Non-Perturbative Quantum Gravity:
The Loop Representation
Tze-C huen Toh
A thesis su b m itted for th e degree
of D octor of Philosophy of
th e A ustralian N ational U niversity
PR EFACE
T his d issertatio n is th e result of my own original research. Any con trib u tio n s
m ade by o th ers are acknowledged explicitly by ap p ro p riate references. T h e contents
p resen ted in C hapters 1 and 2 are review m aterial. C h ap ters 3 and 7 are due
p rim arily to th e efforts of my own unpublished research w hilst any o th er m aterial
present th erein which are contributed by others are duly acknowledged as such.
T he contents of C h ap ter 4 are based on a p ap er published jointly by my su p er
visor D r M. A nderson and myself as:
Toh, T .-C . and A nderson, M. R., K nots and Classical 3 -Geometries, J. M ath. Phys. 36 (1995) 596-604,
as well as on the following conference paper:
Toh, T .-C . and A nderson, M. R., K nots and Gravity, th e In au g u ral A u stralian G eneral R elativity W orkshop, T he A u stralian N ational University, C an b erra, Sep
tem b er, 1994.
T h e m aterial presented in C hapters 5 and 6 is based respectively on th e following
prep rin ts:
Toh, T .-C ., D iffeom orphism -invariant multi-loop measure, (1995).
Toh, T .-C ., A promeasure on the space of A shtekar connection 1-form s, (1995).
C huen Toh
O ctober, 1995
A c k n o w le d g e m e n t
I am g reatly in d eb ted to my principal supervisor Dr M alcolm A nderson for going
into great lengths to help me w ith my research and for his great display of kindness
and patience w hen plagued w ith triv ial questions by me! I also wish to express
my sincere g ra titu d e to my second supervisor Dr Lindsay Tassie and to my adviser
Dr Susan Scott, b o th of w hom were very helpful and especially considerate to set
aside a considerable am o u n t of tim e to assist me w ith my research an d questions.
N ext, I wish to express my deep appreciation to D r B rian Robson, th e form er
Head of D ep artm en t, for having th e welfare of th e stu d en ts at h eart an d for showing
concern for my progress as well as being very helpful and approachable. It is also
a great pleasure to th a n k Prof. R odney B axter, th e H ead of D ep artm en t, an d
M artin a L an d sm an n , th e second D ep artm en tal Secretary, for th eir kind assistance.
It has also been a pleasure to in te ra ct w ith th e staff m em bers of th e D ep artm en t,
an d w ith D r C o n rad B u rd en in p a rtic u la r, who was very helpful.
Well, I can h ard ly leave out m y fellow stu d en ts for brightening up the days and
for generally ju s t being them selves, a fact which alone speaks volumes! I especially
th a n k Jerom e Lewandowski and Alexei K horev for some fruitful an d pleasurable
discussions. It is a delight to convey my appreciations to all my close friends for
th eir great su p p o rt and w e ll... friendship! I w arm ly th a n k b o th my p aren ts an d
my cat Coco for th e ir en th u siastic encouragem ent, care and unfailing su p p o rt for
me an d my research, an d also wish to extend my heartfelt th an k s to Heli Jackson,
th e D e p artm en ta l Secretary, an d to my ethereal friend C lara, for th eir precious
V
A B S T R A C T
T h e loop rep resen tatio n theory of q u an tu m gravity which was developed in the
late 80’s by Rovelli and Smolin is a ra th e r novel approach tow ards unravelling the
stra n d s of puzzle th a t weave ab o u t th e q u an tu m aspects of E in ste in ’s th eo ry of
general relativity. In this thesis, certain aspects of this theory will be explored and
in p a rtic u la r, th e theory will be set fo rth on a rigorous m ath em atical foundation.
Several issues arising from th e loop rep resen tatio n of q u an tu m gravity will be
addressed. Briefly, they are: (i) to establish a relationship betw een knot states
and th e states of 3-geometries; (ii) to show th e existence of a diffeom orphism -
invariant m ulti-loop m easure on th e space of m ulti-loops; (iii) co n stru ctin g a gauge-
invariant prom easure on th e space of A shtekar connection 1-forms; (iv) th e issue
of im plem enting th e reality condition in th e loop rep resen tatio n an d th e question
of a physical in n er p ro d u ct on th e space of m ulti-loop states.
T h e relationship betw een a judiciously chosen subset of loops defined on a fixed
com pact R iem annian 3-m anifold an d th e geom etry of th e 3-m anifold will be es
tab lish ed in this thesis. Loosely p u t, th e subset of loops chosen is a denum erable
set of loops th a t are piecewise geodesic w ith respect to a fixed 3-m etric an d such
th a t th e base points of th e chosen loops form a dense subset in th e 3-manifold.
T he existence of a diffeom orphism -invariant m ulti-loop m easure is d em o n strated
in some d ep th an d th e construction of a gauge-invariant prom easure is also given.
T h e m ulti-loop space will be co n stru cted in detail and its basic topological p ro p
erties analysed. M oreover, th e existence of a m anifold stru c tu re on th e loop space
will be sketched and light is shed on th e inadm issibility of a m anifold stru c tu re on
th e m ulti-loop space. T he space of th e m ulti-loop functionals will also be briefly
studied. T he issue regarding th e d eterm in atio n of th e action of th e H erm itian con
ju g ates of th e q u an tu m T n-operators on th e m ulti-loop functionals will be broached.
