THE SPINORIAL FORMALISM, WITH APPLICATIONS IN PHYSICS · 2019-10-25 · tiu entender sobre a origem...

UNIVERSIDADE FEDERAL DE PERNAMBUCOCENTRO DE CIÊNCIAS EXATAS E DA NATUREZA

DEPARTAMENTO DE FÍSICAPROGRAMA DE PÓS-GRADUAÇÃO EM FÍSICA

JOÁS DA SILVA VENÂNCIO

THE SPINORIAL FORMALISM, WITHAPPLICATIONS IN PHYSICS

Recife2017

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THE SPINORIAL FORMALISM, WITH APPLICATIONS IN PHYSICS

Dissertation presented to the graduationprogram of the Physics Department of Uni-versidade Federal de Pernambuco as part ofthe duties to obtain the degree of Master ofPhilosophy in Physics.

Supervisor: Prof. Dr. Carlos AlbertoBatista da Silva Filho

Recife2017

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Catalogação na fonteBibliotecária Joana D’Arc Leão Salvador CRB 4-572

V448s Venâncio, Joás da Silva.The spinorial formalism, with appllications in physics / . – 2017. 112 f.

Orientador: Carlos Alberto Batista da Silva Filho. Dissertação (Mestrado) – Universidade Federal de Pernambuco. CCEN.

Física, Recife, 2017. Inclui referências e índice.

1. Física matemática. 2. Formalismo espinorial. 3. Relatividade geral. 4. Geometria diferencial. I. Silva Filho, Carlos Alberto Batista da (Orientador). II. Titulo.

530.15 CDD (22. ed.) UFPE-FQ 2017-20

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THE SPINORIAL FORMALISM, WITH APPLICATIONS IN PHYSICS

Dissertation presented to the graduation pro-gram of the Physics Department of Universi-dade Federal de Pernambuco as part of theduties to obtain the degree of Master of Phi-losophy in Physics.

Accepted on: 23/02/2017.

EXAMINING BOARD:

Prof. Dr. Carlos Alberto Batista da Silva FilhoSupervisor

Universidade Federal de Pernambuco

Prof. Dr. Bruno Geraldo Carneiro da CunhaInternal Examiner

Universidade Federal de Pernambuco

Prof. Dr. Marco CarigliaExternal Examiner

Universidade Federal de Ouro Preto

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Abstract

It is well-known that the rotation symmetries play a central role in the development ofall physics. In this dissertation, the material is presented in a way which sets the scene forthe introduction of spinors which are objects that provide the least-dimensional faithfulrepresentation for the group Spin(n), the group that is the universal coverage of the groupSO(n), the group of rotations in n dimensions. With that goal in mind, much of thisdissertation is devoted to studying the Clifford algebra, a special kind of algebra definedon vector spaces endowed with inner products. At the heart of the Clifford algebra liesthe idea of a spinor. With these tools at our disposal, we studied the basic elements of dif-ferential geometry which enabled us to emphasise the more geometrical origin of spinors.In particular, we construct the spinor bundle which immediately lead to the notion of aspinor field which represents spin 1/2 particles, such as protons, electrons, and neutrons.A higher-dimensional generalization of the so-called monogenic multivector functions isalso investigated. In particular, we solved the monogenic equations for spinor fields onconformally flat spaces in arbitrary dimension. Particularly, the massless Dirac field isa type of monogenic. Finally, the spinorial formalism is used to show that the Diracequation minimally coupled to an electromagnetic field is separable in spaces that arethe direct product of bidimensional spaces. In particular, we applied these results on thebackground of black holes whose horizons have topology R× S2 × . . .× S2.

Keywords: Clifford algebra. Spinors. Monogenic. Dirac Equation. Separability.

Resumo

É bem conhecido que as simetrias de rotação desempenham um papel central no de-senvolvimento de toda a física. Nesta dissertação, apresentamos o conteúdo de forma aestabelecer o cenário para a introdução dos chamados spinors, os quais são objetos quefornecem as representações fiéis de menor dimensão para o grupo Spin(n), o grupo que éa cobertura universal do grupo SO(n), o grupo das rotações em n dimensões. Para estefim, grande parte desta dissertação é dedicada ao estudo da álgebra de Clifford, um tipoespecial de álgebra definida em espaços munidos de um produto interno. No coração daálgebra de Clifford está precisa definição de um espinor. Com estas ferramentas à nossadisposição, estudamos os elementos básicos de geometria diferêncial, o que nos permi-tiu entender sobre a origem mais geométrica de espinores. Em particular, construimoso fibrado espinorial, o qual conduziu imediatamente a noção de um campo espinorialque, por sua vez, representa com precição as partículas com spin 1/2 tais como: pró-tons, elétrons e neutrons. Uma generalização para dimensões mais altas do conceito demultivetores monogênicos também é investigada. Em particular, resolvemos a equaçãodos monogênicos para campos espinoriais em espaços conformemente planos em dimen-são arbitrária. Particularmente, o campo de Dirac sem massa é um tipo de monogênico.Finalmente, o formalismo espinorial foi usado para mostrar que a equação de Dirac commassa minimamente acoplada ao campo eletromagnético é separável em espaços que sãoprodutos diretos de espaços bidimensionais. Em particular, aplicamos estes resultados aburacos negros com horizontes topológicos R× S2 × . . .× S2.

Palavras chaves: Álgebra de Clifford. Espinores. Monogenicos. Equação de Dirac. Sepa-rabilidade.

List of Symbols

V Vector space over a field F: Page 11.

V∗ Dual vector space of V : Page 11.

F Field: Page 11.

R Field of the real numbers: Page 11.

C Field of the complex numbers: Page 11.

∂µ Differential operators∂

∂xµ: Page 49.

T[a1a2...ap] Skew-symmetric part of the tensor Ta1a2...ap : Page 12.

∧pV Space of p-vectors: Page 12.

∧V Space of multivectors: Page 12.

〈〉p, 〈A〉p Set of projection operators: Page 12.

∧,A ∧ B Exterior product: Page 13.

< , >,< A,B > Non-degenerate inner product: Page 14.

C`(V) Clifford algebra of V : Page 14.

y,AyB Left contraction: Page 15.

, A Degree involution: Page 16.

, A Reversion: Page 17.,A Clifford conjugation: Page 18.

M(2,F) Algebra of 2× 2 matrices over F : Page 21.

?, ?A Hodge dual: Page 23.

S Reflection operator: Page 26.

R Rotation operator: Page 27.

D Division algebra: Page 38.

( , ), (ψ,φ) Inner product between spinors: Page 38.

g The metric of the manifold: Page 50.

ωab , ωc

ab , ωabc Connection 1-form and its components: Page 57.

T a Torsion 2-form: Page 57.

Rab Curvature 2-form: Page 57.

£K Lie derivative along of K: Page 59.

Γ(SM) Space of the local sections of spinorial bundle: Pages 74.

Γ(CM) Space of the local sections of Clifford bundle: Pages 82.

Contents

1 Introduction 9

2 Clifford Algebra and Spinors 112.1 The Exterior Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 The Clifford Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2.1 Involutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2.2 Pseudoscalar, Duality Transformation and Hodge Dual . . . . . . . . . . . 212.2.3 Periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.3 The Spin Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.4 Spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.4.1 Minimal Left Ideal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.4.2 Pure Spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.4.3 SPin-Invariant Inner Products . . . . . . . . . . . . . . . . . . . . . . . . . 372.5 Spinors, Gamma Matrices and their Symmetries . . . . . . . . . . . . . . . 41

3 Curved Spaces and Spinors 473.1 Manifolds and Tensors Fields . . . . . . . . . . . . . . . . . . . . . . . . . 473.2 The Curvature Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.3 Non-Coordinate Frame and Cartan’s Structure Equations . . . . . . . . . . 563.4 Symmetries, Killing Vectors and Hidden Symmetries . . . . . . . . . . . . 583.4.1 Maximally Symmetric Manifolds . . . . . . . . . . . . . . . . . . . . . . . . 613.4.2 Symmetries of Euclidean Space Rn . . . . . . . . . . . . . . . . . . . . . . 673.5 Fiber Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703.5.1 Spinorial Bundles and Connections . . . . . . . . . . . . . . . . . . . . . . 73

4 Monogenics and Spinors in Curved Spaces 784.1 Monogenics in the Literature . . . . . . . . . . . . . . . . . . . . . . . . . . 784.2 Monogenic Multivector Functions : Definitions and Operator Equalities . . 804.3 Monogenics in Curved Spaces and Some Results . . . . . . . . . . . . . . . 824.4 Monogenic Equation for Spinor Fields . . . . . . . . . . . . . . . . . . . . . 854.5 Direct Product Spaces and the Separability of the Dirac Equation . . . . . 904.6 Black Hole Spacetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964.6.1 The angular part of Dirac’s Equation . . . . . . . . . . . . . . . . . . . . . 1004.6.2 The radial part of Dirac’s Equation . . . . . . . . . . . . . . . . . . . . . . 102

References 105

Index 112

CHAPTER 1. INTRODUCTION 9

1 Introduction

The so-called rotational symmetries play a fundamental role in the development ofphysics. For instance, in classical mechanics and classical field theory the invarianceby rotations gives rise to conserved quantities that allow the analytical integrationof the equations of motion, whereas in quantum mechanics the irreducible repre-sentations of the rotation group are used to label quantum states. Moreover, andforemost, the Lorentz group can be seen as the group of rotations in a space witha metric of Lorentzian signature. It is worth recalling that the invariance underLorentz transformation is the foundation of the standard particle model, the mostsuccessful physical theory of the second half of the twentieth century. In this disser-tation, the material is presented in a way which sets the scene for the introductionof spinors which are objects that provide the least-dimensional faithful representa-tions for the group Spin(n), the universal coverage of the group SO(n), the groupof rotations in n dimensions. To this end, we introduce a special kind of algebradefined by vector spaces endowed with inner products: the Clifford algebra, alsoknown as geometric algebra. This Algebra was created by the English mathemati-cian William Kingdon Clifford around 1880 building on the earlier work of Hamiltonon quaternions and Grassmann about exterior algebra. Although the definition ofa spinor lies at the heart of the Clifford algebra, the spinors were discovered in1913 by Élie Cartan as objects related to linear representations of simple groups;they provides a linear representation of the SO(n) group. Hence, spinors are offundamental importance in several branches of physics and mathematics and thisdissertation sheds light on the role played by spinors on physics and mathematics.This dissertation splits in three chapters. In chapter 1 we introduce the Cliffordalgebras intimately related to the orthogonal transformations, the rotations. Thesealgebras play a central role in the construction of the groups Pin(n) and Spin(n),which are the universal covering groups of the orthogonal groups O(n) and SO(n)respectively. Finally, at the end of chapter, we define the so-called spinors. In chap-ter 2 we study the basic elements of differential geometry, such as the curvaturetensor, the Killing vectors and the fiber bundles which enabled us to emphasize onmore geometrical origin of these objects. In particular, we construct the spinor bun-dle which immediately lead to the notion of a spinor field which represents spin 1/2particles, such as protons, electrons, and neutrons. Finally, in chapter 3 the mono-

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CHAPTER 1. INTRODUCTION 10

genic functions which are multivector functions that are annihilated from the leftby the Dirac operator are reviewed and a higher-dimensional generalization thesemultivetor functions is also investigated. In particular, we solved the monogenicequations for spinor fields on conformally flat space in arbitrary dimension. More-over, in chapter 3 it is present the main results of this dissertation. It is shown thatthe Dirac Equation coupled to a gauge field can be decoupled in even-dimensionalmanifolds that are the direct product of bidimensional spaces.

CHAPTER 2. CLIFFORD ALGEBRA AND SPINORS 11

2 Clifford Algebra and Spinors

A vector space V over a field F is a set of vectors with an operation of addition anda rule of scalar multiplication, which assigns a vector to the product of a vectorwith an element of the field. Elements in F will be called scalars. An algebra is avector space in which an associative multiplication between the vectors is defined.In this chapter we introduce the Clifford algebra, a special kind of algebra defined onvector spaces endowed with inner products. These algebras are intimately relatedto the orthogonal transformations, namely the rotations. This special algebra playsa central role in the construction of the groups Pin(V) and Spin(V), which are theuniversal covering groups of the orthogonal groups O(V) and SO(V) respectively,and also define the so-called spinors, which are elements of the a vector space onwhich a fundamental representation of these groups act.

2.1 The Exterior Algebra

Let V be an n-dimensional vector space over a field F (R or C) and ea an arbitrarybasis for V , where a ∈ 1, 2, ..., n. We can expand any vector V ∈ V in this basisas

V = V a ea , (2.1)where we have started to employ the Einstein summation convention on whichrepeated indices are summed. Associated to V is the dual space of V , denoted byV∗ whose elements are linear functionals, also called co-vectors,

ω : V → FV 7→ ω(V ) ,

(2.2)

which obey the rule of linearity ω(λV + δU ) = λω(V ) + δω(U) ∀ λ , δ ∈ F.If the addition of co-vectors and their multiplication by an element of the field istrivially defined it is straightforward to prove that the dual of V is also a vectorspace.We can define the co-vectors eb ∈ V∗ (b = 1, 2, ..., n) by their action on elements ofV as:

ea(eb) = δab =

1 , if a = b0 , if a 6= b

, (2.3)

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hence the co-vectors ea provide a basis for V∗. So, any co-vector ω can beexpanded in this basis and written as:

ω = ωa ea , (2.4)

where ωa = ω(ea). It follows that if ω ∈ V∗ is co-vector and V ∈ V is a vectorthen ω(V ) = ωaV

a ∈ F. We also may regard elements of V as linear functions onits dual V∗ by defining V (ω) ≡ ω(V ), since V ∗∗ ∼ V .The tensor product refers to way of constructing a big vector space out of two ormore smaller vector spaces. For example, the tensor product of V ∈ V with ω ∈ V∗,denoted as V ⊗ω, gives rise a new tensor t belonging to a vector space V ⊗ V∗ onwhich the n2 elements of the form ea⊗ eb provide a basis, so that the most generalelement of this space can be written as t = tab ea ⊗ eb, where tab = t(ea, eb) ∈ F.Likewise, an arbitrary tensor T has the following abstract representation:

T = Ta1...ap

b1... bqea1 ⊗ . . .⊗ eap ⊗ eb1 ⊗ . . .⊗ ebq , (2.5)

where T a1...apb1... bq ≡ T (ea1 , . . . , eap , eb1 , . . . , ebq). The space of such tensors is de-noted T pq(V).

The so-called p-vectors are totally skew-symmetric tensors of degree p and thevector space generated by all them, namely Aa1...ap = A[a1...ap], is denoted by ∧pV ,where we identify the field F with ∧0V and V with ∧1V . We must assume that allthe indices inside square brackets take different values to ensure that the p-vectorA ∈ ∧pV given by

A = A[a1...ap] ea1 ⊗ . . .⊗ eap , (2.6)

be non-null. Because the antisymmetry, it is a simple matter to prove that thedimension of ∧pV is:

dim(∧pV) =

(n

p

)=

n!

p!(n−p)! if 0 ≤ p ≤ n

0 if p > n, (2.7)

from which we can note that p-vectors of degree greater than n are zero. By takingthe direct sum of the spaces ∧pV , we obtain the 2n-dimensional multivector space,that is

∧V =n⊕p=0

∧pV . (2.8)

Elements in ∧V are called multivectors and we denote them by A. Because of de-composition of ∧V in p-vector subspaces, we can define a set of projection operators,denoted by 〈〉p, whose action is:

〈〉p : ∧V → ∧pVA 7→ 〈A〉p = Ap .

(2.9)


Notice that 〈〉p is in fact a projector, since that it maps an arbitrary multivectorto its p-vector component each of the form (2.6). Thus, an arbitrary multivector Acan be decomposed into a sum of pure degree terms

A = 〈A〉0 + 〈A〉1 + . . .+ 〈A〉n

= A0 +A1 + . . .+An =n∑p=0

Ap . (2.10)

Multivectors containing terms of only one degree are called homogeneous.

An interesting feature of the space of multivectors is that it carries naturally aproduct, denoted by ∧, which maps two tensors of degrees p and q to a totallyantisymmetric tensor of degree (p + q), this is the so-called exterior product.Therefore, if A is a p-vector and B is a q-vector, the exterior product ∧ is a mapsuch that:

∧ : ∧pV × ∧qV → ∧p+qVA ,B 7→ A ∧B , (2.11)

where A ∧B = (−1)pqB ∧A is defined as:

A ∧B =(p+ q)!

p! q!A[a1...apBb1...bq ]ea1 ⊗ . . .⊗ eap ⊗ eb1 ⊗ . . .⊗ ebq . (2.12)

In n dimensions the set 1, ea1 , ea1 ∧ ea2 , ea1 ∧ ea2 ∧ ea3 , . . . , ea1 ∧ . . .∧ ean, whichcontains 2n elements furnish a basis for the space of multivectors ∧V . Therefore, amultivector A ∈ ∧V may be written in this basis as

A = A︸︷︷︸0−vector

+ Aaea︸︷︷︸1−vector

+Aab ea ∧ eb︸︷︷︸2−vector

+Aabc ea ∧ eb ∧ ec︸︷︷︸3−vector

. . .+ Aa1...an ea1 ∧ . . . ∧ ean︸︷︷︸n−vector

.

(2.13)For instance, in V = R3 there exist eight elements that generate this basis. Theseare: one 0-vector which is denoted by 1, three 1-vectors ea, three 2-vectors ea ∧ ebeach of the form ea ∧ eb = ea ⊗ eb − eb ⊗ ea for each choice of a 6= b = 1, 2, 3 and asingle 3-vector. In particular, note that this 3-vector is written as

ea ∧ eb ∧ ec = ea ⊗ eb ⊗ ec + eb ⊗ ec ⊗ ea + ec ⊗ ea ⊗ eb +

− eb ⊗ ea ⊗ ec − ec ⊗ eb ⊗ ea − ea ⊗ ec ⊗ eb.

Definition 1. The exterior algebra is an associative algebra formed from ∧V andthe exterior product on p-vectors.

2.2 The Clifford Algebra

The definitions of the exterior product, and of the multivectors do not dependon any inner product. However, Clifford algebras are a generalization of exterior


algebras, defined in the presence of an inner product. In what follows, we willassume an inner product in our vector space. Let V be a vector space endowed witha non-degenerate inner product < , >, this means that for V ∈ V , < V ,U >= 0for all U ∈ V if and only if V = 0. The pair (V , < , >) is called an orthogonalspace. Then the Clifford product, assumed to be associative and denoted byjuxtaposition, of a pair of vectors is defined to be such that its symmetric partgives the inner product:

V U +UV = 2 < V ,U > ∀ V , U ∈ V . (2.14)

Since the base used is completely arbitrary, it is convenient to adopt ea =e1, e2, . . . , en as an orthonormal basis, since in this case we have < ea, eb >= ±δab and therefore eaeb = −ebea, if a 6= b. Analogously, eaebec is totally skew-symmetric if a 6= b 6= c 6= a. Thus, any element of C`(V , < , >), the Clifford algebraof V , can be written as a linear combination of p-vectors ea1 . . . eap with 1 ≤ p ≤ n.In other words, the vector space of the Clifford algebra associated to V is ∧V , thenit can be written as a direct sum:

C`(V) =n⊕p=0

∧pV , (2.15)

hence the dimension dim(C`(V , < , >)) = 2n. This decomposition introduces amultivector structure into the Clifford algebra C`(V). The multivector structure isunique, that is, an arbitrary element A ∈ C`(V) can be uniquely decomposed into asum of p-vectors A = A0 +A1 + . . .+An . In what follows, for simplicity, instead ofthe previous notation it will be denoted by C`(V) the Clifford algebra associated toV . It follows that the set 1, ea1 , ea1 ea2 , . . . , ea1 . . . ean with the identity element1 ∈ F, forms a basis for C`(V). Therefore, a general element A ∈ C`(V) can bewritten as:

A = A︸︷︷︸scalar

+Aaea︸︷︷︸vector

+Aabeaeb︸︷︷︸2−vector

+Aabceaebec︸︷︷︸3−vector

+ . . .+ Aa1...an ea1 . . . ean︸︷︷︸n−vector

, (2.16)

Now, in the same way that (2.12) was defined in terms of the tensorial product, itis natural to define the exterior product in terms of the Clifford product. So, givena permutation σ : 1, 2, . . . , n → σ(1), σ(2), . . . , σ(2) we define the exteriorproduct as the totally antisymmetric part of the Clifford product:

V 1 ∧ V 2 ∧ . . . ∧ V p :=1

p!

∑σ∈Sp

ε(σ)V σ(1)V σ(2) . . .V σ(p) , (2.17)

where Sp is the symmetric group whose elements are all the permutations and ε(σ)is +1 on even permutations, −1 otherwise. So, for example, given two vectors Vand U we find that:

V ∧U =1

2(V U −UV ) . (2.18)


The equations (2.14) and (2.18) combine to give the Clifford product of two vectors:

V U =< V ,U > +V ∧U , (2.19)

also called the geometric product.

The decomposition of the Clifford product of two vectors into a scalar term and a2-vector term has a natural extension to general multivectors. This may be donethrough the use of the projection operator in terms of which we can convenientlyexpress the inner and exterior products. So, given Ap ∈ ∧pV and Bq ∈ ∧qV theexterior product is written as:

Ap ∧Bq := 〈ApBq〉p+q ,

and the inner product as:

< Ap,Bq > := 〈ApBq〉|p−q| . (2.20)

Another important operation is the left contraction, denoted by y, defined as follows:

ApyBq :=

< Ap,Bq > if p ≤ q

0 if p > q. (2.21)

The products defined above can be extended by bilinearity for the whole algebra.Using the equations (2.18) and (2.17) the Clifford product of a vector and an arbi-trary multivetor is given by:

V A = V yA + V ∧ A ∀ V ∈ V , A ∈ C`(V) . (2.22)

Moreover, one can prove that the contration by a vector satisfies Leibniz’s rule, thismeans that such a contraction is a derivation of the Clifford algebra C`(V). Indeed,a result that is extremely useful in practice is the following:

V y(V 1 ∧ V 2 ∧ . . . ∧ V p) = (V yV 1)V 2 ∧ V 3 ∧ V 4 ∧ . . . ∧ V p + (2.23)− (V yV 2)V 1 ∧ V 3 ∧ V 4 ∧ . . . ∧ V p +

+ (V yV 3)V 1 ∧ V 2 ∧ V 4 ∧ . . . ∧ V p + . . .

=

p∑i=1

(−1)i+1 < V ,V i > V 1 ∧ . . .

∧V i ∧ . . . ∧ V p, (2.24)

where the check on V i denotes that the term should be withdrawn from the series.For instance, when p = 2 we find that:

V y(U ∧X) =< V ,U >X − < V ,X > U .


Using the above result, by means of the equation (2.20), that in its turn ensuresV yλ = 0 for any scalar λ, the equation (2.22) lead us at the following formula forthe Clifford product of three vectors:

V UX =< V ,U >X + < U ,X > V − < V ,X > U + V ∧U ∧X . (2.25)

Note that in (2.22), as expected, multiplication by a vector raises and lowers thedegree of a multivector by 1. But, in general, ifAp is a p-vector andBq is a q-vector

ApBq 6= ApyBp + Ap ∧Bq,

where ApyBp ∈ ∧p−q(V) and Ap ∧Bp ∈ ∧p+q(V). Using (2.22) it is not so hard toprove that the products of two homogeneous multivector decompose as:

ApBq = 〈ApBq〉|p−q| + 〈ApBq〉|p−q|+2 + . . .+ 〈ApBq〉p+q . (2.26)

Definition 2. The Clifford algebra associated to V, denoted by C`(V), consists ofthe vector space ∧V together with the Clifford product.

2.2.1 Involutions

It is well-known that the conjugate of the conjugate of a complex number is thecomplex number itself. An operation, which, when applied to itself, returns theoriginal object is called involution. In particular, the complex conjugation is asimple example of an involution. The Clifford algebra has three involutions similarto complex conjugation. The direct sum decomposition (2.15) gives C`(V) thestructure of a Z-graded algebra. This induces in C`(V) the first involution, denotedby , called the degree involution which is a linear map whose the action isdefined on homogeneous multivectors by:

: ∧V → ∧VAp 7→ Ap = (−1)pAp .

(2.27)

The degree involution is an automorphism1 such that:

AB = AB ∀ A, B ∈ C`(V) . (2.28)1An automorphism is an an invertible mapping from a set in itself. In particular, an automorphism s

of V is said to be orthogonal with respect to < , > if s leaves < , > invariant, i.e, < sV , sV >=< V ,V >∀V ∈ V, see [14].


Since the map applied to itself is identity map, Ap = Ap , the eigenvalues

of are ±1, then the degree involution induces a Z2-grading on C`(V). Underdegree involution the multivectors corresponding to eigenvalue +1 will be called evenand the space of even multivectors will be denoted C`+(V), while the multivectorscorresponding to eigenvalue −1 will be called odd, and belong to the subspaceC`−(V). Thus, given A ∈ C`(V) we have

A = A0 −A1 +A2 −A3 + . . . . (2.29)

In particular, we can write the inner and exterior products in terms of the degreeinvolution. Indeed, we can prove that for an arbitrary multivector A ∈ C`(V) thefollowing relations are satisfied:

V yA =1

2(V A − AV ) (2.30)

V ∧ A =1

2(V A + AV ) , (2.31)

where V ∈ V . Moreover, one can prove that the contration by a vector V ∈ Vobeys the following version of of the Leibniz’s rule

V y(AB) = V yAB + AV yB ∀ A,B ∈ ∧V . (2.32)

The second involution in Clifford algebra is the called reversion, denoted by ˜,which reverses the order of vectors in any product. For instance, for a p-vector thereverse can be formed by a series of swaps of anticommuting vectors, each resultingin a minus sign. The first vector has to swap past p − 1 vectors, the second pastp− 2, and so on. Using this it is easy matter to see that

˜V 1 ∧ V 2 ∧ . . . ∧ V p = V p∧V p−1∧ . . .∧V 2∧V 1 = (−1)p(p−1)/2V 1∧V 2∧ . . .∧V p

(2.33)It follows that the reversion, on a homogeneous vector, is a linear mapping suchthat: ˜ : ∧V → ∧V

Ap 7→ Ap = (−1)p(p−1)/2Ap(2.34)

From the above the degree involution is an anti-automorphism, that is,

AB = BA ∀ A, B ∈ C`(V), (2.35)

thus, we see that the reverse of A ∈ C`(V) is:

A = A0 +A1 −A2 −A3 + . . . . (2.36)


The composition of the two previous involutions is called the Clifford conjuga-

tion, denoted by A = (A) = (A), which is expressed by

: ∧V → ∧VAp 7→ Ap = (−1)p(p+1)/2Ap .

(2.37)

and it is also an anti-automorphism,

AB = BA ∀ A, B ∈ C`(V). (2.38)

Thus,A = A0 −A1 −A2 −A3 + . . . . (2.39)

The Clifford conjugation can be used to determine the inverse. Indeed, ifA ∈ C`(V),then its inverse is given by

A−1 =AAA

. (2.40)

In particular, using the Clifford conjugation of a vector U ∈ V which is given bythe following relation

U−1 =U

U 2 , (2.41)

we can show that the associativity property of the Clifford product ensures that itis now possible to divide by vectors. In fact, defining B = V U ∀ V , U ∈ V wehave that

BU = (V U)U = V (UU ) = V U 2 ,

Now, from (2.41) we can recover V and we eventually arrive at the following ex-pression:

V = BU−1 .

This ability to divide by vectors gives the Clifford algebra considerable power.

