**LOOP QUANTUM GRAVITY (02.04.2002)**

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**LOOP QUANTUM GRAVITY... Agg. 20.07.2004**

**HOME****(****ritorna alla pagina iniziale !****)****Preface by CHANDRAst****Overview****Basic results of loop quantum gravity****Kinematics**:**canonical picture****References(1)****Uniqueness and variations on the space of states****Boundary conditions****References(2)****The cosmological constant****References(3)****Path integral formulations of the dynamics in loop quantum gravity****References(4)****Dynamics****Non-zero cosmological constant****References(5)****Zero cosmological constant****References(****6****)****Current work****Summary****NOTE (1): Quantum geometry****NOTE (2):****Information Theoretic Entropy****NOTE (3):****T-duality****New****ARGOMENTO CORRELATO (1): The Myth of the Beginning of Time****ARGOMENTO CORRELATO (2):****BIG BANG****ARGOMENTO CORRELATO (3):****UNIVERSO INFLAZIONARIO****ARGOMENTO CORRELATO (4):****EXTRA-DIMENSIONI, K-K, brana, (S.I.S.S.A.)****ARGOMENTO CORRELATO (5):****GUT****ARGOMENTO CORRELATO (6):****RINORMALIZZAZIONE di una TEORA di CAMPO****ARGOMENTO CORRELATO (7):****SUPERSIMMETRIA****ARGOMENTO CORRELATO (8):****NUCLEI Galattici Attvii - A.G.N. o Galassie attive**

**La teoria della relatività
generale è stata davvero la più grande elaborazione della fisica
teorica degli ultimi settant'anni ?** **Come
mai non è in grado di spiegare le teorie delle superstringhe**
**?** **Esistono altre teorie fisico-matematiche
che tendono a una spiegazione unitaria** **?** La
teoria della relatività generale è sicuramente una delle tappe
principali della storia della fisica, e in particolare del secolo appena trascorso.
In generale, però, ogni teoria scientifica è aperta, fin dal momento
stesso della sua formulazione, ad essere sostituita da una più completa.
La teoria della relatività spiegava fenomeni che la vecchia **teoria
della gravità** non era in grado di spiegare, e lo faceva con
eleganza matematica ed economia di pensiero; per questo si impose. Ma presto
fu chiaro che, nonostante in accordo con i dati sperimentali, prima o poi anche
la **teoria della relatività** sarebbe stata soppiantata.
**Perché ****?**
Nel **Novecento** ha gradualmente
preso forma anche una teoria della dinamica del mondo microscopico. L'idea forte,
nata dalle osservazioni sperimentali, era che, a livello fondamentale, tutte
le quantità fisiche sono disponibili solo in multipli di una quantità
elementare, detta quanto. **Decenni di studi consentirono
prima di gettare le basi di una spiegazione fisico-matematica di questo fenomeno,
attraverso la cosiddetta** **meccanica
quantistica**, **e poi la graduale applicazione
di quest'idea al complesso mondo delle interazioni microscopiche**.
Queste due teorie, la **quantistica** e la **relativistica**,
nate per spiegare rispettivamente le interazioni microscopiche e la gravità
**(***che, come l'esperienza
quotidiana dice, è rilevante su scale macroscopiche***)**,
hanno conosciuto notevoli successi, e la questione di quale sia **" la
più grande"** va affidata
al gusto del singolo.

**Loop quantum gravity** provides a general framework
for **diffeomorphism** invariant quantum field theories, within
which there is a complete quantization of **Einstein's theory**
of **general relativity**. There is both a canonical and path integral
formalism. In the canonical formalism the **Hilbert space** of
s**patially diffeomorphism invariant states** has been constructed
in closed form. Regularization procedures have been developed which result in
finite, diffeomorphsim **invariant** or **covariant operators
**which represent certain classical diffeomororphism invariant or covariant
functions on the phase space. Among these are the area and volume operators,
which turn out to have finite, computable spectra. These spectra have been computed
in detail. The **Hamiltonian** constraint and **Hamiltonian**
in certain fixed gauges are also among the operators that have been constructed
using these regularization methods, they are represented by finite, diffeomorphism
covariant operators. An infinite dimensional space of solutions to the Hamiltonian
constraint has been constructed. **Some physical operators** **(***those
that commute with all the constraints***)**
have been constructed, using several different methods. These include using
path integral methods to construct the projection operator onto physical states,
making use of matter fields as physical coordinates or explicit operator constructions.
Among those that can be so represented are the area and volume operators, thus
their spectra represent genuine physical predictions of the theory. The path
integral form of **loop quantum gravity** has been constructed
and its relationship to the canonical theory is understood. In the** Euclidean
case** the path integral provides a projection operator onto the space
of solutions of the **Hamiltonian constraints**. This can be used
to construct explicit expressions for the expectation values of physical **(***commuting
with all the constraints***)**
observables. These have been computed explicitly by series and numerical methods
in several **1+1 dimensional examples**. All these results hold
for both the **Euclidean** and **Lorentzian signature cases**,
as well as for extensions of **general relativity** such as **supergravity**,
and coupling to all usual forms of matter is understood. There are also results
in other dimensions such as **1+1** and **2+1**. Many
of these results have been obtained through several different methods, and the
results of these different methods agree up to operator ordering terms, that
are by now well understood. **Many of the results
were first gotten by adopting regularization procedures for products of**
**operators from** **QCD**, **and
were later verified by being proved as rigorous theorems in mathematical**
**quantum field theory**. This applies both to the geometrical
operators such as area and volume and the **hamiltonian constraint**.
There remain open questions about the physical interpretation of the solutions
to the hamiltonion constraint, these are the subject of much current work and
will be discussed in detail below.

