Preface

A fundamental problem with quantum theories of gravity, as opposed to

the other forces of nature, is that in Einstein’s theory of g r avity, general

relativity, there is no background geometry to work with: the ge ometry

of spacetime itself becomes a dynamical variable. This is loosely sum-

marized by saying that the theor y is ‘generally covariant’. The standard

model of the strong , weak, and electromagnetic forces is dominated by the

presence of a ﬁnite-dimensional group, the Poincar´e group, acting as the

symmetries of a spac e time with a ﬁxed geometrical structure. In general

relativity, on the other hand, there is an inﬁnite-dimensional group, the

group of diﬀeomorphisms of spacetime, which permutes diﬀerent mathe-

matical descriptions of any given spacetime geometry satisfying Einstein’s

equations. This causes two diﬃcult problems. On the one hand, one can no

longer apply the usual criterion for ﬁxing the inner product in the Hilbert

space of states, namely that it be invariant under the geometrical symmetry

group. This is the so-called ‘inner product problem’. On the other hand,

dynamics is no longer encoded in the action of the geometrical symmetry

group on the space of states. This is the so-called ‘problem of time’.

The prospects for developing a mathematical framework for quantizing

gravity have been consider ably changed by a number of developments in

the 1980s that at ﬁrst seemed unrelated. On the one hand, Jones discov-

ered a new polynomial invariant of kno ts and links, prompting an intensive

search for generalizations and a unifying framework. It soon became clear

that the new kno t polynomials were intimately related to the physics o f 2-

dimensional systems in many ways. Atiyah, however, raised the challenge

of ﬁnding a manifestly 3-dimensional deﬁnition of these polynomials. This

challenge was met by Witten, who showed that they occur naturally in

Chern–Simons theory. Chern–Simons theory is a quantum ﬁeld theory in

3 dimensions that has the distinction of being generally covariant and also

a gauge theory, that is a theory involving connections on a bundle over

spacetime. Any knot thus determines a quantity called a ‘Wilson loop,’

the trace of the holonomy of the connection around the knot. The knot

polynomials are simply the ex pecta tion values of Wilson loops in the vac-

uum state of Chern–Simons theory, and their diﬀeomorphism invariance

is a consequence of the general covaria nce o f the theory. In the explosion

of work that followed, it became clear that a useful framework for under-

standing this situation is Atiyah’s axiomatic description of a ‘topological

quantum ﬁeld theory’, or TQFT.

On the other hand, at about the same time a s Jones’ initial discov-

ery, Ashtekar discovered a reformulation of general relativity in terms of

v

vi Preface

what are now called the ‘new variables’. This made the theory much more

closely resemble a gauge theory. The work of Ashtekar proceeds from the

‘canonical’ approach to qua ntization, meaning that solutions to Einstein’s

equations are identiﬁed with their initial data on a given spacelike hyper-

surface. In this approach the action of the diﬀeomorphism group gives rise

to two cons traints on initial data: the diﬀeomorphism constraint, which

generates diﬀeomorphisms preserving the spacelike surfa ce, and the Hamil-

tonian constraint, which g e nerates diﬀeomo rphisms that move the surfac e

in a timelike direction. Fo llowing the procedure invented by Dirac, one

exp ects the physical states to be annihilated by certain operators corre-

sp onding to the constraints.

In an eﬀort to ﬁnd such states, Rovelli a nd Smolin turned to a descrip-

tion of quantum general relativity in terms of Wilson loops. In this ‘loop

representation’, they saw that (at least formally) a large space of states

annihilated by the constraints ca n be described in terms of invariants of

knots and links. The fact that link invariants should be annihilated by the

diﬀeomorphism constraint is very natural, since a link invariant is nothing

other than a function from links to the complex numbers tha t is invariant

under diﬀeomorphisms of space. In this sense, the relation between knot

theory and quantum gravity is a natural one. But the fact that link in-

va riants should also be annihilated by the Hamiltonian constraint is deeper

and more intriguing, since it hints at a rela tionship be tween knot theory

and 4-dimensional mathematics.

More evidence for such a relationship was found by Ashtekar and Ko-

dama, who discovered a strong connection between Chern–Simons theory

in 3 dimensions and quantum gravity in 4 dimensio ns. Namely, in terms

of the loop r epresentation, the same link invariant that arises from C hern–

Simons theory—a certain normalization of the Jo nes polynomial called the

Kauﬀman bracket—also represents a state of quantum gravity with cosmo-

logical constant, essentially a quantization of anti-deSitter s pace. The full

ramiﬁcations of this fac t have yet to be explored.

The present volume is the proceedings of a workshop on knots and

quantum gravity held on May 14th–16th, 19 93 at the University o f C ali-

fornia at Riverside, under the auspices of the Depa rtments of Mathematics

and Physics. The goal of the workshop was to bring together researchers

in quantum gravity and knot theory and pursue a dialog between the two

subjects.

