Baez J.C. (Ed.) Knots and Quantum Gravity (Oxford Lecture Series in Mathematics and Its Applications)
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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