The reversion can be used to extend the concept of norm. Now, we are going todefine the norm of an arbitrary multivector A ∈ C`(V) as follows:

|A|2 =⟨AA

⟩0

=⟨AA

⟩0. (2.42)

For example, if V = R3 the most general element A ∈ C`(R3) is given by

A = a︸︷︷︸scalar

+ a1e1 + a2e2 + a3e3︸︷︷︸vector

+ a12e1e2 + a13e1e3 + a23e2e3︸︷︷︸2−vector

+

+ a123e1e2e3︸︷︷︸3−vector

, (2.43)

then, using (2.43) and (2.14) it is straightforward to prove that:

|A|2 = (a)2 + (a1)2 + (a2)2 + (a3)2 + (a12)2 + (a13)2 + (a23)2 + (a123)2 . (2.44)


Since we can split a multivector into those components that, under degree involu-tion, are even and those that are odd, every A ∈ C`(V) can be written as:

A = A+ +A− , (2.45)

where A+ = A0 +A2 +A4 + . . . and A− = A1 +A3 +A5 + . . . . It follows thatwe can write C`(V) = C`+(V)⊕ C`−(V) where

C`±(V) = A ∈ C`(V) | A = ±A . (2.46)

This Z2-grading of the Clifford algebra ensures that elements of even degree forma subalgebra, the called even subalgebra. Indeed, note that, AB ∈ C`+(V) if A ∈C`±(V) and B ∈ C`±(V) while AB ∈ C`−(V) if A ∈ C`±(V) and B ∈ C`∓(V) orA ∈ C`∓(V) and B ∈ C`±(V).

The center Z of an algebra A consists of all those elements z of A such that za = azfor all a in A. We can easily prove that the center of C`(V), denoted by Z(C`(V)),depends on the dimension of space as follows:

Z(C`(V)) =

∧0V , if n is even

∧0V ⊕ ∧nV , if n is odd . (2.47)

Before proceeding let us make as example.

Example 1: The C`(R3) algebra.

Let us work out in the vector space V = R3, whose Clifford algebra C`(R3) is gener-ated by 1, e1, e2, e3, e1e2, e1e3, e2e3, e1e2e3, which contains 23 = 8 elements,where e1e1 = e2e2 = e3e3 = 1 and eaeb = −ebea if a 6= b. A priori just forthe sake of simplicity in notation, let us denote the product e1e2e3 ≡ I and inthe next section we discuss this important symbol. The reversion this symbol isI = −I and their square is equal to

I2 = e1e2e3e1e2e3 = −e1e3e2e1e2e3 = e3e1e2e1e2e3 = −e3e2e1e1e2e3

= −e3e2e2e3 = −e3e3

=⇒ I2 = −1 . (2.48)

The symbol I commutes with all vectors in three dimensions, using this one canprove that IA = AI ∀ A ∈ C`(R3) given by (2.43). C`+(R3), the the evensubalgebra of C`(R3), is generated by 1, e1e2, e1e3, e2e3 which contains 1

223 = 4

elements in terms of which a multivector A+ ∈ C`+(R3) can be written as:

A+ = a︸︷︷︸scalar

+ a12e1e2︸︷︷︸2−vector



.


Defining i = e2e3, j = e1e3, k = ij = e1e2 , we find that i2 = j2 = k2 = −1 ,i,e., is the quaternion algebra H. Thus, the even subalgebra of C`(R3) is isomorphicto the quaternion algebra, as can be seen by the following correspondences:

H C`+(R3)

1 1i e2e3

j e1e3

k e1e2

(2.49)

The basis vectors e1, e2 and e3 can also be represented by matrices σ1 , σ2 and σ3

given by

σ1 =

[0 11 0

]; σ2 =

[0 −ii 0

]; σ3 =

[1 00 −1

],

the so-called Pauli matrices, since these matrices in accordance with this represen-tation satisfy the same algebra, namely σ1σ1 = σ2σ2 = σ3σ3 = I , σaσb = −σbσaif a 6= b. The quaternion algebra admits the following matrix representation

1 ' I ; i = e2e3 ' iσ1 ; j = e1e3 ' iσ2 ; k = ij = e1e2 ' iσ3 ,

where I is the 2 × 2 identity matrix. Equivalently, the multivetor A ∈ C`(R3) canbe represented as:

[A] = a

[1 00 1

]+ a1

[0 11 0

]+ a2

[0 −ii 0

]+ a3

[1 00 −1

]+

+ a12

[i 00 −i

]+ a13

[0 1−1 0

]+ a23

[0 ii 0

]+ a123

[i 00 i

]=⇒ [A] =

[z1 z3

z2 z4

], (2.50)

where z1 = (a + a3) + i(a12 + a123) , z2 = (a1 + a13) + i(a2 + a23) , z3 = (a1 −a13)− i(a2 − a23), and z4 = (a− a3)− i(a12 − a123) . Note also that A+ ∈ C`+(R3)is represented by

[A] =

[w1 −w2

∗

w2 w1∗

], (2.51)

where w1 = a+ ia12, w2 = a13 + ia23 and w∗i is the complex conjugate of wi (i =1, 2). By equation (2.47), the center of C`(R3) is the subalgebra of scalars and 3-vectors, namely ∧0V ⊕ ∧3V = a+ a123I. Note that σ1σ2σ3 = iI and the symbolI can therefore be viewed as the unit imaginary2 i and the combination of a scalar

2The symbol i is an element which commutes with all others, which is not necessarily a property ofI. But, in this case it commutes with all elements and squares to −1. It is therefore a further candidatefor a unit imaginary.


and a pseudoscalar as a complex number. This implies the center of C`(R3) isisomorphic to the complex field C, that is,

Z(C`(R3)) = ∧0V ⊕ ∧3V ' C . (2.52)

The correspondences e1 ' σ1, e2 ' σ2 and e3 ' σ3 establish the isomorphismC`(R3) 'M(2,C)3 with the following correspondences of the basis elements:

M(2,C) C`(R3)

I 1σ1, σ2, σ3 e1, e2, e3

σ1σ2, σ1σ3, σ2σ3 e1e2, e1e3, e2e3

iI I

(2.53)

According to with what was seen above, the Clifford algebra C`(R3) contains twosubalgebras: the center Z(C`(R3)) which is isomorphic to the complex field C andC`+(R3) isomorphic to the quaternion algebra H. Now, since the elements of Cand H commute, zq = qz for z ∈ C , q ∈ H and that, as real algebra, C`(R3) isgenerated by C and H, noting that (dimC)(dimH) = dimC`(R3), we are left withthe following conclude:

C`(R3) ' C⊗H. (2.54)

2.2.2 Pseudoscalar, Duality Transformation and Hodge Dual

The object I mentioned in the previous example is an important element of C`(V).This is the highest degree element in a given algebra, the so-called pseudoscalar4.For a given vector space the highest degree element exists and is unique up to amultiplicative scalar. The exterior product of n vectors is therefore a multiple of theunique pseudoscalar for C`(V). Given an orthonormal basis ea (a = 1, 2, . . . , n),we shall define the pseudoscalar to be I = e1e2 . . . en. This convention is well-defined if we do not change the orientation of the basis. Let us consider that thespace V = Rp,q has dimension n = p + q with p vectors whose square is equal to1 and q vectors whose square is equal to −1 and s = |p− q| the signature of theinner product. Noting that the p+ 1 vector has square −1, the norm of I depends

3M(2,C) denotes the algebra of 2× 2 matrices over C4Directed volume element and volume form are alternative names for the pseudoscalar.


on the dimension of space and the signature in the following form:

|I|2 =⟨II⟩

0= 〈eqeq−1 . . . ep+1ep . . . e2e1e1e2 . . . epep+1 . . . eq−1eq〉0

= e21e

22 . . . e

2pe

2p+1 . . . e

2q−1e

2q = (−1)q

=⇒ |I|2 = (−1)12

(n−s) . (2.55)

In particular |I|2 = 1 if the signature is Euclidean, s = n, and |I|2 = −1if Lorentzian signature, s = (n − 2) . When a multivector A is homogeneuous|A|2 = AA. So, immediately we see that the sign of I2 is specified by:

I2 = (−1)12n(n−1)II = (−1)

12

[n(n−1)+(n−s)] . (2.56)

Another property of the pseudoscalar is that it defines an orientation. Apart from asign, the choice of I is independent of the choice of orthonormal basis. Choosing asign amounts to choosing an orientation for V which is swapped by exchange of anypair of vectors. Since the dimension of λn is 1, given any vectors V 1,V 2, . . . ,V n itfollows that

V 1 ∧ V 2 ∧ . . . ∧ V n = λnI, (2.57)

where5 λn ∈ F. Given the independent vectors V 1,V 2, . . . , and V n their exteriorproduct will either have the same sign as I, or the opposite sign. Those withthe same sign are said to have a positive orientation, otherwise have a negativeorientation. In particular, the pseudoscalar I = e1e2e3 of C`(R3) is always chosento be positive orientation.

An important property is that the pseudoscalar commutes with all vectors in odddimension while in even dimension it anticomutes with all vectors. Using this, it isan immediate consequence that in three dimensions Ie1 = e2e3. Then, I(e1∧e2) =Ie1e2 = e2e3e2 = −e3 = − e1 × e2 where × denote the vector cross product. Itfollows that for any two vectors V ,U ∈ R3 the vector cross product is defined inClifford algebra as:

V ×U = −I(V ∧U) . (2.58)

Note that, in this case, the result of the product of I ∈ ∧3V with the 2-vector∧2V is a vector ∧1V , but this results is not valid in any dimension. This productonly exists in three dimensions. In general we have the product of the pseudoscalarI ∈ ∧nV with a homogeneous multivector Ap ∈ ∧pV is another homogeneousmultivector IAp ∈ ∧n−pV . This operation is called a duality transformation. Inthis language, the equation (2.58) means that the 2-vector was mapped to a vectorby a duality transformation. Note that this is valid just on n = 3 and by this it iscommom refer to the 2-vector V ∧U as a pseudovector in three dimensions.

5One can prove that |λn| is the analogous of the volume of a parallelepiped which is generated by thevectors V 1,V 2, . . . ,V n−1 and V n, see [16, 1, 4].


All properties relative to commutation and anticomutation of I are contained inthe following equation:

IAp = (−1)p(n−1)ApI . (2.59)

The pseudoscalar can be used to define an important operation, denoted by ?,called Hodge dual. This is a linear map from the space of p-vectors to the spaceof (n− p)-vectors:

? : ∧pV → ∧n−pVAp 7→ ?Ap = ApI .

(2.60)

Note that the Hodge map depends of the choice of orientation. By linearity onecan extend this definition to inhomogeneuos multivectors of the C`(V) [1]

?A = AI , (2.61)

and since the dimension dim(∧pV) =(np

)=(nn−p

)= dim(∧n−pV) we have an

isomorphism between ∧pV and ∧n−pV . In particular, in three dimensions, using(2.61) one finds by anticommutation that:

?1 = I = e1e2e3 ; ?e1 = e2e3 ; ?e2 = e3e1 ; ?e3 = e1e2 ;

?(e1e2) = e3 ; ?(e1e3) = e2 ; ?(e2e3) = e1 ; ?I = 1 . (2.62)

By equation (2.56) it follows the (2.61) can be written as:

?A = (−1)12

[n(n−1)+(n−s)]AI (2.63)

with this form the above equation makes clear that the Hodge dual map dependson the signature of the inner product and of the choice of orientation. In particular,the dual of the dual of a p-vector A ∈ ∧pV is proportional to identity

? ?A = (−1)[p(n−p)+ 12

(n−s)]A (2.64)

Example 2: Maxwell’s equations .

The Clifford algebra treatment is useful in several branches of the physics, sinceit provides a more compact formulation, for example, the Eletromagnetism. Thefour Maxwell’s equations can be united into a single equation. Let us define themultivector operator D by

D =∂

∂t+ ∇, (2.65)

where ∇ = e1∂∂x1

+ e2∂∂x2

+ e3∂∂x3

is the usual gradient. The electromagnetic fieldstrength is represented by the multivetor F and the two scalar densities charge ρand vectorial charge current J are combined into a single multivector J given by

F = E + IB ; J = ρ − J , (2.66)


where the vector fields E and B are the eletric field and the magnetic induction,respectively. Since the pseudoscalar commutes with all vectors in three dimensios,using the equations (2.19) and (2.58), the Clifford action of D on F is:

DF = (∂

∂t+ ∇)(E + IB)

=∂E

∂t+∂(IB)

∂t+ ∇E + ∇(IB)

=∂E

∂t+ (< ∇,E > +∇∧E) + I

[∂B

∂t+ (< ∇,B > +∇∧B)

]= < ∇,E >︸︷︷︸

scalar

+

(∂E

∂t− ∇×B

)︸︷︷︸

vector

+ I

(∂B

∂t+ ∇×E

)︸︷︷︸

2−vector

+

+ I < ∇,B >︸︷︷︸3−vector

. (2.67)

From above identity, the Maxwell equations can be written as the following compactformula

DF = J . (2.68)

Indeed, by comparing the both sides we immediately find that:

< ∇,E > = ρ ,∂E

∂t− ∇×B = J ,

∂B

∂t+ ∇×E = 0 ,

< ∇,B > = 0 . (2.69)

2.2.3 Periodicity

Now it is useful to introduce the periodicity of the Clifford algebras from which wecan relate any Clifford algebra to a number of low-dimensional Clifford algebras.The structure of a real Clifford algebra is determined by the dimension of thevector space and the signature of the metric. Then, we take the field F to be thereal numbers R and let us denote the Clifford algebra of the vector space Rp,q byC`p,q, i.e, C`p,q = C`(Rp,q). The Clifford algebra C`0,1 is generated by 1, e2 wheree2

2 = −1, it is therefore isomorphic to the algebra of complex numbers

C`0,1 ' C, (2.70)


but the Clifford algebra C`1,0 with basis 1, e1 where now e21 = 1 is not related

to any known algebra. We shall note that the elements f1 = 12(1 + e1) and f2 =

12(1 − e1) form a new basis f1, f2, with f 2

1 = f1, f22 = f2 and f1f2 = f2f1 = 0, it

follows that f1 and f2 each span mutually orthogonal one-dimensional subalgebras,each of which is isomorphic to the field R. Then,

C`1,0 ' R⊕ R . (2.71)

By multiplication of the elements of 1, e1 and 1, e2 we can construct a basis forClifford algebra C`1,1 for which a possibility is 1, e1, e2, e1e2. Then, the algebraC`1,1 can be naturally associated to algebra C`2,0 which, in its turn, is isomorphictoM(2,R) [2]. So,

C`1,1 ' M(2,R) . (2.72)

In particular, one can easily prove that [12, 1, 2]

C`0,2 ' H .

Actually, these are some general features of associative algebras [1, 14].

The following theorem, called Periodicity theorem and demonstrated in [1], establishimportant isomorphisms between different Clifford algebras.

Theorem 1. Let C`p,q be the Clifford algebra for the vector space Rp,q:

C`p+1,q+1 ' C`1,1 ⊗ C`p,q ;

C`q+2,p ' C`2,0 ⊗ C`p,q ;

C`q,p+2 ' C`0,2 ⊗ C`p,q ,

(2.73)

where p > 0 or q > 0.

This theorem immediately implies the following corollary:

Corollary 1. Any Clifford algebra C`p,q can be determined from the algebras C`0,1,C`1,0, C`1,1, C`0,2 and C`2,0 .

Clifford algebras admit a periodicity of dimension 8 over the real numbers.Indeed, using the latter theorem, we obtain the following relations

C`0,4 ' C`0,2 ⊗ C`2,0 , C`4,0 ' C`2,0 ⊗ C`0,2 , C`0,8 ' C`0,4 ⊗ C`0,4 .

From above identity, we are left with the following final relation

C`p,q+8 ' C`0,4 ⊗ C`0,4 ⊗ C`p,q ' C`0,8 ⊗ C`p,q . (2.74)

thus achieving the periodicity that we were looking for.


2.3 The Spin Groups

In this section we establish the connection between the Clifford algebra and rota-tions, which is the most relevant application of the Clifford algebra. Indeed, thisalgebra provides a very clear and compact method for performing rotations, whichis considerably more powerful than working with matrices. Let n ∈ V be a non-nullvector, n2 =< n,n >6= 0. This elements are invertible in C`(V),

n−1 =n

< n,n >. (2.75)

The set of all the non-null vectors generates a group under Clifford product, this iscalled Clifford group

Γ =np . . .n2n1 ∈ C`(V) |ni is non-null

. (2.76)

Note that n−1 = ±n when n is a normalized vector. Any vector V ∈ V can bedecomposed as:

V = V nn−1

= (< V ,n > +V ∧ n)n−1

= < V ,n > n−1 + (V ∧ n)n−1

=⇒ V = V ‖ + V ⊥ (2.77)

where V ‖ is the component of V along of n and V ⊥ is the component of Vorthogonal to n given by

V ‖ =< V ,n > n−1 , V ⊥ = (V ∧ n)n−1 . (2.78)

One can easily verify that the above relation has the properties that we desire. Inorder to see this, note that the expression for V ‖ is, in fact, the projection of Vonto vector n, since < V ,n > is a constant and the remaining term must be theperpendicular component. Moreover, from the following inner product

< n,V ⊥ >=⟨n(V ∧ n)n−1

⟩0

=⟨V ∧ n−1

⟩0

= 0 ,

we check that V ⊥ is perpendicular to n. We can rewrite (2.77) as:

V − 2V ‖ = V ⊥ − V ‖V − 2 < V ,n > n−1 = (V ∧ n)n−1− < V ,n > n−1 (2.79)

If the linear operator S : V 7→ V is the reflection in the plane orthogonal to n, wehave that:

S(V ⊥ + V ‖) = V ⊥ − V ‖= V − 2 < V ,n > n−1 , (2.80)


and by means of (2.14) and (2.18) we easily obtain that:

S(V ) = (V ∧ n− < V ,n > )n−1

=

(V U − UV

2− V U + UV

2

)n−1

=⇒ S(V ) = −nV n−1 (2.81)

which is valid on spaces of any dimension and S is called reflection operator. Itfollows that for V ∈ V the operation S(V ) ∈ V and it is the reflection of the vectorV with respect to the the plane orthogonal to n. Note that for a normalized vectorn2 = ±1 the action of S with itself is S(S(V )) = S S(V ) = (−1)S(nV n−1) =nnV nn = V , i.e., S S = I. We should check that the expression for thereflection has the desired property of leaving the inner product invariant. A simpleproof is given by:

< S(V ), S(U) > =S(V )S(U) + S(U)S(V )

2=n(V U +UV )n−1

2= < V ,U > . (2.82)

The reflection operator is therefore a linear transformation that preserves the innerproduct. Moreover, it can be proved that this transformation has determinant−1. In fact, defining the action of S on an arbitrary homogeneous multivector byS(V 1V 2 . . .V p) = S(V 1)S(V 2) . . . S(V p), using the following relation betweenthe determinant and the pseudoscalar S(I) = det(S)I, see [4], and the equation(2.59) we find that:

det(S)I = S(I) = S(e1e2 . . . en) = S(e1)S(e2) . . . S(en)

= (−1)nnIn−1 = det(S)I = (−1)n(−1)1−nI = −I=⇒ det(S) = −1 . (2.83)

We say thus that a linear transformation which leaves the inner product invariantand it has determinant −1 is called reflection.

Now, suppose that the vector U is the reflection of the vector V , S1(V ) = U , withrespect to plane orthogonal to normalized vector n1 and the vector X obtained byreflection of U , S2(U) = X, with respect to plane orthogonal to n2. Thus,

X = (S2 S1)(V ) = −n2Un−12 = n2n1V n

−11 n

−12 .

Defining R = n2n1 we can now write the result of the rotation as:

R(V ) = RV R−1 , (2.84)

where R = S2 S1 is a linear operator R : V 7→ V called rotation operator whichrepresent the rotations. We shall note that in this derivation the dimension of the


vector space was never specified, so that it must work in all spaces, whatever theirdimension. Analogously equations (2.82) and (2.83) it is simple matter to provethat the rotation operator preserves the inner product and it has determinant 1.Although this demonstration is completely analogous to the equation (2.82), let usdo it again using the projection operator. Suppose that R(V ) = RV R−1 andR(U) = RUR−1, then

< R(V ), R(U) >=⟨RV R−1RUR−1

⟩0

= 〈V U〉0 =< V ,U > .

Now, let R2 be a rotation operator such that R1(V ) = R1V R−11 = U and R2

another rotation operator such that R2(U) = R2UR−12 = X. It follows that:

X = R2(U) = R2(R1(V )) = (R2 R2)(V ) = R(V )

= R2UR−12 = R2R1V R

−11 R

−12 = RV R−1 , (2.85)

where R = R2 R1 and R = R2R1. In general, for any inhomogeneous multivec-tor A ∈ C`(V) we have that:

R(A) = RAR−1 . (2.86)

We say thus that a linear transformation which leave the inner product invariantand it has determinant 1 is called rotation. In particular, by what was seen above,a rotation in the plane generated by two unit vectors n2 and n1 is achieved bysuccessive reflections with respect to the planes perpendicular n2 and n1. So, wecan construct all rotations and reflections by application of an even number or anodd number of successive reflection operators. Moreover, it can be proved that inn dimensions any rotation can be decomposed as a product of at most n reflections[3].The orthogonal group, denoted by O(V), is the group of linear transformationson (V , < , >) that preserve the inner product and the special orthogonal group,denoted by SO(V), is the subgroup of O(V) restrict to determinant 1. The theorembelow summarizes the results obtained previously

Theorem 2. Any orthogonal transformation T ∈ O(V) can be written as a compo-sition of reflections with respect to the hyperplanes orthogonal to non-null vectors.

The set of all normalized vectors form a group under Clifford product. Denoted byPin(V), this is called pin group

Pin(V) = R ∈ C`(V)|R = np . . .n2n1,ni ∈ V andn2i = ±1 . (2.87)

Due to the Z2-grading, the elements of C`(V) split into those of even degree andthose of odd degree. The even degree elements of the Pin group form a subgroup,denoted by SPin(V), called the spin group

SPin(V) = R ∈ Pin(V)|R = n2p . . .n2n1,ni ∈ V and n2i = ±1 . (2.88)


Note that SPin(V) = Pin(V)∩C`+(V). These groups are naturally defined on theClifford algebra and can be used to implement reflections and rotations in arbitraryvectors V ∈ V . This can be seen very clear and compact below [12].

Rotation+Reflection : (−1)pRV R−1 , R ∈ Pin(V)

Pure Rotation : RV R−1 , R ∈ SPin(V) (2.89)

where p is the degree of R. Since the action of the elements of Γ on vectors isquadratic, R and −R generate the same transformation. So there is a two-to-onemap between elements of Pin(V) and rotations and reflections. Mathematically,Pin(V) and SPin(V) providing a double cover representation of the orthogonalgroups O(V) and SO(V), respectively. Then we establish the following relations

O(V) =Pin(V)

Z2

; SO(V) =SPin(V)

Z2

(2.90)

Note that the group Spin(V) is a subgroup of Pin(V) formed by elements of evendegree. One can also define Pin(V) as being the group of the elements R ∈ Γ suchthat RR = ± 1 , and SPin(V) as being a subgroup of Pin(V) formed by elementsR ∈ Γ+ such that RR = ± 1 . It follows that the group Pin(V) has a subgroup,denoted by Pin+(V), whose elements are R ∈ Γ such RR = 1 . Analogously, thegroup SPin(V) has an even subgroup, denoted by Spin+(V), whose elements areR ∈ Γ+ such RR = 1 . Other subgroups can be obtained by restriction to certainsubgroups of Pin(V) and Spin(V). In fact, the group Spin +(V) has as its elementsR ∈ Γ such that its norm defined by |R| =

⟨RR

⟩0is equal to 1 [8, 2]. By equation

(2.89), an element of the group Spin(V) act in the vector space V and yields anelement on the same vector space. This implies that the vector space V furnish arepresentation for the spin groups. But since the action is quadratic, it follows it isnot faithful [12, 14, 1].

2.4 Spinors

In the previous section it was established that the vector space V provides a repre-sentation for the spins groups, since the action of elements of the spins groups inelements of V results in element of V . However, since elements of the spin groupwhich differ by a sign produce the same orthogonal transformation, it follows thatthis representation is not faithful. In this section we introduce the so-called spinors,that generate a vector space that provides a faithful representation for the spingroup. For simplicity, we will only consider the Clifford algebra of an orthogonalspace (V , < , >) whose dimension is even, n = 2r with r ∈ N.


2.4.1 Minimal Left Ideal

There is an important class of subspaces which are called left ideals. A left idealL ⊂ C`(V) of an algebra C`(V) is a subalgebra of C`(V) such that:

Aφ ∈ L ∀ φ ∈ L ,A ∈ C`(V) , (2.91)

i.e, L is invariant under left multiplication of the whole algebra. Since C`(V) hasa Z2-grading, we can use its even subalgebra C`+(V) as the representation spaceof C`(V) and define ρ : C`(V) 7→ End(C`+(V)). By means of (2.27) we can splitany multivector A ∈ C`(V) as A = A+ + A−, where A± = 1

2(A ± A) ∈ C`±(V).

We should establish in what conditions ρ = ρ+ + ρ− such that ρ(A) = ρ+(A+) +ρ−(A−) will be a representation of C`(V). Note that A+B ∈ C`+(V) if B ∈ C`+(V),this implies that ρ+(A+)(B) = A+B ∀B ∈ C`+(V), while A−B ∈ C`−(V) ifB ∈ C`+(V), this requires that ρ−(A−)(B) 6= A−B when B ∈ C`+(V). If, however,we can take an odd element C ∈ C`−(V) and define

ρ−(A−)(B) = A−BC ∈ ∀B ∈ C`+(V) (2.92)

such that C2 = 1, then ρ = ρ+ + ρ− is a representation in the space C`+(V). Onemight then wonder, how can we know if such representation is reducible? Supposethat there exists an element C1 ∈ C`+(V) such that

C21 = 1 ; C1C = CC1 . (2.93)

Thus we can write C`+(V) = +C`+(V)⊕ −C`+(V), where

±C`+(V) = C`+(V)1

2(1 ± C1) . (2.94)

so that, for B± ∈ ±C`+(V) we have B±C1 = ±B±. Each of these spaces ±C`+(V)is invariant under the action of ρ as we can see by the equation (2.93) and it is alsoa subalgebra of C`+(V). If another element C2 ∈ C`+(V) such that C2

2 = 1 , C2C1 =C1C2 and C2C = CC2 can be found, then the subspaces ±C`+(V) carry no irreduciblerepresentation. So, we define another four subspaces

±C`+(V) = ±C`+(V)1

2(1 ± C1)(1 ± C2), (2.95)

each of which invariant under the action of ρ. We can proceed with this constructionin an analogous fashion if there are other elements Cj commuting with the previousones.

The regular representation ρ : C`(V) 7→ End(C`(V)) given by ρ(A)B = AB pre-serves certain vector subspaces. Theminimal left ideals are just the invariant sub-spaces S ⊂ C`(V) for which the map ρ : C`(V) 7→ End(S) defined by ρ(A)ψ = Aψ


is an irreducible representation. Note that the minimal left ideal contains no leftideals apart from itself and zero, it provides the least-dimensional faithful represen-tation of C`(V), the so-called spinorial representation of the Clifford algebra. Theminimal left ideal S is called, as a vector space, the spinor space, and its elements,denoted by ψ, are called spinors. Note also that these minimal left ideals are al-gebras, the subalgebras of C`(V). Thus, under Clifford product, the space carryinga such representation of C`(V) will be called spinorial algebra and the choice of adifferent minimal left ideal gives other equivalent representations [1, 11, 14]. Wewill therefore establish that the spinor representation of the Clifford algebra inducesa representation of any subset by restricting to left multiplication on the ideal byelements of that set. In particular it induces a representation of the Clifford group[1]. When the dimension of V is n = 2r one can prove that the dimension of thespinor space is 2r. When C`(V) is thought of as a matrix algebra, an example ofa minimal left ideal is the subalgebra of matrices with all columns but the firstvanishing [2, 11, 30]. In particular, C`(V), P in(V), SP in(V), O(V) and SO(V) canbe faithfully represented by 2r × 2r matrices. In this case spinors are representedby the column vectors on which these matrices act.