A nd fu rth erm o re, th e reality-conditions in th e loop rep resen tatio n as well as the
possible construction of a physical inner p ro d u ct for th e m ulti-loop states will be
ten tativ ely delineated. Finally, an a tte m p t will also be m ade to endow th e m u lti
C O N T E N T S
P r e fa c e in
A c k n o w le d g e m e n t iv
A b s tr a c t v
C h a p te r 1. Q u a n tu m G ra v ity : A n O v e rv iew 1
1.1. Introduction... 1
1.2. Supergravity Theories... 3
1.3. Superstring T h eo ry ... 5
1.4. Non-perturbative Canonical Quantum Gravity... 6
1.5. Summary of T hesis... 11
References... 12
C h a p te r 2. T h e A sh te k a r Q u a n tis a tio n P r o g r a m m e 15
2.1. Introduction... 15
2.2. Ashtekar’s Hamiltonian Form ulation... 17
2.3. The Self-Dual Representation... 23
2.4. The Loop Representation... 25
2.5. Discussion... 33
References... 35
C h a p te r 3. T h e S tr u c tu r e o f th e M u lti-lo o p S p a ce 36
3.1. Introduction... 36
3.2. The Topological Structure of
M .... 37
3.3. The Manifold Structure of the Loop S pace... 46
3.4. The Multi-loop Functionals... 47
3.5. Discussion... 50
References... 52
C h a p te r 4. K n o t s a n d C la ssic a l 3 -G e o m e tr ie s 53
4.1. Introduction... 53
4.2. Preliminary Definitions and Notations... 54
viii CONTENTS
4.4. No-Knots and C lassical G e o m e try ... 61
4.5. D iscu ssio n ... 64
R eferen ces... 65
C h a p te r 5. D iffe o m o r p h is m -in v a r ia n t M u lti-lo o p M e a s u r e 66 5.1. In tr o d u c tio n ... 66
5.2. D iffeom orphism -invariant Borel M easure on C£ ... 67
5.3. A Diff+ (£ )-In v a ria n t M easure on A 4... 72
5.4. D iscu ssio n ... 75
R eferen ces... 79
C h a p te r 6. A P r o m e a s u r e o n th e S p a ce o f A sh te k a r C o n n e c tio n 1 -F o r m s 8 0 6.1. In tr o d u c tio n ... 80
6.2. T h e Space of A shtekar C o n n e c tio n s... 82
6.3. A P ro m easu re on A ... 83
6.4. A P ro m easu re on th e A shtekar M oduli S p ace... 92
6.5. D iscu ssio n ... 94
R eferen ces... 97
C h a p te r 7. S p e c u la t io n s a n d C o n c lu d in g R e m a r k s 98 7.1. In tr o d u c tio n ... 98
7.2. T he Loop R ep resen tatio n R e v is ite d ... 98
7.3. R eality C o n d itio n s... 102
7.4. C o n c lu sio n s... 104
R eferen ces... 105
A p p e n d ix 10 6 A. T he C om pact C °°-T o p o lo g y ... 106
B. D ifferential C alculus on A 4 n ... 107
R eferen ces... 114
L ist o f N o t a t io n s 115
CHAPTER I
Q U A N T U M G RAVITY: A N OVERVIEW
1.1. Introduction
It is known since early this century th a t general relativ ity is incom patible w ith
q u a n tu m theory. T h e incom patibility is indeed m ore profound th a n th e fact th a t
gravity is p e rtu rb a tiv e ly non-renorm alisable in th e covariant q u an tisatio n scheme.
It u ltim ately lies in th e role in which space and tim e play in general relativ ity
an d q u an tu m theory. This is a ra th e r subtle issue and is u n d o u b ted ly th e m ain
cu lp rit th a t defies various q u an tisatio n approaches to gravity. A less conceptually
subtle issue is, of course, the non-renorm alisability of gravity. T his is m ore of a
technical issue th a n a conceptual one. It arises from th e a tte m p t to depict gravity
as an o th er field defined on Minkowski space-tim e. Here, th e problem encountered
is prim arily due to th e presence of a dim ensionful coupling c o n sta n t— th e G ravi
ta tio n a l c o n sta n t— (resulting from the Principle of Equivalence) th a t prevents the
con stru ctio n of a predictive q u an tu m theory of gravity. Indeed, th e advent of q u an
tu m field theory led invariably to valiant a tte m p ts in quantising E in ste in ’s theory
of g rav itatio n . All of which proved futile. P erh ap s Isham [25, p. 8] was on the
rig h t track all along w hen he rem arked th a t ra th e r th a n quantising gravity, one
should seek a q u a n tu m theory which yields general relativ ity as its classical lim it.
B ut th en , th e m ain o b stru ctio n here is th e lack of a sta rtin g po in t to co n stru ct
such a q u an tu m theory.
By assum ing th a t q u an tu m theory is th e underlying principle governing th e
behaviour of n a tu re a t th e fun d am en tal level, it is th en alm ost inevitable th a t a
q u a n tu m th eo ry of g rav itatio n should ex ist.1 P erh ap s a m ore p e rtin e n t question to
be raised at th is ju n c tu re is th e following: why quantise gravity in th e first place?
F irst, th ere are issues in q u an tu m cosmology— such as th e q u an tu m effects of black
holes due to th eir intense g rav itatio n al fields— which cannot be fully addressed
w ith o u t a consistent theory of q u an tu m gravity. Second, it is hoped th a t a theory
2 I. QUANTUM GRAVITY: AN OVERVIEW
of q u a n tu m gravity will clear up various enigm atic questions such as th e s tru c tu re
of space-tim e a t a m icroscopic level, causality (an d hence th e arrow of tim e), and
possibly even account for th e presence of singularities in classical space-tim es [36,
C h ap ter 8, p. 256] established by Hawking and Penrose. These questions provide
ra th e r stro n g incentives for co n stru ctin g a theory of q u a n tu m gravity.
An early effort at qu an tisin g gravity was m ade by Rosenfeld in 1930 [48, 49];
needless to say, he headed rap id ly into in su rm o u n tab le technical difficulties! T his
is h ard ly su rp risin g since it is now well known th a t pu re gravity is p ertu rb a tiv e ly
non-renorm alisable a t th e 2-loop level and non-renorm alisable at th e 1-loop level
w hen coupled w ith m a tte r fields. Indeed, a sim ple pow er counting arg u m en t will
quickly p red ict th e non-renorm alisability of gravity. In th e early 1960’s, W einberg
stu d ied th e q u a n tu m aspects of general relativ ity w ithin th e fram ew ork of S-m atrix
th eo ry [61, 62], b u t his work was hindered by hideous non-linearities encountered
in E in ste in ’s field equations. His task was continued by Boulw are an d Deser [22]
who showed in d etail th a t, provided th a t the long range interactions of gravity are
m ed iated by m assless spin-2 particles, in the S -m atrix form ulation, general relativ
ity is indeed th e classical lim it of th e q u an tu m theory. However, th e ir calculations
were done in th e low -frequency dom ain.
In a p a p er by ’t Hooft [57], it was d em o n strated th a t pu re gravity is 1-loop
renorm alisable b u t when coupled w ith m a tte r, the th eo ry ceases to m ake sense
p ertu rb ativ ely . Specifically, Deser and Nieuwenhuizen showed th a t th e E instein-
Maxwell fields diverge a t th e 1-loop level [27] an d th e quantised E instein-D irac
system also diverges at th e 1-loop level [26]. In a recent p ap er by van de Ven
[58], th e 2-loop non-renorm alisability of covariant q u an tu m gravity was proved
explicitly. A nd to m ake m a tte rs even worse, aside from th e technical issues of non-
renorm alisability, m ore conceptually profound questions posed—ju s t to m ention a
few— by W heeler regarding m easurem ent [24, p. 224], and th e issue of causality—
cf. for exam ple, references [15, 40]— m ust also be explained in a satisfactory m an n er by any can d id ate th eo ry of q u a n tu m gravity.
An in itial m otiv atio n for quantising gravity lay in th e hope th a t it m ight elim
in ate th e divergences th a t exist in q u an tu m field th eo ry — u nfortunately, no t only
is such a hope dashed, b u t using p e rtu rb a tiv e m ethods gravity can n o t be ren o r
m alised. T his clearly suggests th a t th e conventional m eans of quantising gravity,
th a t is, th e use of (p e rtu rb a tiv e ) covariant q u an tisatio n , is not th e right approach;
1.2. S U P E R G R A V IT Y T H E O R IE S 3
ap p ro x im ate theory describing th e behaviour of n a tu re a t th e fu n d am en tal level.
