Analysis, Manifolds and Physics, Part II - Revised and Enlarged Edition

By Y. Choquet-Bruhat and C. DeWitt-Morette

Twelve problems have been added to the first edition; four of them are supplements to problems in the first edition. The others deal with issues that have become important, since the first edition of Volume II, in recent developments of various areas of physics. All the problems have their foundations in volume 1 of the 2-Volume set Analysis, Manifolds and Physics. It would have been prohibitively expensive to insert the new problems at their respective places. They are grouped together at the end of this volume, their logical place is indicated by a number of parenthesis following the title.

• Language: English
• Publisher: Elsevier Science
• Release date: Nov 8, 2000
• ISBN: 9780080527154

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ANALYSIS, MANIFOLDS AND PHYSICS

Part II

YVONNE CHOQUET-BRUHAT

Membre de l’ Académie des Sciences, Université de Paris VI, Département de Mécanique, Paris, France

CÉCILE DEWITT-MORETTE

University of Texas, Department of Physics and Center for Relativity, Austin, Texas, USA

Table of Contents

Cover image

Title page

Copyright

PREFACE TO THE SECOND EDITION

PREFACE

CONVENTIONS

Chapter I: REVIEW OF FUNDAMENTAL NOTIONS OF ANALYSIS

1 INTRODUCTION

2 GAMMA MATRICES IN LOW DIMENSIONS

3 GAMMA MATRICES IN ARBITRARY DIMENSIONS

4 PERIODICITY MODULO 8

5 GRADING OF A CLIFFORD ALGEBRA

1 CASE d = n + m EVEN

2 CASE OF m + n = d OF ARBITRARY PARITY

3 EXAMPLES

1 WEYL SPINORS. HELICITY OPERATOR

2 CHARGE CONJUGATION. MAJORANA PINORS

Chapter II: DIFFERENTIAL CALCULUS ON BANACH SPACES

1 HYPERDIFFERENTIABLE MAPPINGS f: A → A

2 SUPERDIFFERENTIABLE MAPPINGS

3 Gp-MAPPING

4 SUPERDIFFERENTIABLE MAPPING f: A ″ → Ap

II COMPUTATION OF GAUSSIAN INTEGRALS

I LAGRANGIANS DEPENDING ON A METRIC

II LAGRANGIAN DEPENDING ON THE METRIC AND OTHER FIELDS

III HYPERBOLIC METRIC G ON A MANIFOLD V OF SIGNATURE ( +, −, …, −)

