Marathe K. Topics in Physical Mathematics

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378 Epilogue

recent lecture

by Curtis McMullen (Fields Medal, ICM 1998, Berlin) was

entitled “From Platonic Solids to Quantum Topology.” McMullen weaves

a fascinating tale from ancient to modern mathematics pointing out unex-

pected links between various areas of mathematics and theoretical physics.

He concludes with the statement of a special case of the volume conjecture,

interpreting it as the equality between a gauge-theoretic invariant and a topo-

logical gravity invariant.

In view of all this activity we might liken this book with a modern tour

through many countries in a few days. When you return home you can look

at the pictures, think of what parts you enjoyed and then decide where you

would like to spend more time. I hope that readers found several parts en-

joyable and perhaps some that they may want to explore further. The vast

and exciting landscape of physical mathematics is open for exploration.

We began this book with a poem dedicated to the memory of my mother

and we conclude with a poem which touches upon some of the great discov-

eries that have shaped our understanding of nature.

In the beginning

God said:

Let there be gauge theories!

And there was light,

clear and bright

followed by particles

fundamental or otherwise.

They jostled along merrily,

sometimes weakly, sometimes strongly,

creating and annihilating things

small and big.

They hurtled along paths

as straight as could be.

And it was called

the miracle gravity.

Three fundamental forces

united in the standard model

now wait patiently to be

tied up with the fourth.

The 2009 Reimar L¨ust lecture in Bonn delivered on June 12.

Appendix A

Correlation of Terminology

As we remarked earlier, gauge theories and the theory of connections were

developed independently by physicists and mathematicians, and as such have

no standard notation. This is also true of other theories. To help the reader

we present a table describing the correlation of terminology between physics

and mathematics, prepared along the lines of Trautman [378]and[409].

Physics Mathematics

Space-time Lorentz 4-manifold M

Euclidean space-time Riemannian 4-manifold M

Gauge group G Structure group of a principal

bundle P (M, G)overM

Space of phase factors Total space of the bundle

Gauge group bundle Ad(P)=P ×

G,whereAdis

the adjoint action of G on itself

Gauge transformation A section of the bundle Ad(P )

Gauge algebra bundle ad(P )=P ×

g,whereadis

the adjoint action of G on g

Inﬁnitesimal gauge A section of the bundle ad(P )

transformation

Gauge potential ω on P Connection 1-form ω ∈ Λ

(P, g)

Global gauge s A section of the bundle P (M, G)

Gauge potential A

on M A = s

∗

(ω)

Local gauge t A section of the bundle P (M,G),

over an open set U ⊂ M.

Local gauge potential A

= t

∗

(ω)

Gauge ﬁeld Ω on P Curvature d

ω = Ω ∈ Λ

(P, g)

Gauge ﬁeld F

on M The 2-form F

∈ Λ

(M,ad(P))

associated to Ω

Group of projectable Diﬀ

(P )={f ∈ Diﬀ(P ) |

transformations f covers f

∈ Diﬀ(M)}

379

380 Appendix A

Correlation of Terminology (continued)

Physics Mathematics

Group of generalized Aut(P )={f ∈ Diﬀ(P ) |

gauge transformations f is G-equivariant}

Group G of gauge Aut

(P )={f ∈ Aut(P ) |

transformations f

= id ∈ Diﬀ(M)}

Group G

⊂G,of Subgroup of Aut

(P )of

based gauge transformations based bundle automorphisms

Group G

, of eﬀective The quotient group of G

gauge transformations by its center Z(G)

Gauge algebra LG Lie algebra Γ (ad P )

Generalized Higgs ﬁeld φ

A section of the associated

bundle E(M,F,r,P)

Higgs ﬁeld φ φ

,withr the fundamental

representation

Bianchi identities for the d

gauge ﬁeld F

Yang–Mills equations for F

Instanton (resp., anti-instanton) ∗F

= ±F

,where∗ is

equations on a 4-manifold M the Hodge operator

Instanton number of P (M,G) The Chern class c

(P )