**Basic results of loop
quantum gravity **

**1)-**
**Kinematics: canonical
picture**

**Loop quantum gravity** is a general framework
for the construction and study of diffeomorphism invariant quantum field theories.
It is now understood in both a canonical and path integral framework. I start
with the former. Given a classical field theory whose configuration space includes
a space of connections and whose gauge invariances include the corresponding
local gauge invariance and diffeomorphism invariance, loop quantum gravity provides
a method which results in the construction of a **Hilbert space**
of spatially **diffeomorphism invariant** and** gauge invariant
states**. The first step is the construction of a **kinematical
Hilbert space** which is a representation of a complete set of basic
operators that is complete in the sense that it coordinatizes the **phase
space**. This is the** loop algebra**. On this space the
spatial diffeomorphisms are represented, without anonalies, by unitary operators.
The quotient gives the space of diffeomorphism invariant states, together with
its inner product. This procedure may be carried out on any theory in the above
class, including general relativity, coupled to arbitrary matter fields, including
**spinors**, **gauge fields** and **antisymmetric
gauge fields**, in any dimension, as well as various extensions such
as **supergravity**. **Loop quantum gravity** then
provides a method for constructing finite operators which represent either diffeomorphism
invariant or covariant operators acting on either the space of diffeomorphism
invariant states or the** kinematical Hilbert space**. This is
a form of point splitting regularization, modified so as to be diffeomorphism
covariant. The main idea of the method is to extract finite parts which, because
they do not depend on the regulator, are also independent of background metrics
introduced to define the regularization. When applied to **spatially diffeomorphism
invariant functions** of the** phase space** it yields finite
operators on the space of **diffeomoprhism invariant states**,
when applied to scalar functions it gives operators on the kinematical state
space that trasform under the action of the unitary representation of the **spatial
diffeomorphsim group**. The operators we have understood by using this
method include the area and volume operators, and these are found to have computable
discrete spectra. We can couple the theory to matter and use the matter degrees
of freedom to label surfaces and then use these to construct, using the regularization
procedures we developed, **diffeomorphism invariant operators**
that measure the areas of the surfaces and the volumes they contain.**These
can also be promoted to real physical operators, as will be described below,
and the result is that the spectra of the area and volume operators are genuine
physical predictions of** **quantum general relativity**.
There are proposals by **Amelino**-**Camelia** and
others to check these predictions using astrophysical observations, based on
the fact that one can check small scale effects on the propagation of light
by observing light that has propagated for cosmological scales. **(**These
are discussed in **G.Amelino**-**Camelia** et al.
**{**\it **Potential Sensitivity
of Gamma-Ray Burster Observations to Wave Dispersion in Vacuo****}**,
**astro-ph/9712103**,**
Nature 393 (1998****)**
**763-765****)**. It
is not possible of course to take any classical observable and turn it into
a **finite quantum operator**. But the operators that have been
understood are sufficeint to give a particular basis -**the spin network
basis**- a complete and precise physical interpretation in terms of quantities
that measure three geometry. **We can also construct
operators that measure the conjugate quantities to all these observables**.
Among the phase space functions we can construct finite representations of are
the hamiltonian constraint and hamiltonian in certain fixed gauges. As in any
**QFT** there are ordering ambiguities in the operators that result
from the regularization procedures, and these have been studied and are by now
largely understood. We also have the spatially diffeomorphism invariant inner
products in all these cases. All this is known for both the **euclidean**
and **lorentzian theories**. The correct way to do the lorentzian
signature theories has been worked out in detail by **Thomas Thiemann**.
At the **kinematical level**, it is possible, given any spatial
metric whose curvature invariants are small in **Planck units**,
to construct a **quantum state** which approximates it as far as
all measurements of geometry are concerned which are averaged over surfaces
or **regions large in Planck units**. Some of these states are
eigenstates of three geometry operators **(***these
are known as* **weaves****)**
while others are coherent states, with uncertainty in both **three geometry**
and **extrinsic curvature**. Thus, kinematically, the theory has
semiclassical states, this is necessary, but not sufficient, for the theory
to have a good semiclassical limit dynamically. All of these results have been
reproduced by the use of a different method, which is mathematically rigorous
and based on the study of diffeomorphism invariant measures on spaces of connections.
**The two methods agree precisely as to the space
of states of the theory and the inner product**. In the rigorous
approach regularization procedures similar to the ones used initially have been
developed, and the results of the two methods agree up to well understood operator
ordering terms. The rigorous results show that inner products can be chosen
at the kinematical level, before imposing diffeomorphism invariance, so that
there are no anomolies in the spatial diffeomorphism constraints. The spatial
diffeomorphisms are represented by unitary operators and one can mod out by
their action and construct completely explicitly the **space of diffeomorphism
invariant states** with inner product **(**** again
in all cases mentioned above**.

**The canonical form
of the loop representation** was introduced in
**C.Rovelli**
and **L.Smolin**, **Nucl Phys** **B133, 80**
**(1990)**,
in complete form, except for the specification of the spin network basis, which
we found later.

**The paper that
introduced the different regularization methods used for the area and volume
was** **LSmolin:** in **{**\it
* Quantum Gravity and Cosmology***}**,
eds **J P\'erez-Mercader** **{**\it
et al**}**, **World Scientific**,
Singapore **1992**

**The spin network
basis and the exact forms of the area and volume operators were presented in**,
**C.Rovelli**
and** L.Smolin** **{**\it
*Discreteness of area and volume in quantum gravity***}**
**Nuclear Physics B 442** **(****1995****)**
**593**. **Erratum**: **Nucl.
Phys. B 456** **(****1995)**
**734**; **"***Spin
networks and quantum gravity***"**
**gr-qc/9505006**, **Physical Review** **D 52**
**(****1995****)**
**5743-5759**.

**For detailed
computations of the spectrum of the volume operator**, see **R.Loll**,
**Nucl. Phys**.** B444** **(**1995**)**
**619**; **B460** **(****1996****)**
**143**; **R**.** DePietri **and **C.Rovelli**,
**{**\it *Geometry eigenvalues
and scalar product from recoupling theory in loop quantum gravity***}**,
**gr-qc/9602023**,
**Phys.Rev**. **D54** **(****1996****)**
**2664**; **Simonetta Frittelli**,
**Luis Lehner**, **Carlo
Rovelli**, **{**\it *The
complete spectrum of the area from recoupling theory in loop quantum gravity*
**}** **gr-qc/9608043**;
**R.Borissov**, Ph.D. thesis, Temple, **(****1996****)**;
**A.Ashtekar** and **J.Lewandowski**, **gr-qc/9711031**

**For the semiclassical
states sharp in three geometry**, see **Ashtekar**,
**Rovelli**
and **Smolin**, **Physical Review Letters 69** **(****1992****)**
**237-240**, for coherent like states see a recent paper by **Andsdorff**
and **Gupta**.

