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 formula11 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 formula17 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 formula21 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
see travel instructions

Scientific Organizers

Dr. Jürgen Tolksdorf
Max Planck Institute for Mathematics in the Sciences
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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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05.04.2017, 12:42