BPST instanton Canonical SU(2)-connection

on the quaternionic Hopf

ﬁbration of S

over S

Dirac monopole Canonical U(1)-connection on

the Hopf ﬁbration of S

Inertial frames on a The bundle O(M,g)of

space-time manifold (M,g) orthonormal frames on M

Gravitational potential Levi-Civita connection λ

on O(M,g)

Gravitational ﬁeld The curvature R

of λ

Gravitational instanton [R

, ∗]=0,where∗ is

equations the Hodge operator

Gravitational instanton Einstein space

Appendix B

Background Notes

This appendix contains material that might have hindered the exposition of

various arguments in the text. It contains some historical observations which

I found interesting and which throw additional light on the origin of concepts,

nomenclature, and notation. It also includes biographical notes on some of

the scientists whose work has inﬂuenced me over the years or are mentioned

in the main body of the text.

1. Bernoulli numbers

Bernoulli numbers B

for n ≥ 0 are deﬁned by the formal power series

− 1

∞



n=0

Expanding the power series on the left and comparing its coeﬃcients with

those on the right, we get the Bernoulli numbers. Note that B

=0for

odd n>1.

=1,B

= −

691

2730

= −

3617

510

43,867

798

Bernoulli numbers satisfy the following recurrence relation

=1,B

= −

n +1

n−1



k=0



n +1



This recurrence relation provides a simple but slow algorithm for comput-

ing B

. Bernoulli numbers seem to attract both amateur and professional

mathematician. Srinavasa Ramanujan’s ﬁrst paper was on Bernoulli num-

bers. He proved several interesting results but also made a conjecture that

381

382 Appendix B

turned out to be false. For more information about the Bernoulli numbers,

see the website (www.bernoulli.org).

An important property of the Bernoulli numbers is their appearence in

the values of Euler’s zeta function for even natural numbers 2n, n ∈ N.

Euler deﬁned and proved a product formula for the zeta function

ζ(s):=

∞



m=1

−s



(1 −p

−s

),s>1,

where the product is over all prime numbers p. He evaluated the zeta

function for all even numbers to obtain the following formula:

ζ(2n)=−

(2πi)

2(2n)!

He also computed approximate values of the zeta function at odd numbers.

There is no known formula for these values similar to the formula for even

numbers. Even the arithmetic nature of ζ(3) was unknown until Apery’s

proof of its irrationality. The nature of other zeta values is unknown.

Riemann generalized Euler’s deﬁnition to complex values of the variable

s. He proved that this zeta function can be extended to the entire complex

plane as a meromorphic function and obtained a functional equation for it.

The extended zeta function has zeros at negative even integers. These zeros

are called the trivial zeros. Riemann conjectured that all the non-trivial

zeros of the zeta function have real part 1/2. This conjecture is known as

the Riemann hypothesis. It is one of the open problems on the Clay

Prize list. For a complete list of Clay mathematics prize problems, see the

website (www.claymath.org).

2. Bourbaki

The ﬁrst world war had a devastating eﬀect on science and mathematics

in France. Jean Dieudonn´e told me that he and his best friend Henri

Cartan followed the course on diﬀerential geometry by Elie Cartan, but

they did not understand much of what he was doing. They turned to their

senior friend Andr´e Weil for help. Even so Dieudonn´e decided to opt for

taking the examination in synthetic geometry. Graduates had large gaps

in their knowledge. After graduation the friends were scatterred around

France. Weil and H. Cartan were teaching calculus in Strasbourg using E.

Goursat’s well known text. Cartan turned to Weil for advice on the course

so frequently that Weil decided it was time to write a new treatise for the

course. Weil called some of his old friends (Jean Delsarte, Dieudonn´e, and

Claude Chevalley) to join him and H. Cartan with the simple idea to write

a new text to replace the text by Goursat.

