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Quantum Algebraic Structures and Models

Periodic Reporting for period 4 - QUEST (Quantum Algebraic Structures and Models)

Reporting period: 2020-06-01 to 2021-11-30

What is the problem/issue being addressed?
The basic problem in the project is to understand the mathematical structure of Quantum Field Theory (QFT). On one hands one wants to understand the relevant mathematical language, develop it and bring back new results in Physics; on the other hands one wants to set up more rigorous mathematical models for QFT, starting from the simpler, low spacetime context. The main mathematical methods concern Operator Algebras.

Why is it important for society?
The project concerns basic science and points towards a deeper understanding of the fundamental structure associated to particle Physics. At the same time, this provides a crucial indication on the important mathematical methods to develop. The reached higher level of knowledge is a strong tool at disposal for solving not only the present physical problems; the mathematical advances are expected to be applied in different contexts, e.g. Quantum Information Theory that actually is an Objectives of the Project, sometimes in an unpredictable way.

What are the overall objectives?
Quantum Field Theory is an evolution of Quantum Mechanics that deals with arbitrarily many particles with interactions and particle creation/annihilation. The construction of non-trivial QFT models that are rigorous from the mathematical view point is one of the most important scientific challenge in the mathematical physics framework. The overall objective of the project is to gain a deeper insight in the fundamental mathematical structure associated with QFT.
Among other results, the project has so far uncovered the structure of Boundary Conformal QFT on the two-dimensional Minkowski spacetime, pointing out the tensor categorical structure and providing a constructive methods to realize the different boundary condition.
Other important advances in the project concerns the relations with different mathematical approaches to QFT, as Vertex Algebras, Scattering matrices and Noncommutative Geometry.
Our methods have recently shown to be powerful concerning quantum information analysis in QFT.
Main advances concern in particular the Minkowskian description and model constructions of Boundary Conformal QFT, the first construction QFT models with minimal length, conditions to construct a Vertex Algebra from a Local Conformal Net and back, the description of the localized states associated with infinite spin particles, showing that states localized in a bounded spacetime regions do not exist.
The important recent advances concerns quantum information for infinite systems, in particular rigorous derivations of entropy bounds in QFT.

Conclusion of the action

The project has been very successful in different respects.
Previous line of research has been brought to completion, in particular the algebraic structure of Conformal Quantum Field Theory has been analyzed and carried to a higher level of comprehension, the relation with Vertex Algebras has understood within a wide general framework.

A new line of research has been opened, it concerns Entropy and Quantum Energy Inequalities. Intense research activity has been performed in this direction during the second part of the project and is still successfully going on.
General derivations of Landauer and Bekenstein inequality have been provided, a new, operator algebraic approach to the Quantum Null Energy Inequality has been set up with model analysis.
A formula for the local massive Modular Hamiltonian has been given, solving a long standing open problem.
Poster of the Final Conference of QUEST