On Friday the 14 th, Dana Fine began with a talk on ‘Chern– Simons

theory and the Wes s–Zumino–Witten model’. Witten’s original work deriv-

ing the Jones invariant of links (or, more precisely, the Kauﬀman bracket)

from Chern–Simons theory used conformal ﬁeld theo ry in 2 dimensions as

a key tool. By now the relationship between co nformal ﬁeld theory and

topological quantum ﬁeld theories in 3 dimensions has been explored from

a number of viewpoints. The path integral approach, however, has not yet

Preface vii

been worked out in full mathematical rigor. In this talk Fine describ e d

work in progress on reducing the Chern–Simons path integral on S

3

to the

path integral for the Wess– Zumino–Witten model.

Oleg Viro spoke on ‘Simplicial topological quantum ﬁeld theories ’. Re-

cently there has been increasing interest in formulating TQFTs in a man-

ner that relies upon triangulating spacetime. In a sense this is an old idea,

going back to the Regge–Ponzano model of Euclidean quantum gravity.

However, this idea was given new life by Turaev and Viro, who rigorously

constructed the Regge–Ponzano model of 3d quantum gravity as a TQFT

based on the 6j symbols for the quantum group SU

q

(2). Viro discussed a

va riety of approaches of presenting manifolds as simplicial complexes, cell

complexes, etc., and methods for constructing TQFTs in terms of these

data.

Saturday began with a talk by Renate Loll on ‘The loop formulation

of g auge theory and gravity’, introduced the loop representation and var-

ious mathematical issues a ssociated with it. She also discussed her work

on applications of the loop representation to lattice gaug e theory. Abhay

Ashtekar’s talk, ‘Loop transforms’, largely concerned the paper with Jer z y

Lewandowski app e aring in this volume. The goal here is to make the loop

representation into rigoro us mathematics. The notion of measures on the

space A/G of connections modulo gauge transformations has long been a

key concept in gauge theory, which, however, has been notoriously diﬃ-

cult to make precise. A key no tion developed by Ashtekar and Isham for

this purpose is that of the ‘holonomy C*-algebra’, an algebra of observ-

ables generated by Wilso n loops. Formalizing the notion of a measure on

A/G as a state on this algebra, Ashtekar and Lewandowski have been able

to construct such a state with remarkable symmetry proper ties, in some

sense analogous to Haar measure on a compact Lie group. Ashtekar also

discussed the implications of this work for the study of quantum gravity.

Pullin spoke on ‘The quantum Einstein equa tions and the Jones poly-

nomial’, describing his work with Bernd Br¨ugmann and Rodolfo Ga mbini

on this subject. He also sketched the proof of a new result, that the coef-

ﬁcient of the 2nd term of the Alexander–Conway polynomial r epr e sents a

state of quantum gravity with zero cosmologic al constant. His paper with

Gambini in this volume is a more detailed proof of this fac t.

On Saturday afternoon, Louis Kauﬀman spoke on ‘Vassiliev invari-

ants and the loop states in quantum gravity’. One aspect of the work

of Br¨uegmann, Gambini, and Pullin that is especially of interest to knot

theorists is that it involves extending the bracket invaria nt to generalized

links admitting certain kinds of self-intersections. This concept also plays a

major role in the study of Vassiliev invariants of knots. Curio us ly, however,

the extensions occurring in the two cases are diﬀerent. Kauﬀman explained

their relationship from the path integral viewpoint.

Sunday morning began with a talk by Ger ald Johnson, ‘Intro duction

viii Preface

to the Feynman integral and Feynman’s operational calculus’, on his work

with Michel Lapidus on rigorous path integral methods. Viktor Ginzburg

then spoke on ‘Vassiliev invariants of knots’, and in the afternoon, Paolo

Cotta-Ramusino spoke on ‘4d quantum gravity and knot theory’. This talk

dealt with his work in progress with Maurizio Martellini. Just as Chern–

Simons theory gives a great deal of infor mation on knots in 3 dimensions,

there appe ars to be a relationship between a certain class of 4-dimensional

theories and so- c alled ‘2-knots’, that is, embedded s urfaces in 4 dimensions.

This class , called ‘BF theories’, includes quantum gravity in the Ashtekar

formulation, as well as Donaldson theory. Cotta-Ramusino and Martellini

have described a way to construct observables as sociated to 2-knots, and

are endeavoring to prove at a pertur bative level that they give 2-knot in-

va riants.