Having said this m uch, this speculative note will not be p u rsu ed any fu rth er in this
dissertatio n . However, th e failure of gravity to be quantised p ertu rb a tiv e ly does
not necessarily m ean th a t a theory of q u an tu m gravity fails to exist.
Q u a n tu m field theory dem ands th a t th e background m etric of space-tim e be
fixed an d th a t Poincare-invariance be preserved.2 M oreover, it assum es th e sm o o th
ness of th e underlying space-tim e m anifold. In q u an tu m gravity, th e m etric itself
becomes a dynam ical variable and the gauge group is no longer th e P oincare group
b u t th e group of sm ooth diffeom orphism s. Also, it is w orthw hile poin tin g out
th a t q u an tu m gravity, should it exist, ought to determ ine (or at least, p red ict) th e
s tru c tu re of space-tim e at the Planck scale an d below— assum ing th e sm oothness
of space-tim e certainly defeats this very purpose. F u rth erm o re, th e presence of
q u a n tu m fluctuations of space-tim e geom etry m ight well destroy its sm ooth stru c
tu re. Indeed, a num ber of researchers in this field, Penrose [30, p. 4] or [47, p. 31]
in p a rtic u la r, are quite convinced th a t th e sm oothness of space-tim e geom etry at
very sm all distances m ust be sacrificed. Some researchers go a step fu rth e r an d toy
w ith th e idea th a t p erh ap s even topology itself ought to be quantised, w hatever
such a statem e n t m ight imply. At least, th e m otivation for such an observation is
th a t perhaps, a t th e Planck scale, fluctuations in th e sp atial topology (of space- tim e) m ight occur, resulting in a space-tim e foam stru c tu re . For an account of
space-tim e foam s, refer to H awking’s p ap er [35]. In itial moves tow ards topological
q u an tisatio n was in itiated in a rigorous way by Isham et a I. [39]. A ra th e r eloquent
(a n d convincing) argum ent outlining th e need for a n o n -p e rtu rb ativ e approach to gravity can be found in a m onograph by A shtekar [1, p. 3]; consult also references
[55, §1], [31, p. 327] and [4].
1 .2 . S u p e r g r a v ity T h e o r ie s
It should be poin ted out th a t p e rtu rb a tiv e covariant q u an tisatio n of gravity
(w hich failed to succeed anyway!) and th e A sh tek ar’s q u an tisatio n p rogram m e are
no t th e only m eans of tackling th e problem of quantising gravity. T h ere are others
besides those two such as the K eluza-K lein th eo ry which curren tly seems to have
gone out of favour am ongst researchers w orking in th e m ain stream of q u an tu m
gravity. P ro b ab ly th e two m ost well known ones are supergravity an d su p erstrin g
theory. Incidentally, they were also candidates for a Unified Field theory. C uriously
4 I. QUANTUM GRAVITY: AN OVERVIEW
enough, strin g th eo ry was originally conceived to provide an exp lan atio n for th e
behaviour of h ad ro n s an d no t to quantise gravity!
S u p ersy m m etry is th e underlying principal ingredient in su p erg rav ity an d su p er
strings. Roughly, it describes a tran sfo rm atio n betw een bosonic fields and ferm ionic
fields. Indeed, su p ersy m m etry can only be im plem ented if space-tim e is curved!
An h eu ristic arg u m en t o u tlin in g th e equivalence betw een th e presence of gravity
and th e im p lem en tatio n of local supersym m etry can be found in [59, p. 201]. T his
fact alone is suggestive th a t p erh ap s quantising gravity requires th e unification of
fu n d am en tal forces of n a tu re . An excellent review article on su p erg rav ity can be
found in reference [59].
In su p erg rav ity theories, each bosonic field has its ferm ionic c o u n terp art (an d
vice versa). T h e ferm ionic p a rtn e r of g rav itatio n al field is a spin | field called th e
gravitino. If th ere are n ^ 8 gravitinos, th e theory is called an TV = n supergravity
theory. TV = 0 corresponds to general relativ ity theory. If TV > 8, fields of spin | (an d higher) en ter into th e p ictu re and this includes several spin 2 fields as well.
However, th e coupling of spin | to gravity an d to fields of different spins are know n
to be inconsistent, an d no satisfactory coupling of fields w ith spins g reater th a n 2
exists. Hence, TV cannot be g reater th a n 8.
In N = 1 su p erg rav ity theory, bosons an d ferm ions (w hich occur in p airs) form irreducible rep resen tatio n s of a supersym m etric algebra3— these are th e spin (2, | )
doublets (i.e., th e g raviton-gravitino system ), th e spin (1, | ) doublets (th e photon-
n eu trin o system ) and th e spin (0, | ) doublets. It is a featu re of th e th eo ry th a t
as m any m a tte r doublets m ay be added to th e spin (2, | ) doublet as desired: in
doing so, say, by adding one or m ore spin (1, | ) doublets to spin (2, | ) doublets,
one o b tain s th e extended (TV = 2 , . . . , 8) supergravity theories. These theories
possess TV Ferm i-B ose sym m etries (plus th e usual space-tim e sym m etries of course),
|TV(TV — 1) spin 1 real vector fields and fields of lower spins. M oreover, they
also have a global U ( N ) group w hereby th e ferm ions ro ta te into them selves, and an O(TV) subgroup which ro ta te s bosonic fields into them selves. In this way, th e
g rav ito n — in TV-extended su p erg rav ity theories— is replaced by a new su p erp article
whose “p o larizatio n s” yield gravitons, quarks, photons, gravitinos, leptons. T his
unification of p articles into one su p erp article leads to th e unification of forces.
T h e u ltra-v io let divergences appearing in supergravity theories seem to be m uch
b e tte r behaved. For instance, the infinities in th e S -m atrix in th e first an d
1.3. SUPERSTRING THEORY 5
ond order q u an tu m corrections cancel due to the sym m etry betw een bosonic and
ferm ionic fields. N onetheless, even th e presence of supersym m etry is not sufficient
to g u aran tee finiteness at all loops— a t least, th ere are no conclusive proofs th a t
supergravity is p ertu rb ativ ely renorm alisable [32]. Indeed, th ere are stro n g reasons
to suspect th a t in 4-dim ensional space-tim e, supergravity theories will diverge at
th e 3-loop level [43]. Hence, it too is not a p articu larly successful th eo ry of q u an
tu m gravity. M oreover, only iV-extended m a tte r m ay be coupled to iV-extended
supergravity.
1 .3 . S u p e r s tr in g T h e o r y
Superstrings p ain t a m ore optim istic p ictu re th a n supergravity theories. How
ever, one now requires a 10 dim ensional space-tim e w ith su p ersy m m etry b u ilt in.
