Mathematical Physics Group

Quantum field theory and Gravitation


Institute for Theoretical Physics, University of Göttingen


Enrico Bothmann
Stephan Bräuer
Prof. Laura Covi
Suman Deb
Christian Gaß
Michael Gustafsson
Prof. Hubert Goenner
Sarif Khan
Prof. Helmut Reeh
Prof. Karl-Henning Rehren
Daniel Reichelt
Prof. Steffen Schumann
Vincent Theeuwes
Simon Luca Villani


Michael Dütsch
Folkert Müller-Hoissen
Manfred Requardt

Master students:

Timo Janssen
Moritz Heep
Karim Shedid Attifa

Bachelor students:

Luis Peters
Felix Tippner
Michael Tsopanopoulos

Diploma and PhD Theses
Research Reports
Local Quantum Physics Crossroads
Mathematics Institute Göttingen
EU Network Noncommutative Geometry
AQFT - 50 years

Department info booklet pages 134/135

Studies of fundamental questions of physics by methods of mathematical physics have a tradition of long standing in Göttingen. In the sixties, G. Lüders and H.-J. Borchers established mathematically oriented quantum field theory and statistical mechanics at the Institute of Theoretical Physics; many papers from Göttingen had great influence in these fields lying at the interface of local quantum physics and functional analysis.

Research along these lines is continued. The focus is on central questions concerning the construction as well as the interpretation of relativistic quantum field theories which cannot be answered by conventional (perturbation or numerical) methods; a complementary approach will - hopefully - lead to a deepened structural understanding of the theory.

The group's projects can be subdivided into five domains:

(1) Constructive aspects:
Questions of existence, construction and classification of quantum field theories.
(2) Symmetries and particles:
Development of methods for analysis and interpretation of quantum field theories in elementary particle physics.
(3) Thermal states:
Structural analysis and development of novel concepts describing thermal properties of macroscopic systems within the framework of quantum field theory.
(4) Gravity and General Relativity:
Quantum structure of space-time, unified theories, history of GR

Further details:

(1) Constructive aspects

A novel approach for the analysis of quantum field theories is based on an advanced Tomita-Takesaki theory, which has been initiated by H.-J. Borchers and H.-W. Wiesbrock. It could be demonstrated that the knowledge of a few subalgebras of the observable algebra suffices to characterize and reconstruct the global theory. A further study of these subalgebras might lead to a new class of invariants, and open new ways of classification and construction of theories. Furthermore, the method is apt to define vacuum-like states in curved space-times by a condition of geometric modular action as proposed by D. Buchholz and S. J. Summers. Work on these questions, which are important for the analysis and construction of quantum field theories, is being continued. (H.-J. Borchers, D. Buchholz)

The implications of conformal symmetry for the kinematical and dynamical structure is analyzed. In two dimensions, an abundance of theories has been constructed, apt to test and to extend the general theory of superselection sectors and statistics. Algebraic methods of construction arise from the application of subfactor theory. These allow for the classification of "boundary CFTs" on a half-space. (K.-H. Rehren) In four dimensions, the restrictivity of the assumption of global conformal invariance is studied, using operator product expansions and partial wave expansions. (K.-H. Rehren, I.T. Todorov)

(2) Symmetries and particles

In view of the quantum equivalence of certain (gauge) theories which are classically distinct theories it may be asked whether the standard interpretation (with the help of gauge theories) of the inner structure of hadronic matter (quarks, gluons, colour) is free of arbitrariness. In order to analyse this problem D. Buchholz and R. Verch established a new method to determine the short distance structure of local observables in quantum field theory (scaling limit). Based on this method a procedure is being developed to determine the symmetry group and the particle content of a theory at small length scales directly from the observables - and thus free of arbitrariness. (D. Buchholz, M. Lutz, R. Verch)

Because of infrared effects the superselection structure of theories with abelian gauge symmetries and corresponding free (electric and magnetic) charges is rather complicated; their particle content and their energy-momentum spectrum cannot be described by the particle notion of Wigner. The spectrum can be analysed by methods of complex analysis developed by H.-J. Borchers; D. Buchholz introduced the notions of charge class and particle weight for the sector analysis. With the help of these notions and methods the sector structure of the physical state space shall be further clarified, and the interesting connection between (bosonic) superselection charges and geometric symmtries shall be investigated systematically. (H.-J. Borchers, D. Buchholz, W. Kunhardt, M. Porrmann, H. Reeh)

Conserved quantities are of great importance for the analysis and, in favourable cases, the complete calculation of the S-matrix in quantum field models. In a programme which has been completed by now, the structure of additive conserved quantities could be clarified fairly completely by rigorous mathematical methods. There are only partial results concerning the interesting cases of charges of higher genus (non-local charges). All possible forms of such charges shall be determined within the general framework of quantum field theory. (H. Reeh)

(3) Thermal states

Thermal equilibrium states can be charcterized by the KMS condition valid for relativistic as well as non-relativistic systems. J. Bros and D. Buchholz gave arguments that in relativistic theories a much stronger form of this condition should hold (relativistic KMS condition). It is to be checked whether this sharpened version can be derived from a relativistic condition of passivity of equilibrium states, and what consequences there are for the structure of relativistic equilibrium states. Generalized KMS conditions should give rise to new kinds of correlation (in)equalities which might be useful in statistical mechanics and Euclidean field theory. (D. Buchholz, M. Requardt, H. Roos)

The concept of a quasiparticle is an important one in statistical mechanics. The development of a general scattering theory for particle-like exitations in thermal states is envisaged on the basis of results by Narnhofer, Requardt and Thirring and recent results of Bros and Buchholz. (D. Buchholz, M. Requardt)

The precise mathematical characterization of thermal non-equilibrium states which are in equilibrium only locally is an important step in the understanding of the structure of thermal processes; furthermore, it is of interest for model calulations. The existing phenomenological ansaetze are partially successful, but they are not at all satisfying given a microscopic point of view. A reformulation of the KMS condition seems necessary and appropriate for this case. With the help of the scaling analysis a generalisation of the notion of temperature ("effective local temperature") is sought for which allows to parametrize local equilibrium states in a consistent way. (M. Requardt, H. Roos)

Dealing with coexisting phases one encounters non-translation invariant equilibrium states. The study of these states is under way in a long time project. The results are of importance for the understanding of phase boundaries, systems in external fields and in contact with container walls ("wetting"). (M. Requardt)

(4) Gravity and General Relativity:

- (Entanglement) entropy of Black Holes, Holographic bounds (M. Requardt)

- Continuum limit of discrete geometries, understanding of various scales of resolution of space-time in quantum gravity (M. Requardt)

- Gravitation and Geometry (H. Goenner (retired)):

Alternative and unified theories of gravity (Kaluza-Klein, torsion, scalar-tensor, Finsler geometry), non-commutative geometry with possible applications (lattice theory, field theory on discrete spaces, stochastic processes on manifolds, non-commutative differential calculus, computer algebra). Foundations of General Relativity and Cosmology.

Members of the Göttingen group are cooperating on the set of problems mentioned above with members of the following universities and research institutes:

University of Florida, Gainesville (1),
RIMS Kyoto (3),
CEA Saclay (2,3),
Università di Pisa (2),
Università di Roma (1,2),
Bulgarian Academy of Sciences, Sofia (1,2)
Universität Wien (1)

The group participates in the topic "Foundations of relativistic quantum physics" within the Graduate Topical Program "Interaction" of the Evangelisches Studienwerk Villigst, along with Theoretical Physics Institutes of U Hamburg, FU Berlin, and U Leipzig.

Publications 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000
Research reports < 2000

Karl-Henning Rehren