By what was seen above, the minimal left ideals are of great relevance in the studyof the spinors. These can be obtained by action of the whole algebra on the so-called primitive idempotent of C`(V), this makes clear the fundamental importanceof the primitive idempotents. An element ξ of an any algebra A is said to be anidempotent if ξ2 = ξ and ξ 6= 0. Particularly, such an idempotent generatesa subalgebra ξAξ. Now, if A is a division algebra, then the single idempotentis the identity, since if ξ 6= 0, then every ξ has an inverse ξ−1, it implies thatξ2 = ξ ∵ ξ−1ξ2 = ξ−1ξ = 1A ∴ ξ = 1A. We can split the element ξ into asum of other two elements ξ1 and ξ2, ξ = ξ1 + ξ2. Hence, ξ will be idempotent ifthe following relations are satisfied: ξ2

1 = ξ1, ξ22 = ξ2 and ξ1ξ2 = −ξ2ξ1. If such

elements satisfying the conditions ξ1ξ2 = 0 and ξ = ξ1 +ξ2 cannot be found, we callξ a primitive idempotent. Therefore, a idempotent ξ is primitive if and only if it isthe single idempotent on ξAξ [2, 14, 17]. Let ξ be a generic primitive idempotentin C`(V), then any minimal left ideal S. has the form:

S = C`(V)ξ = Aξ | A ∈ C`(V) . (2.96)

When the dimension of V is even the square of I depends only on the signature ofthe inner product. So, equation (2.56) is reduced to:

I2 = (−1)12

[n(n−1)+(n−s)] ∀ n

=⇒ I2 = (−1)−s2 , n = 2r . (2.97)

Thus, we can express its square as I2 = ε2, with ε = 1 or ε = i. Since I2 = ±1,


the action of I on S gives a decomposition [12]

S = S+ + S− ; S± = ψ ∈ S | Iψ = ±εψ , (2.98)

where S± are subspaces of dimension 2r−1 whose elements are called semi-spinors.The semi-spinors are also called Weyl spinors of positive and negative chirality.These subspaces are invariant under the action of the even subalgebra. Indeed, bymeans of the equation (2.59) we have that IA+ = A+I for A+ ∈ C`+(V) and neven. Then, due to (2.98) we have:

IA+ψ = A+Iψ

=⇒ IA+ψ = ±εA+ψ ∀ A+ ∈ C`+(V) . (2.99)

This means that the spinorial representation of C`+(V) on S is reducible while thespinorial representation of C`+(V) on S± is irreducible [12, 1]. In paricular, thespinorial representation of C`+(V) splits in two blocks of dimension 2r−1 × 2r−1.It happens that, by equation (2.88), elements of Spin(V) are of degree even. So,by restriction, the spinorial representation of C`+(V) induce a representation ofdimension 2r−1 for Spin(V) [14].

Though we have restricted ourselves to even-dimensional spaces, let us clarify theseconcepts with an example in three dimensions following the line of the previousexamples. We will assume that the dimension of S is 2[n/2], where [ ] denotes theinteger part of the number inside it.

Example 3: Spinors in the minimal left ideal of C`(R3).

Let e1, e2, e2 be an orthonormal basis for vector space V = R3 where e1e1 =e2e2 = e3e3 = 1 , eaeb = −ebea if a 6= b. The set 1, e1, e2, e3, e1e2, e1e3, e2e3, Ispan the Clifford algebra C`(R3) whose the most general element is:

A = a + a1e1 + a2e2 + a3e3 + a12e1e2 + a13e1e3 + a23e2e3 + a123I ,

where I is the pseudoscalar. In three dimensions, we must find a spinor space S ofdimension dim(S) = 2[3/2] = 2. Now, consider the element

ξ1 =1

2(1 + e3).

Note that ξ1, ξ12 = 1

2(1 + e3)1

2(1 + e3) = ξ1, is a primitive idempotent. It follows

that the set of the form

S = C`(R3)ξ1 = Aξ1 | A ∈ C`(R3) ,


is a minimal left ideal of C`(R3). Using the Clifford algebra, one finds after somemanipulations that

Aξ1 = [(a+ a3) + I(a12 + a123)]ξ1 + [(a1 + a13) + I(a2 + a23)]e1ξ1 . (2.100)

Defining ξ2 = e1ξ1, we obtain that:

Aξ1 = [(a+ a3) + I(a12 + a123)]ξ1 + [(a1 + a13) + I(a2 + a23)]ξ2 . (2.101)

In the same way, we obtain

Aξ2 = [(a1 − a13) − I(a2 + a23)]ξ1 + [(a − a3) − I(a12 + a123)]ξ2 . (2.102)

Note that that the pseudoscalar I commutes with all elements and squares to −1,then it can be viewed as the unit imaginary i in C`(R3), Iξ1 = iξ1. Since theaction of A on ξ1 and ξ2 are both linear combination of the set ξ1, ξ2, we seethat:

S = ψ ∈ C`(R3) |ψ = ψ1ξ1 + ψ2ξ2 ∀ ψ1, ψ2 ∈ C , (2.103)

where ψ1 = α1 + iβ1 and ψ2 = α2 + iβ2. This space admits no proper left ideal,so the elements ψ = ψAξA are called spinors and ξA where A ∈ 1, 2 form abasis for S, the called spinor basis. It is simple to prove that S is invariant by theaction of C`(R3). In order to prove this explicitly, we only need to act the elementsthat span V which are e1, e2 and e3 on the elements of S and this implies in thefollowing spinorial representation for the vectors of the basis:

e1ψ = e1(ψ1ξ1 + ψ2ξ2) = ψ1ξ2 + ψ2ξ1 =⇒ e1 '[0 11 0

](2.104)

e2ψ = e2(ψ1ξ1 + ψ2ξ2) = −iψ1ξ2 + iψ2ξ1 =⇒ e2 '[0 −ii 0

](2.105)

e3ψ = e3(ψ1ξ1 + ψ2ξ2) = ψ1ξ1 − ψ2ξ2 =⇒ e3 '[1 00 −1

], (2.106)

i.e., the action of V on S yield elements on S. Thus, as a matrix algebra, usingthese three last equations we can see that the spinor can be represented by:

ψ =

[ψ1

ψ2

]∈ C2 or ψ =

[ψ1 0ψ2 0

]∈M(2,C)ξ1 , (2.107)


where ξ1 =

[1 00 0

]. If we multiply ψ on the left by an arbitrary matrix ∈M(2,C)

we are left with a completely analogous relation [2]. Using these matrices we arriveon the matricial representation of the multivetor A obtained on Example 1.

Defining the normalized vectors n1 = e1 ,n2 = cosθ e1 + sinθ e2, then we canconstruct the following element of the group SPin(V): R1 = n2n1 = cosθ −sinθe1e2. It follows that an element R1 of Spin(R3) acts on the elements of R3 asR1 eaR

−11 (a = 1, 2, 3). Thus,

e1 7→ R1(e1) = n2n1e1n1n2 = cos(2θ)e1 + sin(2θ)e2

e2 7→ R1(e2) = n2n1e2n1n2 = −sin(2θ)e1 + cos(2θ)e2

e3 7→ R1(e3) = n2n1e3n1n2 = e3 . (2.108)

The result of the action of R1 is therefore a rotation of 2θ on the plane generatedby the two unit vectors e1 and e2. Note that from the expression of R1 it can berepresented by R1 = e−θe1e2 . But we can implement rotation in any one of theeaeb planes. So, we can perform others rotation by R = e−θeaeb whose action yieldsa rotation of 2θ on the eaeb plane. In particular, taking θ = π, we have a rotationthrough 2θ on the eaeb plane. In this case, the vectors remain unchanged while thespinors change of sign.

2.4.2 Pure Spinors

From the geometrical point of view, there exists an important class of spinors whichare associated with maximal isotropic vector subspaces, observed by Cartan [3],these are the so-called pure spinors. For example, let V be a null vector, namely< V ,V >= 0. In three dimensions we can write the complex components(V1, V2, V3) of this vector in terms of two elements ψ1, ψ2 as:

V1 = ψ12 − ψ2

2

V2 = i(ψ12 + ψ2

2)

V3 = −2ψ1ψ2 , (2.109)

which automatically guarantees the isotropic character of the vector. Indeed, thelatter equation immediately implies that V 2

1 + V 22 + V 2

3 = 0. Note that theseequations are solved for

ψ21 =

V1 − iV2

2; ψ2

2 = −V1 + iV2

2. (2.110)


So, using the above equation and that V3 = −2ψ1ψ2 we obtain thatV3ψ1 + (V1 − iV2)ψ2 = 0(V1 + iV2)ψ1 − V3ψ2 = 0

=⇒[

V3 V1 − iV2

V1 + iV2 −V3

] [ψ1

ψ1

]= 0 , (2.111)

which is the matrix representation of V ψ = 0, with[V3 V1 − iV2

V1 + iV2 −V3

][V3 V1 − iV2

V1 + iV2 −V3

]=

[1 00 1

]< V ,V >= 0

and < V ,V >= 0. Note that the rotation (2.108) on the vector V implies,

V ′1 = V1 cos(2θ) − V2 sin(2θ)

V ′2 = V1 sin(2θ) + V2 cos(2θ)V ′3 = V3 . (2.112)

Using this, it is simple matter to prove that:[ψ1

ψ2

]=⇒

[ψ′1ψ′2

]=

[e−iθψ1

eiθψ2

]. (2.113)

Note that taking θ = π the vector remains unchanged by the action of the rotationoperator, while the elements ψ1, ψ2 change the sign. Thus, the pair ψ1, ψ2 consti-tutes the components of a spinor ψ which is associated a null subspace spanned bythe vectors that annihilate it, V ψ = 0, whose representation is given by (2.111).The spinor ψ is an example of a pure spinor.

Now, let us make a rigorous definition of a pure spinor. Formally, given a spinorψ ∈ S one can construct a subspace of V , denoted by Nψ, defined by:

Nψ = V ∈ V |V ψ = 0 . (2.114)

Note that if ψ in non-null then for all V ,U ∈ Nψ we have

2 < V ,U > ψ = (V U + UV )ψ = 0

=⇒ < V ,U >= 0 . (2.115)

This subspace is clearly totally null, which means that all vectors belonging to itare orthogonal to each other including to itself.

In what follows, we will only consider the complexified Clifford algebra of an evendimensional orthogonal space. The complexification of a real orthogonal space(V , < , >) is the space (VC, < , >C) whose elements are of the form V + iU forsome V ,U ∈ V and i the unit imaginary. We can define the sum and multiplicationby a complex scalar (λ + iδ) as:

(V 1 + iU 1) + (V 2 + iU 2) = (V 1 + V 2) + i(U 1 + U 2)

(λ + iδ)(V + iU) = (λV − δU) + i(λV + δU) . (2.116)


The complex conjugate of V = V 1 + iU 1 ∈ VC for V 1,U 1 ∈ V is defined byV ∗ = V 1 − iU 1. Since VC can be obtained by the complexification V , VC = C⊗V ,we see that the complex dimension of VC is n and the real dimension 2n. The innerproduct < , > is extended to < , >C on VC by assuming bilinearity of the innerproduct in the complex field. For simplicity, from now on we shall assume thatthe complex Clifford algebra associated to VC endowed is simply C`(VC) and thecomplexification of C`(V , < , >) is C`C(V). From this, we say that theses algebrasare isomorphic, namely:

C`(VC) ' C`C(V) , (2.117)

where C`C(V) = C⊗ C`(V).

We say that a spinor is pure if the null subspace associated to it is maximal [3, 14],namely when the null subspace has the maximum dimension possible. But, what ismaximum dimension of a null subspace? When the complex dimension is n = 2r,the maximal dimension that a null subspace can be have is r. For example, giventhe vectors e1, e2, . . . , er, er+1, . . . , e2r, if we define na = ea + i ea+r, then we havea null subspace of dimension r. In this case we say that it is maximal or maximallynull. Therefore, a spinor associated to a null subspace with this dimension is saidto be pure. Apart from a multiplicative factor, there is a one-to-one associationbetween ψ and Nψ, thus a pure spinor ψ is a representative for Nψ if and only ifV ψ = 0 for all V ∈ Nψ.We should recall that a vector V transforms under rotation as RV R−1 while thespinor ψ transform as Rψ. So, let ψ be a pure spinor and V a null vector thatbelongs to Nψ, then V ψ = 0. If V ′ = RV R−1 and ψ′ = Rψ. It follows that

V ′ψ′ = RV R−1Rψ = RV ψ

=⇒ V ′ψ′ = 0 . (2.118)

i.e., we can transform any null subspace Nψ into any another null subspace N ′ψby a rotation or reflection [3]. It is worth stressing that the sum of two purespinors is not a pure spinor in general, since the purity condition is quadratic onthe spinor [11, 1]. In practice, given a spinor ψ when is it pure? In order toanswer this question, let us workout an example in R3. Let S = ψ ∈ C`(R3) |ψ =ψ1ξ1 + ψ2ξ2 ∀ ψ1, ψ2 ∈ C be the minimal left ideal of the Clifford algebraC`(R3) where ξ1 = (1+ e3)/2, ξ2 = e1ξ1 and let V = V1e1 + V2e2 + V3e3 be a nullvector, i.e., < V ,V >= V 2

1 + V 22 + V 2

3 = 0. Here, the pseudoscalar I = e1e2e3

commutes with all elements and squares equal to −1 and therefore be viewed asthe unit imaginary i. Indeed Iξ1 = iξ1 . Acting V on ψ

V ψ = (V1e1 + V2e2 + V3e3)(ψ1ξ1 + ψ2ξ2)

= (V1ψ2 + V3ψ1 − iV2ψ2)ξ1 + (V1ψ1 − V3ψ2 + iV2ψ1)ξ2 = 0(2.119)


we obtain the following system to be solvedV1ψ2 + V3ψ1 − iV2ψ2 = 0V1ψ1 − V3ψ2 + iV2ψ1 = 0

V 21 + V 2

2 + V 23 = 0

whose solution is

V1 = ψ12 − ψ2

2

V2 = i(ψ12 + ψ2

2)

V3 = −2ψ1ψ2 . (2.120)

Therefore, if the components of the spinor ψ is characterized by a set of quadraticrelations, then the spinor ψ is said to be pure. It turns out that pure spinors arenecessarily semi-spinors. Let us prove this. Since the dimension of VC is 2r we canchoose the first r vectors to be the set e1, e2, . . . , er such that < ea, eb >= 0.Thus, this set forms a basis for a maximal null subspace. To complete this basiswe choose the other r as being the set of the vectors er+1, er+2, . . . , e2r such that< ea, eb >= 0. Theses two sets form a basis for the whole vector space VC andits elements are such that < ea, e

b >= 12δba. The pseudoscalar of C`C(V) has the

following proportionality

I ∝ (e1 ∧ e1)(e2 ∧ e2) . . . (er ∧ er)∝ [e1, e

1][e2, e2] . . . [er, e

r] , (2.121)

where [ea, eb] = eae

b − ebea denote the Clifford commutators. Since the vectorsea span Nψ, by definition that eaψ = 0 . Hence ebeaψ = 0. Thus we only needto know the action of eaeb on ψ, that is,

(eaeb)ψ = (eae

b + ebea)ψ = 2 < ea, eb > ψ = δbaψ . (2.122)

Thus, the action of commutator is

[ea, eb]ψ = (eae

b − ebea)ψ = δbaψ . (2.123)

Therefore, using (2.121) and (2.123) is trivial show that Iψ ∝ ψ. It follows thatevery pure spinor ψ must be a semi-spinor, or Weyl spinor. It turns out that intwo, four and six dimensions all semi-spinors are pure, while in higher dimensionsnot all are semi-spinors.

2.4.3 SPin-Invariant Inner Products

We have referred to a minimal left ideal as the spinor space and its elements asspinors, we now examine some operations and properties that this space possess


with respect to products of two of its elements. We shall construct an invariantproduct under the action of the spin group. For example, the space of spinorsS = C`(R3)ξ1 given by

S =ψ ∈ C`(R3) |ψ = ψ1ξ1 + ψ2ξ2 ∀ ψ1, ψ2 ∈ C

, (2.124)

obtained in the Example 3 can be represented as follows:

S '

[ψ] ∈M(2,C)ξ1 | [ψ] =

[ψ1 0ψ2 0

]∀ ψ1, ψ2 ∈ C

. (2.125)

But since the product ξ1ξ2 = 0, the set

D = ξ1 C`(R3) ξ1 = ψ1ξ1 , ψ1 ∈ C '[

ψ1 00 0

]|ψ1 ∈ C

(2.126)

is a subalgebra of C`(R3) with unity ξ1 such that λξ1 = ξ1λ , for all λ ∈ D.According to equation (2.40), none of its elements is invertible as an element ofC`(R3), however there is a unique λ2 ∈ D with λ2λ1 = ξ1 if λ1 is a non-zeroelement of D. Thus, D is a division algebra and hence the idempotent ξ1 is primitivein C`(R3) as expected. The space S has a natural right D-linear structure definedby

S × D → Sψ,λ 7→ ψλ .

(2.127)

The space S endowed with this right D-linear structure is called, more formally, thespinor space. We must note that on spinorial representation an arbitrary multivec-tor A ∈ C`(R3) and its reverse are given by (see the Example 1):

[A] =

[z1 z3

z2 z4

]; [A] =

[z∗1 z∗2z∗3 z∗4

], zi ∈ C . (2.128)

Then, given two spinors ψ,φ ∈ S the product

[ψ][φ] =

[ψ∗1 ψ∗20 0

] [φ1 0φ2 0

]=

[ψ∗1φ1 + ψ∗2φ2 0

0 0

]∈ D (2.129)

falls in the division algebra D. Now, the inner product can be naturally introducedon the spinor space as

S × S 7→ Dψ,φ 7→ (ψ,φ) = ψφ ,

(2.130)

if for ψ ∈ S and λ ∈ D the following relation holds:

(ψλ) = λψ , (2.131)


where λ→ λ is the complex conjugation of the division algebra D and the reversionψ 7→ ψ is a semi-linear map. Now, let A be an arbitrary element of C`(R3). Then,

(Aψ,Aφ) = AψAφ = ψAAφ

This implies that the elements A ∈ C`(R3) such that AA = 1 preserve the innerproduct, i.e., this inner product between spinors is invariant under the action ofthe group Spin+(R3). In the same way, we can use another involution to define adifferent inner product. Indeed, the spinorial representation of A ∈ C`(R3) furnishthat its Clifford conjugate is:

[A] =

[z4 −z2

−z3 z1

], zi ∈ C . (2.132)

We can use this Clifford conjugation to introduce an inner product for spinors.However, the product

[ψ][φ] =

[0 0−ψ2 ψ1

] [φ1 0φ2 0

]=

[0 0

ψ1φ2 − ψ2φ1 0

]/∈ D (2.133)

does not fall in the division algebra D. But since ψφ = (ψ1φ2 − ψ2φ1)ξ2 its alwayspossible to find an invertible element V = e1 ∈ C`(R3), in this case, such that theproduct e1ψφ = (ψ1φ2 − ψ2φ1)e1ξ2 = (ψ1φ2 − ψ2φ1)ξ1. Thus,

[V ][ψ][φ] =

[0 11 0

] [0 0

ψ1φ2 − ψ2φ1 0

]=

[ψ1φ2 − ψ2φ1 0

0 0

]∈ D . (2.134)

This leads us to the conclusion that,

S × S 7→ Dψ,φ 7→ (ψ,φ) = V ψφ

(2.135)

also defines an inner product. One can easily prove that this latter inner productbetween spinors is invariant under the action of the group Spin(R3).

In what follows, we will deal with a general Clifford algebra on which every spinorspace admits a real, complex or quaternion division algebra. Let η be some involu-tion. If ξ is any primitive idempotent then ξη = JξJ−1 for some element J withJη = εJ and ε = ±1 [1]. Then, if ψ,φ ∈ S, we define:

S × S → Dψ,φ 7→ (ψ,φ) = J−1ψηφ

. (2.136)

If A ∈ C`C(V) is arbitrary element we find that:

(ψ,Aφ) = (Aηψ,φ) . (2.137)


Using the equation (2.127) we find that if λ ∈ D,

(ψ,φλ) = (ψ,φ)λ , (2.138)

the product is D-linear in the second entry. However if we denote λj = J−1ληJfor λ ∈ D it is straightforward to prove that

(ψλ,φ) = λj(ψ,φ) (2.139)

j is an involution of D and it will reverse the order of terms in the inner product,that is,

(ψ,φ) = ε(φ,ψ)j . (2.140)

The inner product ( , ) can be Dj-symmetric or Dj-skew-symmetric depending onthe dimension of the space, i.e., when ε is plus or minus, respectively. Let us definethe D-linear map f from S to D by the following relation

f(ψ)φ = (ψ,φ) . (2.141)

It follows that f(ψ) can be naturally identified with J−1ψη by

f(ψ) = J−1ψη . (2.142)

Note that the function f(ψ) depends on the choice of the invertible element J .

There are some elements that play an important role when we are dealing withspinors. Let us denote by † the involution6 of Hermitian conjugation, ∗ the invo-lution of complex conjugate and the composition †∗ ≡ t stands transpose. So, theinvertible matrices A,B and C are related with these involutions by:

AV A−1 = −V † , BV B−1 = V ∗ , C V C−1 = −V t , V ∈ C`C(V).

(2.143)

In particular the matrix B enables us define the notion of complex conjugation ofa spinor. This is an operation that the spinor space S admits, the so-called chargeconjugation. This is an anti-linear operation given by

c : S 7→ Sψ 7→ ψc = B−1ψ∗ ,

(2.144)

with the property that

(Aψ)c = A∗ψc ∀A ∈ C`C(V) and ψ ∈ S . (2.145)6Note that the dagger of the representation of an arbitrary multivetor is the reversion itself.


The charge conjugation has different froms depending on the signature and on thedimension [12, 11]. In particular, the conjugate of the conjugate can be equal toidentity or equal to minus the identity depending on the signature and on thedimension. The spinors which satisfy

ψc = ±ψ , (2.146)

are called Majorana spinors and the spinors satisfying the above equation and Iψ =±εψ simultaneously are called Majorana-Weyl spinors [11, 1]. We can use thematrix C to rewrite an inner product between spinors. If ψ,φ are two spinors,then

(ψ,φ) = ψtCφ (2.147)

is also invariant under, for example, rotation. We can also define other products thatare invariant under the action of the connected part to identity of the orthogonalgroup, for example, rotation and reflection. These are,

(ψ, Iφ) ; (ψc,φ) ; (ψc, Iφ) . (2.148)

In general, no further inner products can be generated in a trivial way and allthese products are diferent and independent [9]. In physics, to make manipulationsexplicit and less abstract of the Clifford algebra and of the spinorial formalism weuse the so-called Dirac matrices. In next section we will explain this point.

2.5 Spinors, Gamma Matrices and their Symmetries

In this section we shall introduce a matrix realization for the elements of the Cliffordalgebra through the introduction of the Dirac matrices. We will only consider aneven dimensional vector space V of dimension n = 2r. These matrices are therepresentation in dimension n of the Clifford algebra generated by n matrices Γawhich satisfy the anti-commutation relations

ΓaΓb + ΓbΓa = 2δabI , a, b = 1, 2, . . . , n , (2.149)

where I is a 2r × 2r unit matrix and δab is the Kronecker symbol, since repeatedmultiplication of the Dirac matrices leads to a set of 2n matrices ΓA:

ΓA : I,Γa,Γab,Γabc, . . . , (2.150)

where,

Γab = Γ[aΓb] (a < b), Γabc = Γ[aΓbΓc] (a < b < c), . . . . (2.151)


We can define a set of r raising and lowering operators as:

Γa± =1

2(Γa ± iΓ2a) , a = 1, 2, . . . , r , (2.152)

which satisfy the following anticommutation relations

Γa+,Γa− = δab (2.153)Γa+,Γa+ = Γa−,Γa− = 0 . (2.154)

Thus, in particular, note that (Γa±)2 = 0. Let us first give the usual representationof the Clifford algebra in terms of the Pauli matrices. In two dimensions, n = 2,the Pauli matrices σa (a = 1, 2, 3) represent faithfully the Clifford algebra by 2× 2matrices given by

σ1 =

[0 11 0

]; σ2 =

[0 −ii 0

]; σ3 =

[1 00 −1

], (2.155)

where σ1σ2 = iσ3, they square are equal to I and are hermitian. In this case, wefind that

Γ1± = σ± =1

2(σ1 ± i σ2) . (2.156)

If we act Γa− repeatedly times we eventually arrive in a spinor ξ0 such that

Γa−ξ0 = 0 , ∀ a , (2.157)

it is annihilated by all the Γa−. Since (Γa+)2 = 0, we can act Γa+ on the spinorξ0 over all possible ways at most once each. In order to perform this, let us labelthe spinor with the indices s1, s2, . . . , sr. If we assume that every index sa can takethe values 1/2 and −1/2, then, starting from ξ0 one obtains a representation ofdimension 2r by action with the elements of the form

(Γa+)sr+1/2 . . . (Γ1+)s1+1/2 . (2.158)

This action we denote by

ξs1,s2,...,sr = (Γa+)sr+1/2 . . . (Γ1+)s1+1/2ξ0 . (2.159)

Note that ξ−1/2,−1/2,...,−1/2 = ξ0 is the our initial spinor. Regarding the spinors, itis useful to define the following column vectors:

ξ+ =

[10

], ξ− =

[01

]. (2.160)


In terms these column vectors, the spinor ξ0 is written as

ξ0 =

[01

]⊗[

01

]⊗ . . .⊗

[01

]︸︷︷︸

r times

,

and therefore the spinor ξs1,s2,...,sr can be written as

ξs1,s2,...,sr ≡ ξs = ξs1 ⊗ ξs2 ⊗ . . .⊗ ξsr . (2.161)

Defined this, taking the equation (2.161) as a basis, the matrices Γa can be derivedfrom the definitions and the anticommutation relations. For example, when n = 2we find that

σ−ξ− 1

2 = 0 , σ−ξ12 = ξ−

12 , σ+ξ

− 12 = ξ

12 , σ+ξ

12 = ξ

32

=⇒ σ±ξs = ξs±1 . (2.162)

For more clearness on notation we shall explicit the dimensionality of the Diracmatrices, which can be obtained recursively by the relations

Γ(2r)1 = σ1 ⊗ I2(r−1) , Γ

(2r)2 = σ2 ⊗ I2(r−1) ,

Γ(2r)a = σ1 ⊗ Γ

(r−1)a−2 , a = 3, . . . , n , (2.163)

where IN is a N × N unit matrix and Γ(2r)a are 2r × 2r matrices. We can split it

into those that are even or odd. Explicitly they are

Γ(2r)2a−1 = σ3 ⊗ . . .⊗︸︷︷︸

(a−1) times

σ1⊗I⊗ . . .⊗ I︸︷︷︸(r−a) times

Γ(2r)2a = σ3 ⊗ . . .⊗︸︷︷︸

(a−1) times

σ2⊗I⊗ . . .⊗ I︸︷︷︸(r−a) times

. (2.164)

An important fact is that in even dimensions a complete set of 2r × 2r matrices isprovided by ΓA = I,Γa,Γab, . . .. The n(n−1)

2matrices

Σab =1

4[Γa,Γb] , a = 1, 2, . . . , n , (2.165)

satisfy the SO(n) algebra

[Σab,Σcd] = δbcΣad − δacΣbd − δbdΣac + δadΣbc , (2.166)

and generate a representation of dimension 2r of Spin(n) the double cover of SO(n).Note that the operator Σa,2a is hermitian and can be therefore simultaneously di-agonalized. It simple a manner to prove from definition below

Sa ≡ Σa,2a = Γa+Γa− −1

2, (2.167)


that if the spinor ξs is an eigenspinor of the operator Sa its eigenvalues are sa =±1/2. Indeed, if we denote Ma = Γa+Γa−, using the anticommutation relationsof the raising and lowering operators we find that (Ma)

2 = Γa+Γa− = Ma, thenMa(Ma − 1) = 0. If ξs is also an eigenspinor of the operator Ma with eigenvaluema for all a, Ma(Ma − 1)ξs = ma(ma − 1)ξs, it follows that ma can take justthe value, ma = 0, 1. Thus,

Saξs = (Ma −

1

2)ξs = (ma −

1

2)ξs = saξ

s , (2.168)

where sa = ma − 12are the eigenvalues of Sa with ξs as eigenspinor. Note that,

sa =

1/2 , ma = 1

− 1/2 , ma = 0, (2.169)

and the notation s introduced previously reflects the Spin(n) properties of thespinors [33]. The half-integer values of sa show that this is a spinor representation.Then, the spinors ξs form the Dirac representation of the Spin(n) algebra ofdimension 2r.