In spite of th a t, gravity is a necessary ingredient in order for a consistent q u an tu m
th eo ry of su p erstrin g s to exist. From this view point, strings as fu n d am en tal q u a n ta
are strongly su p p o rted by th e presence of gravity. An in tro d u ctio n to S uperstrings
can be found in reference [25, p. 301] by Schwarz or K aku [42]. H ith erto , it is th e
only can d id ate for a Unified Field Theory. Supergravity is now u n d ersto o d to be
th e low-energy lim it of su p erstrin g theory. More on this m a tte r will be broached
in th e next p arag rap h .
In th e th eo ry of Superstrings, th e fun d am en tal objects are extended 1-
dim ensional objects called strings. T he strings can either be open (i.e., a curve) or closed (i.e., a loop). In short, this extension enables ultra-violet divergences
app earin g in th e Feynm an diagram s to be removed. T here are two basic types of
strin g theory: th e type I su p erstrin g theory, w herein th e strings are unoriented, an d type I I in which th e strings are oriented. T he la tte r is also know n as heterotic superstrings. T ype II closed su p erstrin g theories have N = 2 su p ersy m m etry and hence contain N = 8 supergravity m odelled on a 4-dim ensional space-tim e as a lim iting case. Inform ally, supergravity lies in th e zero-m ass sector of closed su
p erstrin g theory. T here, supergravity is q uadratically divergent a t th e 1-loop level
w hereas its corresponding su p erstrin g theory is finite. Strings can in te ra ct by jo in
ing two ends (for open strings), or by breaking at an “in terio r” p oint (in th e case of
a loop) to form an open string. T he la tte r is dem anded by causality sim ply because
two ends of a strin g m ust “decide” to in teract a t once w ith o u t determ ining first
w hether they belong to th e sam e strin g or not.
T he inclusion of supersym m etry to strin g theory m eans th a t, aside from general
6 I. QUANTUM GRAVITY: AN OVERVIEW
also included in th is theory! However, in spite of such grandiose achievem ents,
p e rtu rb a tiv e app ro ach to su p erstrin g theory is plagued w ith problem s [42, p. 285].
O nly th ree m a jo r problem s will be listed here:
(i) th e low energy m ass sp ectru m is still wrong;
(ii) th e th eo ry cannot select th e tru e vacuum am ongst th e host of possible
conform al field theories;
(iii) alth o u g h su p ersy m m etry is preserved to all orders in p e rtu rb a tiv e theory,
it m u st be broken down in th e low energy regime.
To address these problem s, researchers tu rn tow ards a n o n -p ertu rb ativ e a p
proach to su p erstrin g theory. Also, note th a t for bosonic string theory, th e entire
sum of th e p e rtu rb a tiv e expansion diverges [33, 34]. T he A shtekar loop program m e
takes a m ore m odest tu rn : it only seeks to form ulate a consistent theory of q u an
tu m gravity w ith o u t any th o u g h t of unifying th e fu n d am en tal forces. A nd m ore
im p o rtan tly , th e ap p ro ach is n o n -p e rtu rb ativ e from th e outset! Indeed, th e p ro b
lems encountered by su p erstrin g theory, which is h ith e rto th e sole can d id ate for
a “p ro p e r” Unified Field theory, points tow ards a n o n -p ertu rb ativ e approach. A
second im p o rta n t point to observe here is th a t th e A shtekar program m e asserts
th a t th e g rav itatio n al field can be quantised on its own w ithout any o th er fields,
w hereas in su p erstrin g theory, th e very presence of su persym m etry necessitates the
unification of forces in order to produce a consistent theory of q u an tu m gravity.
Q uite a stro n g c o n trast indeed!
1 .4 . N o n -p e r t u r b a t iv e C a n o n ic a l Q u a n tu m G r a v ity
In this section, a cursory account of th e canonical q u an tisatio n of gravity, to
gether w ith th e stre n g th s and shortcom ings of the A shtekar q u an tisatio n p ro
gram m e, will be sketched. To condense th e historical developm ent of q u an tu m
gravity, it is enough to po in t out th a t from th e late 1940’s up to th e m id -1950’s,
B ergm ann em barked on a quest to canonically quantise field theories which are
covariant u n d er general coordinate tran sfo rm atio n s [18, 19, 20, 21]; here general
relativ ity is of course a p a rtic u la r case those theories. He began by doing away
w ith a space-tim e m etric an d considered in stead a m ore fun d am en tal field from
which th e L agrangian of th e th eo ry was constructed. He quickly discovered th a t
th e system possessed co n strain ts. A lthough his q u an tisatio n program m e was not
successfully com pleted, he nonetheless laid some im p o rta n t ground work for la te r
1.4. N O N -P E R T U R B A T IV E C A N O N IC A L Q U A N T U M G R A V IT Y 7
eventually carried ou t by D eW itt [28, 29].
In th e canonical form alism of general relativity, covariance is violated an d space-
tim e is split into space and tim e. T he resulting classical configuration space is the
space of R iem annian 3-geometries. M ore of this will be discussed in C h ap ter 2.
Here, it will suffice to note th a t th e resulting phase space of th e g rav itatio n al
system is constrained. T h a t is, th e physical trajecto ries in th e phase space lie
on a co n strain t surface defined by th e H am iltonian co n strain t an d th e diffeomor-
phism co n strain ts. U pon canonically quantising th is classical system , th e physical
states lie precisely in th e kernel of b o th th e q u an tu m H am iltonian an d diffeomor-
phism co n strain t operators. In fact, this is only tru e for th e case when th e sp atial
3-dim ensional slice is chosen to be com pact; in th e non-com pact case, th e wave-
functionals m ust also satisfy an additional Schrödinger equation [25, E qn (6.1.4),
p. 79]. In this thesis, only th e spatially com pact case will be considered. U n fo rtu
nately, due to th e in trac ta b ility of th e q u an tu m H am iltonian co n strain t equation
arising from involuted non-linearities, not a single explicit solution is known. This
eq u atio n is know n as th e W heeler-D eW itt equation,4 and th e w avefunctional th a t
satisfies it is know n broadly as th e w avefunction of the universe.
A pproxim ate solutions were of course found, b u t this involved tru n c a tin g the
W heeler-D eW itt equation so th a t only a finite num ber of degrees of freedom are
retain ed (in stead of an infinite num ber of degrees of freedom in th e full equation);
th is gave rise to th e theory of baby universes— th e mini-superspace approximation.
At b est, such solutions offer researchers a myopic insight into th e convoluted n a
tu re of gravity. However, it should be rem arked th a t even if th e W heeler-D eW itt
eq u atio n can be solved, th ere rem ains th e question of in te rp re tin g th e solutions.