**For the construction
of diffeomorphism invariant operators using matter fields to locate surfaces**,
see **L. Smolin**, **{**\it
*Finite diffeomorphism invariant observables for quantum gravity*,**}**
**Physical Review** **D 49** **(1994****)**
**4028**,** gr-qc/9302011**. There
are other papers that accomplish the same thing with different matter fields
by **Carlo**
and **Viqar Husain**.

**For the regularization
procedure leading to the precise form of the hamiltonian in a certain gauge**
see**:** **C.Rovelli**
and **L.Smolin**, **{**\em
**The physical hamiltonian in nonperturbative quantum gravity****}**,
**Phys.Rev.Lett**. **72 (1994****)
****446**. This method was used by **Thiemann**,
see below, to also construct the hamiltonian constraint in both the euclidean
and lorentzian theory.

**For Thiemann's
work, which establishes the lorentzian signature canonical theory and many of
its properties**, see **T.Thiemann**, *Quantum
Spin Dynamics I-V, Class.Quant.Grav*. **15** **(****1998****)**
**839-905**, 1**207-1512**. **gr-qc/9606092**,
**gr-qc/9606089**, **gr-qc/9606090**, **gr-qc/9705020**,
**gr-qc/9705021**, **gr-qc/9705019**, **gr-qc/9705018**,
**gr-qc/9705017**.

**A review of loop quantum
gravity is** **C.Rovelli**,
**gr-qc/9710008**, also in the **electronic journal Living
Reviews**

**The rigorous
reformulation of all these results, plus more, were given in**,
**A.Ashtekar** and **C.J.Isham**, **Class**
**and Quant Grav 9** **(****1992****)**
**1069**; **AAshtekar**, **J.Lewandowski**,
**D.Marolf**, **J.T.Thiemann**: **"***Quantization
of diffeomorphism invariant theories of connections with local degrees of freedom***"**,
**gr-qc/9504018**, **JMP 36** **(****1995****)**
**519**; **A.Ashtekar** and **J.Lewandowski**,
**"***Quantum Geometry I:
area operator***" gr-qc/9602046**;
**hep-th/9412073**, **J.Geom.Phys**. **17 (1995****)**
**191-230**, **J.Lewandowski**, **"***Volume
and quantization***" gr-qc/9602035**,
**J. Lewandowski** and **T.Thiemann**, **gr-qc/9901015**,**
Class. Quant. Grav**. **16** **(****1999)**
**2299**; **Abhay Ashtekar**, **Alejandro Corichi**,
**Jose.A.Zapata**, **{**\it
*Quantum Theory of Geometry III: Non-commutativity of Riemannian Structures***}**,
**gr-qc/9806041**, **Class.Quant.Grav**. **15**
**(****1998****)**
**2955-2972**; **J Baez**, **Spin
networks in gauge theory**, **Adv.\ Math.\** **{****\bf
117****}** **(****1996****)**,
**253-272**, **gr-qc/941107**; **Spin
networks in nonperturbative quantum gravity**, in **{**\sl
* The Interface of Knots and Physics*

**2)-
Uniqueness and variations on the space of states**

*It is interesting
to ask the extent to which the space of states used in***loop quantum gravity** * is unique*.
One way to pose the question exactly is to ask whether there are other diffeomorphism
invariant measures on spaces of non-abelian connections besides the ones used.

*We also understand the effects
on the theory of several kinds of***boundary conditions**.
There are results concerning finite, timelike boundary conditions, which were
first studied as a laboratory to connect loop quantum gravity with conformal
**field theory**. I found using them that the **Bekenstein
bound**, which is a relationship between the area of a boundary and the
dimension of the state space of the boundary theory, is a necessary consequence
of the kinematics of loop quantum gravity. One thing this has led to is a detailed
understanding of the quantum geometry of black hole horizons. **Krasnov**,
one of our students, now at **U**niversity** of C**alifornia,**
S**anta **B**arbara** (UCSB)**,
initiated the use of these kinds of boundary conditions, taken on null rather
than timelike surfaces, for the study of **black hole** horizons,
now many people work on it including **Ashtekar**, **Baez**,
**Corichi**, **Beetle**, **Fairhurst**
etc. These results nicely complement the results on **black holes**
in **string theory**, in that they apply to ordinary **black
holes** and not just to the extremal and near extremal varieties. One
important goal of work in progress is to extend the **loop quantum gravity**
description to those **black holes** studied by the methods of
**string theory**. With **Yi Ling** we understand
**loop quantum gravity** now up to **N=2 supergravity**,
and are just at the point when we can attack this problem in detail. There is
also an understanding of asymptotically flat boundary conditions, by **Thiemann
**and **Andsorff **and **Gupta**.

**The finite, timelike
boundary conditions were introduced** in **L.Smolin**,
**{**\it *Linking topological quantum
field theory and nonperturbative quantum gravity***}**
**gr-qc/9505028**, **J.Math. Phys. 36** **(****1995****)**
**6417**. in the euclidean case and **L.Smolin**,
**{**\it *Holographic formulation
of quantum general relativity*. **}**,
**hep-th/9808191**, to appear in **Phys. Rev. D**,
in the lorentzian case. **With Yi Ling** they are now being extended
to **N=1,2 supergravity**.