This is the story of the birth of Bourbaki. The young men eager to ﬁnish

this simple project had no idea what they were getting into. After a few

meetings they realized that they had to start from scratch and present all

Appendix B 383

the essentials of mathematics then known. They believed that they would

have the ﬁrst draft of this work in three years. They met for their ﬁrst

congress in 1935. Eventually, they chose to use the name “Nicholas Bour-

baki” for their group, an invented name. It went on to become the most

famous group of mathematicians in the history of mathematics. Bourbaki’s

founders were among the greatest mathematicians of the twentieth century.

Their work changed the way mathemtics was done and presented. Their

active participation in the Bourbaki group ended when a member turned

50 (this was the only rule of Bourbaki). The ﬁrst chapter of Bourbaki’s

nine volumes (consisting of 40 books) came out four years later, rather

than three. The last volume “Spectral Theory” was published in 1983. For

a long time the membership and the work of the Bourbaki group remained

a well guarded secret. Jean Dieudonn´e broke the silence in his article [100].

All the members were called upon to write various drafts of the chapters

but the ﬁnal version for printing was prepared by Dieudonn´e. (This ex-

plains the uniformity of the style through most of the forty books written

by Bourbaki.) Over the years membership in the Bourbaki group changed

to include some of the most inﬂuential mathematicians of the twentieth

century such as A. Grothendieck, S. Lang, J. P. Serre, and L. Schwartz.

The rapid progress in the physical sciences and increasing abstraction in

mathematics caused an almost complete separation of physics and mathe-

matics. Even the mathematics used in theoretical physics did not have the

rigor required in modern mathematics. Extreme generality in Bourbaki

infurianted many mathematicians. For example, nobody had ever studied

Euclidean geometry as a special case of the theory of Hermitian opera-

tors in Hilbert space even though Dieudonn´esaysin[100] that this is

well known. In his article [99] on the development of modern mathematics

Dieudonn´e was stressing the fact that mathematics in the second half of

the twentieth century had become a self sustaining ﬁeld of knowledge, not

depending on applications to other sciences. Thus, while the development

of mathematics since antiquity to the ﬁrst half of the twentieth century

was strongly inﬂuenced by developments in the physical sciences, a new

chapter in the history of mathematics had now begun.

As we pointed out in the preface, this article is often quoted to show

that the abstraction stressed by Bourbaki was the major cause of the

split between mathematics and the physical sciences. In fact, in the same

article Dieudonn´e clearly stated that a dialogue with other sciences, such

as theoretical physics, may be beneﬁcial to all parties. He did not live to

see such a dialogue or to observe that any dialogue has been far more

beneﬁcial to the mathematics in the last quarter century. I would like to

add that in 2001 a physics seminar “Le S´eminaire Poincar´e” modeled after

the well-known “S´eminaire Bourbaki” was createdbytheAssociationdes

Collaborateurs de Nicolas Bourbaki.

I had several long discussions with Prof. Dieudonne. Most of the time,

I asked the questions and he never tired of giving detailed answers. He

384 Appendix B

was an embodiment of history. He spoke many languages ﬂuently and had

phenomenal memory. He enjoyed Indian food and was a frequent dinner

guest at our home. This created a rather informal setting for our discus-

sions. Early on I asked him about a statement in the preface to the ﬁrst

volume of Bourbaki. It said that no background in university mathematics

was necessary to read these books. I asked him if there was any example

of this. He immediately replied in the aﬃrmative and gave the following

account. There was a high school student in Belgium whom the mathe-

matics teacher found quite a handful. So he gave him Bourbaki’s book

on set theory to read over the holidays. When the school reopened the

student reported that he enjoyed the book very much but found out that

he could prove one of the stated axioms from the others. The teacher was

totally unprepared for this and decided to send a letter to Bourbaki with

his student’s proof. Dieudonn´e (the unoﬃcial secretary of Bourbaki) knew

that the group had not started from a minimal set of axioms. He checked

and found the proof correct. The student’s name is Pierre Degline (b.

