Constructive Algebraic Quantum Field Theory

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The proposer plans to develop operator algebraic and functional-analytic techniques in order to construct models of relativistic quantum field theory both on classical and on noncommutative four-dimensional Minkowski space. The primary focus of the next two years of work will be to construct such models by deforming a given model in a manner recently introduced by D. Buchholz and the proposer. Among other things, this will necessitate the further development of functional-analytic techniques to define multiple operator integrals yielding suitable deformations of quantum field operators or of associated bounded observables, to prove the essential self-adjointness of such integrals, and to prove suitable commutativity properties of the self-adjoint extensions. This would lead to associated von Neumann algebras of observables, whose modular structure would be studied. In addition, will develop further techniques to establish that certain intersections of such von Neumann algebras are sufficiently large (or not), which would allow us to determine the availability of sufficiently many local observables in the constructed models. Such progress is likely to be useful in other applications of functional analysis besides the one discussed here. Relativistic quantum field theory is the most successful theory of elementary particle physics, which itself seeks to describe the fundamental constituents of matter and their laws of interaction, but one of the most serious gaps in our understanding of QFT has been the absence of mathematically rigorously constructed models of nontrivial quantum fields on the standard model of the four-dimensional Minkowski space. The proposed work will provide such models, as well as further mathematical techniques for their analysis and control. Moreover, these methods can also be transferred to quantum field theories on curved spacetimes with a sufficiently large isometry group, such as de Sitter space and anti-de Sitter space. Nontrivial quantum field models on such space--times can then also be constructed. In addition, due to the special nature of the deformation to be studied in this proposal, the proposed work will also result in further rigorous quantum field models on noncommutative Minkowski space, which is expected to be of relevance to the quantization of gravity.