Chapter III: DIFFERENTIABLE MANIFOLDS

1 KOSTANT GRADED BUNDLES

2 GRADED VECTOR OR AFFINE BUNDLES

Chapter IV: INTEGRATION ON MANIFOLDS

1 HOMOLOGY

2 COHOMOLOGY

1 INTRODUCTION: SPINOR FIELDS ON A MANIFOLD; SPIN AND PIN STRUCTURES

2 DEFINITIONS

3 EXISTENCE OF PIN BUNDLES OVER AN O(N) BUNDLE

4 STIEFEL–WHITNEY CLASSES

INTRODUCTION

ACKNOWLEDGEMENTS FOR PROBLEMS IV 1, 2, 3

1 DEFINITIONS AND EXERCISES

1 DIFFERENTIAL EQUATIONS AS EXTERIOR DIFFERENTIAL SYSTEMS

2 INVARIANCE GROUPS OF SETS OF FORMS AND DIFFERENTIAL EQUATIONS

3 PSEUDOPOTENTIALS

4 BÄCKLUND TRANSFORMATIONS

0 DEFINITIONS

1 HAMILTONIAN VECTOR FIELDS

2 SYMPLECTIC MANIFOLDS

3 DUAL OF A LIE ALGEBRA

4 POISSON STRUCTURE DEFINED BY A CONTRAVARIANT TENSOR

5 POISSON MANIFOLDS AND SYMPLECTIC MANIFOLDS

6 HAMILTONIAN VECTOR FIELDS AND AUTOMORPHISMS OF A POISSON MANIFOLD

7 FOLIATION OF POISSON MANIFOLDS BY SYMPLECTIC LEAVES

8 KIRILLOV LOCAL LIE ALGEBRAS

9 KIRILLOV ORBITS

10 LIE ALGEBRA OF OBSERVABLES

INTRODUCTION

1 POISSON MAPS AND POISSON SUBMANIFOLDS

2 INDUCING A POISSON STRUCTURE ON A SUBMANIFOLD M1 OF A POISSON MANIFOLD (M2; ⩘2)

3 LOCAL STRUCTURE OF A POISSON MANIFOLD

4 TRANSVERSE POISSON STRUCTURE TO THE COADJOINT ORBITS IN G*

5 REALIZATIONS OF POISSON MANIFOLDS AND FUNCTION GROUPS

6 DUAL PAIRS

7 TWO IMPORTANT EXAMPLES OF DUAL PAIRS

8 FOR FURTHER STUDY

1 DEFINITIONS AND ARNOLD’S THEOREM

2 CONSTRUCTION OF FUNCTIONS IN INVOLUTION AND THE LAX EQUATION

Chapter V: RIEMANNIAN MANIFOLDS. KÁHLERIAN MANIFOLDS

1 YANG–MILLS OPERATOR

2 DIRAC OPERATOR

3 HIGGS OPERATOR

1 DEFINITIONS

INTRODUCTION

Chapter V BIS: CONNECTIONS ON A PRINCIPAL FIBRE BUNDLE

INTRODUCTION

1 EXISTENCE AND PROPERTIES OF H-INVARIANT VACUA

INTRODUCTION

1 POINCARÉ-HOPF THEOREM

2 VAN HOVE’S SINGULARITIES 2

3 GAUSS-BONNET-CHERN-AVEZ THEOREM (quoted p. 395)

4 SUPERTRACE OF exp(−Δ)

1 PULLBACK OF A BUNDLE

2 UNIVERSAL BUNDLE

3 EXISTENCE OF UNIVERSAL BUNDLES

4 CHARACTERISTIC CLASSES

1 GROUP OF GAUGE TRANSFORMATIONS

2 LIE ALGEBRA OF

3 REPRESENTATION OF G

4 PROJECTIVE REPRESENTATIONS OF G

5 CENTRAL EXTENSIONS OF

6 LIE ALGEBRA OF C∞(S1, G)

7 LIE ALGEBRA OF DIFF S1

8 ANOMALIES

1 FINITE DIMENSIONAL LIE ALGEBRA

2 INFINITE DIMENSIONAL LIE ALGEBRAS

3 SEMI-INFINITE FORMS

4 NORMAL ORDERING. QUANTUM BRST OPERATORS

Chapter VI: DISTRIBUTIONS

I DEFINITIONS

II sobolev embedding theorem

1 DEFINITIONS

2 density theorem

3 embedding and multiplication properties

Chapter VII: SUPPLEMENTS AND ADDITIONAL PROBLEMS

1.1 Definition

Definition

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.1 Definition

2.2

2.3

2.4

2.5

2.6

2.7

2.8

Definition

2.10

3.1

3.2

Definition

3.4

3.5

3.6

4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

Definitions

4.10

4.11

4.12

4.13

INTRODUCTION AND DEFINITIONS

CONCLUSION

INTRODUCTION

CHECK THE GAUSS-BONNET THEOREM FOR A 2-DIMENSIONAL MANIFOLD OF GENUS 0

INTRODUCTION

PROBLEM

THE BERRY PHASE

THE AHARONOV-ANANDAN PHASE

INDEX

ERRATA TO ANALYSIS, MANIFOLDS AND PHYSICS. PART I

Copyright

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[First edition 1989]

Revised and enlarged edition 2000: ISBN 0-444-50473-7

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Printed in The Netherlands.

PREFACE TO THE SECOND EDITION

Twelve problems have been added to the first edition; four of them are supplements to problems in the first edition. The others deal with issues that have become important, since the first edition of Volume II, in recent developments of various areas of physics. All the problems have their foundations in Volume I of the 2-Volume set Analysis, Manifolds, and Physics.

