# Workshop on Mathematical and Physical Aspects of Quantum Gravity

## Abstracts for the talks

### Abstract Quantum Mechanics

**Samson Abramsky** *(Oxford)*

Saturday, July 30, 2005

We introduce a categorical axiomatization of (in the first
instance: finitary, non-relativistic) Quantum Mechanics with the following
main features.

- We expose the underlying structural mathematics - or logic in a suitably broad sense (very different to traditional quantum logic) - of both quantum and classical information flow, allowing effective reasoning about entanglement in compound systems, and information protocols and computation schemes based on this.

- We develop AQM as a typed theory: the types correspond to important conceptual distinctions in the physical theory. This allows e.g. a proper compositional treatment of measurements.

- The theory is `structural' and `qualitative', yet quantitative features (scalars, a Born rule) arise from it automatically.

- Notions such as projectors, which are fundamental building blocks in the usual formalism, are decomposed in AQM. This decomposition plays a key role in the analysis of quantum information flow.

We will also discuss some ideas towards introducing a discrete version of relativistic concepts into AQM.

(joint work with Bob Coecke)

### Physics and Mathematics of Loop Quantum Gravity

**Abhay Ashtekar** *(University Park)*

Saturday, July 30, 2005

This talk will survey some recent advances which illustrate the
underlying ideas, successes and limitations of loop quantum gravity.
It will be addressed to non-specialists in the hope of stimulating
discussion with participants working in other approaches.

### On Noncommutative Differential Geometry and General Relativity

**Ludwik Dabrowski** *(Trieste)*

Sunday, July 31, 2005

A construction of spectral triple associated to a (compact)
Riemannian spin manifold, and the reconstruction theorem is recalled.
General Relativity in terms of Dirac spectrum and the spectral action are
briefly reviewed, and further directions of study indicated.

### A background independent perturbative approach to quantum gravity

**Klaus Fredenhagen** *(Hamburg)*

Sunday, July 31, 2005

Quantum field theory on curved spacetime has recently achieved a
satisfactory formulation in terms of a functor which associates globally
hyperbolic spacetimes to algebras of observables. In this formulation a
generally covariant construction of perturbative quantum field theory on
generic backgrounds can be given. An attractive feature is the
possibility to include dynamical variations of the spacetime metric. The
requirement of background independence amounts to Ward identities which
essentially are equivalent to the field equation for the metric field.
As the theory is nonrenormalizable, it may be interpreted as an effective
theory.

### On the role and determination of mapping-class groups in canonical quantum gravity

**Domenico Giulini** *(Freiburg)*

Friday, July 29, 2005

The diffeomorphism constraint in canonical quantum gravity
eliminates a certain normal subgroup of the group of all
spatial diffeomorphisms. The question remains as to how the
ones left out are to be treated. They include, in particular,
the ones that are not contained in the identity component of
the diffeomorphism group, which may be identified with a
certain mapping-class group of the underlying 3-manifold.
We outline general aspects of their determination and discuss
the problem of whether they should be viewed as gauge
transformations or rather as proper symmetries. This highlights
certain conceptual difficulties with the notion of "eliminating
the constraints".

### Noncommutative Quantum Field theory and Renormalization

**Harald Grosse** *(Wien)*

Sunday, July 31, 2005

The hope that ncQFT will cure the old diseases of undeformed QFT are not
fulfilled, especially due to the occurence of the IR/UV mixing. We first
give the definition of various deformed spaces, give the dictionany from
undeformed to deformed formulations and define nc quantum field theoretic
models. The IR/UV mixing property is mentioned. For the special case of a
duality covariant four dimensional nc scalar QFT we proof perturbative
renormalizability to all orders. This model is related to a dynamical
matrix model. At the self duality point it becomes integrable and
asymptotically free.

### Spin Networks, Anyonic Topological Quantum Computing and Quantum Gravity

**Louis Kauffman** *(Chicago)*

Saturday, July 30, 2005

Spin networks were invented/discovered by Roger Penrose in an
attempt to provide a combinatorial precursor to spacetime. In his
Spin-Geometry Theorem, Penrose showed how angular properties of three
dimensional space would emerge from self-interactions of large spin
networks. The Penrose theory of spin networks eventually was generalized
to a recoupling theory that began with the bracket polynomial skein
relation rather than the Penrose binor identity. This *q*-deformed spin
network theory has been of use in constructing topological
quantum field theories, the Witten invariants of three manifolds and
measurement and spin-foam techniques in loop quantum gravity.