1944, Fields medal, ICM 1978, Helsinki) who is now a professor of math-

ematics at IAS (the Institute for Advanced Study), Princeton. I came to

know Prof. Degline during my visits to IAS, especially during the special

year on QFT and string theory. During a social evening, I ﬁnally found the

courage to ask him about the above story. He was very surprised and asked

me how I knew it. I explained that I had heard it from Prof. Dieudonn´e

during one of his visits to my home. To this day I have never heard of any

other person who started to read Bourbaki before entering the university.

3. Fields Medals

It is well known that there is no Nobel Prize in mathematics. There are

several stories about why this queen of sciences was omitted, most of them

unsupported by evidence. Many mathematicians wanted an internation-

ally recognized prize comparable in importance to the highly regarded No-

bel Prize in the sciences and other areas. Canadian mathematician John

Charles Fields actively pursued this idea as chairman of the committee

set up for the purpose of organizing the ICM 1924 in Toronto. The idea

of awarding a medal was well supported by several countries. The Fields’

committee prepared an outline of principles for awarding the medals. The

current rule that the awardee not be more than 40 years old at the time

the award is granted stems from the principle that the award was to be in

recognition of work already done, as well as to encourage further achieve-

ment on the part of the recipient. Another statement indicated that the

medals should be as purely international and impersonal as possible and

not be attached in any way to the name of any country, institution, or

person.

Fields died before the opening of the ICM 1932 in Z¨urich opened, but

the proposal was accepted by the congress. The medal became known as

the Fields Medal (against Fields’s wishes) since it was awarded for the

ﬁrst time at ICM 1936 in Oslo to Lars Ahlfors and Jesse Douglas. On

Appendix B 385

one side of the medal is engraved a laurel branch and a diagram of a

sphere contained in a cylinder from an engraving thought to have been

on Archimedes’ tomb. The Latin inscription on it may be translated as

“Mathematicians, having congregated from the whole world, awarded (this

medal) because of outstanding writings.” On the obverse is the head of

Archimedes surrounded by the Latin inscription from the ﬁrst century

Roman poet Manilius’s Astronomica. It may be translated as “to pass

beyond your understanding and make yourself master of the universe.” The

complete passage from which this phrase is taken is strikingly similar to

verses in many parts of the Vedas, the ancient Indian scriptures. Manilius

writes:

The object of your quest is god; you are seeking to scale the skies

and though born beneath the rule of fate, to gain knowledge of that

fate; you are seeking to pass beyond your understanding and make

yourself master of the universe. The toil involved matches the reward

to be won, nor are such high attainments secured without a price.

The Fields medalists have certainly fulﬁlled the expectations for contin-

ued achievements. However, the monetary value (currently about $15,000

Canadian) makes it a poor cousin to the Nobel prize, but is consistent with

that old adage: “One does not go into mathematics to become rich.” Here,

I am also reminded of the remark made by the famous American comic

Will Rogers when he was awarded the Congressional Medal of Honor. He

said: “With this medal and 25 cents I can now buy a cup of coﬀee.” In fact,

several other prizes for which mathematicians are eligible far exceed the

monetary value of the Fields Medal. In particular, the Abel prize instituted

in 2003 and given annually is closer in spirit and monetary value to the

Nobel prize. Jean-Pierre Serre (the youngest person to receive the Fields

Medal at ICM 1954, in Amsterdam, at age 27) received the ﬁrst Abel Prize

in 2003. The Abel Prize for 2009 has been awarded to the Fields medalist

M. L. Gromov. More information on the Abel Prize may be found at the

oﬃcial website (www.abelprisen.no/en/).

I would like to add that of the 28 Fields medalists since the ICM

1978 to the ICM 2006, 14 have made signiﬁcant contribution to advanc-

ing the interaction of mathematics and theoretical physics. Ed Witten is

the only physicist to be awarded the Fields Medal (ICM 1990, Keyoto).

More information about the Fields medal can be found at the IMU website

(www.mathunion.org).

