Project C4: Mathematical aspects of quantum field theory



Principal investigator

Participating scientists

Summary

Quantum field theory is the set of principles and methods to deal with quantum systems with infinitely many degrees of freedom. The main example is Yang-Mills theory relevant for high energy physics. An understanding of Yang-Mills theory belongs to the Millenium Prize challenges in mathematics. The project studies quantum field theory of much simpler systems with mathematical rigour. The tools developed in the project might be useful for full Yang-Mills theory.

The main characteristics of Yang-Mills theories are perturbative renormalisability and asymptotic freedem, which suggests non-perturbative existence. During the last 5 years we have proven that the relatively simple φ4-model on a non-commutative geometry is perturbative renormalisable and asymptotically safe, which means that the flow of the coupling constant is bounded. This should suffice to prove non-perturbative existence of non-commutative φ4-theory. One part of the project tries to complete the proof. This involves a Ward identity arising from an U(∞) group action, Schwinger-Dyson equations, and a combinatorial investigation of iterated integrals labelled by rooted trees.
Non-commutative φ4-theory is obtained from a spectral triple related to supersymmetric quantum mechanics. The latter has many applications in topology and relates to problems in topological quantum field theory. An important problem in topological quantum field theory is the refinement of such to theories to local field theories. An approach to local field theories views these as higher functors from a bordism category of manifolds with corners to a higher category. A goal of this project is the investigation and construction of such local field theories. In particular, we will use conformal nets of von Neumann algebras to construct a suitable target category for such local field theories in dimension 3. In the long term, connections to locality in non-commutative geometry defined via residues shall be studied.
The action of infinite-dimensional Lie groups is characteristic to many quantum field theories. For Yang-Mills theory this is the group of vertical automorphisms of an SU(N)-principal fibre bundle ("local gauge transformations"). In contradistinction to the commutative case, noncommutative φ4-theory is characterised by an U(∞)-action. The corresponding Ward identity was the key to prove asymptotic safeness; it is also a main tool in the targeted non-perturbative construction of non-commutative φ4-theory. The mechanism how the U(∞)-action leads to asymptotic safeness is not understood and deserves further investigation in this project.

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