Recently, Freedman, Kitaev and their collaborators have shown how braiding
operators in certain topological quantum field theories are universal for
quantum computation. In particular, one can focus on the topoloogical
quantum field theory called Fibonacci Anyons. (There are two basic
particles call them 1 and 0. The only non-trivial interaction is or 1. The corresponding recoupling theory is intricate.
The braiding is non-trivial and can model quantum computation.) The
purpose of this talk is to give a simple model for the Fibonacci Anyons in
terms of deformed spin networks, and to show how the
structure of the model proceeds from the structure of the bracket model of
the Jones polynomial and the Dubrovnik (Kauffman) skein polynomial.

The point of view of this talk allows discussion of the relatiohship
of quantum information theory and quantum computing with the knot
polynomials. The use of spin networks in these models suggests a deeper
dialogue with quantum gravity. In the first place, the result about
Fibonacci anyons shows that a deformation of the classical spin networks
to a nearby root of unity allows the generation of arbitrarily good
approximations to unitary group transformation in the braiding
representations associated with these spin networks. This means that at
the mathematical level there is a unification of the generation of
background space (spin geometry) and the generation of quantum mechanical
evolutions. In a sense this is a background for a possible unified quantum
geometry. Secondly there is the possiblity of using the machinery of the
Temperley Lieb recoupling theory at a categorified level, using the
framework of Khovanov homology, to generate new four dimensional hybrid
algebraic/state sum models that may impinge on topological approaches to
quantum gravity. All of this part of the research is in a state of flux,
but we will report on its present state at the time of the talk.

### Quantum gravity - What are the problems?

**Claus Kiefer** *(Köln)*

Friday, July 29, 2005

One of the biggest open problems in physics is the consistent
unification of quantum theory with general relativity, resulting in
a theory of "quantum gravity". Such a theory would have an important
bearing on the physics of the early universe and the understanding
of black holes. Moreover, it would possess deep relevance for
our fundamental understanding of nature.
In my talk I shall attempt to present the general problems
in constructing a quantum theory of gravity as well as the main
conceptual ingredients of such a theory. I shall then shortly review the
main approaches and attempt a critical evaluation of their status.

### How can we search for quantum gravity effects?

**Claus Lämmerzahl** *(Bremen)*

Friday, July 29, 2005

Since quantum theory and General Relativity are incompatible
there should be a new theory which consistently applies to all physical
phenomena and in particlar should describe the quantum nature of gravity.
This new theory necessarily must lead to effects which are different from
effects describable within the present day standard theory. We also address
the problem of how to decide whether an anomalous effect to is of quantum
gravity origin. We try to classify the possible and expected non-standard
effects in order to structure the experimental search for quantum gravity
effects and describe the corresponding classes of experiments. One primary
classification comes from the principles underlying the present day physics,
another classification comes from the structure of present approaches to
quantum gravity theories. In both cases, test theories provide a viable
scheme for the description of possible deviations from present day standard
physics and also help to bridge between statements in some abstract
theoretical quantum gravity scheme and a formalism which is more or less
directly applicable to the description of experiments.
According to that scheme, a survey of the present experimental limits for
the validity for present day standard physics or, equivalently, for the
search for quantum gravity effects is given. An outlook on possible future
experimental developments and capabilites will conclude this presentation.

### Quantum gravity on finite sets

**Shahn Majid** *(London)*

Sunday, July 31, 2005

Although finite sets do not have any nontrivial usual manifold
structure, within the more general axioms of noncommuative geometry they
do. A differential structure is defined by a graph on the finite set.
Unlike finite lattice approximations, there are `no truncation errors' but
rather an exact finite geometry with a rich an self-consistent structure.
One can then go on to define bundles, Riemannian curvature etc over the
finite set. In this case funtional integration becomes ordinary integration
and a sum over differentiable structures (which we have proposed before as
required in quantum gravity) becomes a sum over graphs not unlike Feynman
diagrams.

The talk is based on J. Math. Phys 45 (2004) 4596-4627 (with E. Raineri) as
is part of a general programme of gravity on algebras.

### Quantume Fields and Geometric Topology

**Kishore Marathe** *(New York)*

Friday, July 29, 2005

In recent years, several relations between quantization of gauge
fields and geometric topological invariants have been discovered.
These TQFT results may be regarded as toy models for physically
significant QFTs with non-trivial dynamics. We speculate that the
loop variable formulation of gravity may provide a suitable platform
for similar calculations. The results can be considered as furnishing
a toy model for quantum gravity.