The chiral matrix, denoted by Υ, is given by

Υ = (−i)rΓ1Γ2 . . .Γn = σ3 ⊗ . . .⊗ σ3︸︷︷︸r times

, (2.170)

and since Υ is not proportional to identity and commute with all Σab, it follows thatthe representation ξs of Spin(n) with generators Σab must be reducible. Indeed,using (2.163) we find

Σab =

[Σ+ab 00 Σ−ab

], (2.171)

where both Σ±ab satisfy the commutation rules (2.166). Using that Γa+Γa− − 12

=ΓaΓ2a/2 is straightforward to find

Υ = 2rS1S2 . . . Sr , (2.172)

and we see that Υ is diagonal in the basis ξs. Since Υ2 = 1, using (2.172) and(2.168) we see that the action of Υ on ξs is

Υξs = ±ξs . (2.173)

The 2r−1 states with eigenvalue 1 form a Weyl representation of the Spin(n) al-gebra and the 2r−1 states with eigenvalue −1 form a second, inequivalent, Weylrepresentation. For example, denoting the representation by dimension, for n = 4

4Dirac = 2 + 2′ , (2.174)


the Dirac representation split into two inequivalent Weyl representation of dimen-sion 2.

Observing that the matrices ±Γa∗ satisfy the same Clifford algebra obeyed by Γathey must be related to Γa by a similarity transformation. In the basis ξs thematrices Γ1,Γ3, . . . ,Γn−1 are all real while Γ2,Γ4, . . . ,Γn are all imaginary. Thiscan be viewed directly by means of the equation (2.164). If we define the productof r odd Dirac matrices as the following matrices

B− = Γ1Γ3 . . .Γn−1 , B+ = ΥB− , (2.175)

what is the final result of the operations B−ΓaB−1− and B+ΓaB

−1+ ? Let us suppose

that Γa is imaginary, hence its is an even element. So, by anticommutation

B−ΓaB−1− = (−1)rΓa = (−1)r−1(−Γa) = (−1)r−1Γ∗a . (2.176)

Now, if Γa is real, hence its is an odd element. So,

B−ΓaB−1− = (−1)r−1Γa = (−1)r−1Γ∗a . (2.177)

Thus, for any element Γa, be it even or odd, we have

B−ΓaB−1− = (−1)r−1Γ∗a . (2.178)

In the same way we can easily prove that

B+ΓaB−1+ = (−1)rΓ∗a . (2.179)

Performing similar manipulations we obtain the following results,

B±ΣabB−1± = Σ∗ab (2.180)

This relation implies that the charge conjugation of a spinor ψc = B−1ψ∗ trans-forms in the same way as ψ. The action of B± on the chirality matrix results

B±ΥB−1± = (−1)r−1Υ∗ , (2.181)

from which we can see that the charge conjugation can change the eigenvalue of Υdepending on the parity of r−1. For example, when r is odd (n = 2, 6, 10, 14, 18, . . .or n = 2 mod 4) each Weyl representation is its own conjugate, while that for reven (n = 0, 4, 8, 12, 16, . . . or n = 0 mod 4) each Weyl representation is conjugateto the other. In particular, when n = 4

4Dirac = 2 + 2 , (2.182)

where 2 is the conjugate of the representation 2. A Majorana spinor is an eigen-spinor of the charge conjugation operation

ψc = ±ψ , (2.183)


which implies that taking the conjugate ψ∗ = Bψ ∴ (ψ∗)∗ = (Bψ)∗ = B∗ψ∗ =B∗Bψ gives a consistent condition if B∗B = I. But using the reality and anticom-mutation properties of Γ we find

B∗−B− = (−1)r(r−1)/2I , B∗+B+ = (−1)(r−1)(r−2)/2I . (2.184)

If we use B− to define the Majorana spinor, then this operation is possible whenr = 1 (mod 4) or r = 4 (mod 4) while using B+ only if r = 1 (mod 4) orr = 2 (mod 4). The such so-called Majorana-Weyl spinors are the spinors that areMajorana and Weyl. In this case, the Majorana condition and Weyl condition onlycan be imposed if the Weyl representation is conjugate to itself. In particular whenr is even it is not possible to impose both the Majorana and Weyl conditions ona spinor. However, when r = 1 (mod 4) a spinor can simultaneously satisfy theMajorana and Weyl conditions.

The Γ-matrices are all hermitian, thus we get the hermiticity property Γ†a = Γa.So, the operation

AΓaA−1 = −Γ†a (2.185)

is satisfied if AΓa = −ΓaA. The chirality matrix Υ anti-commute with the Diracmatrices. It follows that, in even dimension, we can choose A = Υ.

There is always a charge conjugation matrix C , such that

C± ΓaC−1± = ∓Γta ; Ct = ±C . (2.186)

In the representation (2.163) the two possibilities are the following:

C+ = σ2 ⊗ σ1 ⊗ σ2 ⊗ σ1 ⊗ . . . (2.187)C− = σ1 ⊗ σ2 ⊗ σ1 ⊗ σ2 ⊗ . . . (2.188)

where C− ∝ C+Υ and C = C† = C−1. By means of the matrix C we defineinvariant inner products between spinors as previously seen.

CHAPTER 3. CURVED SPACES AND SPINORS 47

3 Curved Spaces and Spinors

The Euclidean vector space, Rn, is an n-dimensional vector space endowed witha positive-definite symmetric metric whose components are δµν . But, there areother spaces which we intuitively think of as curved on which we would like toperform similar operations to those at Rn: the sphere S2, for example. To treatof spaces that may be curved and may have a complicated topology we introducethe notion of manifolds, which are generalizations of our familiar ideas about curvesand surfaces to arbitrary dimensional objects. In this chapter, we describe the basicelements of differential geometry which is the ideal tool to deal with manifolds. Aswe shall see later, Killing vectors, which characterize the symmetries of a manifold,can be used to construct conserved quantities along geodesics that are curves ofminimum length connecting two points on a manifold. Moreover, we will see thecurvature of a manifold, more precisely of a tangent bundle, is measured by theRiemann tensor which arises from the commutator of the covariant derivatives.The covariant derivative, in formal jargon, is called a connection of the tangentbundle. In particular, we shall uniquely extend it to the spinorial bundle whosesections are the so-called spinor fields.

3.1 Manifolds and Tensors Fields

Manifolds are one of the most fundamental concepts in mathematics and physics.Intuitively, we can think on an n-dimensional manifold M , essentially, as a spacethat that may be curved, but in local regions looks like Euclidean space, Rn. Forexample, a curve in three-dimensional Euclidean space is parametrized locally bya single number τ as (x(τ), y(τ), z(τ)) and a surface can be parametrized by twonumbers u and v as (x(u, v), y(u, v), z(u, v)). A curve is a one dimensional manifoldsince it locally looks like R, while a surface is 2-dimensional manifold since it lookslike R2. In particular, the 2-sphere is a 2-dimensional manifold. The local characterenables us to associate to each point on a manifold a set of n numbers called thelocal coordinate. In general, a manifold may be different from Rn globally, thenit is possible that a single point has two or more coordinates, since generally wehave to introduce several local coordinates in order to cover the whole manifold.

UFPE

Nota como Carimbo


To develop the usual calculus on a manifold we require that the transition fromone coordinate to the other be smooth and the entire manifold be constructed bysmoothly sewing together the local regions. More precisely, a manifold of dimensionn is a topological set such that the neighborhood of each point can be mapped intoa patch of Rn by a coordinate system in a way that the overlapping neighborhoodsare consistently joined [12, 18, 16].

The concept of a vector space is well known to most readers, but when we considercurved geometries the vector space structure is lost. For example, there is no trivialway of add two points on the 2-sphere and end up with a third point on the 2-sphere. Nevertheless, we can recover this structure in the limit of infinitesimaldisplacements. Imagine curves passing through a point p belonging to the surfaceof the 2-sphere. The possible directions that these curves can take generate a plane,this is called the tangent space of p. The intuitive notion of a tangent vector at apoint p ∈ M of a n-dimensional manifold is a vector lying in the tangent plane atpoint p. These vectors generate a vector space of dimension n, denoted by TpM ,called the tangent space of p. Note that, the term vector refer to a vector at a givenpoint p of M . The term vector field, denoted by V , refers to a rule for defininga vector at each point of M . The vector fields generate a new 2n-dimensionalmanifold, denoted by TM , formed by the union of the tangent spaces of all pointsof the manifold M ,

TM ≡⋃p∈M

TpM , (3.1)

this is called the tangent bundle. Now, consider the curves λ defined in the coordi-nates xµ by the equation

xµ(τ) = xµ(p) + V µτ (µ = 1, 2, . . . , n) , (3.2)

for τ in some small interval −ε < τ < ε. Since a curve has a unique parameter, toevery curve there is a unique set V µ with

V µ =dxµ

dτ, (3.3)

evaluated at τ = 0 which are the components of a vector field V tangent to thecurve xµ(τ) on a coordinate system xµ in the neighborhood of p ∈M . The curvesxµ(τ) are called the integral curves of the vector field V . In this coordinate system,a vector field V has the following abstract notation:

V = V µ ∂

∂xµ≡ V µ ∂µ . (3.4)

Sometimes, in a particular problem it might be more convenient to use other coor-dinate system: spherical coordinate, for example. This is simple to implement. For


example, under coordinate transformation xµ 7→ x′µ(xν) we find that the compo-nents of the vector field transform as:

V = V µ ∂µ = V µ

(∂x′ν

∂xµ∂′ν

)⇒ V ′µ =

∂x′µ

∂xνV ν , (3.5)

where V ′µ are the components of V on the coordinate system x′µ. Note that iff belongs to F(M), the space of the functions over the manifold M , then V (f) =V µ ∂µf is the derivative of the function f along of the curve whose tangent vectorfield is V and satisfies the Leibnitz rule when operating on products of functions.Therefore, the vector fields on a manifold define differential operators that act onthe space of the functions over the manifold. In addition, given a local coordinatesystem xµ, then the coordinates define the coordinate frame ∂µ for the tangentspace at each point. For example, if the xµ = θ, φ is the usual coordinatesystem on the 2-sphere, then it defines the coordinate frame ∂θ, ∂φ, with θ beingthe polar angle and φ the azimuthal angle.

Since TpM is a vector space, there exists a dual vector space to TpM , denoted byT ∗pM , called the co-tangent space at p, whose elements are the linear functionalsfrom TpM to R, see the section 2.1. A linear functional is called a co-vector, butin the context of differential forms we always refer to it as 1-form at p ∈ M . Thespace of the 1-form fields is the space TM∗, defined in analogy to construction ofTM , by

TM∗ ≡⋃p∈M

T ∗pM , (3.6)

called the co-tangent bundle. The simplest example of a 1-form field is the differen-tial df of a function f ∈ F(M) which, in terms of the coordinate xµ, is expressedas df = (∂f/∂xµ)dxµ. Note that by the equation (2.3), taking eµ = dxµ andeµ = ∂µ, it follows that:

eµ(eν) = dxµ(∂ν) = δµν .

Thus, it is fruitful to regard that the differentials dxµ provide a local coordinateframe for the co-tangent space T ∗pM , which is the dual to the local coordinate basisprovided by the differential operators ∂µ for the tangent space TpM . An arbitrary1-form field ω is written as:

ω = ωµ dxµ , (3.7)

where the ωµ = ω(∂µ) are the components of ω. Now, if we pass from the coordi-nate system xµ to the coordinate system x′µ, we easily find that:

ω = ωµ dxµ = ωµ

(∂xµ

∂x′νdxν)⇒ ω′µ =

∂xν

∂x′µων . (3.8)


The bases ∂µ and dxµ are sometimes referred to as the canonical bases for thetangent and the cotangent spaces. Notice, in particular, that for every f ∈ F(M)we define the element df ∈ T ∗M by the relation df(V ) = V (f). Indeed, using(3.7), it is immediate see that the action of df on V ∈ TM is given by

df(V ) = V µ ∂f

∂xµ= V (f) . (3.9)

The equations (3.5) and (3.8) express two types of behavior under the arbitrarycoordinate transformation xµ 7→ x′µ(xν). In general, if T µ1...µpν1...νq are the compo-nents of a tensor field T on the coordinate system xµ, then we shall represent iton the canonical basis as follows:

T = T µ1...µpν1...νq ∂µ1 ⊗ . . .⊗ ∂µp ⊗ dxν1 ⊗ . . .⊗ dxνq . (3.10)

Its components on the coordinates x′µ are thus related to its components on thecoordinates system xµ by the following tensor transformation law

T ′µ1...µpν1...νq =

(∂x′µ1

∂xσ1. . .

∂x′µp

∂xσp

) (∂xρ1

∂xν1. . .

∂xρq

∂xνq

)T σ1...σpρ1...ρq , (3.11)

where T µ1...µpν1...νq = T (dxµ1 , . . . , dxµp , ∂ν1 , . . . , ∂νq) . The differential forms are arelevant class of tensors which has all indices down and totally skew-symmetric.For instance, ωµ1...µp = ω[µ1...µp] is called a p-form and space of all p-form fields overa manifold M is denoted ∧pM . Note that we can use several results obtained inthe sections 2.1 and 2.2 by letting eµ = dxµ. In particular, a general p-form fieldcan be written as:

F =1

p!Fµ1...µp dx

µ1 ∧ dxµ2 ∧ . . . ∧ dxµp ,

where ∧ is the exterior product. In this language, functions on the manifold arecalled of 0-forms.

In the section 2.2, we introduced the Clifford algebra as an algebra defined on vectorspaces endowed with a symmetric non-degenerate inner product. A metric g on amanifold M is a tensor field that provides an inner product on the tangent spaceat each point p ∈ M , i.e., it is a map which takes two vector fields and yield afunction over the manifold. For example, given two vector fields V ,U ∈ TpM , wehave that

g(V ,U) =< V ,U >= gµν Vµ Uν , (3.12)

where gµν = g(∂µ, ∂ν) are the components of g in coordinate frame. We can expandthe metric g in terms of its components in the coordinate frame as

g = gµν dxµ ⊗ dxν . (3.13)


Sometimes, the notation ds2 is used in the place of g to represent the metric tensor

ds2 = gµν dxµ dxν . (3.14)

In such case it is called a line element. Intuitivelty, the line element ds2 tells us theinfinitesimal distance between two point on manifold. In physics, we almost alwaysassume that the manifold is endowed with a metric, hence the pair (M, g) willsometimes be called the manifold. In particular, the model of a spacetime in generalrelativity is a differential manifold with a metric g on it [18, 16, 19, 20, 22]. A goodillustration is the Minkowski manifold, (R4,η), the manifold of special theory ofrelativity. According to this theory, we live on the vector space R4 endowed with themetric ds2 = ηµν dx

µ dxν = dt2−dx2−dy2−dz2, where ηµν = diag(1,−1,−1,−1)and xµ = t, x, y, z are the cartesian coordinates. These coordinates define acoordinate frame ∂µ = ∂t, ∂x, ∂y, ∂z in which the components of the metric are±1. Actually, it is always possible at some point p ∈ M , by a convenient choice ofthe coordinates, to put any metric g in a canonical form. In this form the metriccomponents become

gµν = diag(1, 1, . . . , 1,−1,−1, . . . ,−1, 0, 0, . . . , 0) , (3.15)

where the term diag means that gµν is diagonal matrix with the given elements. Ifany of the slots are zero, the metric is degenerate, and its inverse will not exist,otherwise it is non-degenerate. In this last case we can define an inverse metric viagµσ gσν = δµν which allows us to raise and lower indices. In its canonical form, wecall signature s of the metric the modulus of its trace, s = |

∑µ gµµ|. Denoting by n

the dimension of the manifold then when s = n the metric is said to be Euclidean,for s = n− 2 the signature is Lorentzian and s = 0 the metric is said to have splitsignature. In what follows, when we refer about a manifold (M, g) it will always beassumed that the metric g is non-degenerate.

Since the tensor fields are geometric objects which are well defined independently ofthe frame chosen, any set of equations built from tensors will keep its physical mean-ing under arbitrary coordinate transformations. But there is one important objectwhose transformation under coordinate transformation is not trivial, the derivativeof tensors. The partial derivative of a scalar function, ∂µf , is a tensor. However,consider the action of the partial derivative ∂ν in the components of a vector fieldV µ on the coordinate system xµ, ∂ν V µ. Under a change of coordinates, using theequations (3.5) and (3.8), this becomes

∂′ν V′µ =

∂xσ

∂x′ν∂x′µ

∂xρ∂σV

ρ +∂xσ

∂x′ν∂2x′µ

∂xσ∂xρV ρ , (3.16)

obviously, this combination does not transform as a tensor under a arbitrary coor-dinate transformation, since it is not in accordance with the equation (3.11). Thisleads to a natural question: how do we form the derivative of a vector V µ in such


a way that the resulting object is a tensor? To overcome this problem, we define aconnection called Levi-Civita connection or Christoffel symbol

Γµνρ ≡1

2gµσ (∂νgρσ + ∂ρgνσ − ∂σgνρ) , (3.17)

with gµν being the inverse of gµν . This symbol is used to correct the non-tensorialcharacter of the partial derivative. Indeed, after some algebra, it can be provedthat the combination

∇νVµ ≡ ∂ν V

µ + Γµνρ Vρ , (3.18)

does transform as tensor, i.e.,

∇′νV ′µ =∂xσ

∂x′ν∂x′µ

∂xρ∇σV

ρ . (3.19)

To ensure that the covariant derivative transforms as a tensor it must transform as

Γ′µνρ =∂x′µ

∂xσ∂xγ

∂x′ν∂xδ

∂x′ρΓσγδ −

∂2x′µ

∂xγ∂xδ∂xγ

∂x′ν∂xδ

∂x′ρ. (3.20)

Note that, this is not, of course, a tensor transformation law. To prove that thisoccurs, it is enough to use the law of transformation of the metric and the partialderivative. Thus, the Christoffell symbol is not a tensorial object. They are deliber-ately constructed to be non-tensorial, but in such a way that the combination (3.18)transforms as a tensor. The operator ∇ν in (3.18) is called the covariant derivative.This operator shares many properties with the usual partial derivative, it is linearand obeys the Leibniz rule. Particularly, the latter rule can be used to obtain theexpression for covariant derivative of a 1-form ω. In order to perform this, it isenough notice that the action of ω on V provides a scalar, namely, ω(V ) = ωµV

µ.Then, acting the operator ∇µ in this scalar, we are left with the following equation

∇νωµ ≡ ∂ν ωµ − Γρνµ ωρ . (3.21)

It is worth note that the same Christoffel symbol was used as for the covariantderivative of a vector, but now with a minus sign. In general, for an arbitrarytensor field T whose components on the coordinate system are T µ1...µpν1...νq , for eachupper indice we introduce a term with a +Γ, and for each lower indice a term witha −Γ:

∇σTµ1...µp

ν1...νq= ∂σT

µ1...µpν1...νq

+ Γµ1σρ Tρµ2...ip

ν1...νq+ Γµ2σρ T

µ1ρ ...µpν1...νq

+ . . .

− Γρσν1 Tµ1...µp

ρ ν2...νq− Γρσν2 T

µ1...µpν1 ρ...νq

− . . . . (3.22)

One might expect that no reference frame is better than another, all of them areequally arbitrary. In particular, if a tensor vanishes in a coordinate system, then


it must vanishes identically in any coordinate system. A tensor field T is said tobe covariantly constant if its covariant derivative vanishes. In its turn, using thelatter formula it is straightforward to prove that ∇σgµν = 0. Besides, the conceptof moving a tensor along a path, keeping constant all the while, is known as paralleltransport. A tensor field T is thus said to be parallel transported along of a curveλ whose tangent vector field is V if the equation

V σ∇σTµ1...µp

ν1...νq= 0 , (3.23)

is satisfied along the curve. The notion of parallel transport is obviously dependenton the connection, and different connections lead to different answers. If the con-nection is metric-compatible, the metric is always parallel transported with respectto it.

With parallel transport defined, the next logical step is to discuss geodesics. InEuclidean space, the path of shortest distance is the straight line. When we workwith curved manilfods, the generalization of the notion of a straight line in Eu-clidean space is provides by the so-called geodesics. For example, suppose that thecomponents of the tangent vector to curve xµ(τ) on the coordinate system xµ isT µ = dxµ/dτ at τ = 0. The condition that it be parallel transported is thus

T µ∇µTµ = 0 , T µ = dxµ/dτ . (3.24)

Using (3.18), its straightforward to derive the following differential equation:

d2xρ

dτ 2+ Γρµν

dxµ

dτ

dxν

dτ= 0 , (3.25)

known as the geodesic equation. It is the curve of minimum length connecting twopoints p1 and p2 on a manifold (M, g). In other words, it is the curve whose tangentvector T µ is parallel transported along itself. If, however, we had found a curve onwhich T µ∇µT

µ is proportional to the tangent vector T µ

T µ∇µTν = f(τ)T ν , T µ = dxµ/dτ

with f = f(τ) being some function on curve, it is easy to show that we can alwaysreparametrize this curve so that it satisfies the equation (3.24). Thus, without lossof generality we can consider just such a curve. The parameters τ ′ such that f = 0are said to be an affine parameters. In this case, we would be free to rescale theparameter τ → τ ′ = aτ + b with a 6= 0 and b being constant.

To define the covariant derivative we need to put an additional structure on ourmanifold: a connection, which is specified in some coordinate system by a setof n3 Christoffel symbols. However, there exists a relevant differential operationon the space of the forms which does not depends on such a connection, the so-called exterior differentiation, d. Exterior differentiation is effected by an operator


d applied to forms. This operator maps p-forms into (p+ 1)-forms as:

dω =1

p!∂νωµ1...µp dx

ν ∧ dxµ1 ∧ . . . ∧ dxµp . (3.26)

Note that the differential df of a function f ∈ F(M) is exactly the exterior derivativeof a 0-form. The reason why the exterior derivative is so relevant is that it isa tensor, even in curved manifolds, although we have used the partial derivative.Note also that we could have used the covariant derivative, since Christoffel symbolsare symmetric in its indices down Γµνρ = Γµρν . The exterior derivative has someimportant properties. It obeys a modified version of the Leibniz rule when appliedto the product of a p-form ω and a q-form φ:

d(ω ∧ φ) = dω ∧ φ + ω ∧ dφ , (3.27)

where ω = (−1)pω is the degree involution of the p-form ω. Another remarkableproperty of the exterior derivative, which is a consequence of the fact that partialderivatives commute, is that, for any form ω,

d(dω) = 0 . (3.28)

A p-form are said to be closed if dω = 0, and exact if ω = dφ. Clearly, all exactforms are closed, but not all closed forms are exact.

3.2 The Curvature Tensor

Since coordinates are physically meaningless we should always work with tensorialobjects, because they have well-defined geometrical meaning. The Christoffel sym-bol is not a tensor, then it cannot be used as a measure of how much our manifold iscurved. For example, in usual vectorial calculus, the laplacian operator ∇2 in carte-sian coordinates looks different than in spherical coordinates. This happens becausethe laplacian operator is written as ∇2 = ∇µ∇µ, it means that we are implicitlyusing the covariant derivative, but when the cartesian coordinate is employed theChristoffel symbols vanishes and the covariant derivative reduces to partial deriva-tive. Notice, in particular, that the connection defined by the equations (3.17)and (3.22) is named the Levi-Civita connection. The curvature is quantified bythe Riemann tensor, which is derived from the connection. Let us investigate thecurvature tensor formed from an arbitrary connection, not necessarily Levi-Civita,that is, a connection transforming as (3.20) under a change of coordinates. Thoughthe covariant derivative and the partial derivative has some common properties,there exist a big difference between them. The partial derivatives always commute,while the covariant derivative, in general, do not and non-commutativity of thecovariant derivatives implies that the manifold is curved. Then, nothing is fairer


than trying to characterize the curvature of a manifold by the commutator of thecovariant derivatives. Consider the vector field V , after some algebra, one can beproved that:

[∇µ,∇ν ]Vρ = ∇µ∇νV

ρ −∇ν∇µVρ

= RρσµνV

σ − T λµν∇λVρ , (3.29)

where

Rρσµν ≡ ∂µΓρσν − ∂νΓρσµ + ΓρκµΓκσν − ΓρκνΓ

κσµ . (3.30)

T λµν ≡ 2Γλ[µν] . (3.31)

The objects T λµν are the components of a tensor called the torsion tensor. Since theleft hand side of the equation is a tensor, the object Rρ

σµν , although its definitionwas made in terms of a non-tensorial connection symbol, is also a tensor. This isknown as the Riemann tensor. By means of the equation (3.29), we see that theRiemann tensor measures that part of the commutator of covariant derivatives thatis proportional to the vector field, while the torsion tensor measures the part thatis proportional to the covariant derivative of the vector field. The Riemann tensoris also called the curvature tensor, because it is the measure of the curvature oftangent bundle endowed with a connection. In a curved manifoldM , in every pointp ∈ M it is always possible choose a coordinate system on which gµν = δµν andΓµνρ = 0 at this point, this is called Riemann normal coordinates. Besides, whenthe Riemann tensor vanishes we can always construct a coordinate system in whichthe metric components are constant [19, 20, 23]. In particular, a manifold is saidto be flat if, and only if, the Riemann tensor vanishes.

In n dimensions, a general tensor with s indices has ns independent components.But, if it has symmetries not all these components are independent. For example,it is immediate see that the torsion and Riemann tensors have symmetries, that areT λµν = −T λνµ and Rρ

σµν = −Rρσνµ. To find the additional symmetries is useful to

examine the Riemann tensor with all low indices. Then, defining Rρσµν = gρλRλσνµ,

one can prove that this tensor possesses the following symmetry properties:

Rρσµν = R[ρσ][µν] ; Rρσµν = Rµνρσ , (3.32)Rρ[σµν] = 0 ; ∇[λRρσ]µν = 0 . (3.33)

These relations are the complete set of symmetries of the Riemann tensor. In par-ticular, the equation (3.33) is known as the Bianchi identities. These symmetriesreduce the number of independent components of the Riemann tensor. Indeed, inn dimensions, instead of n4, it has 1

12n2(n2 − 1) independent components. For

instance, in four dimensions, the Riemann tensor has 20 independent components.There are other important tensors which are constructed out of the Riemann ten-sor. Let us decompose this tensor into a trace part and a traceless part. By the


antisymmetry properties, the trace of the Riemman tensor over its first two or lastindices vaninhes. However, the following trace is non-vanishing

Rµν ≡ gρσRσµρν = Rρµρν . (3.34)

This is called the Ricci tensor. In particular, as a consequence of the symmetriesof the Riemann tensor, the Ricci tensor associated with the Levi-Civita connectionis automatically symmetric

Rµν = Rνµ . (3.35)

The curvature scalar, R, is defined as the trace of the Ricci tensor:

R ≡ gµνRµν = Rµµ . (3.36)

The Ricci tensor and the curvature scalar contain all of the information about tracesof the Riemann tensor. The traceles part is called the Weyl tensor. It is given indimension n ≥ 3 by

Cρσµν ≡ Rρσµν −2

n− 2

(gρ[µRν]σ − gσ[µRν]ρ

)+

2

(n− 1)(n− 2)Rgρ[µgν]σ . (3.37)

This tensor is basically the Riemann tensor with all of its contractions removedand vanishes for a flat manifold. Particularly, in dimensions 3 the Weyl tensorvanishes identically, while in dimensions n ≥ 4 it is generally non-null. One of themost important properties of the Weyl tensor is that it is invariant under conformaltransformations [21]. For instance, if in the neighborhood of a point p of a manifold(M, g),

gµν = Ω−2δµν Ω ∈ F(M) , (3.38)

then (M, g) is said to be a conformally flat manifold. Then, by the conformalinvariance of the Weyl tensor, it vanishes on (M, g). This tensor is one among themost relevant tensors in general relativity. It can be associated to it an algebraicclassification and relate such classification with integrability properties of the fieldequations. For instance, the Petrov classification is an algebraic classification forthe Weyl tensor of a 4-dimensional curved manifold which help us on the search ofexact solutions for Einstein’s equation, the most relevant example being the Kerrmetric [23, 12, 25, 27].

