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Towards an operator algebraic construction of quantum field theories on de Sitter space

Grant number: 18/09613-9
Support type:Regular Research Grants
Duration: September 01, 2018 - August 31, 2020
Field of knowledge:Physical Sciences and Mathematics - Mathematics
Principal Investigator:Christian Dieter Jäkel
Grantee:Christian Dieter Jäkel
Home Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil

Abstract

The aim of this research project is to investigate a wide variety of physical phenomena, with mostly functional analytical methods. We will provide theorems and proofs, and all assumptions will be stated explicitly. The physics questions we are interested in arise in quantum field theory on de Sitter space. The mathematical tools we use include spectral theory, complex analysis, groups and their representations, and, in particular, operator algebras. Hence, from a methodical point of view, this proposal should be seen (and evaluated) as a program in applied mathematics, directed to open problems in physics. Mathematical physics is one of the oldest interdisciplinary subjects in science, and without doubt one of the most successful ones. To cast the laws of nature in a clear mathematical form with rigorously deducible consequences has been one of the greatest intellectual accomplishments of our era. Mathematical physics has also played a crucial role in the development of both mathematics and physics, especially when the path to knowledge has hit unforeseen obstacles. In fact, the mathematical analysis of physics problems frequently leads to a drastic improvement of our understanding of nature. It is also documented that entire areas of mathematics were (and are) created to resolve physics problems. In contrast to wide-spread believe, the identification and analysis of mathematical difficulties (often called subtle, and frequently disregarded by the majority of theoretical physicist) has a proven record to be an effect strategy to discover new physics. The possibility to achieve this at low costs, without the enormous resources needed to realises modern high precision experiments, makes financing research in mathematical physics an attractive option, especially in times of limited resources. The objective of my personal research is to invent an purely mathematical framework for (relativistic) quantum physics, based on advanced mathematical tools: non-commutative harmonic analysis, the theory of (inclusions of) von Neumann algebras, and the non-commutative integration theory. The latter has been developed only recently. My expectation is that my work will initiate a new subject within applied mathematics, which I believe, will be explored by my colleagues and myself in years to come. To be more precise, my objective is to establish (interacting) quantum field theory as a subject in applied mathematics, in some way similar, at least in spirit, to the construction and classification of unitary representations of groups. This has already been achieved in some special cases of quantum field theory, namely for non-interacting fields with the classification of positive energy representations of the Poincaré group by Wigner, and for conformal fields, in terms of representations of the Virasoro algebra. I am convinced that all two-dimensional quantum theories, which satisfy the Haag-Kastler axioms and respect the structure of the space-time symmetries, and which are locally Fock, can be constructed and classified by the methods described in more detail in the main text of this proposal. (AU)