Operator-algebraic construction of integrable field theories
Lead Research Organisation:
University of York
Department Name: Mathematics
Abstract
In the recent paper "A C*-algebraic approach to interacting quantum field theories" (https://arxiv.org/abs/1902.06062) Buchholz and Fredenhagen proposed a new way to describe interacting field theories using C*-algebras. In his project, Berend will explore how integrable models fit into this framework. The starting point will be the construction of a net of C* algebras of the sine-Gordon model, following the paradigm of Buchholz and Fredenhagen, and comparison with the earlier result of Bahns and Rejzner in "The Quantum Sine Gordon model in perturbative AQFT" (https://arxiv.org/abs/1609.08530).
Organisations
People |
ORCID iD |
| Katarzyna Anna Rejzner (Primary Supervisor) | |
| Berend Visser (Student) |
Studentship Projects
| Project Reference | Relationship | Related To | Start | End | Student Name |
|---|---|---|---|---|---|
| EP/V52010X/1 | 01/10/2020 | 31/10/2025 | |||
| 2440735 | Studentship | EP/V52010X/1 | 01/10/2020 | 30/09/2024 | Berend Visser |
| Description | In the framework of perturbative algebraic quantum field theory, a key concept that is used is the notion of functions: Smooth functions on some space of smooth functions. These spaces form the set of objects that underlie both the classical and the quantum observables in models of quantum field theory. In classical mechanics, a system is described by a phase space, which is some finite dimensional spaces describing the degrees of freedom of the system. Observables are given by smooth functions on this space: They give a definite prediction of some quantity, depending on the configuration the state is in. In going to a quantum theory, it's possible to keep the same vector space to label the observables, but just change the algebraic structure from a commutative product to a so-called star-product. One would like to lift this picture to field theories as well, which are the theories that underlie our understanding of the fundamental forces of the universe. This introduces two technical complications however. The first one is that the space of configurations will no longer be a manifold, but an infinite dimensional analogue thereof. In general, it will be given by the space of sections of some vector bundle, but for simplicity one can think of smooth functions on some spacetime M. This complicates defining what we mean by a smooth function on the configuration space, as there are several inequivalent ways of expressing the smoothness of a function in an infinite dimensional setting. The second complication is that, in defining the Poisson-bracket (or the star-product), the full space of smooth functionals is too big. The derivatives of smooth functionals are distributions, and in general the multiplication of two distributions is ill-defined. This leads one to only consider distributions that satisfy some conditions on their singular structure, so that the algebraic structure can be introduced. This is the reason why the microcausal functionals are of particular interest in what follows: These are functionals that satisfy a robust wavefront set condition, so that a wide range of physically desirable operations is well defined. It's an interesting to see what properties the algebra they generate has. In particular, I've been trying to see if the proof of the time-slice property for Wick polynomials generalizes to this setting. I have shown that this problem is solvable for the case of all smooth functionals (ie. forgetting about wavefront set conditions for a moment) through an application of the fundamental theorem of calculus for smooth functions on Frèchet spaces. However, it turns out that this procedure does not respect the additional requirements we put on the singular structure. In a sense, this is due to the fact that the definition of microcausal functionals only imposes 'pointwise` wavefront set conditions. The main body of work that I've performed to this point is proving, through an explicit counterexample, that the obvious construction does not respect wavefront set conditions. On top of that, I've been researching ways to obtain a slightly smaller class of functionals, that is well-behaved with respect to the operations we would like to perform. This is ongoing work, and at this point there have not been definite answers to any of the research questions. |
| Exploitation Route | The main obvious use of this work is in trying to study non-perturbative effects in quantum field theory. The space of microcausal functionals has been instrumental mainly in the setting of perturbative models, but out approach of viewing them as honest smooth functions, rather than just polynomials, would allow one to strike a bridge between perturbative and non-perturbative treatments. |
| Sectors | Other |