nLab
quantum field theory

Higher categories in quantum field theory

Even though quantum field theory has been around for decades and has been very successful both as a phenomenological model in experimental physics as well as a source of deep mathematical structures and theorems, from a mathematical perspective it is still to a large extent mysterious.

There are essentially two alternative approaches for formalizing quantum field theory and making it accessible to mathematical treatment:

• algebraic quantum field theory: AQFT

• functorial quantum field theory: FQFT.

(Other structures which are used to define quantum field theories, such as vertex operator algebras are now more or less understood to be special cases of these two approaches.)

Both AQFT and FQFT lead to higher categorical structures. In fact, a couple of important higher categorical structures were motivated from and first considered in the context of quantum field theory. For instance

• John Roberts first conceived the idea of $\mathrm{\infty }$-categorical nonabelian cohomology in the context of AQFT.

• the notion of the ”$\mathrm{\infty }$-category of cobordisms”, which is thought to play a role analogous to and as fundamental as the sphere spectrum? was motivated from FQFT;

• the generalized tangle hypothesis was formulated by Baez and Dolan in the context of extended topological FQFT.

There are some indications that such higher categorical structures, such as those appearing in groupoidification, are essential for clarifying some of the mysteries of quantum field theory, such as the path integral. While this is far from being clarified, this is what motivates research in higher categorical structures in QFT.

Ours is the age to figure this out.

References

For references see FQFT and AQFT.