It would have been prohibitively expensive to insert the new problems at their respective places. They are grouped together at the end of this volume, their logical place is indicated by a number in parenthesis following the title.

The new problems are:

M). A supplement to Problem I.4 and I.3 (I.17)." Its logical place is the seventeenth problem of Chapter I.

• The problem Lie derivative of spinor fields (III. 15) belongs to Chapter III.

• Poisson–Lie groups, Lie bialgebras, and the generalized classical Yang–Baxter equation (IV.14) has been contributed by Carlos Moreno and Luis Valero. It belongs to Chapter IV.

Additions to Chapter V on Riemannian and Kählerian manifolds include:

• "Volume of the sphere Sn. A supplement to Problem V.4 (V.15)"

• Teichmuller spaces (V.16)

• Yamabe property on compact manifolds (V.17)

To Chapter V bis on Connections are added:

• The Euler class. A supplement to Problem V bis 6 (V bis 13)

• Formula of Laplacians at a point of the frame bundle (V bis 14)

• The Berry and Aharanov–Anandan phases (V bis 15) based on notes by Ali Mostafazadeh.

To Chapter VI on Distributions:

• "A density theorem. A supplement to Problem VI.6 on ‘spaces Hsm)’ (VI.17)"

• Tensor distributions on submanifolds, multiple layers, and shocks (VI.18)"

• Discrete Boltzman equation (VI.19)

A fair number of misprints have been corrected. An updated list of errata for Volume I is included.

Naturally more problems are on our drawing boards. We would like to think of them as contributions to a third edition.

Most of the new problems were completed during a visit of Y. Choquet-Bruhat to the Center for Relativity of the University of Texas, made possible by the Jane and Roland Blumberg Centennial Professorship in Physics held by C. DeWitt-Morette. Help and comments from M. Berg, M. Blau, M. Godina, S. Gutt, M. Smith, R. Stora, X. Wu-Morrow and A. Wurm are gratefully acknowledged.

PREFACE

This book is a companion volume to our first book, Analysis, Manifolds and Physics (Revised Edition 1982). In the context of applications of current interest in physics, we develop concepts and theorems, and present topics closely related to those of the first book. The first book is not necessary to the reader interested in Chapters I–V bis and already familiar with differential geometry nor to the reader interested in Chapter VI and already familiar with distribution theory. The first book emphasizes basics; the second, recent applications.

Applications are the lifeblood of concepts and theorems. They answer questions and raise questions. We have used them to provide motivation for concepts and to present new subjects that are still in the developmental stage. We have presented the applications in the forms of the problems with solutions in order to stress the questions we wish to answer and the fundamental ideas underlying applications. The reader may also wish to read only the questions and work out for himself the answers, one of the best ways to learn how to use a new tool. Occasionally we had to give a longer-than-usual introduction before presenting the questions. The organization of questions and answers does not follow a rigid scheme but is adapted to each problem.

This book is coordinated with the first one as follows:

1. The chapter headings are the same – but in this book, there is no Chapter VII devoted to infinite dimensional manifolds per se. Instead, the infinite dimensional applications are treated together with the corresponding finite dimensional ones and can be found throughout the book.

2. The subheadings of the first book have not been reproduced in the second one because applications often use properties from several sections of a chapter. They may even, occasionally, use properties from subsequent chapters and have been placed according to their dominant contribution.

3. Page numbers in parentheses refer to the first book. References to other problems in the present book are indicated [Problem Chapter Number First Word of Title].

The choice of problems was guided by recent applications of differential geometry to fundamental problems of physics, as well as by our personal interests. It is, in part, arbitrary and limited by time, space, and our desire to bring this project to a close.

The references are not to be construed as an exhaustive bibliography; they are mainly those that we used while we were preparing a problem or that we came across shortly after its completion.

The book has been enriched by contributions of Charles Doering, Harold Grosse, B. Kent Harrison, N.H Ibragimov, and Carlos Moreno, and collaborations with Ioannis Bakas, Steven Carlip, Gary Hamrick, Humberto La Roche and Gary Sammelmann. Discussions with S. Blau, M. Dubois-Violette, S.G. Low, L.C. Shepley, R. Stora, A. H. Taub, J. Tits and Jahja Trisnadi are gratefully acknowledged.