3.3 Non-Coordinate Frame and Cartan’s Structure Equations

By what was seen previously, the tangent space TpM and its dual T ∗pM are spannedby the coordinate frames ∂µ and dxµ, respectively. However, when the manifoldM is endowed with a metric g it is convenient to use a non-coordinate frame. Anelement of this frame is a vector field given by the following linear combination,

ea = e µa ∂µ , (3.39)


where the index a = 1, 2, . . . , n is only a label for the n vector fields, it is not atensorial index. Moreover, the coefficients of the linear combination are such thatdet e µ

a 6= 0. Under the metric g, we desire that ea be orthonormal, that is

g(ea, eb) = e µa e ν

b gµν = δab , (3.40)

whose inverse relation is given by

gµν = eaµ ebν δab . (3.41)

where eaµ is the inverse of e µa where eaµ is the inverse of e µ

a when we see it as an×n matrix. For example, if V a are the components of the a vector field V with torespect to non-coordinate frame ea, it follows that V a = ea(V ) = eaµV

µ, whereV µ are the components on the frame ∂µ. Associated to non-coordinate vectorfield frame ea is the so-called dual frame of 1-forms ea, defined to be such thatits action on the frame ea is: ea(eb) = δab, the elements of such frame are 1-formsfields given by

ea = eaµ dxµ . (3.42)

The components of a 1-form field ω with to respect this frame are ωa = ω(ea) =e µa ωµ. The convenient choose of the non-coordinate frame ea enables us workwith the metric on a useful way. Using (3.41), it immediate see that

g = gµν dxµ ⊗ dxν = δab e

a ⊗ eb . (3.43)

The frames ea and its dual ea constitute what are called non-coordinate basis.It is worth note that, on a coordinate frame we have d(dxa) = 0, while that ona non-coordinate frame we have d(dea) 6= 0. Once fixed the frame ea, we candefine a set of n2 spin-connection 1-forms ωab by

V µ∇µea = −ωab(V )eb ∀ V ∈ TM , (3.44)

where ωcb = ω cab e

a. Using (3.39)and (3.30), after some algebra, we can prove thatspin connection 1-forms are related with the torsion and curvature by the followingidentities [20, 21, 22]

T a = dea + ωab ∧ eb , Rab = dωab + ωac ∧ ωcb . (3.45)

The object T a = 12T abc e

b ∧ ec is called the torsion 2-form of the connection ∇and Ra

b = 12Ra

bcd ec ∧ ed the curvature 2-form of the connection ∇, where T abc

and Rabcd are the components of the torsion and Riemann tensors with to respect

to the non-coordinate frame ea. We should keep in mind that, for example,Ra

b represents the entire Riemann tensor, with Greek indices suppressed. Theseequations are known as the Cartan’s structure equations. The coefficients of thespin connection 1-forms with all indices down, ωabc = ω d

ab gdc, are called Ricci


rotation coefficients. If ea is orthonormal then the components of the metric gabin this frame are constants. We can choose a connection according to which themetric is covariantly constant tensor, ∇σgµν = 0. In particular, in considering this,we can see that coefficients of the spin connection 1-form with all low indices areantisymmetric in their two last indices, ωabc = −ωacb. In terms of spin connection1-forms ωab = gacω

cb this is expressed as

ωab = −ωba , (3.46)

which is the antisymmetry condition. Moreover, and foremost, if the connectionsatisfies ∇σgµν = 0 and the torsion-free condition, which is expressed as

dea + ωab ∧ eb = 0 , (3.47)

the connection is uniquely determined. Note that this condition implies immedi-ately the symmetry of the Christoffel symbols. Actually, one can prove that theLevi-Civita connection is the unique among the connections which satisfies the twoproperties simultaneously. In general relativity, we always assume that the torsionvanishes. Cartan’s structure equations provides a fruitful and quicker way to com-pute the Riemann tensor of a manifold, see [20, 28, 21] for some applications orwait for the next section in which we make an interesting application.

In physics, the relevance of these equations stems from description of the funda-mental interactions of nature as a gauge theory, more precisely Yang-Mills theory.This theory is a generalization, with non-abelian symmetry group, of the eletromag-netism which is, in turn, an abelian gauge theory under the group U(1). Moreover,the gauge theory is a type of field theory in which the Lagrangian is invariant undera continuous group of local transformations. This means that the physics shouldnot depend on how we describe it, which in accordance with the principles of generalrelativity [4, 21].

3.4 Symmetries, Killing Vectors and Hidden Symmetries

A manifold (M, g) is said to possess a symmetry if its geometry is invariant underany transformation that mapsM to itself. A map which realizes this transformation,denoted by φ, is called a diffeomorphism. Let φτ be a family of such maps on M ,generated by a vector field V whose integral curves are dxµ(τ)/dτ = V µ,

φτ : M → Mxµ 7→ φτ (x

µ) ≡ xµ(τ) .(3.48)

The set of such maps on M forms an abelian group of M if for each τ ∈ R we havea diffeomorphism φτ satisfying

φτ φs = φτ+s , φ0 is the identity . (3.49)


A diffeomorphism on the 2-sphere, for example, is given by φτ (θ, φ) = (θ, φ + τ).This map provides a natural way of comparing tensors at different points on amanifold. For example, any tensor at initial point τ0 is dragged along the curvexµ(τ) by this map. The set theses curves fill1 the manifold M . If the metric tensoris the same from one point to another on a manifold (M, g) irrespective of the valueof the coordinate xµ, it means that the geometry of (M, g) does not change. Whenthis happens we say that the diffeomorphism φτ generated by a vector field V isa symmetry of this tensor field. Isometries are diffeomorphisms that preserve themetric. But, given an isometry, who are its generators? Let us find it now. Let εbe an infinitesimal parameter and K a vector field. Suppose the map φε drags thepoint xµ up to the point φε(xµ) = x′µ in the direction ofK by a parameter ε alongof the integral curves of K. Then,

x′µ = xµ + εKµ(x) + O(ε2) ⇒ ∂x′µ

∂xρ= δµρ + ε ∂ρK

µ + O(ε2) . (3.50)

Now, expanding the metric gµν(x′) in Taylor series, we have

gµν(x′) = gµν(x) + εKσ∂σgµν + O(ε2) , (3.51)

and using (3.50) and (3.11), it is easy to prove that the components of the metricsg′µν and gµν are related by the following equation:

gµν(x′) = gµν(x) + ε£K gµν + O(ε2) , (3.52)

where£K gµν ≡ Kσ∂σgµν + (∂µK

σ)gσν + (∂νKσ)gµσ . (3.53)

Suppose now that the manifold (M, g) is symmetric in the direction of the vectorfield K = ∂σ which in component notation is given by Kµ = (∂σ)µ = δµσ . Thismeans that the components of a metric gµν are invariant by the transformationxµ → x′µ = xµ + εKµ(x). Then, from the equation (3.51) we have that:

gµν(x′) = gµν(x)⇒ £K gµν = 0 . (3.54)

The operator £K is the so-called Lie derivative along ofK. It is worth note that,in this case, the Lie derivative is simply the partial derivative, ∂σgµν = 0. AKillingvector field is defined to be a vector field K along of which the Lie derivative ofthe metric vanishes. We say that the vector K generates the isometry, i.e., thetransformation under which the geometry is invariant is expressed infinitesimallyas a motion in the direction of K. Even though the independence of the metriccomponents on one or more coordinates implies the existence of isometries, the

1Formally, this is the idea of a congruence. A congruence is set of curves that fill the manifold M , orsome part of it, without intersecting and it is such that each point in the region of M is on one and onlyone curve [16, 21].


converse does not necessarily hold. Sometimes the existence of isometries is nottrivial. For instance, the Minkowski manifold (R4,η), has 10 independent Killingvector fields, although only 4 symmetries are obvious from the usual expression ofthis metric in Cartesian coordinates. In the next section this will be best illustratedwith an example. In this sense, the Lie derivative characterizes the symmetries ofa manifold without explicitly use coordinates. Moreover, we can rewrite it in aconvenient form. First, note that the operator £K has the remarkable propertythat when acting on a tensor it yields another tensor. Its action on a general tensoris given by [20, 16]

£K T µ1µ2...µpν1ν2...νq = Kσ∂σTµ1µ2...µp

ν1ν2...νq

− (∂σKµ1)T σµ2...µpν1ν2...νq

− (∂σKµ2)T µ1σ...µpν1ν2...νq − . . .

+ (∂ν1Kσ)T µ1µ2...µpσν2...νq

+ (∂ν2Kσ)T µ1µ2...µpν1σ...νq + . . . . (3.55)

Note also that although we have used the partial derivative, we could have usedthe covariant derivative and the result would be the same, since all of the termsthat would involve Christoffel symbol would cancel because of its the symmetry,Γρµν = Γρνµ. Therefore,

£K gµν = Kσ∇σgµν + ∇µKν + ∇νKµ , (3.56)

and using that the metric is covariantly constant, we obtain that:

£K gµν = 0⇒ ∇µKν + ∇νKµ = ∇(µKν) = 0 . (3.57)

The above equation is the so-called Killing equation and the vector fields K thatsatisfy it are known as Killing vector fields.

The Noether theorem is one important theorem in physics which states that con-tinuous symmetries are associated to conserved charges. So, since Killing vectorsare generators of symmetries of the space (M, g), we can use it to construct scalarsthat are conserved along geodesic curves. For example, if K is Killing vector fieldand x(τ), with τ an affine parameter, a geodesic curve whose tangent vector fieldis p, then following scalar is conserved along such geodesic.

c = pµKµ = pµKµ , pµ =

dxµ(τ)

dτ. (3.58)

Indeed, any vector Kµ that satisfies ∇(µKν) = 0 implies that

pµ∇µ(Kνpν) = pµpν∇(µKν) = 0 .

This can be understood physically. In the classical dynamics of particles, supposethat the potential is independent of the Cartesian coordinate x, a free particle will


not feel any forces in this direction, then it known that the linear momentum isconserved along the direction x. Since a Killing vector field generates an isometry,this shows that symmetry transformations of the metric give rise to conservationlaws.

It is immediate to conclude that a linear combination with constant coefficientsof two Killing vectors is again a Killing vector, that is, the Killing vectors forma linear space. But, given two Killing vector fields K and H , is the final resultof the operation [K,H ] = KH −HK a Killing vector? The answer is yes. Inparticular, one can prove that this commutator is also a vector field given by

[K,H ]µ = Kν∂νHµ − Hν∂νK

µ . (3.59)

One can easily prove that acting on functions and fields, the Lie derivative along[K,H ] satisfy the equation £[K,H] = [£K ,£H ] = £K£H − £H£K . Applyingthis relation to the metric one obtains £[K,H] gµν = 0. Thus, the commutator[K,H ] is again a Killing vector. This commutator is sometimes called the Liebracket. Since the commutator of Lie derivatives is the Lie derivative with respectto a Lie bracket of vector fields, in the neighbourhood of any point inM the Killingvector fields form a Lie algebra under the Lie bracket. Let Ki (i = 1, 2, . . . , r) bea basis in the linear space of the Killing vectors, then the algebra Lie is completelycharacterized in this basis by its structure constants Ck

ij defined by the followingrelation

[Ki,Kj] = C kij Kk . (3.60)

By the Frobenius theorem, is well-known that a distribution generated by the vectorfields Ki is integrable if, and only if, there exists a set of structure constants C k

ij

such that the equation (3.60) is satisfied. In particular, when all the structureconstants vanish, the lie algebra is said to be Abelian. Thus, the set of Killingvectors fields Ki generates a r-dimensional integrable distribution, that is, theyspan a r-dimensional vector subspace ∆p ⊂ TpM on every point p ∈ M and thereexist a smooth family of submanifolds of (M, g) such that the tangent spaces ofthese submanifolds are ∆p.

3.4.1 Maximally Symmetric Manifolds

A manifold M is said to be homogeneous and isotropic if it is invariant under anytranslation along a coordinate and if it is invariant under any rotation of a coor-dinate into another coordinate, respectively. In general, a manifold will admit nosymmetry, and hence possesses no Killing vectors. Furthermore, there is a maxi-mum number of linearly independent Killing fields that can exist for any metric onM . The manifolds with highest degree of symmetry are said to be homogeneousand isotropic manifolds. A maximally symmetric manifold is a manifold pos-sessing the maximum number of isometries, which is n(n+ 1)/2 in n dimensions.


In particular, as we will see, these manifolds has constant curvature and manifoldswith constant curvature can be shown to be conformally flat. Indeed, in the caseof Euclidean signature, one can find a coordinate system xµ in which the lineelement is:

ds2 =1

Ω2[(dx1)2 + (dx2)2 + . . .+ (dxn)2] , (3.61)

with a yet unknown arbitrary function Ω = Ω(xµ). Often, it is convenient to use anon-coordinate frame. Identifying the coefficients of the linear combination (3.42)with eaµ = 1

Ωδaµ, it leads us to introduce the following dual frame

ea =1

Ωdxa , (3.62)

from which we find

dea = −∂bΩΩ2

dxb ∧ dxa = (∂bΩ) ea ∧ eb . (3.63)

In this frame, the line element can be written as

ds2 = (e1)2 + (e2)2 + . . .+ (en)2 = δab ea eb . (3.64)

Since δab = g(ea, eb), we do not need to distinguish between raised and loweredindices. Ordinarily, Cartan’s structure equations surpasses in efficiency every otherknown method for calculating the curvature of a manifold. As an exercise let uscompute the curvature in terms of Ω for this space using the Cartan equations andthe fact that maximally symmetric solutions are regarded as solutions of Rab =Λ gab with Λ = R/n. Manifolds whose Ricci tensor satisfies the latter equation arecalled an Einstein manifolds. Let us find the function Ω which provides a solutionfor this latter equation. We will assume that the torsion vanishes. Once chosenthe metric and the frame, we should calculate spin connection 1-forms ωab. Using(3.63), the Cartan’s first equation (3.45) implies the spin connection 1-forms

ωab = (∂aΩ)eb − (∂bΩ)ea . (3.65)

According to Cartan’s second structure equation, the curvature 2-forms are

Rab = dωab + ωac ∧ ωcb= Ω (∂a∂cΩ e

c ∧ eb − ∂b∂cΩ ec ∧ ea) − (∂cΩ∂

cΩ) ea ∧ eb , (3.66)

This implies that the Riemann tensor is given by

Rabcd = 2Ω (∂a∂[cΩδd]b + ∂b∂[dΩδc]b) − 2(∂Ω)2 δa[cδd]b . (3.67)

where (∂Ω)2 ≡ ∂cΩ∂cΩ. Contracting the first and third indices we obtain the Riccitensor,

Rab = (n− 2)Ω ∂a∂bΩ + δab[Ω ∆Ω − (n− 1) (∂Ω)2] . (3.68)


Here ∆Ω ≡ ∂c∂cΩ. Hence, the trace of the Ricci tensor yields the curvature scalar,

R = 2(n− 1) Ω ∆Ω − n(n− 1)(∂Ω)2 . (3.69)

Now, using that our manifold is an Einstein manifold, Rab = Λ δab, this immediatelyimplies

∂a∂bΩ = 0 (a 6= b) , (3.70)

thus, Ω must be a sum of functions depending just on the coordinates xa

Ω = Ω1(x1) + Ω2(x2) + . . . + Ωn(xn) , (3.71)

because otherwise the mixed derivatives could not vanish. However, for a = b weshould solve the following equation

(n− 2)Ω ∂a∂aΩ = Ω ∆Ω − (n− 1) (∂Ω)2 + Λ . (3.72)

Since the right-hand side does not depend on index a, all the second derivative∂a∂aΩ must be equal and constant, which we choose conveniently as

∂a∂aΩ = 2κ , (3.73)

and Ω must be quadratic in xa with a coefficient of (xa)2 which is independent ofxa. Therefore, we can write

Ω = c + κ r2 , (3.74)

where r2 = [(x1)2 + (x2)2 + . . . + (xn)2] = xaxa. Note that if the linear term isnon-null, it can be made zero by translating the coordinate origin, and a constantfactor on Ω is irrelevant because it simply scales the coordinates, hence we can fixc = 1 without loss of generality. By means of the equation (3.74), we have that(∂Ω)2 = 4κ2r2 and ∆Ω = 2nκ. Then, inserting these latter identities into (3.72),we find that the constant κ must be equal to

Λ = 4κ(n− 1) ⇒ κ =R

4n(n− 1), (3.75)

where we use that Λ = R/n. The constant κ is the so-called the curvatureparameter. Finally, an important property of maximally symmetric manifolds isthat it has constant curvature. Moreover, by substitution of (3.74) on the equation(3.67) we easily find that:

Rabcd = 4κ (δacδdb − δadδcb) . (3.76)

Conversely, if the Riemann tensor satisfies this condition the manifold (M, g) willbe maximally symmetric. In general, Maximally symmetric manifolds of dimensionn can be classified by their signature, their curvature parameter and discrete in-formation related to the global topology. But locally, if we ignore questions about


global topology, maximally symmetric manifolds with Euclidean signature are fullyspecified by their curvature parameter of the following form:

M =

Sn , κ > 0Rn , κ = 0Hn , κ < 0

, (3.77)

where Sn is the n-sphere, Rn the Euclidean space and Hn the hyperbolic space. Notethat, when considering Lorentzian signature, the maximally symmetric manifoldwith κ = 0 is the Minkowski manifold. The metric of maximally symmetric spaceswith other signatures, like de Sitter and Anti-de Sitter spacetimes, can be obtainedfrom (3.61) by means of analytical continuations of the form xa → ixa [29]. Beforeproceeding, let us clarify the previous concepts in this section with an example.

Example 4: The Schwarzschild-AdS black hole in arbitrary dimensions.

In Lorentzian signature, the anti-de Sitter (AdS) manifold is a solution with Λ < 0,of the equation Rab = Λ gab. We want to find the solution in arbitrary dimensionof the latter equation for the Schwarzschild-AdS space whose line element can bewritten as follows:

ds2 = −e2α(r)dt2 + e2β(r)dr2 +r2

(1 + κ r2)2[(dx1)2 + (dx2)2 + . . .+ (dxn−1)2]

= −e2α(r)dt2 + e2β(r)dr2 + r2 δab eaeb , (3.78)

where ea (a = 1, 2, . . . , n − 1) is an orthonormal frame on the (n − 1)-sphere.Note that all metric components are independent of the coordinate t, thereforethis metric possesses the Killing vector K = ∂t. This is the general form of ametric describing a static spherically symmetric spacetime geometry. Here followsthe stepwise algorithm we use to determine the components of the Ricci tensor.

1. A suitable orthonormal frame for this a space is given by

et = eα(r) dt , er = eβ(r) dr , ea = r ea . (3.79)

In this frame, the line element is given by

ds2 = −(et)2 + (er)2 + δab eaeb = ηαβ e

αeβ (α = t, r, a) , (3.80)

where ηαβ = diag(−1, 1, . . . , 1) is the Minkowski manifold in n+ 1 dimensions.

2. Computing the spin connection 1-forms by applying Cartan’s first structureequation, namely

dea + ωab ∧ eb = 0 .


Taking the exterior derivative of et, it is simple to show that:

det = −eβ(r)α′(r) et ∧ er ⇒ ωtr = −eβ(r)α′(r) et ,

where α′(r) stands for the derivative of α(r) with respect to its variable r. Con-tinuing in the same manner, all the non-vanishing spin connection 1-forms are thefollowing:

ωtr = −eβ(r)α′(r) et , ωar = −eβ(r)

rea , ωab = ωa

b. (3.81)

3. Determining the curvature 2-forms by use of Cartan’s second structure equation

Rab = dωab + ωac ∧ ωcb .

Let us do the less trivial case. From (3.81), we have that ωab = ωab, then

Rab = dωab + ωab ∧ ωbc + ωat ∧ ωtb + ωar ∧ ωrb

= dωab + ωab ∧ ωbc +e−2β(r)

r2ea ∧ eb

= Rab −

e−2β(r)

r2ea ∧ eb , (3.82)

where Rab is the curvature 2-forms derived from the connection 1-forms ωab with

respect to the frame of 1-form ea defined on the (n− 1)-sphere.

4. By applying the the relation

Rab =

1

2Ra

bcd ec ∧ ed

and using the skew-symmetry of the exterior product, we find that:

Rabcd =

Rabcd

r2− e−2β(r)

r2(δacδdb − δadδcb) ,

where Rabcd is the Riemann tensor with respect to the frame of 1-forms ea. Now,

using the fact that the sphere is a maximally symmetric manifold and therefore itsRiemann tensor satisfies the equation

Rabcd = 4κ (δacδdb − δadδcb) ,

we get the components

Rabcd =1

r2

(4κ − e−2β(r)

)(δacδdb − δadδcb) .


Here, we must choose the curvature parameter as being κ = 1/4, since this choicemeans the term of the metric δab eaeb is an unit (n− 1)sphere. Thus, the Riemanntensor is given by

Rabcd =1

r2

(1 − e−2β(r)

)(δacδdb − δadδcb)

⇒ Rabab = δacRa

bcb =(n− 2)

r2

(1 − e−2β(r)

), (3.83)

where for the index b Einstein’s convention is not being used. Analogously, it isstraightforward obtain the other components of the Riemann tensor that are non-vanishing, they are,

Rtrtr = e−2β[α′′(r) − β′(r)α′(r) + α′(r)2] , Rr

ara =e−2ββ′(r)

r,

Rtata = −e

−2βα′(r)

r, (3.84)

where for the index a we do not employ Einstein’s convention.

5. Contraction of these components gives the components of the Ricci tensor,Rγδ = ηαβRαγβδ, and using the four symmetries of the Riemann curvature tensor,we arrive at the following equations:

Rtt = Rαtαt = e−2β

[α′′(r) − β′(r)α′(r) + α′(r)2 +

(n− 1)α′(r)

r

](3.85)

Rrr = Rαrαr = − e−2β

[α′′(r) − β′(r)α′(r) + α′(r)2 − (n− 1)β′(r)

r

](3.86)

Raa = Rαaαa =

[e−2β β

′(r)− α′(r)r

+(n− 2)

r2(1 − e−2β(r))

]. (3.87)

6. In the end we use the fact that our spacetime is an Einstein manifold, that is,

Rαβ = Λ ηαβ ⇒

Rtt = −ΛRrr = ΛRaa = Λ

.

Using this condition, the equations (3.85) and (3.86) immediately lead to

α′(r) = −β′(r) ⇒ α + β = λ , (3.88)

where λ is a constant. Note that, by choosing a suitable coordinate time t → eλt,we can achieve λ = 0 and it follows that α = −β. Inserting the equation (3.88)into (3.87) and noting that

[rn−2(1− e−2β(r))]′ = (n− 2)rn−3(1− e−2β(r)) + 2 rn−2 e−2β(r)β′(r) ,


it is straightforward to prove that from Raa = Λ, it follows the relation:

e−2β(r) = 1 − C

rn−2− Λ r2

n= e2α(r) , (3.89)

where C is an arbitrary constant. Therefore, the metric (3.78) becomes

ds2 = −(

1 − C

rn−2− Λ r2

n

)dt2 +

(1 − C

rn−2− Λ r2

n

)−1

dr2

+ r2 δab eaeb . (3.90)

This metric, known as the Schwarzschild-AdS metric for Λ < 0, looks like theSchwarzschild black hole for small r and approaches the AdS space for large r. TheSchwarzschild-AdS black hole is the unique static, spherically symmetric solutionof the equation Rαβ = Ληαβ.

3.4.2 Symmetries of Euclidean Space Rn

An example of a space with the highest possible degree of symmetry is Rn with theflat Euclidean metric. Let us examine its symmetries. Consider the n-dimensionalflat space, for which the metric is Euclidean. In cartesian coordinates xµ, themetric is given by

δµν = diag(1, 1, . . . , 1) . (3.91)

The Taylor series of a function f(x) : Rn → R near the origin of the coordinatesystem is a power series of the form

f(x) = q + qµxµ +

1

2!qµνx

µxν +1

3!qµνσx

µxνxσ + . . . +1

n!qµ1...µnx

µ1 . . . xµn ,

where, for simplicity, q means the function evaluated at xµ = 0 and qν1...νn itspartial derivatives evaluated at the latter point. In the same way, the Taylor seriesexpansion for a vector field K = Kµ∂µ can be represented near point x as:

Kµ = qµ + qµνxν +

1

2!qµν1ν2x

ν1xν2 +1

3!qµν1ν2ν3x

ν1xν2xν3 +. . .+1

n!qµν1... νnx

ν1 . . . xνn ,

(3.92)Since that the indices ν1, ν2, . . . , νn are completely arbitrary, swapping any twocoordiantes, xν1 and xν2 for example, the result must be the same. This, in its turn,implies that qµν1... νn is symmetric in the indices ν1, . . . , νn. The choice of Cartesiancoordinates is very convenient, since all Christoffel symbols vanishes and we may


replace the covariant derivatives by partial derivatives. Then, the Killing equationis simply

∂(µKν) = 0 .

Substituting (3.92) into the above Killing equation we arrive at the following ex-pression

q(µν) + q(µν)ν2xν2 + q(µν)ν2ν3x

ν2xν3 + . . . + q(µν)ν2... νnxν2 . . . xνn = 0 ,

where indices inside round brackets are symmetrized. Since each term in the sum re-late different powers in the coordinates and that the functions xν , xν2xν3 , . . . , xν2 . . .xνn are linearly independent, it follows that for any xµ each terms of this sum mustvanishes,

q(µν) = 0 , q(µν)ν2 = 0 , . . . , q(µν)ν2... νn = 0 . (3.93)

The symmetry of the first two indices in the above equation implies immediatelythat qµν must be antisymmetric, qµν = −qνµ. Moreover, it is immediate see thatqµνν2... νn = −qνµν2... νn . In particular, this latter symmetry and by means of thethe symmetry of qµνν2... νn in the indices ν1, ν2, . . . , νn all the derivative of q, exceptqµν , must vanishes. For example, we know that qµνα = −qνµα. However, by thethe previous symmetries we obtain the following relation qνµα = −qνµα which issatisfied if and only if qµνα = 0. We can continue with this same procedure untilreaching the term q(µν)ν2... νn which, by the same reason, must also vanishes, as waspreviously stated. We can thus conclude that Kµ must be linear in the coordinates,

Kµ = qµ + q[µν] xν . (3.94)

Since that qµ has n independent components and q[µν] has n(n− 1)/2, the equation(3.94) lead us to the conclusion that the Killing vector field has n(n+1)/2 indepen-dent components. We have, thus, n(n+ 1)/2 independent vector fields, each of theform (3.94) for independent choices of the n(n + 1)/2 constants qµ and q[µν]. Thisimplies that the set of all solutions of the Killing equation forms a vector space ofdimension n(n + 1)/2. We can expand the Killing vector field K in terms of itscomponents Kµ on the frame and we arrive at the following equation:

K = Kµ ∂µ = qµ ∂µ + qµν x[µ∂ν] = qµpµ + qµνLµν , (3.95)

where pµ = ∂µ is the linear momentum andLµν = 2x[µpν] is the angular momentum.The basis pµ,Lµν is the so-called Killing basis. Once we identify a basis, anysolution of the Killing equation is therefore a linear combination with constantcoefficients of the elements of this basis. The simplest choice of the n(n + 1)/2vector fields is obtained taking only the constants qµ non-null and letting qµν = 0.In this case, the components of pµ non-null are n constant vector fields of the form

K = ∂ν ⇒ Kµ = (∂ν)µ = δµν . (3.96)


This represents a unit vector in each of the coordinate directions and are calledtranslational isometries. Indeed, from equation (3.50) we see that x′µ = xµ +εδµν . Since they are constant, the integral curves are just the Cartesian coordinateaxes, and the metric is indeed independent of each of these coordinates. Now,setting qµ = 0, we have n(n − 1)/2 rotational isometries. Indeed, choosing oneof the n(n − 1)/2 antisymmetric matrices qµν , q12 = 1/2 for example, and all therest zero, it is straightforward to prove that:

K = q12 x[1p2] = xpy − y px = Lxy , (3.97)

where we identify x1, x2 = x, y. One can obtain easily all the n(n− 1)/2 L[µν]

in analogous ways. In particular, its the algebra is given by the Lie brackets

[Lµν ,Lρσ] = δµρLνσ − δµσLνρ − δνρLµσ + δνσLµρ . (3.98)

The group of all these, translations and rotations, isometries is known as thesymmetry group of n-dimensional Euclidean space. So, the most general sym-metry transformation which includes both translations and rotations has the formxµ → x′µ = Λµ

νxν + aµ, where aµ is a constant and δρσΛρ

µΛσν = δµν . Physically,

the equation (3.98) is the angular momentum operator algebra. These operatorsperform rotations in a physical system which do not commute. For instance, aninfinitesimal rotation around the 1-axis followed by an infinitesimal rotation aroundthe 2-axis is not the same as rotating around 2 and then 1. The Lie brakets areexactly those of SO(n), the group of rotations in n dimensions. This is no coin-cidence, of course, but we won’t pursue this here. All we need to know here isthat a spherically symmetric manifold is one which possesses n(n + 1)/2 Killingvector fields with the Lie brakets (3.98) [16, 20, 18]. It is possible to establishLie brackets of this sort for all of the Euclidean Killing fields. Indeed, one canprove that the following relation holds: The group of all these, translations androtations, isometries is known as the symmetry group of n-dimensional Euclideanspace. So, the most general symmetry transformation which includes both trans-lations and rotations has the form xµ → x′µ = Λµ

νxν + aµ, where aµ is a constant

and δρσΛρµΛσ

ν = δµν . Physically, the equation (3.98) is the angular momentumoperator algebra. These operators perform rotations in a physical system which donot commute. For instance, an infinitesimal rotation around the 1-axis followed byan infinitesimal rotation around the 2-axis is not the same as rotating around 2 andthen 1. The Lie brakets are exactly those of SO(n), the group of rotations in ndimensions. This is no coincidence, of course, but we won’t pursue this here. All weneed to know here is that a spherically symmetric manifold is one which possessesn(n+1)/2 Killing vector fields with the Lie brakets (3.98) [16, 20, 18]. It is possibleto establish Lie brackets of this sort for all of the Euclidean Killing fields. Indeed,one can prove that the following relation holds:[

pµ,pν]

= 0 ,[Lµν ,pρ

]= δµρ pν − δνρ pν . (3.99)


We therefore find exactly n(n + 1)/2 isometries in Euclidean space Rn. Actually,this is the maximum number of independent solutions to the Killing equation in an-dimensional space.