**The extension
to black
hole event horizons** **(**i.e.**
null boundaries****)**
was developed in **K.Krasnov**, **gr-qc/9603025**,
**Phys**.**Rev**. **D55** **(****1997****)**
**3505-3513**; **gr-qc/9605047**,
**Gen.Rel.Grav**. **30** **(****1998****)**
**53-68**; **gr-qc/9710006**,
**Class.Quant.Grav**. **16** **(****1999****)
****563-578**; **A.Ashtekar**, **J.
Baez**, **A.Corichi**, **K.Krasnov**, **Phys.Rev.Lett**.
**80** **(****1998)**
**904-907**, **gr-qc/9710007**;
**gr-qc/9902015**;**
J.Lewandowski**, **gr-qc/9907058**;
**A. Ashtekar** and **K.Krasnov**, **gr-qc/9804039**,
published in **"****Black
Holes**, Gravitational Radiation and the Universe**"**,
Essays in honor of **C.V.Vishveshwara**, **Ed. B.R.Iyer**
and **B.Bhawal**, **Kluwer**, **Netherlands**;
**A. Ashtekar**, **A.Corichi**, **K.Krasnov**,
**gr-qc/9905089**.

**The asymptotically
flat boundary conditions** are described in **T.Thiemann**,
**Class.Quant.Grav**. **15** **(****1998****)**
**1463-1485**; **Class. Quantum Grav**.
**{**\bf 12**}**
**(****1995****)**
**181-198**, **gr-qc/9705020**,
and a recent paper by** Andsdorf **and **Gupta** **(gr-qc/9909053)**,
** (gr-qc/9910084)
**.

We also understand the role of the cosmological constant in
the kinemtical framework, which is to force the **spin network states**
to be constructed from the representation theory of **quantum deformed
SU**(**2**). All this has been worked out in detail, with
**Borissov** and **Major** with also contributions
from **Baez**. **In the case of time
like boundary conditions it leads to a complete understanding of the boundary
theory in terms of conformal field theory**.

**Quantum deformed
spin networks for the case of non-zero cosmological **case
were constructed in **S.Major** and **L.Smolin**,
**{**\it *Quantum deformation of
quantum gravity***}** **gr-qc/9512020**,
**Nucl. Phys**. **B 473 (1996****)**
**267**; **R. Borissov**, **S.Major**
and **L.Smolin**, **{**\it
*The geometry of quantum spin networks***}**
**gr-qc/9512043**,
**Class. and Quant. Grav**. **12** **(1996)**
**3183** and in **L.Smolin JMP**, above.

**5)-** **Path
integral formulations of the dynamics in loop quantum gravity**

The path integral framework for this large class of theories
has been worked out during the last three years. In the **Euclidean case**
it is called** spin foam**. The basic framework is by now understood,
so that we have a framework to do path integral calculations to find, given
a form of the action or of the **hamiltonian constraint**, the
amplitude to evolve from any **spin network** to **any spin
network**. As in **QCD**, several different forms of the
action have been proposed, and are under study. These differ by terms which
in some cases correspond to the operator ordering ambiguities of the * canonical
theory*. It turns out that spin foam is closely connected with
a well-understood mathematical subject, which is

**The basic papers in
spin foam are:** **M.Reisenberger**,
**{**\it *Worldsheet formulations
of gauge theories and gravity***}**,
**gr-qc/9412035**;
**{**\it *A lattice worldsheet
sum for 4-d Euclidean general relativity***}**
, **gr-qc/9711052**;
**M. Reisenberger** and **C.Rovelli**,
**{**\it **"***Sum
over Surfaces***''** *form
of Loop Quantum Gravity***}**,
**gr-qc/9612035**,
**Phys**.**Rev**. **D56** **(****1997****)**
**3490-3508**; **J.Barrett** and **L.Crane**,
**{**\it *Relativistic spin networks
and quantum gravity***}**; **gr-qc/9709028**,
**J.Math.Phys**. **39** **(****1998****)**
**3296-3302**;

**The relationship
between spin foam and the canonical theory** is described in detail
in **C.Rovelli**,
**{**\it *The projector on physical
states in loop quantum gravity***}**,
**gr-qc/9806121**,
**Phys**.**Rev**. **D59** **(****1999****)**
**104015**.

**A recent review**
by **Baez** is **{**\it
*An Introduction to Spin Foam Models of Quantum Gravity and BF Theory***}**,
**gr-qc/9905087**;
an older review is **Baez**, **{**\it
*Spin foam models***}**, **gr-qc/9709052**.

**A selection of
recent papers on spin foam is:** **Laurent Freidel**,
**Kirill Krasnov**,**{**\it
so(4) *Plebanski Action and Relativistic Spin Foam Model***}**,
**gr-qc/9804071**,
**Class**.**Quant**.**Grav**. **16**
**(****1999****)****
2187-2196**; **{**\it *Spin
Foam Models and the Classical Action Principle***}**
**hep-th/9807092**,
**Adv**.**Theor**.**Math**.**Phys**.
**2** **(****1999****)**
**1183-1247**; **John C.Baez**, **R
De Pietri**, **{**\it *Canonical*
**"***Loop***"**
*Quantum Gravity and Spin Foam Models***}**,
**gr-qc/9903076**,
To appear in the proceeding of the **XXIII SIGRAV conference**,
Monopoli **(****ITALY****)**,**
September 21st**-**25th**,
**1998**; **R. De Pietri**,
**L.Freidel**, **K.Krasnov**, **C.Rovelli**,
**{**\it *Barrett-Crane model
from a Boulatov-Ooguri field theory over a homogeneous space***}**,
**hep-th/9907154**;

**The basic papers
on causally evolving spin networks are:** **F.Markopoulou**,
**{**\it *Dual formulation of
spin network evolution***}** preprint,
**March 1997**, **gr-qc/9704013**;
**F.Markopoulou** and **L.Smolin** **{**\it
*Quantum geometry with intrinsic local causality***}**
preprint, **Dec. 1997**, **gr-qc/9712067**,
**Phys**.**Rev**. **D58** **(****1998****)**
**084032**.

