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Quantum fields and Curvature--Novel Constructive Approach via Operator Product Expansion

Objective

It was realized already from the beginning that the theory of quantized fields (QFT) does not easily fit into known mathematical structures, and the quest for a satisfactory mathematical foundation continues to-day. Parts of this theory have already been tremendously successful, e.g. in the quantitative description of elementary particles, and ideas from QFT have revolutionized entire fields of mathematics. But the non-perturbative construction of the most important QFT s, namely renormalizable theories in 4d, remains unsolved. The aim of this project is to make a substantial contribution to this quest for the mathematical construction of such QFT s (on curved manifolds), and the exploration of their mathematical structure. We want to pursue a novel ansatz to achieve this goal. The essence of our novel approach is to focus attention on the algebraic backbone of the theory, which manifests itself in the so-called operator-product-expansion. The study of such algebraic structures related to operator products has already been tremendously useful in the study of conformal field theories in low dimensions, but we here propose that a suitable version of it also has great potential to be used as a constructive tool for the much more complicated quantum gauge theories in four dimensions. It is not expected that an explicit solution can be obtained for such models-especially so in curved space-but the idea is instead to analyze powerful consistency conditions on the quantum field theory arising from the OPE ( associativity conditions ) and to use them to prove that the theory exists in a mathematically rigorous sense. Our approach will be complemented by other powerful and deep mathematical tools that have been developed over the past decades, such as the sophisticated non-perturbative expansions uncovered in the school of constructive quantum fields theory , Hochschild cohomology, RG-flow equation techniques, microlocal analysis, curvature expansions, and many more.

Field of science

  • /natural sciences/physical sciences/theoretical physics/particle physics
  • /natural sciences/physical sciences/quantum physics/quantum field theory

Call for proposal

ERC-2010-StG_20091028
See other projects for this call

Funding Scheme

ERC-SG - ERC Starting Grant

Host institution

UNIVERSITAET LEIPZIG
Address
Ritterstrasse 26
04109 Leipzig
Germany
Activity type
Higher or Secondary Education Establishments
EU contribution
€ 537 226,49
Principal investigator
Stefan Hollands (Dr.)
Administrative Contact
Gerhard Fuchs (Mr.)

Beneficiaries (2)

UNIVERSITAET LEIPZIG
Germany
EU contribution
€ 537 226,49
Address
Ritterstrasse 26
04109 Leipzig
Activity type
Higher or Secondary Education Establishments
Principal investigator
Stefan Hollands (Dr.)
Administrative Contact
Gerhard Fuchs (Mr.)
CARDIFF UNIVERSITY
United Kingdom
EU contribution
€ 280 872,11
Address
Newport Road 30-36
CF24 ODE Cardiff
Activity type
Higher or Secondary Education Establishments
Administrative Contact
Eevi Laukkanen (Ms.)