In general relativity, it is well-known that is a manifold admits admits Killingvectors, in general, the corresponding scalars associated with theses vectors can beused to find the geodesics trajectories without integrating the non-linear geodesicequation directly. However, spacetimes with geometries more complicated admitsless Killing vectors than dimensions. In this cases, the work to find the geodesictrajectories is much harder. For example, 4-dimensional Kerr spacetime has justtwo independent Killing vectors and therefore it is not possible to find its geodesicstrajectories using only these symmetries. But, there are other tensors associatedwith symmetries of a manifold. For example, differently from the conserved scalarsassociated to Killing vectors, which are linear on the momentum, the constantfound by Carter in 1968 which enabled him to solve the geodesic equation of theKerr metric is quadratic on the momentum and it is a consequence of the fact thatKerr metric admits a Killing tensor of order two [26]. This is one example of theso-called the hidden symmetries. We say that a manifold has a hidden symmetriesif its geodesic equation possess conserved quantities of higher than first order inmomentum. The geometric structure responsible for this kind of symmetry is theKilling tensor [19, 29]. Killing tensors are symmetric generalizations of the conceptof Killing vectors. There are also anti-symmetric generalizations of the Killingvectors called Killing-Yano tensors (KY). Theses are more fundamental objectsthan the Killing tensors. They can be used to construct Killing tensors, since thesquare of a Killing-Yano is a Killing tensor of order two. For scalars conversed alongnull geodesic, the tensor responsible for such conservation law is conformal Killingtensor (CKT). One can also find the so-called conformal Killing-Yano tensor (CKY)which is completely anti-symmetric tensor that also lead to conserved quantitiesalong null geodesics.

3.5 Fiber Bundles

In the previous sections it was seen that a manifold is a space which looks locally likeRn, not necessarily globally. Now, we will study certain kind of manifolds whichlooks locally like a direct product of two manifolds. Such manifolds are calleda fiber bundles. A particularly interesting manifold is formed by combining amanifold M with all its tangent spaces TpM . For motivation, let us construct thefiber bundle TM , previously called tangent bundle. It was formed by collectingtogether all the tangent spaces TpM from all points of M . The manifold M overwhich TM is defined is called the base space. Now, suppose that there exist acovering Ui of M by open sets and diffeomorphism φi such that xµ = φi(p) isthe coordinate on Ui to each point p ∈M at which one calculates the tangent space


TpM . The union of TpM over all points of Ui gives

TUi ≡⋃p∈Ui

TpM . (3.100)

We specify an element q of TUi by a point p ∈M and a vector V ∈ TpM decomposedon frame ∂µ at p. Note that Ui can be continuously mapped to an open subsetφ(Ui) of Rn, and TpM mapped to Rn and vice versa. Then, we can think on TUias the direct product Rn × Rn, i.e., TUi is a 2n-dimensional manifold decomposedinto a direct product Ui × Rn. It means that each point q on TUi has coordinates(xµ, V µ) and contains information of a point p ∈ M and of a vector V ∈ TpM .Then, we can introduce a map π : TUi → Ui called projection such that to anypoint q ∈ TUi the object π(q) is a point p ∈ Ui at which the vector is defined. Now,we can introduce the idea of fibers of a fiber bundle. A fiber above p is the resultof the action of the map π−1 on point p ∈M , π−1(p). In our case, when p is fixed,it straightforward see that the fibers are exactly the spaces π−1(p) = TpM for eachp. Let Uj be a chart as Ui such that Ui ∩ Uj 6= ∅ and x′µ = φ′j(p) its coordinates.If V ∈ TpM is a vector at p ∈ Ui ∩ Uj, it is then simple matter to arrive at thefollowing relation

V = V µ∂µ = V ′µ∂′µ ⇒ V ′µ = Aµν Vν , (3.101)

i.e., their components are related by a non-singular matrix Aµν ≡ ∂x′µ/∂xν ofGL(n,R) which act on TpM by the left with the matrix Aµν . The group GL(n,R) isthen called the structure group of TM . Now, a curve in the fiber bundle identifies aparticular vector at each point p ofM , and so the curve defines a vector field onM .Such a curve which is nowhere parallel to a fiber is called a section σ of TM . Notethat it does not make sense to ask for the length, since here we have an exampleof a manifold on which is not necessary to define a metric. A local section of TUiis a smooth map σi : Ui → TUi defined on a chart Ui. However, the projectionπ : TM → M can be defined globally with no reference to local charts, since theequation π(q) = p does not depend on a special coordinate chosen. Therefore, wedefine a section σ of TM as a map σ : M → TM such that π σ = e, where e isthe identity on M . The set of sections on M is denoted by Γ(TM). Consider nowtwo coordinate neighbourhood (Ui, φi(p)) and (Uj, φj(p)). The transition functiontij ∈ Ui ∩ Uj : TpM → TpM is given by tij(p) = (φ−1

i φj)(p). Since that φ−1i φj is a

diffeomorphism its derivative is invertible, tij ∈ Ui ∩ Ui ∈ GL(n,R). To summarizeall this informations, one says that TM is a fiber bundle above M , with fibre TpMand structure group GL(n,R).

The tangent bundle is an example of a more general framework called a fibre bundle.A fiber bundle E over a smooth manifold M , with fibre F and structure group G,is a set (E,M,F,G, π) such that

(i) E is a smooth manifold called the total space.


(ii) M is a smooth manifold called the base space.

(iii) F is a smooth manifold called the fibre.

(iv) G is a Lie group acting smoothly on F on the left, called the struture group.

(v) π is a smooth surjection2 from E to M , called the projection. The set π−1(p),where p ∈M , is called the fibre over p, and is denoted by Fp.

(vi) There exist a covering Ui of M by open sets, and diffeomorphisms φimapping Ui × F to π−1(Ui) such that π φi(p, f) = p. The map φi is called thelocal trivialization since φ−1

i maps π−1(Ui) onto the direct product Ui × F .(vii) If we write φi(p, f) = φi,p(f), the map φi,p(f) : F → Fp is a diffeomorphism.On Ui ∩ Uj 6= ∅, we require that tij(p) ≡ φ−1

i,p φj,p : F → F be an element of G.Then φi and φj are related by a smooth map tij : Ui ∩ Uj → G as

φj(p, f) = φi(p, tij(p)f) .

The maps tij are called the transition functions.

We need to clarify several points about the transition functions. For convenience,we will often use a shorthand notation π : E → M to denote a fibre bundle. So,let π : E → M be a fibre bundle and Ui a chart of the base space M . The localtrivialization φ−1

i : π−1(Ui)→ Ui×F is a diffeomorphism. Now, let us take a chartUj such that Ui ∩ Uj 6= ∅. Then, we have two maps φi and φj on Ui ∩ Uj. Letq ∈ E such p = π(q) ∈ Ui ∩ Ui. Each local trivialization φ−1

i and φ−1j carries the

point q to different elements of F :

φ−1i (q) = (p, fi) , φ−1

j (q) = (p, fj) . (3.102)

The central role of the transition function tij : Ui ∩ Uj → G consists on establishinga relation between these two elements: fi = tij(p)fj. Since tij(p) = φ−1

i,p φ−1j,p one

can prove that the following consistency conditions are satisfied:

tii(p) = e (p ∈ Ui)tij(p)tji(p) = e (p ∈ Ui ∩ Uj)

tij(p)tjk(p)tki(p) = e (p ∈ Ui ∩ Uj ∩ Uk), (3.103)

where e : E → E is the identity element.

The tangent bundle TM constructed in the beginning of this section is a fibre bundlewhose fibre is a vector space. These are special types of fibre bundles which arecalled vector bundles [21, 16]. Another example of a fibre bundle is the bundle oflinear frames overM which is closely related to TM , the frame bundle LM . Insteadof using the tangent space TpM as a fibre above a point p of M which was done

2A map f from a set X to a set Y is said to be a surjection, if every element y in Y has a correspondingelement x in X such that f(x) = y.


in the construction of TM , we can take as a fibre at p the set of all the frames ofTpM . Therefore, a point of the fibre Fp above p is a particular frame of TpM whichis given by n linearly independent vectors of TpM . It means that each point q ofLM may be assigned by the coordinates (xµ, V µ

a ) which contains information of apoint p ∈M whose coordinates are xµ and of n2 components V µ

a (a = 1, 2, . . . , n) ofthe vectors V a of a frame of TpM in the coordinate frame ∂µ. Since the elementV µa form a frame, it is invertible. Hence, V µ

a is an element GL(n,R), so that thefibre of LM is GL(n,R). Moreover, given any two frames V a and U b thereexists an element of GL(n,R) that relate them and therefore the structure group ofLM is GL(n,R). Such a bundle is called a principal fiber bundle, which is a fiberbundle whose fibre F is the same as the structure group G. We establish, thus,that a vector bundle naturally induces a principal fiber bundle over the manifoldby employing the same transition functions. In particular, when the frame V a isorthonormal, the elements V µ

a becomes the coefficients of the non-coordinate frameand the structure group reduces to O(n). Furthermore, if the frame V a haspositive orientation throughout the whole manifold, the corresponding structuregroup is SO(n).

3.5.1 Spinorial Bundles and Connections

In the chapter 1 we defined spinors algebraically, which arose within the study ofClifford algebra, its Clifford group, and some subgroups of the latter, especiallythe spin group. It now becomes possible to attack the problem of differentiation ofspinor fields. We shall here construct the spinorial bundles which will immediatelylead to notion of a spinor fields, the concept of covariant derivative of a spinor fieldsand at the end we will compute the curvature of the spinorial bundle.

In what follows, the m-dimensional manifold is assumed to have an inner productdefined on the bundle TM . Let π : TM → M be a tangent bundle above M , withstructure group O(m) or SO(m) when M is orientable and let LM be the framebundle associated with TM . Given the transition functions tij ∈ SO(m) of LM , itis meaningful to consider the set of functions tij ∈ Spin(m) defined by

φ(tij) = tij , (3.104)

where φ is the double covering φ : Spin(m) → SO(m). Since the transition func-tions tij satisfy tijtjktki = e and tii = e and that the mapping φ is a representa-tion of the spin group, there are solutions of the equation (3.104) for tij satisfyingtij tjk tki = ±e and tii = ±e among which we select those tij satisfying

tij tjk tki = e and tii = e . (3.105)

A spin structure on M is defined by the transition functions tij ∈ Spin(m)satisfying (3.104) and (3.105). The principal bundle LSM of the spin frames over


M is defined as that principal bundle with M as base, with Spin(m) as structuregroup, and tij satisfying (3.105). A spinor above p ∈ M is a linear combinationof the elements of the spin frame given by sections of LSM at p. Now, we canintroduce of a spinor field in a way completely analogous to definition of a vectorfield of TM . Once defined the notion of a spinor at a point p, we can construct thespinorial bundle SM above p as the bundle which admits as fibre above the pointp of M the set of all spinors at p. A spinor field, denoted by ψ, over M is thena map the associates to every point p of the manifold M a spinor, it is a sectionof SM . The set of all sections of SM we denote by Γ(SM). It is interesting tonote that not all manifolds admit spin structures. For examples of the latter caseof spin structures, see [30, 21]. In our case the manifold M is said to admit a spinstructure.

From now on we are going to work on a manifold (M, g) of dimension m = 2nendowed with a non-degenerate metric g. Furthermore, the tangent bundle TM isassumed to be endowed with a torsion-free and metric-compatible derivative, theLevi-Civita connection, and it will be assumed that (M, g) admits a spin structure,namely it has a well-defined spinorial bundle.

Consider ea a local frame such that the components of the metric in this frameare constants,

g(ea, eb) = gab , ∂c gab = 0 . (3.106)

Now, let us introduce a connection ∇a on the spinorial bundle SM . Let ∇a be thethe Levi-Civita connection of the tangent bundle TM . The action of this connectionin the frame vectors ea is given by

∇aeb = ω cab ec , (3.107)

where ω cab ≡ ωcb(ec). This connection can be uniquely extended to the spinorial

bundle. For such, one might impose for it to satisfy the Leibniz rule with respectto the Clifford action, here denoted by juxtaposition:

∇a(V ψ) = (∇aV )ψ + V (∇aψ) ∀ V ∈ Γ(TM) ,ψ ∈ Γ(SM) , (3.108)

we refer to it as covariant differentiation of Clifford Products. We must also requireit to be compatible with the natural inner products on the spinorial bundle,

∇a < ψ,φ >=< ∇aψ,φ > + < ψ, ∇aφ > ∀ ψ ,φ ∈ Γ(SM) . (3.109)

Let ξA be a local frame of the spinorial bundle SM , where A = 1, 2, . . . , 2n

are spinorial indices. The spinors ξA give a linear and faithful representation forthe Clifford algebra, the so-called spinorial representation. Indeed, we can chooseconveniently the spinorial frame so that the Clifford action of the frame ea onthe spinors ξA is constant in a given patch of M ,

ea ξA ≡ (ea)BAξB , (3.110)


where the matrices (ea)BA are constants. In the physics literature the matrices

(ea)BA are the so-called Dirac matrices. These matrices satisfy the following anti-

commutation relation

(ea)AB (eb)

BC + (eb)

AB (ea)

BC = 2 gabδ

AC , (3.111)

which is the definition of the Clifford algebra itself. In the same way as (3.107), wecan define the action of the connection on this frame of spinors ξA as

∇a ξA ≡ (Ωa)BAξB , (3.112)

where Ωa is the so-called the spinorial connection. We must find a expression forΩa for which the equation (3.108) and (3.110) holds. Then, computing the actionof ∇a on eb ξA by means of the equations (3.107) and (3.112) we find that

ω cab ec = Ωaeb − ebΩa , (3.113)

on which the spinorial indices have been omitted for simplicity. One can show that,if we choose the spinorial frame satisfying the equation (3.110), the general solutionfor this equation is [9]:

Ωa = − 1

4ω bca eb ec . (3.114)

Thus, if ψA are the components of ψ on the frame ξA, then an arbitrary spinorfield has the following abstract representation

ψ = ψAξA , (3.115)

and its covariant derivative is given by:

∇aψ =(∂aψ

A + (Ωa)ABψ

B)ξA =

(∂a −

1

4ω bca eb ec

)ψ . (3.116)

In terms of the components ψA the latter equation is written as

∇aψA = ∂aψ

A + (Ωa)ABψ

B . (3.117)

We can also construct a section in a dual spinorial bundle. Let ξA be a framefor the latter bundle. If χA are the components of a dual spinor field χ on frameξA, then its covariant derivative is given by

∇aχA = ∂aχA − (Ωa)BA χB . (3.118)

Thus, the operator ∇a when acting on objects with several spinorial indices followstrivially from the latter equations. Its action on a such object is given by

∇a ΨA1...Ap

B1...Bq= ∂a Ψ

A1...Ap

B1...Bq

+ (Ωa)A1C Ψ

CA2...Ap

B1...Bq+ (Ωa)

A2C Ψ

A1C...Ap

B1...Bq+ . . .

− ΨA1...Ap

CB2...Bq(Ωa)

CB1

− ΨA1...Ap

B1C...Bq(Ωa)

CB2− . . . . (3.119)


In particular, using the equations (3.112) and (3.113) one can easily prove that theaction of operator ∇a on a vector field V = V beb lead us to the following result:

∇aV = (∂aVb)eb + V b(Ωa eb − eb Ωa) = ∂aV + [Ωa,V ] . (3.120)

where [Ωa,V ] = ΩaV − V Ωa is the commutator of the Clifford algebra. It meansthat, in spinorial language, a vector is an object with two spinorial indices. Moreprecisely, each vectorial index is equivalent to two spinorial indices, one up and onedown. In particular, in general relativity, the spinorial formalism of 4-dimensionalLorentzian manifolds furnishes two types of indices, the ones associated with Weylspinors of positive chirality and ones related to Weyl spinors of negative chirality.In this case, a vectorial index is equivalent to the product of two spinorial indices,one of positive chirality and one of negative chirality. This is the principle of thePenrose’s method [32, 31, 23].Now, since the Riemann tensor measures the curvature of a tangent bundle, we canrelate the curvature of the spinorial connection with the Riemann tensor. First, notethat the vector fields ea are written in terms of the coordinate frames eµ = ∂µ. Inthis frame, the spinorial representation of the coordinate frame eµ will be denotedby the matrices Γµ,

eµ ξA ≡ (Γµ)BAξB . (3.121)

Although, sometimes, it is convenient to omit the spinorial indices, when we com-pute the covariant derivative of Γµ, we must account for the covariant derivativeof both the vectorial index µ as well as the omitted spinorial indices. Taking thisinto account and noting that these contributions cancel each other we eventuallyobtain: ∇µΓν = 0. The curvature of the spinorial bundle is defined by the actionof the commutator [∇µ, ∇ν ] on spinor ψ and its relation with the Riemann tensoris given by [1]:

[∇µ, ∇ν ]ψ =1

4R ρσµν Γρσψ , (3.122)

where Γµν ≡ 12

[Γµ,Γν ]. The spinorial space is defined to be the space where anirreducible and minimal representation of the Clifford algebra acts. Then, if eµ is asection whose spinorial representation is the matrix Γµ, we can define an operatorD as D = Γµ∇µ, called Dirac operator. A spinor field ψ satisfy the followingequation:

Dψ = mψ , (3.123)

where m is some complex constant. What, in fact, determine if the eigenvalue m isreal or imaginary is the nature of the manifold. After some algebra, we can provethat

D2ψ =

(∇µ∇µ −

1

4R

)ψ , (3.124)


where R is the curvature scalar. The square of the Dirac operator is known as thespinorial Laplacian. Finally, if we define the constant α by the following relation:

∇µψ = αΓµψ (3.125)

the corresponding spinor field ψ is a Killing spinor field. The constant α is knownas Killing constant. Note that by means of (3.123), a Killing spinor field is a Diracspinor field withm = αn, but the converse is not true. In practical terms, the utilityof the Killing spinor fields is that they can be used to generate symmetry tensorswhich eventually lead to conservation laws as well as to the integrability of fieldequations using the manifold as a background [9]. Indeed, if Υ is the chiral matrix,ψ is a Killing spinor and ψ its conjugate then each of the tensors [35, 36, 37]

< ψ,Γµ1µ2...µpψ > , < ψ,Γµ1µ2...µp Υψ >

< ψ,Γµ1µ2...µpψ > , < ψ,Γµ1µ2...µp Υψ > , (3.126)

is either a Killing-Yano (KY) tensor or a closed conformal Killing-Yano (CCKY)tensor [37, 38].

CHAPTER 4. MONOGENICS AND SPINORS IN CURVED SPACES 78

4 Monogenics and Spinors in CurvedSpaces

In the previous chapters the Clifford algebra was introduced in the geometry, thatis the reason why the Clifford algebra is also called the geometric algebra. Now, it iswell known that holomorphic functions, also called conformal maps, are the centralobjects of study in complex analysis [39]. Holomorphic functions of complex vari-ables satisfy Cauchy-Riemann’s equations and, hence, they also satisfy Laplace’sequation [2]. In this chapter we introduce some basic concepts in Clifford analysis,the Clifford algebra in analysis, which furnish one approach of the complex analysisin dimensions higher than two. One of the main subjects studied in Clifford analysisis the function theory of monogenic functions and its interaction with the represen-tation theory of the group Spin(m). In Clifford analysis one consider multivectorfunctions that solve Cauchy-Riemann or Dirac equations on some manifolds. Inparticular, the idea of higher dimensional holomorphic functions is given by the so-called monogenic functions which are multivector functions that are annihilatedfrom the left by the Dirac operator. Also, we will show that the Dirac equationminimally coupled to an electromagnetic field is separable in spaces that are thedirect product of bidimensional spaces. In particular, we applied on the backgroundof black holes whose horizons have topology R× S2 × . . .× S2.

4.1 Monogenics in the Literature

For a review of many of the basic properties of Clifford analysis including somehistorical remarks we refer to [42, 43]. Clifford analysis may also be regarded as aspecial function theory within harmonic analysis since the Dirac operator factorizesthe Laplace operator, so it may be seen as a refinement of harmonic analysis. Oneimportant tool in the analysis of the Dirac operator is the Cauchy integral for-mula. This formula allows us to calculate the values of functions in terms of givenboundary data arising from practical measurements. Based on these representa-tions important existence and uniqueness theorems for the solutions of boundaryvalue problems on manifolds could be established. The so-called monogenic func-

UFPE

Nota como Carimbo


tions are those that are in the kernel of the Dirac operator and they can be seen ashigher dimensional holomorphic functions, they also satisfy the Laplace’s equation.In the simplest case of one dimensional space a monogenic function is a functionwith null derivative, that is, a constant with no structure; in two dimensional spacemonogenic functions are equivalent to analytic functions of a complex variable andincreasing the space dimensionality one finds that those functions can generatemany kinds of interesting geometrical structures. The monogenic functions wereintroduced by F. Brackx, R. Delanghe and F. Sommen in [46] and in the past fortyyears have been successfully and intensively studied. The literature is very rich ofresults but studies on the topic are ongoing, see [47, 48, 49]. In [50] the monogenicfunctions were applied on some conformally flat manifolds such as cylinders, toriand some conformally flat manifolds of genus bigger or equal to two. Lounestoobtains in reference [2] the monogenic functions on flat space in arbitrary dimen-sion and emphasizes that the monogenic functions take their value in a Cliffordalgebra which is a natural environment to represent internal degrees of freedom ofelementary particles such as spin.

It is well known that the spherical harmonics play an important role in the har-monic analysis of the Laplace operator, in Clifford analysis a similar role is played byspherical monogenics, also called spin weighted spherical harmonics, which are poly-nomial solutions of the Dirac equation without mass defined on the sphere. Thesefunctions form a complete orthonormal set for each value of spin weight and are arefinement of the notion of spherical harmonics. In reference [51] was made a studyof the generating functions for the standard orthogonal bases of spherical harmon-ics and spherical monogenics in arbitrary dimensions. The spin weighted sphericalharmonics were introduced by Newman and Penrose in [53] as a means to describecertain quantities exhibiting a particular ”spin-gauge” behavior. For instance, forparticular choice of spin-gauge, the spin-weighted spherical harmonics reduce tothe monopole harmonics which arise as solutions of the Schrodinger equation foran electron in the field of a magnetic monopole introduced by Wu and Yangin in[54], see also [23, 55, 56]. Several angular functions, including scalar, vector, andtensor spherical harmonics, are used to perform separation of variables in the gen-eral relativity literature. These functions include the Regge-Wheeler harmonics,the symmetric, trace-free tensors of Sachs and Pirani, the Newman-Penrose spinweighted spherical harmonics. A good review article of these functions by Thorneis the reference [57]. The spin weighted spherical harmonics can be expressed interms of the Wigner matrix since its formulas in spherical coordinates are identicalto formulas for Wigner matrices for certain values of Euler angles, also in termsof the Jacobi Polynomials, the generalized associated Legendre functions and thehypergeometric functions [58]. Applications of these functions in the solution byseparation of variables of various systems of partial differential equations includessource-free Maxwell equations in flat space-time and in the Schwarzschild space-time, the Einstein vacuum field equations linearized about the flat space-time and


the Dirac equation [59]. Finally, to solve the Teukolsky Master Equation, whichdescribes the dynamics of various fields of different spins as perturbations to Kerrmetric, by separation of variables there are two non-trivial equations obtained thatare the angular equation and the radial equation whose solutions for the angularequation are the called spin weighted spheroidal harmonics, see [60, 61, 62, 63, 64]for more details, introduced by Teukolsky in the context of black hole physics,which can be reduced to spin weighted spherical harmonics for a particular case.So, the spin weighted spherical harmonics have innumerable applications in severalbranches of the physics as was mentioned.