4. Harmonic Oscillator

The quantization of the classical harmonic oscillator was one of the earliest

results in quantum mechanics. It admits an exact closed-form solution in

terms of the well-known Hermite polynomials. The oscillator’s discrete en-

ergy spectrum also showed that the lowest energy level need not have zero

energy and it can be used to explain the spectrum of diatomic molecules

and to provide a tool for approximation of the spectra of more complex

molecules. It was used in the early study of the black body radiation in the

386 Appendix B

theory of heat and it can be applied to understand the motion of atoms in

lattice models of crystals. We remark that the classical harmonic oscillator

was the ﬁrst dynamical system that was quantized through the canonical

quantization principle.

We now discuss the canonical quantization of a single classical harmonic

oscillator. The Hamiltonian H of the particle of mass m vibrating with

frequency ω is given by

H(p, q)=

mω

where p, q are the conjugate variables momentum and position. In canon-

ical quatization they are replaced by operators ˆp, ˆq on the space of wave

functions deﬁned by

ˆp(ψ):=−i

dψ

, ˆq(ψ):=q(ψ),ψ∈ L

(q).

These operators satisfy the cononical commutation relation

[ˆq, ˆp]=i.

This prescription gives us the Schr¨odinger equation

H(ψ)=E(ψ)forthe

wave function ψ (eigenfunction of

H) with energy level E (eigenvalue of

H). It is a second order ordinary diﬀerential equation

−

ψ +

mω

ψ = Eψ.

The set of values of E for which the Schr¨odinger equation admits a solution

is called the spectrum of the quantum harmonic oscillator. It can be shown

that the spectrum is discrete and is given by

= ω(n +

),n≥ 0.

The corresponding wave function ψ

is given by the explicit formula

(x)=





1/4

(x)e

−x

√

, where α = mω/,x= q

√

α.

In the above formula H

(x)isthenth Hermite polynomial, deﬁned by

(x)=(−1)

(

)

−x

Dirac deﬁned creation (or raising) and annihilation (or lowering) op-

erators, collectively called the ladder operators, which allow one to ﬁnd the

spectrum without solving the Schr¨odinger equation. (Feynman used this

Appendix B 387

result to test his path integral quantization method.) These operators can

be deﬁned for more general quantum mechanical systems and also gener-

alize to quantum ﬁeld theory. We now indicate their use for the harmonic

oscillator. The creation and annihilation operators a

†

and a are related to

the position and momentum operators by the relations

†

+ a =



α/2ˆq, a

†

− a =



2/αˆp

i

From these relations it is easy to show that

†

(ψ

√

n +1ψ

n+1

,a(ψ

√

nψ

n−1

It is these relations that led to the names “creation” (or raising) and

“annihilation” (or lowering) operators. It follows that

[a, a

†

]=I, (a

†

a)(ψ

)=nψ

In view of the second relation the operator (a

†

a) is called the number

operator. Using the deﬁnitions of a

†

and a we can write the quantum

Hamiltonian

H as

H = ω(a

†

a +

The energy spectrum formula follows immediately from the above expres-

sion. It is these energy levels that appear in Witten’s supersymmetric

Hamiltonian and its relation to the classical Morse theory. The harmonic

oscillator serves as a test case for other methods of quantization as well.

The geometric quantization method applied to coupled harmonic oscilla-

tors leads to the same results as shown in [271,272]. This method requires

the use of manifolds with singularities.

5. Parallel Postulate The parallel postulate was the ﬁfth postulate or ax-

iom in Euclid’s geometry. For those readers who have not seen it recently,

here is the statement:

Let L be a straight line in a plane P . Through every point x ∈ P

not lying on the line L, there passes one and only one straight line

that is parallel to L.

The parallel postulate is much more complicated and non-intuitive than

the other postulates of Euclidean geometry. Euclid attempted to prove this

postulate on the basis of other postulates. He did not use it in his early

books. Eventually, however, it was necessary for him to use it, but he wrote

his 13 books without ﬁnding a proof. Finding its proof became a major

challenge in geometry. Over the next 2000 years, many leading mathemati-

cians attempted to prove it and announced their “proofs.” There are more

false proofs of this statement than any other statement in the history of

mathematics. It turns out that most of these attempts contain some hid-

den assumption which itself is equivalent to the parallel postulate. Perhaps