**The relation
between spin foam and causally evolving spin networks** is shown
in **Sameer Gupta**, **{**\it
*Causality in Spin Foam Models***}**,
**gr-qc/9908018**;

**Detailed numerical
results concerning 1+1 systems which can be thought
of as simple models of causally evolving spin networks
are found in:** **J.Ambjorn** and **R.Loll**,
**{**\it *Non-perturbative Lorentzian
Quantum Gravity, Causality and Topology Change***}**,
**Nucl**.**Phys**. **B536** **(****1998****)**
**407-434**, **hep-th/9805108**;
**J.Ambjorn**, **J.L.Nielsen**, **J.Rolf**,
**R.Loll**, **{**\it
*Euclidean and Lorentzian Quantum Gravity* - *Lessons from Two Dimensions***}**,
**hep-th/9806241**,
**Chaos Solitons Fractals** **10** **(****1999****)**
**177-195**; **J.Ambjorn**, **K.Anagnostopoulos**
and **R.Loll**, **{**\it
*Crossing the c=1 barrier in 2d lorentzian quantum gravity***}**,
**hep-lat/9909129**;
**{**\it *A new perspective on
matter coupling in 2d quantum gravity***}**
**hep-th/9904012**.

**Numerical results
for 2+1 evolving spin networks**
are in** R.Borissov** and **S.Gupta**, **{***Propagating
spin modes in canonical quantum gravity*},** gr-qc/9810024**.

** Let me now turn to the study
of dynamics**. This must be prefaced by a remark about what
we expect to happen in a quantum theory of gravity at the dynamical level, when
the theory has been formulated completely non-perturbatively. The main questions
are what should be the role of renormalization and how should we expect the
classical theory to emerge from the correct quantum theory.

**Non-zero cosmological constant**

In this case there is a good physical state, discovered by
**Kodama**, which is an exact solution to all the constraints and
also has a good semiclassical limit. That limit gives rise to a semiclassical
interpretation of the state, which is as the ground state in **DeSitter
spacetime** or **AntiDesitter spacetime**, depending on
the sign. When boundaries are introduced this becomes a finite dimensional space
of states, whose number grows as the exponential of the area of the boundary,
exactly as stipulated by the **Bekenstein bound**. This all works
in supergravity also, at least up to **N=2**. Different aspects
of these states have been studied by **Brugmann**, **Pullin**,
**Gambini** and collaborators, **Soo**, **Borissov**,
**Major**, -*myself*- and others. **The
study of these states seems to require the** **quantum deformation
of the theory**, **as mentioned above, in
order to make the** **Kodama state normalizable**.

**{****H.Kodama**,
**Prog.Theor**.**Phys**.**80**:**1024**,**1988****}**;**
B.Bruegmann**, **R.Gambini** and **J.Pullin**,
**Phys**.**Rev**.** Lett**. **68**,
**431** **(****1992****)**;
**Gen**. **Rel**. and **Grav**. **251**
**(****1993****)**;
**L.Smolin** and **C.Soo**, **{**\it
*The Chern-Simons invariant as the natural time variable for classical and
quantum cosmology***}** **Nucl**.
**Phys**. **B449** **(****1995****)****
289**, **gr-qc/9405015**.

*With zero cosmological constant, there are two general
classes of results*;

**A)-** those that
follow from the methods of **Rovelli**, **Borrisov**
and **Theimann**, and

**B)**- those that follow from the
methods of **Gambini**, **Pullin** et al.

* As mentioned these differ
by operator ordering terms*.

**A)** In the case
of the ordering studied by **Rovelli**
-* and myself*-,

In the case **B)**
of the results of **Gambini**, **Pullin** etc. some
of these problems are resolved **(**see
**gr-qc/9909063****)**.
**The algebra of the hamiltonian constraints is
no longer ultralocal, and close on the diffeomorphism generators**.
There are some remaining issues concerning the regularization dependence of
the** commutators**. Whether their ordering resolves the other
problems is not yet known. * I do not think
we should be too disturbed by these result*. No one is surprised
if the quantization of the

**1-**
Find an operator ordering which fixes all the problems. It is in fact easy to
see that were a term of a certain form to be added to the quantum hamiltonian
constraint the problems of being ultralocal and of there being no transmission
of information would be avoided. **The problem is
that so far no one can derive this missing term from a rigorous regularization
of the** **classical hamiltonian constraint**, **or
show that its addition results in a good constraint algebra**.
Several people believe this can be accomplished by changing the inner product,
or equivalently the precise space of states on which the operator acts. This
is partly the motivation for the approach of **Gambini **and **Pullin**.

**2****-**
Fix the problem in the path integral formalism by putting the missing term directly
in the path integral action. This has been advocated by **Rovelli**,
who showed in his paper with **Reisenberger** **(**r**eferenced
above****)** that it emerges
naturally in the path integral framework by changing the time ordering of the
term which comes from the regularization of the hamiltonian constraint. What
this means is that the regularization procedure must be carried out in a way
that is covariant in space and time to get the right dynamics, and this is much
harder to do in the hamiltonian framework than in the path integral framework,
because operators at different times are very complicated to represent in the
**canonical framework. Reisenberger**, and **Barrett**
and **Crane** **(***referenced
above***)** also have
found forms for the action in spin foam that they get by direct translation
of the classical action, without going throught the canonical formalism. In
these the missing term needed for long ranged propagatation appears, which is
evidence for **Rovelli**'s view.

**3-**
Fix the problem in the path integral framework by searching directly for the
form of the action that leads to a good classical limit. This is analogous to
the way one uses the renormalization group in statistical physics to search
for the hamiltonians that describe phase transitions. **I
find this particlarly interesting because it allows us to investigate the general
question of how path integrals are to be defined in a** **lorentzian
theory** **in which the causal structure
is a degree of freedom**. **Because**
of this there is not necessarily any well defined continuation from a **Euclidean
theory**. If the limit exists such that **2****)**
is good there must be a different kind of critical phenomena than the **usual
2nd order phase transitions** that are necessary to define **conventional
Euclidean quantum field theories**. We have identified the appropriate
analogoue, which is certain non-equilibrium critical phenomena associated with
problems like directed percolation and the growth of soap bubbles. Markopoulou
and I have worked in collaboration with people in those areas such as **Per
Bak**, **Maya Paczuski** and **Stuart Kauffman**
to apply the methods in that field to quantum gravity. **Borissov**
and **Gupta**, and **Gupta** have results that confirm
this general idea in **2+1** and **1+1** models. Very
significant results along these lines were found recently by **Abjorn**,
**Loll** and **Anagnastopolous** **(***referenced
above***)**. They find
that one can define a **convergent path integral** for**
lorentzian 1+1 quantum gravity**, which can be thought of as simple low
dimensional examples of evolving** spin network theory**. Their
results are quite surprising as they show a qualitatively different kind of
behavior **(***technically, a new
universality class***)** than
exists in **2d euclidean quantum gravity models**. Extensions of
their results to higher dimension are under study by several people. **(***I
might note that their results were characterized as the most interesting resutls
of the***last year** i**n
the summary talk of the** **lattic** **QFT**
**conference in Pisa** **2001)**.
In very recent results they show that these **lorentizan methods**
allow one to cross **"the c=1"**
barrier, which has been a major stumbling block for **non-perturbative**
approaches to **string theory**.