4.2 Monogenic Multivector Functions : Definitions and OperatorEqualities

Let e1, e2, . . . , en be an orthonormal frame of Euclidean space Rp,q of dimensionn = p+ q endowed with a non-degenerate symmetric metric of signature s = |p− q|and C`p,q the Clifford algebra over Rp,q. It is possible to extract a certain kindof square root of the n-dimensional Laplacian operator ∆ and consider instead afirst-order differential operator, called the Dirac operator D. The Dirac operatorgets its name from its appearance in Dirac’s wave equation for the electron. Withrespect to this basis, the Dirac operator on Rn,0 is given by

D = e1∂1 + e2∂2 + . . . + en∂n = δabea∂b , a,b = 1, 2, . . . , n , (4.1)

where ∂a represents the partial derivative with respect to the variable xa. If wewant the Dirac operator to be the square root of Laplacian, DD = ∆, the elementsof the frame ea must obey the following algebra

eaeb + ebea = g(ea, eb) , (4.2)

which is just the very definition of Clifford algebra C`n,0. It means that the Diracoperator is seen as a vector in C`n,0. An important variation of the Dirac operatorwhich provides a closer contact with the classical function theory, denoted by DW ,is the Weyl (or Cauchy-Riemann) operator obtained as follows. Let X be a vectorof the Euclidean space Rn,0, X = Xaea. We can take a special vector of the frameea, e1 for example, and then consider its left Clifford multiplication on the vectorX, that is,

Z = e1X = Xae1ea

= X1 + X2e1e2 + . . . + Xne1en , (4.3)

where the elements e1ea (a = 2, 3, . . . , n) are unit bivetors. The combination of ascalar and a bivector, which is formed naturally via the Clifford product, is therefore


a multivector of the C`+n,0, the even subalgebra of C`n,0, which is generated by the set

1, e1e2, e1e3, . . . , e1en. In two dimensions, for example, defining X1 = x,X2 = y,the element Z = x + Iy can be viewed as a complex number, since the bivectorI = e1e2, the pseudoscalar of C`+

2,0, is such that I2 = −1 and can be viewed asthe unit imaginary i. The complex number Z = x + iv represents a vector inthe complex plane, with Cartesian components x and y. Since the bivectors e1eaanticommute and square equal to −1, they form an orthonormal basis for the vectorspace R0,n−1. Thus, the even subalgebra C`+

n,0 is isomorphic to C`0,n−1. Definingia = e1ea, a vector X ∈ Rn,0 can be replaced by a sum of a vector and a scalar.Such an element will be called a paravector Z ∈ R⊕ R0,n−1,

Z = X1 + X2i2 + . . . + Xnin = X1 + X1 , (4.4)

where X1 = X2i2 + . . . + Xnin is a vector on the vector space R0,n−1. Therefore,there is a natural map betweenX and Z achieved by means of multiplying the firstby e1 the left. The role of the preferred vector e1 is clear, it is the ”real axis” ofthe higher dimensional analogue of complex analysis. From the geometrical pointof view, we could say that when using the paravector formalism, we have chosen a”real axis” namely the e1 axis by multiplying the vector X ∈ C`n,0 on the left bye1. This established a correspondence between the following two mappings [2]:

f : Rn,0 → C`+n,0

X 7→ f(X),

f : R⊕ R0,n−1 → C`0,n−1

Z 7→ f(Z). (4.5)

For convenience, we denote both by f . In this way, the Dirac operator is replacedby the Weyl operator DW on R⊕ R0,n−1 defined by

DW = ∂1 + i2∂2 + i3∂3 + . . . + in∂n = ∂1 + D1 , (4.6)

where here D1 is the Dirac operator on R0,n−1. A function f : U ⊂ Rp,q →C`p,q is seen as a function f(X) of X ∈ Rp,q. There are in the literature severalways to define a notion of generalized holomorphic function with values in theClifford algebra. The most successful is the so-called monogenicity which has beenintensively studied during the past forty years. Formally, let V be a vector spaceover a real or complex field F. A smooth multivector function f(X) defined onan open set U ⊂ V and having values in the Clifford algebra C`(V) is said to bemonogenic if and only if it is in the kernel of the Dirac operator on V

Df(X) = 0 , (4.7)

for each X ∈ U . The theory of the functions in the kernel of the Dirac operator orof the Weyl operator are equivalent and the word monogenic is used for functionsin the kernel of either of them. One of the simplest example is the case n = 2 onwhich the Clifford algebra C`0,1 is isomorphic to the complex field C. In this case,


the Weyl operator is given explicitly byDW = ∂∂x

+ i ∂∂y

and a multivector functionf(Z) = u(x, y) + i v(x, y) ∈ C`0,1 of Z = x + iy, which is equivalent to analyticfunctions, is monogenic if and only if the following equations of holomorphy hold:

∂u

∂x=

∂v

∂yand

∂v

∂x= −∂u

∂y. (4.8)

These equations are the well-known Cauchy-Riemann equations. The statementthat f(Z) is an analytic function, i.e., a function that satisfies the Cauchy-Riemannequations, reduces to a monogenic equation. In the case n = 3, the Clifford algebraC`0,2 corresponds to the quaternion algebra H. The Weyl operator is given byDW = ∂

∂x+i ∂

∂y+j ∂

∂z+k ∂

∂w. Then, identifying i = i1, j = i2,k = i3, the quaternion

Q = qx + qyi + qzj + qwk ∈ H is monogenic in U if DWQ = 0, thus giving rise toa generalized Cauchy-Riemann system. Note that the holomorphic functionsare easily transferable to higher dimensions when considering Clifford algebra.

In the next sections, we will offer an approach of monogenic sections on manifoldswith values in the Clifford bundle or Spinor bundle. A lot of the formulas thusestablished for the Dirac operator remain valid in the case of the Weyl operatorDW by formally replacing e1 by 1 and ea by ia , a = 2, . . . , n.

4.3 Monogenics in Curved Spaces and Some Results

In the section 2.2 we identified the Clifford algebra with the vector space of multi-vectors ∧V with the Clifford product between vectors given in (2.14), namely thefollowing algebra holds

UV + UV = 2 g(V ,U ) ∀ V ,U ∈ V .

In the context of bundles, the vectors are elements of Γ(V), the space of sections ofthe bundle V . So, in this approach a multivector is a multisection of the exteriorbundle ∧V . Since we are concerned only with local results, we are allowed to identifythe complexified tangent spaces C⊗ TqM , at point q ∈ M , with a vector space V ,so that all the results of the previous chapters can be used. Thus, it follows thaton a manifold (M, g) we have the structure of a Clifford algebra on each fibre of∧M =

⋃p ∧pM , the exterior bundle. The exterior bundle ∧M equipped with the

Clifford product in the fibres will be called the Clifford bundle and denoted by CM .We use the term Clifford forms refer to the multisections as elements of the Cliffordbundle rather than exterior algebra. The set of the Clifford forms on CM is denotedby Γ(CM).

In a formal way, in curved spaces, the Dirac operator D = ea∇a is a first orderdifferential operator that acts on a vector bundle over the manifold (M, g). Thestudy of Dirac and Laplace operators on manifolds, in particular on Riemannian


manifolds, has lead to a profound understanding of many geometric aspects relatedto these manifolds, in particular some applications for the Dirac operators on man-ifolds have been developed in [41, 44, 45, 40]. In turn, Riemannian manifolds playa central role in several branches of modern physics. They appear in importantcosmological models, in general relativity theory, in the standard model of particlephysics, in string theory and in general quantum field theory [23, 19, 33]. Here, weconsider only Clifford forms A whose components are smooth, that is, its partialderivatives up to a certain order are all continuous in some domain U of (M, g).Occasionally we will focus on smooth sections V = V aea ∈ Γ(TM) of the bundleTM on which ∇a is the Levi-Civita connection. Saying that a section V ∈ Γ(TM)is monogenic, which means that Clifford action of the Dirac operator D on V isnull,

DV = 0 ,

is equivalent to saying that its components V a , (a = 1, ..., n) satisfy the followingsystem of equations

DV = <D,V > +D ∧ V = ∇aVa + ∇[aV b]ea ∧ eb = 0

⇒ ∇aVa = 0 and ∇[aV b] = 0 . (4.9)

When restricted to the flat space, the equations (4.9) are the generalized Cauchy-Riemann equations ∂aV a = 0 and ∂[aV b] = 0. Introducing the co-derivative opera-tor, δ : Γ(∧pM)→ Γ(∧p−1M), defined by the action on Clifford forms as:

δω = ?−1d ? ω , ω ∈ Γ(∧pM) , (4.10)

with ? being the dual Hodge, d the exterior derivative and ω is the degree involutionof a p-vector ω ∈ Γ(∧pM), the equation (4.9) can be conveniently written as follows:

δV = 0 and dV = 0 . (4.11)

By the Poincarè lemma, locally we can assert that dV = 0 implies that in someneighborhood of every point p on M there exists a function φ such that V = dφ,hence δV = δdφ = 0. Note that, while d increases the degree of a Clifford formby one, δ decreases the degree by one. This implies that dδφ = 0, since δφ = 0 bydefinition. So, the action of the operator (dδ + δd) on φ = 0 also vanishes. Thisoperator is exactly the negative of the Laplacian operator

∆ = −(dδ + δd) . (4.12)

Thus, the function φ is said to be harmonic, that is satisfy ∆φ = 0. Therefore,given φ ∈ Γ(∧0M) such that ∆φ = 0, any monogenic section V is obtained byjudiciously choosing

V = dφ ∀ V ∈ Γ(TM) . (4.13)


The previous results about monogenicity can be generalized immediately to higherdimensions. These generalisations invariably turn out to be of mathematical andphysical importance, and it is no exaggeration to say that equations of the type ofequation (4.7) are amongst the most studied in physics.

Instead of sections V ∈ Γ(TM), we can be more ambitious and pass to considermore general elements A ∈ Γ(∧M) of the Clifford bundle ∧M . Let us considera Clifford form of degree p, F = F a1a2...apea1 ∧ ea2 ∧ . . . ∧ eap ∈ Γ(∧pM), withF a1a2...ap = F [a1a2...ap] totally skew-symmetric a section on Γ(∧pM). Then, theelement F is monogenic if the following equation holds:

DF = <D,F > +D ∧ F= ∇aF

a1a2... ap < ea, ea1 ∧ ea2 ∧ . . . ∧ eap >+ ∇[aF a1a2... ap]ea1 ∧ ea2 ∧ . . . ∧ eap = 0 . (4.14)

Using an important relation obtained in the equation (2.23) which we express hereas

< ea, ea1 ∧ ea2 ∧ . . . ∧ eap >=p∑i=1

(−1)i+1 < ea, eai > ea1 ∧ . . . ∧ eai ∧ . . . ∧ eap , (4.15)

where the check on eai select the term will be removed from the series when thesum is expanded, we can easily prove that F ∈ Γ(∧pM) is said to be a monogenicsection if and only if

∇aFaa2...,ap = 0 ⇔ δF = 0

∇[aF a1a2...,ap] = 0 ⇔ dF = 0. (4.16)

Many equations in physics can be elegantly formulated in terms of the exteriorderivative d and the co-derivative δ. In particular, letting p = 2 the equationsδF = 0 and dF = 0 are the source-free Maxwell’s equations which rise from theitself definition of monogenic sections. In this case, the section F = F ab ea ∧eb ∈ Γ(∧2M), which represents the electromagnetic field, is nothing more than theexterior derivative of the section A = Aaea ∈ Γ(TM), F = dA.

Instead of sections F ∈ Γ(∧pM), we could instead consider the sections on thesubset Γ(C+M) of Γ(CM), the space of the even sections of the even sub-bundleC+M . The bundle C+M is a direct sum of ∧pM with even p. If we require an evensection F =

∑p 〈F〉p =

∑pFp ∈ Γ(C+M), where Fp = F a1a2...apea1 ∧ea2 ∧ . . .∧eap ,

to be monogenic,

DF =∑p

DFp

=∑p

(<D,Fp > +D ∧ Fp) = 0 , (4.17)


then we have a system of coupled equations:

<D,Fp > +D ∧ Fp−2 = 0 , <D,Fp+2 > +D ∧ Fp = 0 , (4.18)

where Fp is the homogeneous part of degree p of F . Note that, if a Clifford form Acontains all grades it is clear that both the homogeneous part of even degree andodd degree must be monogenics independently. Without loss of generality, we cantherefore assume that A has even degree. Neverless, if we consider an arbitraryClifford form A ∈ Γ(CM) and its decomposition in pure degree terms

A = A + Aaea + Aabeaeb + Aabceaebec + . . . + Aa1...anea1 . . . ean

=n∑p=0

〈A〉p =n∑p=0

Ap (4.19)

Then claiming that DAp = 0, we obtain

n∑p=0

(<D,Ap > +D ∧ Ap) = 0 . (4.20)

Hence, A satisfies the condition of monogenicity, DA = 0, if and only if

<D,Ap+1 > +D ∧ Ap−1 = 0 , p = 0, 1, . . . , n , (4.21)

with A−1 = An+1 = 0. The main difference with complex analysis is that wecannot derive new monogenics simply from power series in this solution, due tonon-commutativity. Instead, we can construct monogenic functions from series ofgeometric products, but a more instructive route is to classify monogenics via theirangular properties, but in this dissertation we do not analyze this approach. See[4, 2, 40, 51] for a more detailed study.

Instead of Clifford forms on the Clifford bundle, in the next section, we will examinethe solutions lying in minimal left ideals, these carrying irreducible representationsof the Clifford algebra.

4.4 Monogenic Equation for Spinor Fields

In section 2.4 spinors were defined as the elements that generate a vector space thatprovides a faithful representation for the spin group and carry an irreducible repre-sentation of the Clifford algebra. Any such irreducible representation is equivalentto that carried by a minimal left ideal of the Clifford algebra, the called spinorialrepresentation. Hence we thus took the minimal left ideals as the space of spinors,see subsection 2.4.1. The Clifford bundle of a manifold (M, g) has as fibre at p, theClifford algebra of the tangent space of (M, g) at p the Clifford algebra C`(TpM).


Any minimal left ideal of this fibre algebra carries the spinor representation. Amanifold (M, g) is said to be a spin manifold if it has a well-defined spin structure.However, a spin manifold can allow different spin structures. Then, in the following,when talking about a d-dimensional spin manifold (M, g) it will always be assumedthat a spin structure is already chosen and fixed. This means that we can smoothlyassign a minimal left ideal of the fibre algebra to each p in (M, g), then we havea bundle over (M, g) with each fibre carrying an irreducible representation of thecorresponding fibre of the Clifford bundle. This is the spinor bundle SM , sectionsbeing the spinor fields on Γ(SM), the space of the sections of SM . Such a bundle ofspinor spaces is a sub-bundle of the Clifford bundle. Note that the spin structure,the spinor bundle and the Dirac operator depend on the metric g of M .

Below, let us investigate the behavior of the Dirac operator under conformaltransformation. In order to accomplish this, let us introduce a local coordinatesystem xµ (µ = 0, 1, . . . , d) on the manifold (M, g). The line element is written asds2 = gµν dx

µ dxν , where gµν = g(∂µ, ∂ν) are the components of g in the coordinateframe. However, given an orthonormal tangent frame eα, we can define the dualframe of 1-forms eα defined by eα(eβ) = δαβ and such that the previous lineelement can be written as

ds2 = δαβ eαeβ α, β = 1, . . . , d , (4.22)

where δαβ = g(eα, eβ) are now the constant components of the metric relative tothis orthonormal frame. Now, let g be a metric that is conformally related to themetric g, i.e., there is a function Ω ∈ F(M) over the manifold with Ω > 0 andg = Ω2g. Then, if eα is an orthonormal frame with respect to the metric g thefollowing relations hold

eα = Ω eα , eα =1

Ωeα . (4.23)

Remember that, the derivatives of the frame vector fields eα determine the coef-ficients of the spin connection according to the equation (3.44), namely ∇αeβ =

ω εαβ eε. Since we are dealing with metric-compatible connections, a change of met-

ric leads to a different spin connection. With respect to the metric g it is given bythe following relation

∇αeβ = ω εαβ eε . (4.24)

Given the frames of 1-forms eα, we recall that the connection 1-forms ωαβ aredetermined by the first Cartan structure equation and since the relation betweenthe frames eα and eα is known, we can find the relation between the two spinconnection one-forms ωαβ corresponding to the frame of 1-forms eα and ωαβ relativeto the frame eα. Indeed, after some algebra, from the equation

deα + ωαβ ∧ eβ = 0 ,


using (4.23) we eventually arrive at the following expression:

ωαβ = ωαβ +1

Ω( ∂βΩ eα − ∂αΩ eβ ) . (4.25)

Defining ωεβ(eα) ≡ ω εαβ and using the expression eα(eβ) = δαβ, from the equation

(4.25) we see that the coefficients of the spin connection 1-forms ω εαβ and ω ε

αβ arerelated by

ω βεα =

1

Ωω βεα +

2

Ω2∂[βΩ δε]α , (4.26)

where indices inside square brackets are anti-symmetrized.

We now consider a conformal transformation of the Dirac operator, which willbe of future relevance in the next section. Let eα be an orthonormal tangentframe for a bundle carrying the irreducible representation of the Clifford bundle,g(eα, eβ) = δαβ. The monogenic equation for a spinor field ψ is

Dψ = 0 , ∀ ψ ∈ Γ(SM) , (4.27)

where D = δαβeα∇β is the Dirac operator and ∇α is the Levi-Civita connection ofthe spinor bundle, that is the Levi-Civita connection satisfing the Leibniz rule withrespect to the Clifford product and with respect to the natural inner products onspace of spinorial sections. When we deal with spinors, the physical concepts canbe understood in a less abstract way by making use of the Dirac matrices. In evendimension d = 2n, the Dirac matrices Γα represent faithfully the Clifford algebraby 2n × 2n matrices obeying the relation

ΓαΓβ + ΓβΓα = 2δαβ . (4.28)

In this case, the spinors are represented by the column vectors on which thesematrices act and their spinorial covariant derivative are given by:

∇αψ = ∂αψ −1

4ω βεα ΓβΓεψ , (4.29)

with ∂α denoting the partial derivative along the vector field eα. We should observethat the Dirac matrices are unchanged under conformal tranformations, Γα = Γα.If ψ ∈ Γ(SM) is a spinor field, let us define

ψ = Ωpψ , (4.30)

with p being a constant parameter that will be conveniently chosen in the sequel.Then, using the equation (4.26), we obtain that the Dirac operator D = Γα∇α

behaves under conformal transformations as follows

D ψ = D (Ωpψ)

= Ωp−1Dψ +

(p + n − 1

2

)Ωp−2(∂αΩ)Γαψ . (4.31)


Thus, choosing p = −n+ 12, it follows the below relations

Dψ = Ωn+ 12 D ψ , ψ = Ωn− 1

2 ψ . (4.32)

It means that the monogenic equation is invariant under conformal transformations.Indeed, if ψ ∈ Γ(SM) is a spinor field satisfying the monogenic equation on themanifold (M, g) then

D ψ = 0 ⇒ Dψ = 0 , (4.33)

since Ω is a positive definite function throughout the manifold. Thus, the problemof finding solutions of the monogenic equation in the manifold (M, g) is reduced tofinding monogenic spinor fields on (M, g).In the recent years much success has been provided in the particular context ofconformally flat spin manifolds, which are Riemannian manifolds with a vanishingWeyl tensor. Let (M, g) be a d-dimensional conformally flat manifolds, that is

gαβ = Ω2δαβ , Ω ∈ F(M) , (4.34)

in the neighborhood of a point p of (M, g). Let us present a simple constructionof a solution to the monogenic equation in this manifold. Due to the conformalinvariance of the monogenic equation, it is sufficient to find the monogenic spinorfields in the flat space which, in the case of Euclidean signature, is the space Rd

generated by the standard basis e1, e2, . . . , ed. In order to accomplish this, let ususe a suitable representation for the Dirac matrices introduced in the section 2.5,but here with a slight modification. We recall that

σ1 =

[0 11 0

]; σ2 =

[0 −ii 0

]; σ3 =

[1 00 −1

], (4.35)

are the hermitian Pauli matrices and we will denote the 2× 2 identity matrix by I.Instead of spliting the representation of the Dirac matrices into those that are evenor odd as Γ2a or Γ2a−1, let us use the following notation

Γa = σ3 ⊗ . . .⊗︸︷︷︸(a−1) times

σ1⊗I⊗ . . .⊗ I︸︷︷︸(n−a) times

Γa = σ3 ⊗ . . .⊗︸︷︷︸(a−1) times

σ2⊗I⊗ . . .⊗ I︸︷︷︸(n−a) times

, (4.36)

where a and a are indices that range from 1 to n = d/2. Indeed, we can easily checkthat the Clifford algebra given in equation (4.28) is properly satisfied by the abovematrices1. The Euclidean Dirac operator is then represented by

D =n∑a=1

(Γa∂a + Γa∂a) =n∑a=1

σ3 ⊗ . . .⊗︸︷︷︸(a−1) times

Da⊗I⊗ . . .⊗ I︸︷︷︸(n−a) times

, (4.37)

1In d = 2n + 1, besides the 2n Dirac matrices Γa and Γa we need to add one further matrix, whichwill be denoted by Γn+1 given by Γn+1 = σ3 ⊗ σ3 . . .⊗ σ3︸︷︷︸

n times

.


whereDa = σ1∂a + σ2∂a ,

is the Dirac operator on R2 with coordinates xa, ya. Let us assume that the spinorfield ψ can be written as the following direct product of two component spinors

ψ = ψ1 ⊗ψ2 ⊗ . . .⊗ψn , (4.38)

where each spinor ψa (a = 1, 2, . . . , n) has the form

ψa =∑s

ψsaa ξsa , (4.39)

with their components ψsaa (sa = ±) depending just on the coordinates xa and ya,that is ψsaa = ψsaa (xa, ya). Here, we have introduced as the basis of the spinors thecolumn vectors

ξ+ =

[10

], ξ− =

[01

]. (4.40)

From equation (4.27), using (4.37) we obtain that the spinor field of expression(4.38) satisfies

D1ψ1 ⊗ψ2 ⊗ . . .⊗ψn + σ3ψ1 ⊗D2ψ2 ⊗ . . .⊗ψn

+σ3ψ1 ⊗ σ3ψ2 ⊗ . . .⊗Dnψn = 0 , (4.41)

which is satisfied if we choose each ψa such that it belongs to the kernel of theDirac operator Da on R2 and whose form can be easily obtained. It is given by

Daψa = 0 ⇒ ψa(xa, ya) = ψa(x

a + isaya) .

Thus, the monogenic equation on Rd for a spinor fields ψ can be reduced to amonogenic equations on R2 for each spinor fieldψa. It gives a kind of holomorphicityconditions and allows us to construct monogenic spinor fields on flat space as tensorproduct of spinor fields defined on 2-dimensional flat space with the components ofψ given by holomorphic functions ψsaa (xa + isay

a) of the plane coordinates.

The monogenic equations for spinor fields are a particular case of the Dirac equationwith zero eigenvalue, massless Dirac equations. In particular, the relation (4.33)enables us to investigate the conformal invariance of the massive Dirac equation.Indeed, if ψ is a spinorial field of mass m that obeys Dirac equation in the manifoldwith metric g, that is Dψ = mψ, it is straightforward see that

D ψ = m ψ , m = Ω−1m. (4.42)

Since, generally, Ω is a non-constant function, it follows that the massive Diracequation is not conformally invariant, whereas the massless Dirac equation is in-variant under conformal transformations. Such massless equations for the Weyl


spinors are known as Weyl equations. In reference [70] it is made the separation ofthe Neutrino Equations in a Kerr Background. The called zero-mode of the Diracoperator is a non-trivial global solution to the euclidean massless Dirac equation.In reference [65] it is made a study Dirac operator zero-modes on a torus for gaugebackground with uniform field strengths. In particular it is shown that under thebasic translations of the torus coordinates the components of the spinor are sub-ject to twisted periodic conditions and by a suitable choice of coordinates in thetorus the zero-mode wave functions can be related to holomorphic functions of thecomplex torus coordinates and finally it shown that the chirality and the degener-acy of the zero-modes are uniquely determined by the gauge background and areconsistent with the index theorem.

In the next section we will present the main results of this dissertation. It will beshown that the Dirac Equation coupled to a gauge field can be decoupled in even-dimensional manifolds that are the direct product of bidimensional spaces. We usethe conformal invariance of the massless Dirac equation to decouple the equationof motion of a charged test field of spin 1/2 propagating in a particular black holesolution background.

4.5 Direct Product Spaces and the Separability of the Dirac Equation

Our goal in this section is to show that the Dirac equation minimally coupledto an electromagnetic field is separable in spaces that are the direct product ofbidimensional spaces. In order to accomplish this, let us first fix some notations.The Greek letters from the beginning of the alphabet (α, β, ε) run from 1 to d = 2nand label, as previously, the vector fields of an orthonormal frame eα; lowercaseLatin indices with and without tildes (a, b, . . . , a, b, . . .) range from 1 to n and arealso used to label the vector fields of an orthonormal frame ea, ea in a pairwiseform, which will be quite suitable to our intent, as will be clear in the sequel; theindices (`, ˜) run from 2 to n and serve to label the angular directions of the blackhole spacetime considered here; finally, the indices (s, s1, s2, . . .) can take the values±1 and label spinorial degrees of freedom. We continue to use Einstein’s summationconvention, but when equal indices are in the same position, both up or both down,they should not be summed in principle, unless an explicit sign of sum is included.In what follows, we shall deal with a d-dimensional spin manifold (M, g) endowedwith a metric g of arbitrary signature.

Using the above notation, if we introduce an orthonormal frame ea for Γ(TM)we have

g(eα, eβ) = δαβ ↔

g(ea, eb) = δabg(ea, eb) = 0g(ea, eb) = δab

. (4.43)


The manifold (M, g) is a direct product of n bidimensional spaces which can be cov-ered by coordinates x1, y1, x2, y2, . . . , xn, yn such that the line element is writtenas

ds2 =n∑a=1

ds2a =

n∑a=1

(ea ea + ea ea) , (4.44)

where each 2-dimensional line elements ds2a and the 1-forms ea = e aµdx

µ andea = e aµdx

µ should depend just on the two coordinates corresponding to theirbidimensional spaces. Actually, this is our hypothesis. Note, for instance, that ds2

1,e1 and e1 depend just on the differentials dx1 and dy1 and their components shoulddepend just on the coordinates x1 and y1. In such a case, the only components ofthe spin connection that are potentially non-vanishing are

ωaaa = −ωaaa , ωaaa = −ωaaa . (4.45)

Thus, for example, ωaba = 0 and ωaab = 0 if a 6= b. Furthermore, the non-nullspin connections for some index a depend just on the coordinates xa and ya. Forinstance, ω111 depends just on x1 and y1.In order to accomplish the separability of the Dirac equation, let us use theconvenient representation (4.36) for the Dirac matrices. We can introduce a basisof this representation by the direct products of spinors ξs (s = ±) given in (4.40)which, under the action of the Pauli matrices, satisfy concisely the relations

σ1ξs = ξ−s , σ2ξ

s = i s ξ−s , σ3ξs = s ξs . (4.46)

The basis in d = 2n dimensions for the spinor space, the space on which the Diracmatrices act, is then spanned by

ξs1 ⊗ ξs2 ⊗ . . .⊗ ξsn .Once the base is defined, any spinor field can be expanded on this basis as

ψ =∑s

ψs1s2...sn ξs1 ⊗ ξs2 ⊗ . . .⊗ ξsn , (4.47)

where the sum over s means the sum over all possible values of s1, s2, . . . , snand ψs1s2...sn stands for the components of ψ. Remember that every sa can take twovalues. It means, in turn, that this sum comprises 2n terms, which is the dimensionof the spinor space in d dimensions as viewed in the section 2.4. This basis is veryconvenient, since the action of the Dirac matrices on the spinor fields can be easilycomptuted. Indeed, using the equations (4.36), (4.46) and (4.47) that

Γaψ =∑s

(s1s2 . . . sa−1)ψs1s2...sn ξs1 ⊗ ξs2 ⊗ . . .⊗ ξsa−1 ⊗ ξ−sa ⊗ ξsa+1 ⊗

. . . ⊗ ξsn =∑s

(s1s2 . . . sa) ψs1s2...sa−1(−sa)sa+1... sn ξs1 ⊗ ξs2 ⊗

. . . ⊗ ξsa−1 ⊗ ξsa ⊗ ξsa+1 ⊗ . . .⊗ ξsn , (4.48)


where from the first to the second line we have changed the index sa to −sa, whichdoes not change the final result, since we are summing over all values of sa, whichcomprise the same list of the values of −sa. Moreover, we have used that (sa)

2 = 1.Analogously, we have:

Γaψ =∑s

(s1s2 . . . sa−1) (isa) ψs1s2...sn ξs1 ⊗ ξs2 ⊗ . . .⊗ ξsa−1 ⊗ ξ−sa ⊗ ξsa+1 ⊗

. . . ⊗ ξsn = − i∑s

(s1s2 . . . sa) sa ψs1s2...sa−1(−sa)sa+1... sn ξs1 ⊗ ξs2 ⊗

. . . ⊗ ξsa−1 ⊗ ξsa ⊗ ξsa+1 ⊗ . . .⊗ ξsn . (4.49)

Here, the Dirac operator has the same form of (4.37) just replacing the partialderivative by the covariant derivative, that is

D =n∑a=1

(Γa∇a + Γa∇a) . (4.50)

All that was seen above are necessary tools to attack our initial problem of sepa-rating the general equation[

D − (ΓaAa + ΓaAa)]ψ = m ψ (4.51)

in its 2-dimensional blocks, where Aa, Aa and m are arbitrary functions of thecoordinates. In order to perform this separation, for the spinor field (4.47) let ususe the ansatz take the following separable form

ψs1s2... sn = ψs11 (x1, y1) ψs22 (x2, y2) . . . ψsnn (xn, yn) . (4.52)

From this hypothesis and using the equations (4.48) and (4.49) it follows that theequation (4.51) is given by

n∑a=1

∑s

(s1s2 . . . sa) ψs11 . . . ψ

sa−1

a−1 ψsa+1

a+1 . . . ψsnn[

i sa

(∂a +

1

2ωaaa − Aa

)+

(∂a +

1

2ωaaa − Aa

)]ψ (−sa)a ξs1 ⊗ ξs2 ⊗ . . .⊗ ξsn

= i m∑s

ψs11 ψs22 . . . ψsnn ξs1 ⊗ ξs2 ⊗ . . .⊗ ξsn , (4.53)

where ∂a and ∂a are the derivatives along the vector fields ea and ea, respectively.Let us write this latter equation in a more convenient form. Factorizing the com-


ponents ψs11 ψs22 . . . ψsnn of the spinor, we have

∑s

(n∑a=1

(s1s2 . . . sa)1

ψsaa

[i sa

(∂a +

1

2ωaaa − Aa

)+

(∂a +

1

2ωaaa − Aa

)]ψ (−sa)a

− i m ψs11 ψs22 . . . ψsnn ξs1 ⊗ ξs2 ⊗ . . .⊗ ξsn = 0 dotted

(4.54)

We can thus conclude thatn∑a=1

(s1s2 . . . sa)1

ψsaa

[i sa

(∂a +

1

2ωaaa − Aa

)+

(∂a +

1

2ωaaa − Aa

)]ψ (−sa)a

= i m . (4.55)

In order for the latter equation to be separable in blocks depending only on thecoordinates xa, ya for each value of a, we assume that the functions Aa and Aamust depend only on the two coordinates xa, ya and using this assumption, sincenow the left hand side depend only on these pairs of coordinates the function mmust be a sum over a of functions depending also on these pairs of coordinates. Wecan summarize these results by the relations

Aa = Aa(xa, ya) , Aa = Aa(x

a, ya) , m =n∑a=1

ma(xa, ya) . (4.56)

These assumptions lead us to the following equation:

n∑a=1

[(s1s2 . . . sa)

1

ψsaaDsaa ψ

(−sa)a − i ma

]= 0 , (4.57)

where the operator Dsaa is defined by

Dsaa = i sa

(∂a +

1

2ωaaa − Aa

)+

(∂a +

1

2ωaaa − Aa

). (4.58)

Note that each term in the sum over a depends just on the two coordinates xa, ya.In order to the sum of functions depending on distinct variables to be zero, is thusnecessary that each term of these sum be a constant, here called of separationconstant, with the sum of the constants being null. However, it is worth notingthat the separation constants can depend on the choice of s ≡ s1, s2, . . . , sn.Indeed, the equation (4.57) provides not only one equation but rather a total of 2n

independent equations, since for each choice of s ≡ s1, s2, . . . , sn we have one


equation. Let us denote this separation constant conveniently by iηsa . Then, wefind the following set of coupled first orders for ψsaa and ψ (−sa)

a :

(s1s2 . . . sa)Dsaa ψ

(−sa)a = i (ma + iηsa ) ψsaa ,

n∑a=1

ηsa = 0 . (4.59)

For each of these equations we can have different separation constants. Theseequations enable us to integrate the fields ψsaa and, therefore, find the solutions forthe generalized Dirac equation (4.51). We obtain thus that in the manifold with lineelement (4.44) the generalized Dirac equation (4.51) reduces to pair of first orderdifferential equations. However, although these equations are first order differentialequations, they are coupled in pairs, namely the equations involving the field ψsaahave ψ (−sa)

a as source and vice-versa. Therefore, we can eliminate ψsaa or ψ (−sa)a from

the of equations (4.59) to obtain a decoupled second order differential equation foreach component ψsaa , thus achieving the separability that we were looking for.