The manuscript has been prepared by Ms. Serot Almeras, Peggy Caffrey, Jan Duffy and Elizabeth Stepherd.

This work has been supprted in part by a grant from the National Science Foundation PHY 8404931 and a grant INT 8513727 of the U.S.–France Cooperative Science Program, jointly supported by the NSF and the Centre National de la Recherche Scientifique.

CONVENTIONS

(1) {fn}N := {fn : n ∈ N}.

.

(3) Integer part: if d/2 = 3.5, then [d/2] = 3.

(4) A\B and A/B sometimes mean left and right coset, respectively; but usage varies and is determined in each context.

(5) Exterior product, exterior derivative, interior product

When operating on a p.

(6) Riemann tensor, Ricci tensor

i.e.

These conventions agree with Misner, Thorne, and Wheeler and differ from those of our first book Analysis, Manifolds and Physics.

(7) The Dirac representation of the gamma matrices

Majorana representation of the gamma matrices

Note that in Vol. I, p. 176, we give the Dirac representation of the gamma matrices for ημv = diag(+, −, −, −).

I

REVIEW OF FUNDAMENTAL NOTIONS OF ANALYSIS

1

GRADED ALGEBRAS

For applications and references see, for instance, Problems II 1, Supersmooth mappings and III 14, Graded bundles.

2 graded algebra A is a vector space over the field of real or complex numbers which is the direct sum of two subspaces A+ (called even) and A− (called odd)

endowed with an associative and distributive operation, called product, such that

graded algebra

2 graded algebra is called graded commutative if any two odd elements anticommute and if even elements commute with all others:

where d(a) = r if a ∈ Ar is the parity of a.

graded commutative

We shall consider in this section only graded commutative algebras, so we shall omit the word commutative.

parity

The algebras we shall use will be endowed with a locally convex Hausdorff topology for which sum and product are continuous operations. For example, the exterior (Grassmann) algebra over a finite dimensional vector space X (p. 196) is a graded algebra.

A generalization used in physics, which we shall call a (Bryce) DeWitt algebra is the algebra B of formal series with a unit e and an infinite number of generators zI, I , with the usual sum and product laws and the anticommutation property

(Bryce) DeWitt algebra B

An element a ∈ B is written (notion of convergence is irrelevant)

a(0) = a0e is called the body of a, aS = ∑p 1 a(p) its soul. The numbers a0, aI1 … Ip are real or complex, aI1 … Ip is totally antisymmetric in I1 … Ip; the degree of a(p) is p.

body

soul

degree

B+ consists of the formal series which contain only terms of even degree, B− consists of those with only terms of odd degree. B+ is a subalgebra of B, while B− is not.

Show that if ab = 0 for all b ∈ B+ [resp. b ∈ B−] then a = 0.

Are these properties true in a finitely generated Grassmann algebra?

Answer

If ab = 0 for all b belonging to B+, or to the even part of a finitely generated Grassmann algebra we see that a = 0 by taking b = e. Suppose now ab = 0 for all b ∈ B−. In particular azI = 0 for each zI, I . Suppose a coefficient aI1 … Ip ≠ 0. Choose zJ ∉ (zI1,…, zIp). We have

If there is a finite number N of generators the hypothesis ab = 0 for all odd b implies only

B is endowed with a locally convex, metrizable, Hausdorff topology by the countable family of seminorms (cf. for instance, p. 424)

The sum of formal series (in particular the decomposition B = B+ ⨁ B−) and their product have the required continuity.

Show that: The partial sums

converge to a, in the B-topology, when m tends to infinity.

Answer

If || ||I1 … Ip is a seminorm on B we have exactly

Let f(x) = ∑∞n=0 cnxn be a numerical series with radius of convergence ρ. Show that f(a) = ∑ cnan is a well-defined formal series in B, depending continuously on a, if |a0| < ρ.

