commutativity that

γi1 γi2 ... γip

is totally anti-symmetric in the indices (i1,..., ip) for any p. This, in turn, implies that p is at most equal to n = dimV. With the abbreviation

γi1...ip := γi1 ... γip

each Clifford element can be written in the form

i i1i2 i1...ip 12...n α = α0 + α γi + α γi1i2 + ··· + α γi1...ip + α γ1...n.

Such an element α is known exactly when all coefﬁcients are speciﬁed. These are all totally anti-symmetric in their indices so that there the above linear combination contains exactly

1 1 n n 1 + (n − 1) + n(n − 1) + ... + n(n − 1) + (n − 1) + 1 = = 2n ∑ p 2 2 p=0

coefﬁcients. This implies that the dimension of the Clifford algebra corresponding to the n-dimensional vector space V is generated by 2n linearly independent elements, i.e., it has dimension 2n. Let us now assume that this algebra is represented as a matrix algebra of an N-dimensional vector space S. Then it is clear that the formally independent elements may not be rep- resentable necessarily independently from each other. For, we had seen that due to the

48 irreducibility for odd n the element η must be represented as a multiple of the identity. Therefore, η and the identity are no longer independent. Similarly, for the other γi-products (recall odd n!)

ηγi1...ip ∼ γip+1...in , where (i1,..., in) is a permutation of (1, . . . , n). For example,

ηγ12 ∼ γ3...n.

In the representation in which η is a multiple of the identity the products γi1...ip and

γip+1...in are represented by the same matrix up to multiples. This reduces the number of linearly independent generators for even n by half, to 2n−1. One can show, that the Clifford algebra represented in this way corresponds to the full matrix algebra of the representation space. So, if N is the dimension of S then n2 is the dimension of its matrix algebra and 2 n−1 n−1 N = 2 ⇐⇒ N = 2 2 . When n is even then there are no more relations between the generators and one gets

2 n n N = 2 ⇐⇒ N = 2 2 .

For instance, the dimension of the space of Dirac spinors in n = 4 dimensions is also N = 4, while the space of Weyl spinors (the reduced spinors) is 2-dimensional. In three dimensions (n = 3), the dimension of spin space is also N = 2. We see that the dimensions of spin space for n = 2k and n = 2k + 1 are equal, namely N = 2k, and that N grows exponentially with n. What does the Clifford algebra for a 1-dimensional vector space look like? So let V = Ce and g(e, e) = 1. Then γ = γ(e) = η and γ2 = −1. The Clifford algebra is generated by γ, i.e., it consists of all ﬁrst degree polynomials in γ

a + bγ.

On the other hand, the corresponding irreducible representation is 1-dimensional. If wanted to represent it by 2 × 2 matrices then we would have to represent γ by

i 0 γ = ± . 0 ±i

However, this representation is not irreducible because there are sub-spaces which are invariant under all elements.

For n = 2 and g = diag(−1, −1) there are two generators γ1 and γ2, which anti- 2 commute and satisfy γi = 1. The representation of the Clifford algebra (which is 4- dimensional) can be done using 2 × 2 matrices. If we choose

0 1 γ = = σ 1 1 0 1

49 then 0 i γ = = σ 2 −i 0 2 can be chosen and it follows that i 0 η = γ γ = = iσ . 1 2 0 −i 3 We see here again the general structure. The matrix iη has eigen-values ±1 and the reduced spin spaces are spanned by 1 0 bzw. . 0 1 We can also see that the reduced spaces contain a representation of the Clifford algebra for n = 1. For n = 3 and g = diag(−1, −1, −1) the corresponding Clifford algebra is 8-dimensional. However, its irreducible representation lives in a 2-dimensional space as 2 × 2 matrices. For the generators we need three 2 × 2 matrices whose squares are the negative identity 2 matrix. We can take the matrices γ1 and γ2 as above. We observe that (γ1γ2) = −1 and that η anti-commutes with γ1 and γ2. Therefore, it is useful to deﬁne γ3 = η. for the matrix γ1γ2γ3 we get 2 γ1γ2γ3 = (γ1γ2) = −1 as it should be for s = −3. Similarly, we may calculate explicitly that

γ1γ2 = γ3, γ3γ1 = γ2, γ2γ3 = γ1. The process can be continued successively so that the representation of the Clifford algebra for n dimensions one can construct the one for n + 1.

6.4 Null vectors, null ﬂags, etc.

In chapter 4 we introduced null flags in order to have available an interpretation of spin vectors. We now ask the question as to how these can be reconstructed from the spinor algebra? So let a spin vector κA ∈ SA be given, which space-time objects (i.e., space-time tensors) can be constructed solely from this information? The first possibility is 0 Ka = κAκ¯ A . (6.4.1) 0 This defines a real null vector since it is clear that κAκ¯ A0 = κAκ¯ A and it is also easy to a A 2 see that KaK = |κAκ | = 0. This defines a map which assigns a real null vector to each spin vector. However, this map is not invertible. It is not even surjective. Instead, for every real null vectors Ka 0 0 either Ka = κAκ¯ A or Ka = −κAκ¯ A holds. This follows from the theorem

50 0 Theorem 6.1. For every complex null vector Xa there are spin vectors ξ A and ζ A , so that 0 Xa = ξ Aζ A .