**4-**
Hypothesize that strict quantization of the field equations of general relativity
does not yield a theory whose classical limit is general relativity but that
there may be a theory in the large class of d**iffeomorphism invariant**
**QFT**'s that can be expressed in loop quantum gravity that will
have a good classical limit that includes general relativity. One approach to
this is to hypothesize that this can happen if the description of gravitons
that results from the semiclassical limit matches that given by one of the infinite
list of **perturbative string theories**. **There
are, in fact, general arguments that string-like excitations must appear as
weakly coupled modes around any classical limit of****loop
quantum gravity**.

This is the approach I have used to try to invent a form of
the theory that also solves the main problem in **string theory**,
which is to invent a background indepednent approach. **The
idea is that as string theory seems to provide the only consistent description
of gravitons at a perturbative level, and as there are many**
**perturbatively consistent string theories**, perhaps any **quantum
theory of qravity** that solves **2)**
will do it in a way that reproduces the main features of **perturbative
string theory**. In a recent paper I showed that this strategy works,
in the sense that there are choices of kinematics and dynamics for** evolving
spin networks** such that one can show that, IF there is a **semiclassical
limit**, that limit reproduces **string theory**, in the
form of the **matrix model**.

**The behavior of the
solutions to the hamiltonian constraint with zero cosmological constant, with
relevance to the classical limit were discussed from different points of view
in:** **L.Smolin**,
**{**\it *The classical limit and
the form of the hamiltonian constraint in nonperturbative quantum gravity***}**
preprint **CGPG-96/9-4**, **gr-qc/9609034**;
**R.Loll**,** gr-qc/9708025**,
**Class**.**Quant**.**Grav**. **15**
**(****1998****)****
799-809**; **R.Gambini**, **J.Lewandowski**,
**D.Marolf**, **J.Pullin**, **{**\it
*On the consistency of the constraint algebra in spin network quantum gravity***}**,
**gr-qc/9710018**,
**Int**.**J.Mod**.**Phys**. **D7**
**(****1998)
97-109**; **J.Lewandowski** and **D.Marolf**,
**Int.J.Mod.Phys**. **D7** **(****1998****)****
299-330**, **gr-qc/9710016**;
**D.Neville**, **Phys**.**Rev**. **D59**
**(1999****)**
**044032**, **gr-qc/9803066**.

**The connection
between classical limits of lorentzian path integrals and non-equilibrium critical
phenomena** has been discussed in, **F.Markopoulou**
and **L.Smolin** **{**\it
*Causal evolution of spin networks***}**
**gr-qc/9702025**. **CGPG** preprint **(****1997****)**,
**Nuclear Physics** **B**, **Nucl**.**Phys**.
**B508** **(****1997****)**
**409-430**; **R.Borissov** and **S.Gupta**,
**{***Propagating spin modes in
canonical quantum gravity***}**,
**gr-qc/9810024**; **D.P.Rideout**, **R.D.Sorkin**,
**{**\it *A Classical Sequential
Growth Dynamics for Causal Sets***}**,
**gr-qc/9904062**; **A.Criscuolo**, **H.Waelbroeck**,
**{**\it *Causal Set Dynamics:
A Toy Model***}**, **gr-qc/9811088**;
**Class**.**Quant**.**Grav**. **16**
**(****1999****)**
**1817-1832**.

**The connections
to string theory** are discussed in **L.Smolin**,
**{**\em *Strings as perturbations
of evolving spin-networks***}**,**
hep-th/9801022**,
**L.Smolin**, **{**\it
*A candidate for a background independent formulation of* **M theory}**,
**hep-th/9903166**.

*The direction of research
in***loop quantum gravity** * has
changed rather strongly in the last few years*.

**1**- **How
to take the semi-classical limit and how to choose the dynamics so that there
is a semi-classical limit**. This includes the different approaches
I mentioned above. **This is a hard problem**,
we should note that it is still not completely solved for **QCD**.
But it is now as well defined in quantum gravity as it is in **QCD**
and it is under attack from several directions, as I indicated. The recent results
of **Ambjorn**, **Loll**, **Anagnastopolou**,
**Borissov**, **Gupta** and others shows that definite
results can be achieved for low dimensional systems and more work in this direction
is underway. The recent paper of **Gambini**, **Pullin**
and collaborators shows that this issue is very much alive also within the **hamiltonian
framework**.

**2-**
**Working out of different examples**,
including different versions of **supergravity**, including **antisymmetric
tensor gauge fields**, and **quantum gravity** in various
dimensions, both lower and higher than **3+1**.

**3-
Application to various problems of physical interest**. The framework
is sufficiently well understood to study various problems about which physical
predictions can be made. Two under investigation are the description of **black
hole** **horizons** and **predictions
for corrections to propagation of light in vacua**, coming from
the discrete stucture, because of claims that these could be observed astrophysically.