Louis C rane sp oke on ‘Quantum gr avity, spin geometry, and categor-

ical physics’. This was a review of his work on the relation between

Chern–Simons theory and 4 -dimensional quantum gravity, his construc-

tion with David Yetter of a 4-dimensional TQFT ba sed on the 15 j symbols

for SU

q

(2), and his work with Igor Frenkel on certain braided tensor 2-

categories . In the ﬁnal talk of the workshop, John Baez spo ke on ‘Strings ,

loops, knots and g auge ﬁelds’. He attempted to clarify the similarity be-

tween the loop representation of quantum gravity and string theory. At

a ﬁxed time both involve loops or knots in space, but the string-theoretic

approach is also related to the study of 2-kno ts.

As some of the speakers did not submit papers fo r the procee ding s, pa-

pers were also solicited from Steve Carlip and also from J. Scott Carter and

Masahico Saito. Carlip’s paper, ‘Geometric structur e s and loop variables’,

treats the thorny issue of relating the loop representation of quantum grav-

ity to the traditional formulation of gravity in terms of a metric. He treats

quantum gravity in 3 dimensions, which is an exac tly soluble test case. The

paper by Carter and Saito, ‘Knotted surfa c es, braid movies, and beyond’,

contains a review of their work on 2-kno ts as well as a number of new re-

sults on 2-braids. They also discuss the r ole in 4-dimensional topo logy of

new algebraic structures such as braided tensor 2-categories.

The editor would like to thank many people for making the workshop

a success. First and foremost, the speakers and other participants are to

be congratulated for making it such a lively and interesting event. The

workshop was funded by the Departments of Mathematics and Physics

of U. C. Riverside, and the chairs of these departments, Albert Stralka

and Benjamin Shen, were crucial in bringing this ab out. Michel Lapidus

deserves warm thanks for his help in planning the workshop. Invaluable

help in organizing the workshop and setting things up was provided by the

staﬀ of the Department of Mathematics, and particularly Susan Spranger,

Linda Terry, and Chris Truett. Arthur Greenspoon kindly volunteered to

help edit the proceedings. Lastly, thanks go to all the participants in the

Preface ix

Knots and Quantum Gravity Seminar at U. C. Riverside, and especially

Jim Gilliam, Javier Muniain, and Mou Roy, who helped set things up.

x Preface

Contents

The Loop Formulation of Gauge Theory and Gravity 1

Renate Loll

1 Introduction 1

2 Yang–Mills theory and general relativity as

dynamics on connection space 1

3 Introducing loops! 6

4 Equivalence between the connection and loop

formulations 8

5 Some lattice results on loops 11

6 Quantization in the loop approach 13

Acknowledgements 18

Bibliography 18

Representation Theory of Analytic Holonomy C*-

algebras 21

Abhay Ashtekar and Jerzy Lewandowski

1 Introduction 21

2 Preliminaries 26

3 The spectrum 30

3.1 Loop decomposition 30

3.2 Characterization of A/G 34

3.3 Examples of

¯

A 37

4 Integration o n A/G 39

4.1 Cylindrical functions 40

4.2 A na tur al measure 45

5 Discussion 49

Appendix A: C

1

loops and U(1) connections 51

Appendix B: C

ω

loops, U (N) and SU(N) connections 56

Acknowledgements 61

Bibliography 61

The Gauss Linking Number in Quantum Gravity 63

Rodolfo Gambini and Jorge Pullin

1 Introduction 63

2 The Wheeler–DeWitt equation in terms of loops 67

3 The Gauss (self-)linking numbe r as a s olution 71

4 Discussion 73

Acknowledgements 74

xi

xii Contents

Bibliography 74

Vassiliev Invariants and the Loop States in Quantum

Gravity 77

Louis H. Kauﬀman

1 Introduction 77

2 Quantum mechanics and topology 78

3 Links and the Wilso n loop 78

4 Graph invariants and Vass iliev invariants 87

5 Vassiliev invariants from the functional integral 89

6 Quantum gravity—loop states 92

Acknowledgements 93

Bibliography 93

Geometric Structures and Loop Variables in (2+1)-

dimensio nal Gravity 97

Steven Carlip

1 Introduction 97

2 From geometry to holonomies 98

3 Geometric structures: from holonomies

to geometry 104

4 Quantization and geometrical observables 107

Acknowledgements 109

Bibliography 109

From Chern–Simons to WZW via Path

Integrals 113

Dana S. Fine

1 Introduction 113

2 The main result 114

3 A/G

n

as a bundle over Ω

2

G 115

4 An explicit reduction from Chern–Simons to WZW 118

Acknowledgements 119

Bibliography 119

Topol ogical Field Theory as the Key to Quantum

Gravity 121

Louis Crane

1 Introduction 121

2 Quantum mechanics o f the universe. Categorical physics 123

3 Ideas from TQFT 125

4 Hilbert space is dead—long live Hilbert space

or the observer gas approximation 128

5 The problem of time 129

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