Answer

We have a = a0e + aS, so

Since f(x) is convergent for |x| < ρ, the numerical series ∑n p cn Cpn an−p0 are convergent for |a0| < ρ. We denote their sum by αp and we write

Each term on the right-hand side is well defined: b(q) is obtained by finite sums and products since a term of order q arises from apS only when p q.

In a similar spirit one proves that the inverse in B of an element a with a0 ≠ 0 is the formal series:

2

BEREZINIAN

A graded matrix on a graded algebra A is a rectangular array of elements of A, together with a parity attached to each row and column. A square graded matrix with p even and q odd rows and columns is said to be of order (p, q).

graded matrix

A graded matrix X= (xij) is called even [resp. odd] if for all i, j:

one then says that d(X) = 0[resp. d(X) = 1].

order (p, q) even, odd

1) We shall always suppose that in a graded matrix X of order (p, q)

the p even rows and columns are written first.

Give the conditions on the parities of the elements of R, S, T, U for X to be even [resp. odd].

Answer

The parities of the columns of R, T are even, those of S, U odd, while the rows of R, S are even and of T, U odd. Thus d(X) = 0 if and only if the elements of R, U are even and the elements of T, S odd. The opposite condition holds for d(X) = 1.

2) Show that the space Matp,q(A) of graded matrices of order (p, q) forms a 2 graded algebra.

Answer

The space Matp,q(A(like A), and each element can be written as the sum (usual sum of matrices) of an even and an odd one.

The elements in the product are defined by the usual law

It is easy to check that if X and X′ have a parity, then

3) Let B be a DeWitt algebra. Denote by GLp,q(B) the multiplicative group of even invertible graded matrices of order (p, q).

a) Let

.

Show that X is invertible if and only if R and U are invertible.

b) The determinant of a square matrix with even elements in B is well defined by the usual polynomial. The Berezinian of a matrix X ∈ GLp,q(B) is the mapping GLp,q(B) → B given by

Show that Ber X is even valued and invertible.

Berezinian

c) Show that

Answer a

Under the hypothesis the body of X is

which is invertible if R0 and U0 are invertible.

Answer b

Ber X is even because R and U have even elements, S and T odd elements. It is invertible because

Answer c

The proof is straightforward, in a number of steps (cf. for instance, Leites, p. 16) using in particular the decomposition

and the fact that any matrix of the form

is a product of matrices of the same type, but with a matrix A having only one nonzero element.

REFERENCES

DeWitt, B. S. Supermanifolds. London: Cambridge University Press, 1984.

The spacetime approach to quantum field theory. In: DeWitt B. S., Stora R., eds. Relativité, Groupes et Topologie II. Amsterdam: North-Holland, 1984. [and Appendix of].

Leites, D. A. Introduction to the theory of supermanifolds. Russian Mathematical Surveys. 1980; 35:1.

3

TENSOR PRODUCT OF ALGEBRAS

A real algebra A endowed with an associative product, A × A → A, bilinear with respect to the vector space structure (cf. a more general definition of algebra, p. 9).

1) Suppose A and B are finite dimensional (as vector spaces) real algebras. Find a natural structure for A ⨂ B.

Answer

Let (ei) and (eα) be basis for A and B respectively. Then e1 ⨂ eα is a basis for A ⨂ B. We define products of such elements by

where juxtaposition denotes product in the relevant algebra.

The product of arbitrary elements c = ciα ei ⨂ eα, d = djβ ej ⨂ eβ is given by

It is easy to show that this product has the required properties and is independent of the choice of basis in A and B.

2) Show that if A is a real algebra, then the complexified algebra A is generated by the complexified vector space A , that is, the vector space spanned by aiei, ei basis of A, ai .

Answer

as a real vector space is (1, i), and the algebra structure is determined by i² = −1; a basis of A is (ej ⨂ 1, ej ⨂ i), which we can denote (ej, iej) without breaking the product law.

3) Example: Tensor products of matrices (see Problem I 4, Clifford algebras). Let A be the space of n × n matrices and B be the space of m × m matrices. Construct a ⨂ b for a ∈ A, b ∈ B.