**4-**
**Relationship to other approaches**.
Now that the kimematical frameowrk is understood and a lot is known about the
general form of the dynamics, we can investigate relationships to other approaches
to **quantum gravity**. **Several of
these are in progress now****:**

**-****relationship
to** **T**opological** Q**uantum**
F**ield** T**heory** (TQFT)**.
This is turning out to be quite important as I indicated, both because certain
boundary conditions of interest induce **boundary hilbert spaces**
connected to **TQFT**'s and because the general structure of the
actions for **GR** and **supergravity **is closely
connected to certain **TQFT**'s. One important outcome of this
work is the idenficiation of a mathematical langauge which is common to **loop
quantum gravity**, **perturbative string theory** and **topological
quantum field theory**. This is the representation theory of **quantum
groups** or, as it is sometimes called, the** theory of modular
tensor categories**. * This gives a
mathematical langauge within which to seek the unification of*

-**relationship to causal sets**.
This connection, invented by **Markopoulou** has led to the new
attack on** lorentzian path integrals**, the first fruit of which
we see in the results of **Ambjorn**, **Anagnastopolous**
and **Loll**. There are also new results in **causal set
dynamics** **(**by **Rideout
**and **Sorkin**, and **Criscuolo** and **Waelbroeck)**
* that confirm our discovery of the analogy
between the classical limit of theories with fluctuating causal structure and
non-equilibrium critical phenomena such as directed percolation models*.

-**relationship to non-commutative
geometry**. This has been approached by **Ashtekar**
and **Zapata**, who noticed that there is a sense in which **quantum
geometry** in **loop quantum gravity** is **non-commutative**.
It has also been studied by **Rovelli**
and several collaborators.

-**relationship to consistent histories**.
Once a path integral framework for** loop quantum gravity** was
established it becomes natural to examine whether the consistent histories approach
developed by **Gell-Mann**, **Hartle** and others
can be applied to it. It turns out that this requires the precise mathematical
formulation of consistent histories given by **Isham** and collaborators,
which makes use of the langauge of **topos theory**. **Markopoulou**
has shown that a related histories approach, called **quantum causal histories**,
formulated in the same mathematical langauge, describes the structure of causally
evolving **spin networks**. What both approaches have in common
is that there is a structure **(****technically
a pre-sheaf****)** consisting
of many **hilbert spaces** and **observables algebras**,
each connected with a subset of observables of the **quantum cosmology**.
In** Isham** et. al. these correspond to the different consistent
sets of histories, in **Markopoulou** these correspond to what
is observable in the causal pasts of the different local observables in the
**spacetime**. * Under way is the
examination of the relationship between these two appraoches*.

**It should be noted that the mathematics
involved in both approaches is related to the langauge of** **modular
tensor categories**. I mentioned above, **giving
more evidence that this may be the right mathematical framework for the**
**full theory of quantum gravity**. I should note also that the
approaches of **Markopoulou** and **Isham**, while
different, each show some promise of giving a new formulation of **quantum
cosmology **which resolves the foundational problems in that subject.
**I think it very striking that a mathematical
structure that emerges from trying to make a sensible formulation of**
**quantum cosmology** t**hat would
apply to observations made by observers inside the universe is closely related
to that which emerged from**** loop quantum gravity**,
**TQFT** and **string theory**.

-**relationship to string theory**.
This is my main occupation now, there are some striking results already, such
as the background independent formulation of **(p,q)** **strings**
and **SL(2,Z)** duality we did with **Markopoulou**,
**(****hep-th/9712148**,
**Phys.Rev**. **D58** **(****1998)**
**084033)**, the general argument
mentioned above that the perturbations of evolving spin networks involve stringlike
degrees of freedom and a particular proposal for a background independent form
of **M theory**.

**Let me close by coming back to
the two issues you mentioned, to make clear how they have been addressed:**

**The problem of gauge invariant
gravitational observables:**

**First**, at the
level of **spatially diffeomorphism invariant observables**, **a
sufficient set has been constructed and** **diagonalized**,**
in closed form, to label a complete basis of states in terms of their eigenvalues,
for each of a large set of theories**.

*Concerning***spacetime diffeomorphism invaraint observables**, * one
does not expect to be able to do the same in closed form*.

On the other hand, it is not difficult to construct as many
physical, **spacetime difeomorphism invariant**, observables as
one likes in a histories framework. **(***This
point has been emphasized for a long time by* **Hartle)**.
This is the case in spin foam, as emphasized by** Rovelli**,
**Reisenberger**, **Baez** and others. Once that is
done one can work backwards and express those physical observables as canonical
operators using **Rovelli**'s
formulation of projection operators on physical states **(****referenced
above****)**. The expectation
values of the resulting operators are expressed as infinite series which, while
not in closed form, are completely well defined.

**Most importantly**,
this means that in spin foam or evolving **spin networks**, expectation
values of physical, gauge invariant observables can be written down and explicitly
computed, using either analytic or numerical approximation techniques. Examples
in which this is carried out in detail in **1+1** cases are discussed
in the papers of **Ambjorn**, **Loll **etc **referenced
above**. **Work is in progress to extend
this first to**** 2+1** **then
to** **3+1**.

One can also construct physical operators by two other methods
**1****)** fix the time
gauge and then construct the normal hamiltonian theory for evolution in that
time gauge. Diffeomorphism invariant observables are then promoted to physical
observables, defined on spacelike slices picked out by the gauge conditions.
This approach is described in detail in **L. Smolin**, **{**\it
*Time, measurement and information loss in quantum cosmology***}**
in **{**\it *Directions in general
relativity*,** vol. 2 Papers** in honor of **Dieter Brill}**
ed. **B. L. Hu **and** T. A. Jacobson** **(****Cambridge
University Press**, Cambridge, **1993****)**,
**gr-qc/9301016**;
**C. Rovelli**
and **L. Smolin,** **{**\it
*The physical hamiltonian for non-perturbative quantum gravity***}**,
with **Carlo
Rovelli**, **Phys. Rev**. **Lett. 72**
**(****1994****)**
**446-449**, **gr-qc/9308002**.

Alternatively one can construct evolving constants of motion,
as described in detail in several papers of **Rovelli**,
and then evaluate them for the time variable set equal to zero, when they are
equal to ordinary spacially diffeomorphism invariant observables. **This
is sufficient to promote the spectrum of the area and volume operators to physical
observables**.

**The precise statement of the
operator valued nonlinar field equations: **

*As mentioned above, we have
precise regularization techniques that, given the***classical
hamiltonian constraint**, **return finite
well defined****quantum constraint operators**.
Those that arise from different regularization procedures differ by now understood
operator ordering terms. **There are no anomolies**,
in the sense of terms proportional to **1** that arise in their
algebra; thus they have infinite dimensional spaces of solutions. The resulting
theory is like **QCD,** completely well defined, but certain forms
appear not to have a **classical limit** which reproduces the classical
field equations. The operators that result from regularizations of the classical
expression appear to be missing a certain term which it is known would lead
to long ranged propagation.