Loosely p u t, th e w avefunctionals describe th e physical states of space-tim e as
p ro b ab ility am plitudes of possible histories. B ut this im plies a t once th a t the
concept of tim e seems to have vanished in this picture; th a t is, th ere is th e u n
p a latab le absence of dynam ics, of evolution, of tim e. T his d istu rb in g dissonance
is seemingly overcome by identifying p a rt of th e geom etry as an “in trin sic” tim e;
th en , th e W heeler-D eW itt equation is in terp reted as encoding inform ation th a t re
lates to how a w avefunctional changes w ith respect to this newly in tro d u ced notion
of “tim e ” . B ut alas, by introducing a physical inner p ro d u ct on th e H ilbert space of
physical states, th e in teg ratio n integrates over “tim e” as well! Hence, th e problem
of tim e is really not resolved at all. Tim e, however it m ight be in te rp re te d here, is
8 I. QUANTUM GRAVITY: AN OVERVIEW
tre a te d very differently from q u an tu m theory. See Isham [25, §6, p. 78] for a lucid
b u t laconic account relatin g to th e problem of tim e in th is canonical form ulation
and o th er related problem s arising from quantising in th e canonical form alism .
It should be p o in ted out th a t the riddle of tim elessness only occurs for th e sp a
tially com pact case. W hen th e sp atial slice of space-tim e is non-com pact, tim e is
defined by a Schrödinger evolution equation. For a lively assessm ent of th e canon
ical approach, refer to [31, §2, p. 330] by A shtekar. Before concluding th is sorry
tale, a b rief w ord m u st be m entioned on th e H artle-H aw king functional integ ral
approach to W heeler-D eW itt equation. Aside from com m enting th a t it yields,
heuristically a t le ast,5 explicit solutions to th e W heeler-D eW itt equation, it fails
to provide any inform ation w hatsoever at th e in sta n t of creation. Also, th ere is
th e confounded issue of tim e cropping up tim e and tim e again! It is th u s a fervent
hope th a t th e problem of tim e will be resolved w ith th e form ulation of a consistent
theory of q u a n tu m gravity.
If (2-(-l)-quantum gravity was not m entioned earlier, th e n it is sim ply because it
is essentially an open book! M uch work has been done on it. In p a rtic u la r, (2 + 1 )-
q u an tu m gravity is often used as a toy-m odel for th e seem ingly in trac ta b le (3 + 1 )-
q u an tu m gravity. For m ore details, see for exam ple reference [63] by W itte n —
as well as a com plem entary p ap er by M oncrief [45] w ho m ade some constructive
criticism s regarding th e conclusions draw n by W itte n in his p a p er— an d m ore recent
ones such as [44, 11], or a som ew hat refreshing article by W aelbroeck [60] to nam e
ju st a few out of th e p le th o ra of literatu res on (2 + l) -q u a n tu m gravity.
This section will end w ith some com m ents on A sh tek ar’s approach to quan tisin g
gravity: th e connection rep resen tatio n and th e loop rep resen tatio n of gravity. A
concise su m m ary and m otivation for A shtekar’s altern ativ e H am iltonian form ula
tion of general relativ ity [2, 3]— which in itia te d w hat is now known as th e A shtekar quantisation programme— can be found in th e in tro d u ctio n of C h ap ter 2. It should suffice to m ention here th a t A shtekar’s form ulation of “com plex” general re la tiv
ity [3] led im m ediately to the connection rep resen tatio n of q u an tu m g rav ity — th e
general relativ ity fo rm u lated in [3] is really “real” general relativ ity in term s of a
com plex and a real variable, the A shtekar connection and its conjugate m o m en tu m respectively.
An advantage of form ulating general relativ ity in term s of connections (th e
A shtekar connection 1-forms) an d th eir conjugates— these are th e soldering forms;
1.4. NON-PERTURBATIVE CANONICAL QUANTUM GRAVITY 9
i.e., “square ro o ts” of m etrics— is th a t th a t th e conjugate variable need not be invertible! T his differs greatly from general relativ ity which dem ands th a t th e
m etric be non-degenerate. An obvious conclusion to be draw n from A sh tek ar’s
form ulation is th a t it yields solutions th a t are m ore general th a n those ob tain ed
via E in ste in ’s field equations. It is p erh ap s a som ew hat tan talisin g speculation
th a t A sh tek ar’s form ulation will yield a profound insight into th e relatio n betw een
sig n atu re changes in th e space-tim e m etric and th e changes in sp atial topology of
space-tim e, an d p erh ap s even m ore interestingly, how these affect q u a n tu m gravity.
An in stru ctiv e prelim inary analysis regarding sp atial topological changes and the
degeneracies of L orentzian m etrics can be found in an article by Horow itz [38]. A
related com m ent, if som ew hat p rem atu re at this stage as it p e rta in s to th e loop rep
resen tatio n to be m entioned shortly below, relates to an intriguing p ap er by Smolin
[56]: he d em o n strated th a t, using th e loop rep resen tatio n of q u an tu m gravity, th e
sp atial topological changes effected by creating or an n ih ilatin g a special class of
w orm holes— w h at he calls m inim alist wormholes, which are created by identifying p airs of d istin ct points on th e sp atial 3-m anifold— is equivalent to general relativ ity
coupled to a single Weyl ferm ion field!
A nother positive spin-off from A shtekar’s form ulation of general relativ ity is th a t
in th e connection represen tatio n , th e H am iltonian co n strain t is greatly sim plified—
indeed, to th e ex ten t th a t some nontrivial solutions can now be found: they are
ju st th e W ilson loops. U nfortunately, W ilson loops are not invariant u n d er diffeo-
m orphism s. For m ore details, see [46, p. 12-13] or [41, §7, p. 333]. T his startlin g
hitch led to th e developm ent of th e loop rep resen tatio n of q u an tu m gravity by
Rovelli and Sm olin [52]. In th e loop rep resen tatio n , solutions to all th e q u an tu m
co n strain ts were found— refer to [41, 52] again.
T h e loop rep resen tatio n theory was applied to free Maxwell theory w ith reso u n d
ing success [12]. It was later applied to linearised q u a n tu m gravity [13] an d was
shown to correctly reproduce gravitons. A pplications were also m ade to (2+ 1)-
dim ensional q u an tu m gravity prim arily on tori [44] using th e connection as well as
th e loop rep resen tatio n — th e D irac tran sfo rm atio n reveals th a t th ey are all equiv
alent. In th e case of (2 + l)-q u a n tu m gravity, th e loop rep resen tatio n yields a com
b in ato rial p ictu re w hereas th e connection rep resen tatio n depicts a “tim eless” one.
Of course, going over to (3 + l)-q u a n tu m gravity is a different m a tte r altogether.
T here are no local degrees of freedom in (2 + l)-d im e n sio n s (due to th e vanishing of
10 I. QUA NTUM GRAVITY: AN OVERVIEW
on (2-f-l)-quantum gravity in th e loop rep resen tatio n , refer to p ap ers by A shtekar
et a1. [6, 11].
U nfortunately, like m ost theories in th e real world, th e loop rep resen tatio n of
q u an tu m gravity is no t free of problem s. T here are a n u m b er of unresolved issues.