For the proof we use essentially the same argument as in the previous section. In a a AA0 BB0 000 110 010 100 spin frame we have 0 = X Xa = ε ABε A0 B0 X X = 2 X X − X X . This 0 means that the determinant of the matrix XAA vanishes. Hence, the column vectors are linearly dependent and this yields the statement.

0 Thus, if Ka is a non-vanishing real null vector, then Ka = αAβA for some spin vectors 0 0 0 0 0 αA and βA and, due to the reality of Ka i.e., αAβA = β¯ Aα¯ A it follows that βA = λα¯ A 0 p for a real λ. Hence, Ka = λαAα¯ A . Putting κA = |λ|αA, gives the representation of a real null vector as stated above. This result also yields the fact that the existence of a spin-structure divides the set of all real non-vanishing null vectors into two disjoint sets and therefore provides the possibility to deﬁne a consistent time orientation. This is done by labeling those null vectors which can be written as “squares” as future-pointing and those which are negative squares as past-pointed. But even when we restrict ourselves to the future-pointing null vectors as in (6.4.1), this does not imply the invertibility of that map. Each spin vector of the form eiφκA produces the same null vector Ka. But, of course, this is not the only space-time tensor that can be constructed from κA alone. Consider κAκB: in order to reach the necessary 0 0 indices we multiply with εA B . This yields a complex space-time tensor and we take its real part 0 0 0 0 Pab = κAκBεA B + κ¯ A κ¯B εAB. (6.4.2) Then Pab is real and anti-symmetric, Pab = −Pba. Furthermore, as one can see imme- ab A A A A diately, P Kb = 0. We choose a spin vector τ so that κAτ = 1, so that (κ , τ ) is a spin frame. Due to (??) we have εAB = κAτB − κBτA and inserted into (6.4.2) it follows that

0 0 0 0 0 0 Pab = κAκB κ¯ A τ¯ B − κ¯B τ¯ A + κ¯ A κ¯B κAτB − κBτA

0 0 0 0 0 0 = κAκ¯ A κBτ¯ B + κ¯B τB − κBκ¯B κAτ¯ A + κ¯ A τA = Ka Mb − Kb Ma.

a A A0 A0 A a a Here, M = κ τ¯ + κ¯ τ is a real, space-like vector with MaK = 0 and Ma M = −2. But τA is not uniquely determined since with τA also every other spin vector of the form τA + λκA for complex λ could have been used. Hence, also Ma is not unique. Changing τA as above changes Ma to Ma + (λ + λ¯ )Ka, without changing Pab. The positive multiples of the vectors Ma + rKa, r ∈ R thus defined form a half-plane which touches the future light-cone at the origin along the null direction defined by Ka, the searched for null flag.

51 With the formulae we have found thus far one can also quickly determine the behaviour 0 of the null ﬂag under rotations. We deﬁne a second real null vector Na = iκAτ¯ A − 0 iκ¯ A τA. Rotate κA with an angle φ: κA 7→ eiφκA. Due to the normalisation condition this implies τA 7→ e−iφτA and this implies

0 0 Ma 7→ eiφκAeiφτ¯ A + e−iφκ¯ A e−iφτA = cos(2φ)Ma + sin(2φ)Na.

Therefore, when κA is turned through an angle φ then Ma rotates by an angle 2φ in the plane spanned by Ma and Na.

52 Index

2-component spinors, 44 Lorentz transformation, 13 Lorentzian space, 11 aberration formula, 22 abstract index, 30 Möbius transformation, 19 affine group, 10 metric, 11 affine transformation, 9 Minkowski space-time, 8, 11 antipodal map, 17 Minkowski tetrad, 12 basis, 34 null direction, 17 null flag, 26 causal, 13 null rotations, 22 celestial sphere, 17 null tetrad, 41 chirality, 48 null vector, 17 Clifford algebra, 44, 46 complex 4-vectors, 40 orientation, 12 contravariant, 12 orthochronous, 13 coordinate system, 9 orthonormal, 12 coordinates, 9 past pointing, 13 dimension, 34 Pauli matrices, 44 Dirac matrices, 44 Poincaré group, 11 Dirac spinor, 46 Poincaré transformation, 13 dual basis, 34 point, 8 Euclidean space, 11 position vector, 9 extent, 25 projective space, 16 four-screw, 22 real 4-vector, 40 future pointing, 13 regular, 10 Riemann sphere, 16 half space, 26 right-handed, 13 homogeneity, 9 signature, 11 invariance group, 10 spin bundle, 28 isometries, 11 spin frame, 28 Kronecker delta, 33 spin matrix, 19 Kronecker symbol, 34 spin space, 27 spin transformation, 19 Lorentz group, 11 spinorial object, 27

53 squared distance, 11 tensor, 32 tensor operations, 32 tensor product, 32 tetrad, 12 time orientation, 13 triad, 13 van-der-Waerden symbols, 42, 43 vector, 8