There could be several reasons for this, which are all under
investigation. **1****)**
The missing term will result from a modification of the inner product on diffeomorphism
invariant states **(***this is
being investigated by* **Theimann**, **Gambini**,
**Pullin** and collaborators**)**.
**2****)** The missing
term was lost by a regularization method that was not sufficiently covariant
in both space and time; it can be easily reinserted by symmetrizing the path
integral amplitude that corresponds to the canonical operator and could in principle
be found also be a complicated procedure in the canonical theory that would
involve point spliting in time and space **(**this
is** Rovelli**'s
approach**)**** 3)**
the condition that the quantum constraint operator must arise as the limit of
the regularization of the classical expression is too strong, what is required
is only that the expectation value of the expression reproduce the classical
constraint, and the missing term may be added without changing the fact that
this property is satisfied. In this case we are free to add it and study its
consequences. **4****)**
**the theory of** **quantum
gr** **does not in fact have a semiclassical
limit which includes** **classical gr**, **but
theories may exist which can be expressed in the same kinematical langauge which
that will do this, and they may accomplish this by having not only ****classical
gr**,** but a** **perturbative
string theory**, **as their semiclassical
limit**.

** As I indicated, research
is going on in all of these directions**. So to summarize
the situation in

**Finally**, as in
**QCD**, **there are hard problems**,
still under investigation, concerning the behavior of the solutions to **different
versions of the theory at large scales**.

**NOTE
(1):** **Quantum
geometry****:** In
theoretical physics, quantum geometry is the set of new mathematical concepts
generalizing the concepts of geometry whose understanding is necessary to describe
the physical phenomena at very short distance scales **(**comparable
to **Planck length****)**.
At these distances, quantum mechanics has a profound effect on physics. **Each
****theory of quantum gravity** **uses
the term quantum geometry** **in a
slightly different fashion**. **String theory**,
the leading candidate for a **quantum theory of gravity**, uses
the term **quantum geometry** to describe exotic phenomena such
as **T-duality** and **other geometric dualities**,
mirror symmetry, topology-changing transitions, minimal possible distance scale,
and other effects that challenge our usual geometrical intuition. **More
technically**, **quantum geometry** refers to the
shape of the spacetime manifold as seen by **D-branes** which includes
the quantum corrections to the metric tensor, such as the worldsheet instantons.
**For example**, the **quantum
volume** of a cycle is computed from the mass of a brane wrapped on this
cycle. **In an alternative approach**
to quantum gravity called **loop quantum gravity** **(****LQG****)**,
the phrase** quantum geometry** usually refers to the calculation
within **LQG** that shows that certain** physical observables**,
such as the **area**, have a discrete spectrum of eigenvalues.
This phrase is often used as synonymous with** loop quantum gravity**
as a whole.** It is possible but considered
unlikely by many that this strictly quantized understanding of geometry will
be confirmed by****string ****theory**.

**NOTE
(2)**: **
Information Theoretic Entropy:
**The basic concept of entropy in **Information
Theor**y has to do with how much randomness there is in a signal or in
a random event. An alternative way to look at this is to talk about how much
information is carried by the signal. As an example consider some English text,
encoded as a string of letters, spaces and punctuation
**(****so our signal is a string of characters****)**.
Since some characters are not very likely **(**e.g.
**'****z****')**
while others are very common **(**e.g.
**'****e****')**
the string of characters is not really as random as it might be. On the other
hand, since we cannot predict what the next character will be, it does have
some 'randomness'. Entropy is a measure of this randomness, suggested by **C.E.Shannon**
in his **1949** paper **"A
Mathematical Theory of Communication"**.
**Shannon derives his definition of entropy from
the assumptions that****:**

The measure should be continuous. I.e. changing
the value of one of the probabilities by a very small amount should only change
the entropy by a small amount. **If all the outcomes**
**(****letters
in the example above****)**
**are equally likely then increasing the
number of letters should always increase the entropy**.
We should be able to make the choice **(****in
our example of a letter****)**
in** two steps**, in which case the entropy of the final result
should be a weighted sum of the entropies of the final result. **Shannon**
defines **Entropy** in terms of a discrete random event **x**,
with possible states **1**..**n** as**:**

That is the entropy of the event x is the sum over all possible
outcomes i of the product of the probability of outcome i times the log of the
probability of i. We can equally apply this to a probability distribution rather
than a discrete-valued event.** Shannon** shows that any definition
of entropy satisfying his assumptions will be of the form**:**

Where **K** is a constant**
(****and is really just a choice of measurement units****)**.

**NOTE
(3):** **T-duality:**
**T-duality** is a symmetry of **string theory**,
relating **type IIA** and **type IIB string theory**,
and the two **heterotic string theories**. **T-duality transformations**
act on spaces in which at least one direction has the topology of a circle.**
Under the transformation**, the radius **R** of that
direction will be changed to **1/R**, and **"wrapped"**
string states will be exchanged with **high-momentum string states**
in the **dual theory**. **For example**,
one might begin with a** IIA** **string** wrapped
once around the direction in question. Under **T-duality**, it
will be mapped to a** IIB **string which has momentum in that direction.
A **IIA** **string** with a winding number of two**
(**i.e., **wrapped twice****)**
will be mapped to a **IIB string** with **two units of momentum**,
and so on.

**The total squared mass of a closed
string**

is invariant under the exchange ,

and the interactions and all other physical
phenomena can be proved invariant under this operation, too. **T-duality
**acting on **D-branes** changes their dimension by **+1**
or **-1**.

**Andrew Strominger**, **Shing-Tung
Yau**, and **Eric Zaslow** have showed that **mirror
symmetry** can be understood as **T-duality** applied to
**three-dimensional toroidal fibres of the**
**Calabi-Yau space**.

**Utenti connessi**...:

**CHANDRAst (midi
by CHANDRAst 2004)**