Yuto Moriwaki

Mathematics of quantum field theory (QFT) is vast, mysterious and fascinating. In mathematics, we often define a concept mathematically and study a subject based on that definition, but in the case of quantum field theory, it is fair to say that there is no complete definition yet. Quantum field theory has many faces and corresponding mathematical formulations. To give a few examples:

[Path integral formulation/probabilistic or analytic approach]
Osterwalder-Schrader axioms, Glimm-Jaffe axioms, prefactorization algebra, etc.
(where QFT is formulated as a probability measure on the space of distributions )

[Hamiltonian formulation/algebraic approach]
Garding-Wightman axioms, Haag-Kastler axioms, Operator product expansion, etc.
(formulated by unbounded operators, net of vonNeumann algebras or representation theory of vertex operator algebras)

Each of these axioms looks at a partial aspect of the vast mathematics of quantum field theory, and they are connected (in ways that are not necessarily equivalent).
In recent years, an idea called “Functorial quantum field theory,” a philosophy that unifies these various approaches using (higher) category theory, has been discovered in studies of topological quantum field theory and two-dimensional conformal field theory.
I am interested in the mathematical formulation of this idea and applications of these mathematical structures to theoretical physics.