Answer

Let a = (aij), b = (bαβ) be respectively an n × n and an m × m matrix. Then a ⨂ b = ((a ⨂ b)IJ), where the indices I and J stand for a pair of indices (i, α) or (j, β) and (a ⨂ b)IJ = aij bαβ. Usually one orders pairs of indices as follows: (1, 1), (1, 2),…, (2, 1), (2, 2),….

Note

In Problem IV 2, Obstruction, following Atiyah, Bott and Shapiro, we shall use the graded tensor product of two graded algebras defined as follows. Let A = ∑i=0, 1 Ai and B = ∑i=0, 1 Bi be two graded algebras. The graded tensor product is, by definition, the algebra whose underlying vector space is ∑i, j=0, 1 Ai ⨂ Bj with multiplication defined by

where xi, [resp. yj] is an element of Bi [resp. Aj], u [resp. v] is an arbitrary element of A [resp. B].

graded tensor product

The graded tensor product is again a graded algebra

For example, consider the odd element e1 ⨂ 1 + 1 ⨂ e2, its square e²1 ⨂ 1 + 1 ⨂ e²2 is even.

4

CLIFFORD ALGEBRAS

A supplement to this problem entitled "The isomorphism H ⨂ H M)" can be found near the end of the book.

1 INTRODUCTION

Let V be a real d = n + m dimensional vector space with a pseudo-euclidean scalar product g, invariant under the group O(n, m), given by g = (gAB), gAB = 0 if A ≠ B, gAA = 1, A = 1,…, n, gAA = −1 if A = n + 1,…, n + m. The Clifford algebra (p. is the real vector space endowed with an associative product, distributive with respect to addition, generated by d symbols¹ γA, and their products which satisfy

(1)

has dimension 2dis

We set

thus γA1 … Ap = γA1 γA2 … γAp if Ai < Aj for i < j. We set also

It has been proved (p. , when d when d have the same complex extension if n + m = n′ + m′.

Case d = n + m = 2p even

is isomorphic to the algebra M2p) over the complex numbers of 2p × 2p ²p. A set of such 2p × 2p matrices ΓA representing the fundamental elements γA are called gamma matrices of O(n, m) or gamma matrices for . They satisfy the identities

gamma matrices of O(n, m

It can be proved that if (ΓA) and (Γ′A) are two sets of gamma matrices of O(n, m) there exists an invertible 2p × 2p matrix M such that

The algebra over the real numbers generated by the (eventually complex) matrices ΓA , n + m = 2p.

Case d = 2p + 1 odd

on a complex vector space of dimension 2p, n + m = 2p + 1, is isomorphic to M2p ) + M2p may or may not be unfaithful (cf. 2c) in the following subsection).

In all cases it is interesting to know whether there exists a representation on a real vector space of the real Clifford algebra; the problem is to find, if possible, real matrices ΓA. In this problem we shall show how the gamma matrices can be constructed recursively, and we shall study their structure, which will give a direct proof of properties quoted before, as well as other details such as the periodicity, in (m − n.

2 GAMMA MATRICES IN LOW DIMENSIONS

a) Show that is isomorphic to , and give an algebra isomorphic to .

b) We recall the Pauli matrices (−σ2 is also used)

Give a faithful representation of and a real, faithful representation of and .

Pauli matrices

c) Is it possible to deduce from the Pauli matrices a faithful representation of ? Of ?

Answer 2a

.

.

Answer 2b

The Pauli matrices anticommute and satisfy

(2)

and two of the matrices iσ1, iσ2, iσ3; a basis of this 4-dimensional real vector space is

The product laws of quaternions (see also Problem V 10, Invariant geometries).

, σ1, σ3, iσ2. It is identical to the vector space Mis isomorphic to the algebra M).

, one of the two real matrices σ1 or σ3, and iσ2; it is also isomorphic to M).

Answer 2c

they generate an algebra which is, by formula .