The second equation of (4.59) are constraints which the separation constants mustobey. Let us solve these constraints. What makes our task nontrivial is that sinceeach one of the n spinorial indices sa (a = 1, 2, . . . , n) can take two values, thecollective "index" s can take 2n values, it follows that the equation

n∑a=1

ηsa = 0 (4.60)

comprise 2n constraints. But, it is worth noting that ηsa cannot depend of sa+1, sa+2,. . . , sn, since the first equation of (4.59) by consistency is independent on these in-dices. In particular, by writing the quoted equation in the form

iηsa = (s1s2 . . . sa)1

ψsaaDsaa ψ

(−sa)a − i ma (4.61)

we can see that ηs1 depends just on s1, so that we can write it as

ηs1 = s1 κ

s11 , (4.62)

where κs11 is a pair of constants that depends just on s1. Now, multiplying theequation (4.61) by s1s2 . . . sa and using that s2

a = 1 we can write

1

ψsaaDsaa ψ

(−sa)a = i (s1s2 . . . sa)(ma + ηsa ) ψsaa . (4.63)

Note that the right hand side depends on s1, s2, . . . , sa while the left side dependsonly on sa. In order to the right hand side of this equation, likewise, depend juston sa we must have

ma + ηsa = (s1s2 . . . sa)κsaa , a ≥ 2 , (4.64)


where differently of κs11 , here κsaa is a pair of parameters in principle not constant thatdepends just on sa for each a to determine in the sequel. Since ηsa are constants,taking the derivative of both sides of the latter equation, we have

∂µma = (s1s2 . . . sa) ∂µκsaa . (4.65)

Since the left hand side of the latter equation does not depend on the elementsof the set s, it follows that ∂µκsaa must vanish for the equation to be consis-tent. This, in turn, implies that ma should be constant for a ≥ 2. But, ifm2(x2, y2), m3(x3, y3), . . . , mn(xn, yn) are constants we can, without loss of gen-erality, make all of them zero and absorb these constants in m1(x1, y1). Therefore,we can say that a consistent separability process requires that

m2 = m3 = . . . mn = 0 . (4.66)

Once assumed the previous conditions, the equations (4.62) and (4.64) immediatelylead to

ηsa = (s1s2 . . . sa)κsaa , (4.67)

where now κsaa is a pair of constants as κs11 . Since the sum of the constants ηsavasinhes, we are left with the following equation

n∑a=1

(s1s2 . . . sa)κsaa = 0 , (4.68)

which must hold for all possible choices of s. We thus wish know who are theconstants κsaa . In order to perform this we need manipulate this equation to solvesuch constraint. From now on, our task is restricted to identifying on both sides ofthe equation their corresponding dependencies with respect to the elements of theset s. Indeed, isolating κs11 we obtain that

κs11 = −n∑a=2

(s2 . . . sa)κsaa .

On the left hand side of the equation we have the term corresponding to the valueof a equal to 1 which depends just on s1 and on the right hand side a sum of termsthat, in principle, may be dependent on s1. However, since the sum start countingfrom a equal to 2 forward, none of the terms in the sum depend on s1. We canconclude that this sum is constant, that is

n∑a=2

(s2 . . . sa)κsaa = c1 , κs11 = − c1 ,


where c1 is a constant that does not depend on s. Following the same reasoning,let us write the latter equation as follows

κs22 − s2 c1 = −n∑a=3

(s3 . . . sa)κsaa .

Noting that the left hand side of the above equation depends just on s2 and thatthe sum of the terms on the right hand side clearly do not depend on s2, we thuscan conclude that

n∑a=3

(s3 . . . sa)κsaa = c2 , κs22 = s2 c1 − c2 ,

where c2 is a constant that does not depend on s. We can continue with this sameprocedure until reaching the term κsnn . Eventually, we are left with the followingfinal equation

κsaa = saca−1 − ca , c0 = cn = 0 , (4.69)

where c1, c2, . . . , cn−1 are arbitrary constants. Hence, this problem admits (n − 1)constants of separation. The latter equation is the general solution that solve theconstraint (4.68). With the results obtained on the equations (4.66), (4.67) and(4.69), we can use them in the first equation of (4.59) to find, finally, the followingset of coupled first order differential equations:

Ds11 ψ

(−s1)1 = i (s1 m1 − c1) ψs11

Dsaa ψ (−sa)

a = i (sa ca−1 − ca) ψsaa , a ≥ 2 . (4.70)

We have thus reduced the solution of the generalized Dirac equation (4.51) in themanifold with line element (4.44) n pairs of first order differential equations. Elimi-nating ψsaa or ψ (−sa)

a gives us a second order equation for ψsaa and the general solutioncan be expressed as a linear combination of the of the particular solutions belong-ing to the different values of c1, c2, . . . , cn−1 which can only take discrete valuesfor appropriate boundary conditions. In this sense, the constants c1, c2, . . . , cn−1can be viewed as eigenvalues and determined by the condition of regularity of thesolutions ψsaa .

In the next section, we shall use these results to separate the Dirac equation insome black hole spacetimes

4.6 Black Hole Spacetimes

The study of scalar fields, spin 1/2 fields and gauge fields (abelian and non-abelian)propagating in curved spacetimes plays a central role on the study of General rela-tivity and any other theory of gravity. The main reason is that besides the detection


of gravitational radiation and observation of the direct interaction between objectsvia gravitation, the most natural and simple way to probe the gravitational fieldpermeating our spacetime is by letting other fields interact with it. In this sec-tion we shall use the previous results to separate the Dirac equation correspondingto a massive and electrically charged field of spin 1/2 in the background of blackholes described in [78], which is a static black hole whose horizon have topologyR × S2 × . . . × S2. These black hole solutions possessing electric and magneticcharge, have been also obtained in references [87] and [86]. These are spacetimeswhose line element in even dimensions d = 2n is

ds2 = −f(r)2 dt2 +dr2

f(r)2+ r2

n∑`=2

(dθ2` + sin2θ` dφ

2`) , (4.71)

where f = f(r) is a function of the coordinate r as follows

f(r) =

√1

d− 3+

2M

r d−3+

Q2e(d− 3)

2(d− 2)r 2(d−3)− Q2

m

4(d− 5)r 2− Λ r 2

d− 1, (4.72)

with M , Qe and Qm being the mass, the arbitrary electric and magnetic charges,respectively, of the black hole and Λ the cosmological constant. The details of thissolution can be consulted in reference [78]. A suitable orthonormal frame of 1-formsfor such spacetime is the following:

e1 = i f(r) dt , e1 =1

f(r)dr , e` = r sin θ` dφ` , e

˜= r dθ` , (4.73)

where the index l ranges from 2 to n. In this frame, the line element is given by

ds2 =n∑a=1

(ea ea + ea ea) . (4.74)

This spacetime is the solution of Einstein-Maxwell equations with a cosmologicalconstant Λ and electromagnetic field F = dA, where the gauge field A given by

A =Qe

rd−3dt + Qm

n∑`=2

cos θ` dφ` ,

which, in the orthonormal frame, can be written as

A = Aa ea + Aa e

a = A1 e1 +

n∑`=2

A` e` , (4.75)

where

A1 = − i Qe

f(r) r d−3, A` =

Qm

rcot θ` , Aa = 0 . (4.76)


The Dirac equation is written in the form

Dψ = mψ , (4.77)

where D = Γα∇α is the dirac operator. The field ψ of spin 1/2 with electric chargeq and mass m in this spacetime and minimally coupled to the electromagnetic fieldobeys the following version of the Dirac equation

(D − i qA)ψ = mψ , (4.78)

where A = AαΓα is the representation of the gauge field. Using that Aa = 0, itfollows that the above equation is written as

Dψ = (m+ i qAaΓa)ψ . (4.79)

Our goal in this section is integrate this equation. Making a connection with theprevious section, we recall that an equation analogous to this has been separatedwhen the space is a direct product of bidimensional spaces. Clearly this is not thecase, since in front of the angular part of the black hole line element (4.71) there isthe multiplicative factor r2. However, we can factor out the function r2 in the lineelement (4.71) and define ds2 that is conformally related to our initial line element

ds2 =ds2

r2,

where

ds2 = −f2

r2dt2 +

dr2

(rf)2+

n∑`=2


2`) =

n∑a=1

ds 2a , (4.80)

with

ds 21 = −f

2

r2dt2 +

dr2

(rf)2, ds 2

` =n∑`=2


2`) .

The conformally transformed space with line element ds2 is a direct product of bidi-mensional spaces. It is worth noting that ds 2

1 depends just on the coordinates t andr, while ds 2

` depends on the angular coordinates θ` and φ`. A suitable orthonormalframe of 1-form for this space is given by

e1 =if

rdt , e1 =

1

rfdr , e ` = sin θ` dφ` , e

˜= dθ` . (4.81)

In this case, the only non-vanishing components of the connection ωabc in the frameea are

ω111 = − ω111 = r f ′ − f , ω` ˜ = − ω``˜ = cot θ` , (4.82)


where f ′ stands for the derivative of f with respect to its variable r. The conformaltransformation of the Dirac operator given by the equation (4.42) enable us relatethe quantities concerning to the spacetime with line elements ds2 to the quantitiesdefined on ds2. Indeed, we can write the field equation (4.79) in terms of an equationin the space with line element ds2, so that the separability results of the previoussection can be fully used. In particular, we recall that the relevant conformaltransformations are the following

D ψ = Ω−(n+ 12

)Dψ , ψ = Ω12−nψ , m = Ω−1m with Ω = r−1 .

From the latter equation follows that the field equation (4.79) can be written as

D ψ = Ω−1(m+ i qAaΓa) ψ (4.83)

in its turn, defining

m = m1 = rm , Aa = i q rAa , Aa = 0 , (4.84)

we are left with the following equation

( D − ΓaAa ) ψ = m ψ , (4.85)

which is exactly the form of equation studied in previous section and we have beenable so separate. Moreover, and foremost, defining the coordinates

x1 = t , y1 = r , x` = φ` , y` = θ` , (4.86)

it follows that the function m and the gauge field are exactly of the form necessaryto attain separability, namely the constraints (4.56) and (4.66) are obeyed. Inparticular, the solutions of the equation (4.79) in the black hole background isgiven by

ψ = r ( 12−n)∑s

ψs11 (t, r)ψs22 (φ2, θ2) . . . ψsnn (φn, θn) ξs1 ⊗ ξs2 ⊗ . . .⊗ ξsn . (4.87)

From equations (4.58), (4.70), (4.76) and (4.82) - (4.84), it follows that the functionsψsaa must be solutions of the following differential equations[

i s1

(r

i f∂t −

q Qe

fr d−4

)+

(r f ∂r +

1

2(r f ′ − f)

)]ψ

(−s1)1 = i (s1 rm− c1)ψs11[

i s`

(1

sin θ`∂φ` − i q Qm cot θ`

)+

(∂θ` +

1

2cot θ`

)]ψ

(−s`)` = i (s` c`−1 − c`)ψs`` ,

(4.88)

where we recall that the constants of separation c1, c2, . . . , cn−1 take discrete valuesonce boundary conditions and regularity requirements are imposed and the con-stant cn is zero. It is fruitful noting that the coefficients of the latter equation are


independents of the coordinate t and φ`, therefore both the metrics g and g possessthe Killing vector fields ∂t and ∂φ` . This stems from the fact that the coordinates tand φ` are cyclic coordinates of this metric. It is thus convenient to decompose thedependence of the fields ψsaa on these coordinates in the Fourier basis, namely,

ψs11 (t, r) = eiωt Ψs11 (r) , ψs`` (φ`, θ`) = eiω`φ` Ψs`

` (θ`) . (4.89)

The final general solution for the field ψ must, then, include a "sum" over allvalues of the Fourier frequencies ω and ω` with arbitrary Fourier coefficients. Whileω can be interpreted as related to the energy of the field, ω` are related to angularmomentum. Note that in order to avoid conical singularities in the spacetime, thecoordinates φ` must have period 2π, namely φ` and φ` + 2π should be identified[88]. As it is well-known, a spin 1/2 field changes its sign after a 2π rotation,which implies that the angular frequencies ω` must be half-integers, see [88] formore details:

ω` = ± 1

2,± 3

2,± 5

2, . . . . (4.90)

Finally, inserting the decomposition (4.89) into equation (4.88), we end up with thefollowing pairwise coupled system of differential equations:[

r fd

dr+

1

2(r f ′ − f) + i s1

(ω r

f− q Qe

fr d−4

)+

]Ψ

(−s1)1 = i (s1 rm− c1) Ψs1

1[d

dθ`+

1

2cot θ` − s`

(ω`

sin θ`− q Qm cot θ`+

)]Ψ

(−s`)` = i (s` c`−1 − c`) Ψs`

` .(4.91)

4.6.1 The angular part of Dirac’s Equation

Since we have been able so separate the generalized Dirac equation into radial andangular coordinates, we now shall investigate a little further these equations. Letus start with the angular part of the equations, namely the equations for Ψs`

` . Onecan make a simplification on these equations by performing a field redefinition alongwith a redefinition of the separation constants, as we show in the sequel. Insteadof using the (n− 1) separation constants c1, c2, . . . , cn−1, we shall use the constantsλ2, λ3, . . . , λn, defined by

λ` ≡√c 2`−1 − c 2

` , (4.92)

where we recall that cn = 0, by definition. Inverting these relations, we find thatthe old constants can be written in terms of the new constants as follows:

c `−1 =√λ2` + λ2

`+1 + . . .+ λ2n . (4.93)


Then, defining the parameter

ζ ` = arctanh(

c`c`−1

),

we find thatc`−1 = λ` cosh ζ` , c` = λ` sinh ζ` ,

so that the following relation holds:

s` c`−1 − c` = s` λ` e− s`ζ` .

Thus, performing the field redefinition given by

Ψs`` (θ) = e s` ζ`/2 Φs`

` (θ) , (4.94)

it turns out that the angular part of (4.91) can be written in a simpler way in termsof the fields Φs`

` (θ). Indeed, using the latter equation on the second equation of(4.91) we find that[

d

dθ`+

1

2cot θ` − s`

(ω`

sin θ`− q Qm cot θ`

)]Φ

(−s`)` = i s` λ` Φs`

` . (4.95)

Although it may seem that we did not achieve much simplification by the redefini-tion of the fields and separation constants, it turns out that in the case in whichthe black hole has vanishing magnetic charge, Qm = 0, these equations reduce to[

d

dθ`+

1

2cot θ` −

s` ω`sin θ`

]Φ

(−s`)` = i s` λ` Φs`

` . (4.96)

The above equation is exactly the Dirac equation on the 2-sphere S2, that is aeigenvalue equation DS2Φ = i λΦ, where by DS2 we mean the Dirac operator onthe 2-sphere and Φ a 2-component spinor. Indeed, using the frame e1 = sinθdφe2 = dθ along with the Dirac matrices γ1 = σ1 and γ2 = σ2 one can easily derivethe latter equation. The problem of finding eigenstates of Dirac operator on thesphere S2 is a well posed mathematical problem that can be solved exactly. Thesolution Φ has its components written in terms of Jacobi polynomials [88, 90] andgeometrically, these solutions can be understood in terms of the Wigner elements ofthe group Spin(R3), that give rise to the so-called spin weighted spherical harmonics[93, 94], which are tensorial generalizations of the spherical harmonics. Moreover,not all values are allowed for λ`. Indeed, once regularity requirements are imposeon 2-sphere, λ` must fulfill the condition to be non-zero integers

λ` = ±1,±2,±3, . . . . (4.97)

Solutions with non-integer eigenvalues are not well-defined on the whole sphere,while a vanishing eigenvalue is forbidden by the Lichnerowicz theorem [92], sincethe sphere is a compact manifold with positive curvature.


Regarding the general case in which the black hole magnetic charge is non-vanishing,Qm 6= 0, we have tried to make a redefinition of the fields Φs`

` by means of ageneral linear combination of the fields Φ+

` and Φ−` , with non-constant coefficients,in order to convert equation (4.95) into the eigenvalue equation DS2Φ = i λΦ.However, it turns out that the coefficients of the linear combination must obeyfourth-order differential equations, whose solutions seem to be quite dificult toattain analytically. In spite of this, we can make an important progress regardingthe system of equations (4.95) by decoupling the fields Φ+

` and Φ−` , which, after all,is our goal at this section. The final result is that the fields Φs`

` satisfy the followingsecond order differential equation:

1

sin θ`d

dθ`

(sin θ`

dΦs``

dθ`

)+[

(1 + 2qQm)ω` cos θ`sin2 θ`

− 1 + 2qQm + 2ω2`

2 sin2 θ`+

(1− 4q2Q2m) cos2 θ`

4 cos2 θ`− λ2

`

]Φs`` = 0 .

(4.98)

It is worth stressing that the latter equation must be supplemented by the require-ment of regularity of the fields Φs`

` at the points θ = 0 and θ = π, where ourcoordinate system breaks down. These regularity conditions transform the task ofsolving the latter equation in a Sturm-Liouville problem, so that the possible valuesassumed by the separation constants λl form a discrete set. Since the case Qm = 0in equation (4.98) has a known solution, as described above, it follows that we canlook for solutions for the case Qm 6= 0 by means of perturbation methods, with Qm

being the perturbation parameter. Indeed, in the celebrated paper [74], a similarpath has been taken by Press and Teukolsky in order find the solutions and theireigenvalues for the angular part of the equations of motion for fields with arbitraryspin on Kerr spacetime, in which case the angular momentum of the black hole wasthe order parameter. In this respect, see also the reference [96].

4.6.2 The radial part of Dirac’s Equation

Now, we shall investigate a little further about the pair of radial equations in expres-sion (4.91). In order to acomplish to solve such equations we should first decouplethe fields Ψ+

l and Ψ−l . The form of radial equation in (4.91) suggests that instead ofthe factors inside of the brackets we define the following functions of the coordinater,

Bs1(r) =1

r f

[1

2(r f ′ − f)− i s1

(ω r

f− q Qe

fr d−4

)]Cs1(r) = − i

r f(s1 rm+ c1) ,


in terms of which the radial equation in (4.91) can be written as

dΨs11

dr= −Bs1 Ψs1

1 + Cs1 Ψ−s11 . (4.99)

Then, deriving this equation with respect to r and eliminating Ψ−s11 of the aboveequation to obtain

Ψ−s11 =1

Cs1

(dΨs1

1

dr+Bs1 Ψs1

1

),

using this, we eventually arrive at the following expression:

d2Ψs11

dr2+

(Bs1 +B−s1 −

1

Cs1

dCs1dr

)(dΨs1

1

dr+Bs1 Ψs1

1

)+(

dBs1

dr−B2

s1− Cs1C−s1

)Ψs1

1 = 0 , (4.100)

which is the decoupled second order differential equation for Ψs11 . An analytical

exact solution of the latter differential equation is, probably, unapproachable. Nev-ertheless, we can use (4.100) to infer the asymptotic forms of the solution near theinfinity, r → ∞, as well as near the horizon r → r∗ , where r∗ is a root of thefunction f , namely f(r∗) = 0. In order to perform this analysis, we shall write theequation (4.100) as:

d2Ψs11

dr2+ hs11 (r)

dΨs11

dr+ hs10 (r) Ψs1

1 = 0 , (4.101)

where functions hs11 (r) and hs10 (r) can be obtained without great effort by comparingequations (4.100) and (4.102). Before proceeding, we should note that consideringthe solutions of the equation (4.102) in the limit r → ∞ we must expand thefunction hs11 (r) up to order r−(p+1) and hs10 (r) up to order r−(p+2) if we want toknow Ψs1

1 up to order r−p. This ensures that our research in this limit will beconsistent. Once called attention to this important detail, we now can work out theasymptotic forms of the functions hs11 (r) and hs10 (r) in the region of interest.Let us separate the analysis of these two cases. The motivation for this division isthat the function f(r) in the black hole line element (4.71) has a term that multi-plies Λ becomes the dominant one as we approach the infinity, r →∞. Therefore,it is natural to guess that the cases of vanishing and non-vanishing Λ should bequalitatively different. Let us then consider first the case Λ 6= 0. Collecting thefunctions that that accompany d2Ψ

s11

dr2and Ψs1

1 in equation (4.100) and then expand-ing them in powers of r−1, we can find, after some algebra, the following asymptoticforms

hs10 (r) =(d− 1)m2

Λ r2+i (d− 1) s1 ω

Λ r3+O

(1

r4

),

hs11 (r) =1

r− s1 c1

mr2+

[2 (d− 1)

(d− 3)Λ− c2

1

m2

]1

r3+O

(1

r4

). (4.102)


Particularly, if we consider the expansion of hs10 (r) up to order r−2 and the expan-sion of hs11 (r) up to order r−1, the integration of the equation (4.102) gives us thefollowing asymptotic form:

Ψs11 ∼ C0 sin

[m√d− 1√Λ

log (r) + ψ0

], (4.103)

where C0 and ψ0 are arbitrary integration constants.

Now, let us consider the second case on which the cosmological constant is null,Λ = 0. In this case, when d > 6 one can see, after performing a subtle algebra, thatthe asymptotic forms of the functions hs11 (r) and hs10 (r)are the following

hs10 (r) =[

(d− 3)2 ω2 − (d− 3)m2]

+

[3

4+ (d− 3)

(c2

1 −i ω c1

m

)− (d− 3)2Q2

mm2

4 (d− 5)+

(d− 3)3Q2mω

2

2(d− 5)

]1

r2

+ O

(1

r3

),

hs11 (r) = −1

r− s1 c1

mr2+O

(1

r3

).

Note, however, that the case of vanishing cosmological constant, the well-known cased = 4 is qualitatively different from the higher-dimensional cases d ≥ 6. Indeed,these formulas do not apply to the well-studied situation d = 4, in which case hs10 (r)has a term of order r−1 depending on M and Qe and the function corresponding tothe term of order r−2 has additional terms also depending on these constants. Acompletely analogous analysis can be done for hs11 (r). These considerations lead usto the following conclusion. For Λ = 0, the spinor field that represents a chargedparticle of spin 1/2 moving in the black hole (4.71) has qualitatively different falloff properties in the asymptotic infinity depending on whether d = 4 or d ≥ 6.

Above, we describe the asymptotic behavior in the limit when the coordinate rassumes values near the infinity. However, a similar analysis can be done near thehorizon. This is the case of the values of r for which the function f vanishes. In suchcase the coordinate r ceases to be reliable and we should change the radial coordi-nate to tortoise-like coordinates. In [74] these coordinates are useful in the studyof the dynamical stability of the Kerr Metric. Besides such asymptotic behaviours,one can also look for approximate solutions valid in a broader domain by meansof other approximation methods. In particular the Wentzel-Kramers-Brillouin ap-proximation method ( WKB ) has been widely used in the pursuit of such solutions.In [95] the WKB method is used to obtain an approximate solution for the Diracfield on the Kerr spacetime, while on [99] it is utilized to solve the radial part ofthe Dirac equation in the Schwarzschild geometry. In both references, the WKBmethod is applied after transforming the radial second order differential equationinto a Schrodinger equation. To know more, consult the references [97, 82].

REFERENCES 105

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Index

Algebra of 2× 2 matrices over F, 21

Cartan’s structure equations, 57Clifford algebra, 14Clifford conjugation, 18Clifford Product, 14Connection 1-form and its components, 57Curvature 2-form, 57

Degree involution, 16Differential operators, 49Dirac equation, 90Division algebra, 38

Exterior derivative, 53

Frobenius theorem, 61

Hodge dual of a Clifford form, 23

Inner product between spinors, 38Isometries, 59

Killing vector field, 59

Left contraction, 15Left ideals, 30Lie derivative, 59, 60

Maximally symmetric manifold, 61maximally symmetric manifold, 61Metric of the manifold, 50Minimal left ideals, 30

Non-coordinate frame, 56, 62

Periodicity, 24Pin group, 28Pseudoscalar, 21

Reflection operator, 26Reversion, 17Rotation operator, 27

Separability, 91Simple form, 50Space of the local sections of Clifford bundle, 82Space of the local sections of spinorial bundle, 74Spin group, 28Spinor space, 31Spinorial bundle, 74

The Riemann tensor, 55Torsion 2-form, 57

112

UFPE

Nota como Carimbo

UFPE

Nota como Carimbo

UFPE

Nota como Carimbo

THE SPINORIAL FORMALISM, WITH APPLICATIONS IN PHYSICS · 2019-10-25 · tiu entender sobre a origem...

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