One of th e problem s of th e loop rep resen tatio n was discovered by B riigm ann and
P u llin [23, §4, p. 239]. T hey noticed w ith some co n stern atio n th a t solutions of th e
q u an tu m H am ilto n ian o p e ra to r represented by p ro d u cts of W ilson loops were also
an n ih ilated by a m etric d eterm in an t o p erato r in term s of th e A shtekar variables. It
follows as a corollary th a t th e solutions will also satisfy th e H am iltonian co n strain ts
for a rb itra ry cosm ological constant! A concise account can be found in [46, p. 13-
14].
A nother d istu rb in g problem of th e loop rep resen tatio n lies in th e physical in
te rp re ta tio n of th e theory. A ttem p ts have been m ade a t in terp retin g th e th eo ry in
term s of knots an d weaves by Rovelli and Smolin [53, 10]. See also references [64,
65, 66] by Zegw aard. Also, a physical inner p ro d u ct on th e m ulti-loop states is not
known: this is a problem th a t is intim ately tied w ith th e physical in te rp re ta tio n of
th e theory. M oreover, th ere is th e pressing issue of defining physical observables
[5, 40, 51]. O nce again, all of these issues are intertw ined; plus, th e fact th a t very
little is know n ab o u t classical observables in general relativ ity does very little by
way of lighting a p a th for ard en t researchers.
In spite of th is setback, Smolin [53, 54] has co n stru cted a num ber of in terestin g
observables in q u a n tu m gravity: a surface area o p erato r, a volume o p e ra to r an d an
o p e ra to r th a t m easures th e “len g th ” of a 1-form on th e sp atial slice of space-tim e.
T he sp ectacu lar resu lts arising from th e first two o p erato rs are th a t a rea an d volum e
in q u a n tu m gravity are quantised in some m ultiple of th e Planck a rea and P lanck
volum e respectively! T his seems to vindicate th e conjecture th a t th e stru c tu re
of space-tim e is discrete a t th e Planck scale— a conjecture th a t was established
heuristically by Rovelli [49, §4, p. 1648]. Along this note, Rovelli and Sm olin
[54] also c o n stru cted a physical H am iltonian o p erato r (w ith a cosmological term
included) which acts in essence by breaking an d rejoining th e points of intersections
of loops in different ways. M oreover, it is also finite as well as diffeom orphism -
invariant. Hope is expressed th a t th e H am iltonian o p erato r m ight encode th e full
contents of E in ste in ’s field equations in a diffeom orphism -invariant m anner.
R etu rn in g to o th er obstacles present in th e theory, th ere are technical m a t
1.5. SU M M A R Y O F T H E S IS 11
1-form s— prelim inary studies have been m ade by A shtekar et a 1. [7, 8, 9, 16]6 a n d by Baez [16, 17]. The construction of a diffeom orphism -invariant m easure on
th e m ulti-loop space is an o th er issue th a t needs addressing: th is is a problem re
la te d to th e absence of a physical inner p ro d u c t to date. O n a whole, th e fu tu re
to th e A shtekar quan tisatio n program m e is not as bleak as it seems, an d aside
from its m ath em atical beauty, it is a t present, a novel approach tow ards a non-
p e rtu rb a tiv e q u an tu m gravity th a t has yet to reach an im passe. Indeed, recently, fu rth e r progress in the connection rep resen tatio n is m ade. A shtekar et a 1. [10] perform ed a detailed study of diffeom orphism -invariant theories in th e connection
re p re se n ta tio n an d they found com plete solutions to the G auss an d diffeom orphism
co n strain ts for th e following class of theories in th e connection rep resen tatio n : th e
H usain-K uchar m odel, R iem annian general relativ ity and C hern-Sim ons theories.
F u rth erm o re, they were able to endow th e space of such states w ith a H ilbert space
s tru c tu re , w here the inner p ro d u ct of th e H ilbert space is com patible w ith th e
reality conditions im posed on th e theories.
1 .5 . S u m m a r y o f T h e sis
In C h ap ter 2, A shtekar’s H am iltonian form ulation of general relativ ity will be
review ed and th e loop representation presented in an inform al setting. In C h ap ter
3, th e topological stru c tu re of the m ulti-loop space will be exam ined. It will be
estab lish ed th ere th a t th e loop space is second countable and it m oreover adm its
a m anifold stru c tu re . U nfortunately, it will also be shown th a t th e m ulti-loop
space does not ad m it any m anifold stru c tu re alth o u g h it is second countable and
m etrizable. T he space of th e m ulti-loop functionals will also be stu d ied briefly
a n d th e action of th e q u an tu m T °-o p erato r on th e m ulti-loop functionals will be
discussed at length.
An exact relationship betw een th e knot classes of a subset of Ko-l°°Ps (m u lti
loops w ith denum erably infinite loop com ponents) and th e 3-geom etries defined
on a com pact 3-m anifold will be described in C h ap ter 4, and in C h ap ter 5, a
diffeom orphism -invariant m easure on th e space of m ulti-loops will be constructed.
M oreover, questions regarding th e H erm itian conjugates of th e q u a n tu m T
n-o p eratn-o rs will alsn-o be discussed therein. T his th en fn-olln-owed by th e cn-o n stru ctin-o n n-of a
gauge-invariant prom easure on th e space of A shtekar connection 1-form s described
in th e following chapter. The construction differs som ew hat from th a t developed
12 I. QUANTUM GRAVITY: AN OVERVIEW
by A shtekar an d Lewandowski [8], and so provides an altern ativ e construction. U n
fortunately, th e prom easure was not co n stru cted to be diffeom orphism -invariant,
unlike th e co n stru ctio n carried out in reference [8]. Finally, in th e concluding
ch ap ter, th e loop rep resen tatio n will be briefly reviewed from a m ore rigorous p e r
spective an d issues relatin g to th e im plem entation of th e reality conditions in th e
loop rep re se n ta tio n will be touched upon.
To conclude this in tro d u cto ry chapter, some conventions used th ro u g h o u t th is
thesis will be defined below.
(1) £ will always denote a com pact, orientable, sm ooth, closed R iem annian
3-manifold;
(2) a R iem a n n ia n 3-m etric q on £ is defined as a positive-definite (i.e., has sig n atu re ( + , + , + ) ) , sym m etric, covariant 2-tensor field on £ ;
(3) th e sig n atu re of a (sm ooth) Lorentzian m etric g is tak en to be ( _ 5+ 5 + 5 +)>
(4) u n its will be chosen so th a t th e speed of light c and th e G rav itatio n al co n stan t G are set to un ity for n o tatio n al convenience, alth o u g h a t tim es they will be w ritten down explicitly to highlight certain points.