3 GAMMA MATRICES IN ARBITRARY DIMENSIONS

a) Set, if n + m = d = 2p

show that

Thus if ΓA , and

then

Answer 3a

Straightforward computation using the fundamental relation (1).

b). Show that if Γa, a = 1,…, d, is a set of gamma matrices of O(n, m), n + m = 2p then (Γa, kΓd+1) is a set of gamma matrices of O(n, m + 1) if k² = (−1)n+p [resp. of O(n + 1, m) if k² = (−1)n+p+1].

Answer 3b

Γd+1 anticommutes with all Γa if d if k² = (−1)n+p if k² = (−1)n+p+1].

c) Show that if n + m = d = 2p is even then we have (the sign means isomorphic to)

Construct a representation by real matrices of .

Answer 3c

The d + d′ elements

(3)

satisfy the relation (cf. Problem I 3, Tensor product)

and anticommute, because γa anticommute with γd], depending on the sign of k² = (−1)n+phas dimension 2d × 2d′ = 2d+d′.

We have in particular (here n = 1, p = 1)

by using formulas ; these give the Dirac matrices in the so-called Majorana representation see [Problem I 8, Weyl]. (Note that the tensor product of two matrices (Aij), (Bi′j′) has elements AijBi′j′ = (A ⨂ B)ii′jj′.)

Majorana representation

d) The previous construction shows that gamma matrices for an arbitrary group O(n, m) can be obtained from the Pauli matrices σj or iσj. Show that:

i) All the constructed gamma matrices, as well as the algebra they generate not included) have a zero trace.

Prove this property directly.

ii) The matrices are linearly independent on .

iii) The matrices ΓA of O(n, m) are antihermitian for A = 1,…, n and hermitian for A = n + 1,…, d, in the choice of Clifford algebra defined by (1).

Answer 3d i

The Pauli matrices have a zero trace, and the trace of a tensor product is the tensor product of the traces.

To prove directly that the trace of ΓA1…Ap is zero we proceed as follows. If p is even

but

If p is odd

and the same argument as in the case p even gives

Answer 3d ii

Suppose there exist complex numbers not all zero such that

By taking the trace one obtains λ = 0; by taking the trace with product of the gamma matrix ΓA one obtains λA = 0, and so on.

Answer 3d iii

, are all hermitian, while the iσj are antihermitian. A tensor product of matrices which are either hermitian or antihermitian is either hermitian or antihermitian. The conclusion follows by inspection.

4 PERIODICITY MODULO 8

a) Express , for arbitrary n, in terms of tensor products of and .

Show that

(4)

where M) is the algebra of real 16 × 16 matrices (using the relation M), where is the algebra of quaternions).

b) Show that, if m > n

(5)

These two relations show that all the Clifford algebras are determined through the properly euclidean ones, and their classification depends upon n − m modulo 8. Write the classification table.

Answer 4a

Using the result of paragraph 4 with k² = −1 we obtain:

The tensor product of vector spaces is commutative, up to an isomorphism. It is easy to see that if Mn ) denotes the algebra of n × n real matrices

and it is known that (the proof is given in a problem at the end of the book)

formula (4) follows.

Answer 4b

The proof of (5) is analogous:

The table reads, with d = n + m, abbreviating Mp(p(p) denoting the algebra of p × p quaternionic matrices:

Note that for d even the Clifford algebra is isomorphic to either the algebra of real matrices or the algebra of quaternionic matrices: for n admits a real representation for m = 1, 2, 3 mod. 8, thus d = 2, 3, 4, 10, 11, 12, etc.

are not isomorphic, unless |m − n| = 0 mod 4.

The explicit construction of the table is done in [M. Berg et al., see p. 39] where the Clifford algebra is not .

5 GRADING OF A CLIFFORD ALGEBRA

a) Give a 2-grading to a Clifford algebra.

b) Show that the even parts of and are isomorphic.

c) Show that

Answer 5a

is a direct sum

(6)

, is generated by the even products of elements of a basis:

, called the odd part, is generated by the odd products:

if d if d is odd. We have obviously

(7)

(8)

The laws 2 commutative (Problem I 1).

