Tensor types and their use in physics
Abstract
The content of this paper can be roughly organized into a threelevel hierarchy of generality. At the first, most general level, we introduce a new language which allows us to express various categorical structures in a systematic and explicit manner in terms of socalled 2schemes. Although 2schemes can formalize categorical structures such as symmetric monoidal categories, they are not limited to this, and can be used to define structures with no categorical analogue. Most categorical structures come with an effective graphical calculus such as string diagrams for symmetric monoidal categories, and the same is true more generally for interesting 2schemes. In this work, we focus on one particular noncategorical 2scheme, whose instances we refer to as tensor types. At the second level of the hierarchy, we work out different flavors of this 2scheme in detail. The effective graphical calculus of tensor types is that of tensor networks or Penrose diagrams, that is, string diagrams without a flow of time. As such, tensor types are similar to compact closed categories, though there are various small but potentially important differences. Also, the two definitions use completely different mechanisms despite both being examples of 2schemes. At the third level of the hierarchy, we provide a long list of different families of concrete tensor types, in a way which makes them accessible to concrete computations, motivated by their potential use in physics. Different tensor types describe different types of physical models, such as classical or quantum physics, deterministic or statistical physics, manybody or singlebody physics, or matter with or without symmetries or fermions.
 Publication:

arXiv eprints
 Pub Date:
 August 2022
 arXiv:
 arXiv:2208.01135
 Bibcode:
 2022arXiv220801135B
 Keywords:

 Mathematical Physics;
 Mathematics  Category Theory