### From quantum field theory to gerbes and twisted K-theory

**Jouko Mickelsson** *(Stockholm)*

Sunday, July 31, 2005

A breakdown of classical symmetries (gauge or reparametrization)
can occur in a system of fermions in external fields. The topology of this
mechanism is understood in terms of families index theory in the
path integral formalism. In the Hamiltonian approach
the families index theory leads to odd cohomology classes as obstructions
to covariant quantization. In particlular, the 3-cohomology class
known as the Dixmier-Douady class plays a central role; it gives a
topological classification of gerbes over the configuration space
of the quantum system.

I give a review how the quantum mechanical symmetry breaking leads to
the concept of gerbe in terms of concerete examples from gauge theory.
Also, more recent developments leading to constructions of twisted K
theory classes on a manifold from a field theory model will be
covered.

### Strings, higher curvature corrections, and black hole entropy

**Thomas Mohaupt** *(Jena)*

Friday, July 29, 2005

We review old and recent results about the correspondence
between the states of black holes and string states. Due to quantum
corrections the entropy of extremal black holes is corrected in two ways:
first, there is an explicit modification of the geometry due to higher
curvature terms in the effective action. Second, the area law itself is
modified, as was first realized by R. Wald. When taking both kinds of
corrections into account, it can be shown that not only the leading
term but also subleading corrections to the black hole entropy match
with the asymptotic state degeneracy predicted by string theory.
Recently, H. Ooguri, A. Strominger and C. Vafa used this result to
propose a non-perturbative definition of topological string theory.
This bold idea has catalysed a renewed interest in the role of
higher curvature corrections, and leads to several interesting
open questions.

### Quantum gravity as quantum field theory of simplicial geometry?

**Daniele Oriti** *(Cambridge)*

Saturday, July 30, 2005

I present a brief overview of the group field theory approach
to quantum gravity, leading to a complete definition of spin foam models
including a sum over triangulations of spacetimes of all topologies. These
models are non-perturbative and fully background independent.
Here gravity, and therefore spacetime geometry, is described in purely
algebraic (representation-theoretic) and combinatorial terms.
Most important, I discuss what are, in my opinion, the pleasent features
of this approach and what are the next steps needed in order to make these
models not only promising but solid candidates to the final theory of
quantum gravity.

### Stringtheory: Facts and expectations

**Stefan Theisen** *(Golm)*

Friday, July 29, 2005

After a brief review of string theory as a framework of
unification I will
discuss the role of gravity in string theory. I will explain the claim
that it is a theory of quantum gravity. Finally, the notion of string
geometry will be mentioned.

### Gravitational Waves and Energy-Momentum Quanta

**Robin Tucker** *(Lancaster)*

Sunday, July 31, 2005

The notion of energy-momentum in any relativistic description
of gravitation is a subtle one and requires careful analysis in
any cogent attempt to quantize the gravitational field.
By embedding Einstein's original formulation of General Relativity into
a broader context it will be show that a dynamic covariant description
of purely gravitational stress-energy emerges naturally from a variational
principle. A new tensor will be constructed (from a contraction of the
Bel tensor with a symmetric covariant second degree tensor)
that has a form analogous to the stress-energy tensor of the
Maxwell field in an arbitrary space-time. For plane-fronted
gravitational waves, helicity-2 polarised (graviton) states can be
identified carrying non-zero energy and momentum.

The content of this talk will survey conventional wisdom on pure gravitational stress-energy, offer comments on the need for a tensorial formulation and construct a model in which such a formulation can be achieved. Some implications for the nature of gravitational quanta will be deduced from this model.

The talk will be based on published work by T Dereli and R W Tucker.

## Date and Location

**July 28 - August 01, 2005**

Heinrich-Fabri-Institut

Blaubeuren

Germany

see travel instructions

## Scientific Organizers

**Dr. Jürgen Tolksdorf**

Max Planck Institute for Mathematics in the Sciences

Contact by Email

**Dr. Bertfried Fauser**

Max Planck Institute for Mathematics in the Sciences

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**Prof. Dr. Eberhard Zeidler**

Max Planck Institute for Mathematics in the Sciences

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## Administrative Contact

**Dr. Jürgen Tolksdorf**

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**Dr. Bertfried Fauser**

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