T h e te rm diffeom orphic will m ean sm oothly diffeom orphic unless explicitly s ta te d
and th e E in stein su m m atio n convention will be used th ro u g h o u t: i.e., a sum is
im plied w henever identical u p p er and lower indices are encountered. D iff(£ ) will
denote th e topological group of 3-diffeom orphism s endowed w ith th e com pact C°°-
topology— refer to §A of th e A ppendix for a description of this topology— and
I d= [
0
,1
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50. Rovelli, C ., W h a t is observable in classical and q u a n tu m gravity?, C lass. Q u a n tu m G rav. 8 (1991), 297-316.
51. ______ , Q u a n tu m reference system s, C lass. Q u a n tu m G rav. 8 (1991), 317-331.
52. Rovelli, C. a n d S m olin, L., Loop space rep resen ta tio n o f q u a n tu m general relativity, N ucl. P h y s. B 3 3 1 (1990), 80-152.
53. ______ , K n o t theory and q u a n tu m gravity, P h y s. Rev. L ett. 61 (1988), 1155-1158.
54. ______ , T h e physical H a m ilto n ia n in n o n -p ertu rb a tive q u a n tu m gravity, LANL archives p re p rin t gr-qc/9308002 (1993).
55. S m olin, L., R ecen t d evelo p m en ts in no n -p ertu rb a tive q u a n tu m gravity, LANL archives p re p rin t h e p -th /9 2 0 2 0 2 2 (1992).
56. ______ , F erm io n s and topology, LANL archives p re p rin t gr-qc/9404010 (1994).
57. ’t H ooft, G ., A n algorithm fo r the poles at d im e n sio n fo u r in d im en sio n a l regularization procedure, N ucl. P h y s. B 62 (1973), 444-460.
58. van de Ven, A ., Tw o-loop q u a n tu m gravity, N ucl. P h y s. B 3 7 8 (1992), 309-366. 59. van N ieuw enhuiszen, P., Supergravity, P h y s. R ep. 68 (1981), 189-398.
60. W aelbroeck, H., S olving the tim e -e v o lu tio n problem in 2+ 1 gravity, Nucl. P h y s. B 3 6 4 (1991), 475-494.
61. W einberg, S., P h o to n s and gravitons in p ertu rb a tio n theory: d eriva tio n o f M a x w e ll’s and E in s te in ’s equations, P h y s. Rev. B 1 3 8 , 988-1002.
62. ______ , P h o to n s and gravitons in S -m a trix theory: d eriva tio n o f charge co n serva tio n and equality o f g ra vita tio n a l and in e rtia l m ass, P h y s. Rev. B 13 5 (1964), 1049-1056.
63. W itte n , E., 2+ 1 d im e n sio n a l gravity as an exactly soluble system , N ucl. P h y s. B 3 1 1 (1988), 46-78.
64. Z egw aard, J ., P h ysica l in te rp re ta tio n o f loop rep resen ta tio n fo r n o n -p ertu rb a tive q u a n tu m gravity, C lass. Q u a n tu m G rav. 10 (1993), S273-S276.
65. ______ , G ravitons in loop q u a n tu m gravity, N ucl. P h y s. B 3 7 8 (1992), 288-308.
CHAPTER II
T H E A SH T E K A R Q U A N T ISA T IO N P R O G R A M M E
2.1. Introduction
In this c h ap ter, a n o n -p ertu rb ativ e q u an tisatio n of canonical gravity in term s of
th e A shtekar connections and loops will be review ed.1 In this thesis, only vacuum
general relativ ity will be considered: i.e., vacuum E in ste in ’s field equations. T he
tra d itio n a l H am iltonian approach to general relativity, begins w ith a g rav itatio n al
phase space defined in term s of a R iem annian 3-m etric an d its conjugate m om en
tu m . However, under this p air of canonical variables, th e co n strain ts of general
relativ ity were non-polynom ial in th eir dependence on th e 3-m etric. T his led to
technical difficulties in finding solutions th a t satisfy th e co n strain ts. T his problem ,
to g eth er w ith works on conical singularities of th e reduced phase space of sp atially
com pact space-tim es by Arm s et aI. [1], an d a connection 1-form in tro d u ced by Sen [20], m o tiv ated A shtekar [3] to co n stru ct w hat is now referred to as th e A shtekar variables.
In essence, A shtekar shifted th e em phasis of tra d itio n a l canonical form ulation of
general relativ ity from th e m etric rep resen tatio n to th e connection rep resen tatio n .
Recall briefly th a t in th e m etric rep resen tatio n , th e fu n d am en tal (canonical) vari
ables are th e 3-m etric an d its conjugate m om entum (a covector in th e cotangent
b u ndle over th e space of R iem annian 3-m etrics), w hilst in th e connection rep resen
ta tio n , th e canonical p air is th e connection 1-form an d its conjugate m om entum .
T h e advantages arising from this shift in view point are many. However, for th e
purpose of this in tro d u ctio n , it will suffice to highlight th e m ain benefits of such
an approach. For a m ore detailed explanation, refer to references [2, p. 19], [16]
an d of course, A shtekar’s original article [3] on th e new H am iltonian form ulation
of general relativity. Some of the m ajo r advantages are listed below.
(a) T he co n strain t equations in A shtekar’s new variables are m uch sim pler in
16 II. T H E A S H T E K A R Q U A N T IS A T IO N P R O G R A M M E
appearance: they are polynom ial in th eir dependence on th e new “canon
ical” p a ir of variables
(A ,E),
whereA
is th e A shtekar connection 1-forman d
E
is its conjugate m om entum . To w it, the H am iltonian co n strain t isq u a d ra tic in
E
, w hilst th e o th er co n strain ts are linear inE . 2
(b) T h e co n strain ed phase space of general relativ ity m ay be im bedded into th e
phase space of com plex Yang-Mills th eo ry and hence bringing E in ste in ’s
th eo ry of gravity in line w ith th e theories th a t describe fu n d am en tal in te r
actions of n a tu re — gauge theories. Explicitly, every initial d a tu m (A,
E)
ofE in ste in ’s th eo ry is also an in itial d a tu m for Yang-Mills th eo ry — it satis
fies a vector an d a scalar co n strain t (in ad d itio n to th e G auss co n strain t
m en tio n ed in (a) which is satisfied by th e Yang-Mills th e o ry ).3
(c) T h e co n strain ts do not depend on th e inverse
E ~ l
of th e conjugate m om entu m
E\
consequently, A shtekar’s form ulation is an extension of E in ste in ’sth eo ry of gravity as degenerate m etrics are also possible solutions u n d er
A sh tek ar’s form alism . Hence, restrictin g th e m etrics to be non-degenerate
in A sh tek ar’s form alism yields precisely general relativity. This has possible
im plications in q u an tu m gravity w here p erhaps th e change in sig n atu re of
th e m etric m ight becom e significant, and m ore im portantly, it m ight also
play a crucial role in th e stu d y of singularities of classical space-tim e an d
th e topological changes in th e sp atial slice of space-tim e.