Answer 5b

and γA, A = 1,…, n + m, with a product such that

Consider a vector space V , elements denoted iγA, A = 1,…, n + mand the products of complex numbers, namely,

(9)

We endow the vector space V with the structure of an algebra through the product so defined. We deduce from (9) that

hence V. It results also from are identical:

(10)

Answer 5c

and (γA) = (γ0, γa), a = 1,…, p + q . Then

and the γaand the even products γ0γa, γaγb. We deduce the isomorphism

(11)

from the relation

The equalities (11) and (10) imply

which can also be read from the table of §4.

5

CLIFFORD ALGEBRA AS A COSET OF THE TENSOR ALGEBRA

Let V be a real vector space with a symmetric bilinear form V × V given by (v, wb(v, w). Let us denote by the tensor algebra of V. We say that the subset of the algebra is generated by the subset {τa, a ∈ A family of indices} of if the elements of are the finite sums

1) Show that the subset of is a left ideal of .

2) Let be generated by elements of the form

Show that the coset space is isomorphic to

a) The exterior algebra of V if b = 0.

b) The Clifford algebra if b is a nondegenerate pseudo-euclidean scalar product g.

Answer 1

is a left ideal (p. 8).

left ideal

Answer 2

There is a natural map from V , namely v [v], where [v. It is easy to see that the equivalence class [v ⨂ w] depends only on [v] and [w] and thus defines a product for these elements of C\T:

To prove the isomorphisms stated in a) and b) we compute

(1)

thus, since b is bilinear and symmetric, we have

(2)

a) If b = 0 relation and the exterior algebra of V.

b) If b is an inner product, there is an isomorphism between Vdefined by {[v], v ∈ V} and the subspace {γv, v ∈ V.

6

FIERZ IDENTITY

Here we use the generic term spinors for both spinors and pinors. For the definition of spinors and cospinors see pp. 415, 418 and [Problem IV 2, Obstruction].

The following identity is useful in many computations, particularly in supergravity. Let φ, ψ be arbitrary spinors on a d = 2p dimensional vector space with pseudo-euclidean product g, and arbitrary cospinors. Prove the equality of the two scalars (juxtaposition denotes the duality product of a spinor and a cospinor)

(1)

where ΓI, I = 1,…, 2d by 2p × 2p complex (or real) matrices and ΓI = (ΓI)²ΓI.

Answer

Let S be the (complex) vector space of spinors, Sis isomorphic to the algebra S* ⨂ S of linear mappings S → S.

The left-hand side of (1) defines a quadrilinear map (S* × S) × (S* × S. It can thus be written as

where the αIJ are numbers which we shall determine. The above equality can be written, if a, b denote indices for components in S and S* respectively, as

We determine the αIJ by taking traces of products with various ΓK, as follows:

(2)

We know that

so (2) reduces to

from which we deduce

Each matrix ΓK . If

then

The conclusion follows.

Example d = 4: we take as matrices ΓI

Then

The formula can then be written:

7

PIN AND SPIN GROUPS

To be in agreement with modern physics notation we shall modify somewhat the terminology used (p. 67). We shall give the fundamental properties of Spin(n, m) and Pin(n, m), for arbitrary dimensions. We treat in the first paragraph the case n + m even, which is easier, using a matrix representation. In paragraph two, we treat the general case by using the grading of the Clifford algebra, as done by Atiyah, Bott and Shapiro in the euclidean case.

Recall the following definitions. Let GL(V) denote the group of isomorphisms (linear, invertible maps) from a real vector space V onto itself; V has dimension d = n + m and g is a pseudo-euclidean scalar product on V of signature (n, m) (i.e., with n plus signs and m minus signs):

Orthogonal group O(n, m) = {L ∈ GL(V), g(Lu, Lv) = g(u, v)}.

(special) orthogonal group identity component

Special orthogonal group SO(n, m) = {L ∈ O(n, m), det L = 1}.

Identity (connected) component of O(n, m): SO0(n, m).

In the euclidean case (n or m = 0), SO(n, m) = SO0(n, m) but in the general case they are not equal. In the Lorentz case, SO(m, 1) and