A sh tek ar’s new variables not only h ad a profound im pact on q u an tu m gravity,
they also provided a deeper insight into the classical solutions of E in ste in ’s field
equations. For instance, his variables led to an altern ativ e characterizatio n of half
flat solutions to E in ste in ’s field equations [4].4 It relies essentially on th e fact th a t
th e A shtekar connection
A
can be either th e p o ten tial~A
for th e self-dual p a rt ofth e Weyl te n so r or th e p o ten tial
+A
for th e anti-self-dual p a rt of th e Weyl tensor.T he self-dual solutions are th en obtained by setting
+A =
0, and vice versa— seereference [4].
T h e use of loops in physics is not a new idea. A b rief historical account can
be found in [16, p. 1635] and th e references cited therein. Suffice to n ote th a t
Jacobson a n d Sm olin [11] discovered nontrivial solutions to th e H am iltonian
con-2In th is fo rm u la tio n , an a d d itio n a l c o n stra in t, th e G au ss c o n s tra in t, is in tro d u c e d d u e to th e a d d itio n a l deg rees of freedom in tro d u c e d by th e form alism . However, th e s tru c tu re of th is c o n s tra in t is n o t co m p licated : it is lin ear in th e c o n ju g ate m o m en tu m .
3M ore d e ta ils co n cern in g th e rela tio n betw een th e E in ste in a n d Yang-M ills eq u a tio n s can be fou n d in a p a p e r by M ason an d N ew m an [12].
2.2. ASHTEKAR’S HAMILTONIAN FORMULATION 17
stra in t of general relativ ity in th e connection rep resen tatio n and this in tu rn m o
tiv ated Rovelli an d Smolin [17] to construct th e loop rep resen tatio n of q u a n tu m
gravity. A nd because th e th e loop form alism au to m atically cap tu res SU(2) gauge
invariance, th e ad d itio n al co n strain t— the G auss c o n strain t— present in th e con
nection rep resen tatio n is elim inated in th e loop rep resen tatio n . F u rth erm o re, in
th e loop rep resen tatio n , all solutions of th e diffeom orphism co n strain ts are known:
th ey are noth in g b u t loop functionals defined on th e space of equivalent classes of
loops, w here two loops are said to be equivalent if they are related by a sm ooth o rientation-preserving diffeomorphism ; th a t is, if th e two loops are k n o tte d in th e
sam e way.
W ilson loops an d th e conjugate m om entum of th e A shtekar connection 1-form s
play an essential role in th e theory of loop rep resen tatio n . By tak in g th e traces
of suitable com binations of th e complexified SU(2) holonom ies and th e conjugate
m om enta, a class of observables called the T-observables are ob tain ed . T h e con
stra in ts of general relativ ity can th en be recast in term s of suitably defined lim its of
these T-observables. In short, th is yields th e loop rep resen tatio n . U nfortunately,
th e physical in te rp re ta tio n of th e loop rep resen tatio n is far from triv ial. For a
com prehensive (b u t intuitive) insight into how th e way loops are k n o tte d to give
rise to gravity, refer to a com prehensive review article by Rovelli [16, §4, p. 1648].
2 .2 . A s h te k a r ’s H a m ilto n ia n F o r m u la tio n
In this section, th e tra d itio n a l canonical form ulation (or th e A D M fo rm a lism) of general relativ ity will be outlined in order to m otivate A sh tek ar’s H am ilto n ian
form ulation of general relativ ity [3]. Incidentally, ADM stan d s for A rnow itt-D eser-
M isner. For m ore details regarding th e ADM form alism , consult reference [9, p.
138] for a detailed exposition by Fischer and M arsden, or to reference [8, C h a p te r 7,
p. 226] for th e in itial value form ulation of general relativity. T he ADM form alism
given below is based on a laconic exposition by R om ano [15, §2, p. 765].
Let (X , g) be a sm ooth, globally hyperbolic, L orentzian 4-m anifold (w hich is b o th space- an d tim e-orientable), and i : E » X be a spacelike sm o o th
def
im bedding— th a t is, q = i*g is a R iem annian 3-m etric on S . T hen, S is a C auchy surface for X [5, theorem 1, p. 88]. In fact, it m ay be assum ed w ith o u t any loss of generality th a t X is diffeomorphic to E x R. Let t : X —> R be a sm o o th fun ctio n defining a spacelike foliation (of codim ension 1) such th a t for each fixed A E R ,
18 II. T H E A SH TEK A R QUANTISATION PR O G R A M M E
on Ea for different A’s an d it defines evolution via th e Lie derivative C Vt along th e
integral curve of v t .
2.2.1. Rem ark. T h ere are strong reasons to su p p o rt th e restrictio n of th e topology of E to be com pact. If E were not com pact, strong conditions m ust be im posed on
X in ord er for it to ad m it a Cauchy spacelike surface [5, §IV, p. 94].
Let n be a norm alised, tim elike vector field— th a t is, norm alised relative to th e
def
Lorentzian 4-m etric gon X : gx ( n ( x ) , n ( x ) ) = — 1
V x
G X — and set q% = 6%+narib.T his defines a p ro jectio n o p erato r onto Ea for each fixed A £ R . If ^a <==f # | E a,
th en th e induced m etric q \ on Sa is given by
q\ab =* q \ a Q l\ b 9 X k l = gxab + n \ an\b .
T he tim elike vector field vt m ay be decom posed as
v f = N n a + N a,
w here N = —n av \q ab is th e lapse fu n ctio n which determ ines th e infinitesim al deform ation of Ea to Ea+<$a in X , and N a q^v\ is th e shift vector an d is responsible for g enerating a 1-param eter fam ily of 3-diffeom orphism s on Ea- For n o ta tio n a l sim plicity, identify E w ith its im age in X in all th a t follows. T h e
Einstein-H ilbert action Se h =
Jx V~~
det g 4R, w here 4R is th e scalar cu rv atu re w ith respect to th e L orentzian 4-m etric <7, can be w ritten in term s of th e inducedR iem an n ian 3-m etric q on E as
See = (det
q
) 2
N ( R
+K abKa
—K 2)
+ surface integral,w here R is th e scalar cu rv atu re of 5 , K ab = f qaqlb(£ng)kl is th e extrinsic curva tu re of E and K K abqab. For m ore details, see reference [15, §2, p. 765].
Let r j denote th e space of R iem annian 3-m etrics
q
on E an dT*
its cotangentb u n d le . 5 In th e ADM form alism , th e evolution of th e in itial d a ta (q, p) 6 T * r J , w here p =
SL
e e/ hCVtq
an dL
ee^
(det <7 ) 2 N ( R +K abKah
—K
2)
is th e E instein- H ilbert L agrangian, is studied. However, in order to satisfy E in ste in ’s field eq u ations, no t every po in t in
T
*
is accessible: th ere exist co n strain ts. Theseco n strain ts— th e diffeom orphism and H am iltonian constraints respectively— are
(2.2.1) C b(q, p) = D ap 0,
(2-2.2) C( q, p) =~ \ p 2 - = 0,
